Variable exponent spaces are a generalization of Lebesgue spaces in which the exponent is a measurable function. Most of the research done in this topic has been situated under the context of real functions. In this work, we present two examples of variable exponent spaces of analytic functions: variable exponent Hardy spaces and variable exponent Bergman spaces. We will introduce the spaces together with some basic properties and the main techniques used in the context. We will show that in both cases, the boundedness of the evaluation functionals plays a key role in the theory. We also present a section of possible directions of research in this topic.

**Keywords:** variable exponents, Hardy spaces, Bergman spaces, Carleson measures,

In recent years the interest for nonstandard function spaces has risen due to several considerations including their need in some fields of applied mathematics, differential equations, and simply the possibility to study extensions and general-

One of such type of spaces is the variable exponent Lebesgue spaces. A first introduction to them was due to Orlicz in 1931. He considered measurable functions

j j *u x*ð Þ *p x*ð Þ d*x*< ∞,

the usual exponent *p* from the classical theory of Lebesgue spaces is replaced by a suitable function *p*ð Þ� . Orlicz was interested in the summability of Fourier series

j j *λan pn* < ∞,

Later generalizations consisted on considering real functions *u* on a domain Ω

*φ*ð Þ *x*, j*u x*ð Þj d*x*< ∞,

ð1 0

> X∞ *n*¼1

which led him to consider spaces of sequences ð Þ *an* so that

ð Ω

of Analytic Functions

#### **Chapter 5**

## Variable Exponent Spaces of Analytic Functions

*Gerardo A. Chacón and Gerardo R. Chacón*

#### **Abstract**

Variable exponent spaces are a generalization of Lebesgue spaces in which the exponent is a measurable function. Most of the research done in this topic has been situated under the context of real functions. In this work, we present two examples of variable exponent spaces of analytic functions: variable exponent Hardy spaces and variable exponent Bergman spaces. We will introduce the spaces together with some basic properties and the main techniques used in the context. We will show that in both cases, the boundedness of the evaluation functionals plays a key role in the theory. We also present a section of possible directions of research in this topic.

**Keywords:** variable exponents, Hardy spaces, Bergman spaces, Carleson measures, analytic functions

#### **1. Introduction**

In recent years the interest for nonstandard function spaces has risen due to several considerations including their need in some fields of applied mathematics, differential equations, and simply the possibility to study extensions and generalizations of classical spaces.

One of such type of spaces is the variable exponent Lebesgue spaces. A first introduction to them was due to Orlicz in 1931. He considered measurable functions *u* such that

$$\int\_0^1 |u(\mathbf{x})|^{p(\mathbf{x})} \, \mathbf{d}\mathbf{x} < \infty,$$

the usual exponent *p* from the classical theory of Lebesgue spaces is replaced by a suitable function *p*ð Þ� . Orlicz was interested in the summability of Fourier series which led him to consider spaces of sequences ð Þ *an* so that

$$\sum\_{n=1}^{\infty} |\lambda a\_n|^{p\_n} < \infty,$$

for *λ*>0 and *pn* ∈½ Þ 1, ∞ .

Later generalizations consisted on considering real functions *u* on a domain Ω such that

$$\int\_{\Omega} \rho(\mathfrak{x}, |u(\mathfrak{x})|) \, \mathrm{d}\mathfrak{x} < \infty,$$

where

$$\varphi: \mathfrak{Q} \times [\mathfrak{O}, \mathfrak{so}) \to [\mathfrak{O}, \mathfrak{so})$$

**2. Variable exponents**

For Ω ⊂ *d* we put *p*þ

abbreviations *p*þ ¼ *p*þ

*ρp*ð Þ� ,*μ* by

such that

Hölder continuous.

**67**

*Lp*ð Þ� ð Þ Ω . Such operator is defined as

defining parameter *p* is allowed to vary.

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

and the *Luxemburg-Nakano norm* by

Perhaps the most known function spaces in mathematical analysis are the *Lebesgue spaces Lp*. These are a fundamental basis of measure theory, functional analysis, harmonic analysis, and differential equations, among other areas. A recent generalization is the so-called *variable exponent Lebesgue spaces*. In this case, the

Ω ≔ ess sup*x*∈ Ω *p x*ð Þ and *p*�

*ρp*ð Þ� ,*μ*ð Þ *φ* ≔

Ω and *p*� ¼ *p*�

The basics on variable Lebesgue spaces may be found in the monographs [1, 17, 18].

it a *variable exponent* and define the set of all variable exponents with *p*þ < ∞ as Pð Þ Ω . For a complex-valued measurable function *φ* : Ω ! , we define the *modular*

j j *φ*ð Þ *x p x*ð Þ d*μ*ð Þ *x*

*C*log log 1ð Þ *=*j*x* � *y*j ,

> 2ℓ*C*log log 2ℓ ∣*x*�*y*∣ � �

for all *x*, *y* ∈ Ω such that ∣*x* � *y*∣< ℓ. We will write *p*ð Þ� ∈Plogð Þ Ω when *p*ð Þ� is log -

This condition has proven to be very useful in the theory since, among other things, it implies the boundedness of the Hardy-Littlewood maximal operator in

> 1 ∣*B x*ð Þ ,*r* ∣

and represents an important tool in harmonic analysis. We used it in [19] to show

ð

*B x*ð Þ ,*r*

∣*f y*ð Þ∣d*y*

*φ λ* � � ⩽1 n o*:* (1)

ð

Ω

Given *p*ð Þ� ∈Pð Þ Ω , the *variable Lebesgue space Lp*ð Þ� ð Þ Ω, *μ* is defined as the set of all complex-valued measurable functions *φ* : Ω ! for which the modular is finite, i.e., *ρp*ð Þ� ,*μ*ð Þ *φ* < ∞*:* Equipped with the Luxemburg-Nakano norm (1) is a Banach space. Most of the research being done on variable exponent spaces makes use of a regularity condition in the variable exponent in order to have a "fruitful" theory. **Definition 1.** A function *p* : Ω ! is said to be log *-Hölder continuous—*or satisfy the *Dini-Lipschitz condition* on Ω if there exists a positive constant *C*log

∥*φ*∥*Lp*ð Þ� ð Þ Ω,*μ* ≔ inf *λ*>0 : *ρp*ð Þ� ,*μ*

∣*p x*ð Þ� *p y*ð Þ∣⩽

for all *x*, *y*∈ Ω such that ∣*x* � *y*∣<1*=*2, from which we obtain

*Mf x*ð Þ ≔ sup

*r*>0

that the Bergman projection is bounded on variable exponent Bergman spaces. Another consequence of the log -Hölder condition is the following inequality

∣*p x*ð Þ� *p y*ð Þ∣ ≤

Ω ≔ ess inf *x*∈ Ω *p x*ð Þ; we use the

Ω. For a measurable function *p* : Ω ! ½ Þ 1, ∞ , we call

is a suitable function. Such spaces are known nowadays as Musielak-Orlicz spaces. We are interested in the case

$$
\rho(\mathfrak{x}, \mathfrak{y}) = \mathfrak{y}^{p(\mathfrak{x})},
$$

which corresponds to variable exponent Lebesgue spaces, with some adequate restrictions for the *exponent p*.

The study of problems in spaces with nonstandard growth has been mainly related to Lebesgue spaces with variable order *p x*ð Þ, to the corresponding Sobolev spaces, and to generalized Orlicz spaces (see surveys [1, 2]). The investigation of structural properties of variable exponent spaces and of operator theory in such spaces is of interest not only due to their intriguing mathematical structure—also worthy of investigation—but because such spaces appear in applications. One of these applications is the mathematical modeling of inhomogeneous materials, like electrorheological fluids. These are special viscous liquids that have the ability to change their mechanical properties significantly when they come in contact with an electric field. This can be explored in technological applications, like photonic crystals, smart inks, heterogeneous polymer composites, etc. [3–6].

Another known application is related to image restoration. Based on variable exponent spaces, the authors of [7] proposed a new model of image restoration combining the strength of the total variation approach and the isotropic diffusion approach. This model uses one value of *p* near the edge of images and another one near smooth regions, which leads to the variable exponent setting of the problem. Other references discussing image processing are [8–13]. This theory has also been applied to differential equations with nonstandard growth (see [14, 15]).

The theory of spaces of analytic functions on domains of the complex plane, or in general of *n*, happens in parallel. Starting from the classical Hardy *Hp*ð Þ and Bergman *Ap*ð Þ spaces, the functional Banach spaces have had a preferential study.

A Banach space X of complex-valued functions on a domain Ω is said to be *a functional Banach space* (cf. [16]) if the following hold:

Classical Hardy and Bergman spaces are examples of functional Banach spaces. In the case of spaces of analytic functions, the evaluation functionals *Kx*ð Þ¼ *f f x*ð Þ (*x*∈ Ω) are an extension of the well-known *reproducing kernels* in Hilbert spaces of analytic functions. The analyticity property of the functions in the space interacts with the Banach space structure by means of the continuity of evaluation functionals raising a fruitful relation between function theory and operator theory.

The setting of nonstandard spaces of analytic functions is a young area of research; a lot is yet to be explored. In this chapter, an introduction to the subject of variable exponent spaces of analytic functions will be given, making emphasis in the special place that evaluation functionals have in the theory.

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

#### **2. Variable exponents**

where

in general of *n*

**66**

spaces. We are interested in the case

*Advances in Complex Analysis and Applications*

restrictions for the *exponent p*.

*φ* : Ω � ½ Þ! 0, ∞ ½ Þ 0, ∞

is a suitable function. Such spaces are known nowadays as Musielak-Orlicz

*φ*ð Þ¼ *x*, *y yp x*ð Þ,

which corresponds to variable exponent Lebesgue spaces, with some adequate

The study of problems in spaces with nonstandard growth has been mainly related to Lebesgue spaces with variable order *p x*ð Þ, to the corresponding Sobolev spaces, and to generalized Orlicz spaces (see surveys [1, 2]). The investigation of structural properties of variable exponent spaces and of operator theory in such spaces is of interest not only due to their intriguing mathematical structure—also worthy of investigation—but because such spaces appear in applications. One of these applications is the mathematical modeling of inhomogeneous materials, like electrorheological fluids. These are special viscous liquids that have the ability to change their mechanical properties significantly when they come in contact with an electric field. This can be explored in technological applications, like photonic

Another known application is related to image restoration. Based on variable exponent spaces, the authors of [7] proposed a new model of image restoration combining the strength of the total variation approach and the isotropic diffusion approach. This model uses one value of *p* near the edge of images and another one near smooth regions, which leads to the variable exponent setting of the problem. Other references discussing image processing are [8–13]. This theory has also been

The theory of spaces of analytic functions on domains of the complex plane, or

Bergman *Ap*ð Þ spaces, the functional Banach spaces have had a preferential study. A Banach space X of complex-valued functions on a domain Ω is said to be *a*

Classical Hardy and Bergman spaces are examples of functional Banach spaces. In the case of spaces of analytic functions, the evaluation functionals *Kx*ð Þ¼ *f f x*ð Þ (*x*∈ Ω) are an extension of the well-known *reproducing kernels* in Hilbert spaces of analytic functions. The analyticity property of the functions in the space interacts with the Banach space structure by means of the continuity of evaluation functionals raising a fruitful relation between function theory and operator theory. The setting of nonstandard spaces of analytic functions is a young area of research; a lot is yet to be explored. In this chapter, an introduction to the subject of variable exponent spaces of analytic functions will be given, making emphasis in the

, happens in parallel. Starting from the classical Hardy *Hp*ð Þ and

crystals, smart inks, heterogeneous polymer composites, etc. [3–6].

applied to differential equations with nonstandard growth (see [14, 15]).

• Vectorial operations are precisely the pointwise operations on Ω.

*functional Banach space* (cf. [16]) if the following hold:

• If *f x*ð Þ¼ *g x*ð Þ for every *x*∈ Ω, then *f* ¼ *g*.

• If *f x*ð Þ¼ *f y*ð Þ for every *f* ∈ X, then *x* ¼ *y*.

• Evaluation functionals *x*↦*f x*ð Þ are continuous on X.

special place that evaluation functionals have in the theory.

Perhaps the most known function spaces in mathematical analysis are the *Lebesgue spaces Lp*. These are a fundamental basis of measure theory, functional analysis, harmonic analysis, and differential equations, among other areas. A recent generalization is the so-called *variable exponent Lebesgue spaces*. In this case, the defining parameter *p* is allowed to vary.

The basics on variable Lebesgue spaces may be found in the monographs [1, 17, 18]. For Ω ⊂ *d* we put *p*þ Ω ≔ ess sup*x*∈ Ω *p x*ð Þ and *p*� Ω ≔ ess inf *x*∈ Ω *p x*ð Þ; we use the abbreviations *p*þ ¼ *p*þ Ω and *p*� ¼ *p*� Ω. For a measurable function *p* : Ω ! ½ Þ 1, ∞ , we call it a *variable exponent* and define the set of all variable exponents with *p*þ < ∞ as Pð Þ Ω .

For a complex-valued measurable function *φ* : Ω ! , we define the *modular ρp*ð Þ� ,*μ* by

$$\rho\_{p(\cdot),\mu}(\rho) := \int\_{\Omega} |\rho(\mathfrak{x})|^{p(\mathfrak{x})} \cdot \mathrm{d}\mu(\mathfrak{x}),$$

and the *Luxemburg-Nakano norm* by

$$\|\|\rho\|\|\_{L^{p(\cdot)}(\Omega,\mu)} := \inf \left\{ \lambda > 0 : \rho\_{p(\cdot),\mu} \left( \frac{\rho}{\lambda} \right) \leqslant 1 \right\}.\tag{1}$$

Given *p*ð Þ� ∈Pð Þ Ω , the *variable Lebesgue space Lp*ð Þ� ð Þ Ω, *μ* is defined as the set of all complex-valued measurable functions *φ* : Ω ! for which the modular is finite, i.e., *ρp*ð Þ� ,*μ*ð Þ *φ* < ∞*:* Equipped with the Luxemburg-Nakano norm (1) is a Banach space.

Most of the research being done on variable exponent spaces makes use of a regularity condition in the variable exponent in order to have a "fruitful" theory.

**Definition 1.** A function *p* : Ω ! is said to be log *-Hölder continuous—*or satisfy the *Dini-Lipschitz condition* on Ω if there exists a positive constant *C*log such that

$$|p(\mathbf{x}) - p(\mathbf{y})| \lesssim \frac{C\_{\log}}{\log \left( 1/|\mathbf{x} - \mathbf{y}| \right)},$$

for all *x*, *y*∈ Ω such that ∣*x* � *y*∣<1*=*2, from which we obtain

$$|p(\mathbf{x}) - p(\mathbf{y})| \le \frac{2\ell C\_{\log}}{\log\left(\frac{2\ell}{|\mathbf{x} - \mathbf{y}|}\right)}$$

for all *x*, *y* ∈ Ω such that ∣*x* � *y*∣< ℓ. We will write *p*ð Þ� ∈Plogð Þ Ω when *p*ð Þ� is log - Hölder continuous.

This condition has proven to be very useful in the theory since, among other things, it implies the boundedness of the Hardy-Littlewood maximal operator in *Lp*ð Þ� ð Þ Ω . Such operator is defined as

$$M\!f^\circ(\mathbf{x}) \coloneqq \sup\_{r>0} \frac{1}{|B(\mathbf{x},r)|} \int\_{B(\mathbf{x},r)} |f^\circ(\mathbf{y})| \mathbf{d}\mathbf{y}d\mathbf{y}$$

and represents an important tool in harmonic analysis. We used it in [19] to show that the Bergman projection is bounded on variable exponent Bergman spaces.

Another consequence of the log -Hölder condition is the following inequality

$$|B|^{p\_{-}(B)-p\_{+}(B)} \leq \mathcal{C}\_\*$$

∥*f* ∥*hp*ð Þ� ð Þ ¼ sup

*every U* ∈ *hp*ð Þ� ð Þ *, there exists u* ∈*Lp*ð Þ� ð Þ *such that Pu* ¼ *U and moreover,*

*f z*ð Þ¼ ð

For each *z*∈ , the functions *Kz* : ! defined as

Consequently, the set of polynomials is also dense in *Hp*ð Þ� ð Þ .

*Kz*ð Þ¼ *w*

*as γz*ð Þ¼ *f f z*ð Þ*. Then γz is a bounded operator for every z* ¼ ∣*z*∣*eiθ* ∈ *and*

∥*γz*∥≲ 1

lowing theorem analogous to the constant exponent context.

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

**Theorem 6.** *The following chain of inclusions holds*

and moreover, the inclusions are continuous.

of analytic functions *f* : ! such that *f* ∈*hp*ð Þ� ð Þ .

as such, *Hp*ð Þ� ð Þ is a Banach space. Recall that for all functions *f* ∈ *H*1

*on compact subsets of .*

1

**69**

½ �! 0, 2*π* þ *as*

Hölder continuity plays a key role.

*hp*þ

∥*U*∥*hp*ð Þ� ð Þ � ∥*u*∥*Lp*ð Þ� ð Þ *.*

0 ≤*r* <1

Notice that, since *Pf* is harmonic for *f* ∈*Lp*ð Þ� ð Þ , then Theorem 3 shows that the Poisson transform *P* : *Lp*ð Þ� ð Þ! *hp*ð Þ� ð Þ is bounded. Moreover, we have the fol-

**Theorem 5.** *Suppose that p* : ½ �! 0, 2*π* ½ Þ 1, ∞ *is* log *-Hölder continuous. Then for*

ð Þ ⊂*hp*ð Þ� ð Þ ⊂*hp*�

**Definition 7.** The variable exponent Hardy space *Hp*ð Þ� ð Þ is defined as the space

In an analogous way to the classical setting *Hp*ð Þ� ð Þ can be identified with the subspace of functions in *Lp*ð Þ� ð Þ whose negative Fourier coefficients are zero, and

> *f*ð Þ*ζ* 1 � *ζz*

> > 1 1 � *zw* ,

are called *reproducing kernels*. They are bounded on and consequently belong to every space *Hp*ð Þ� ð Þ . Moreover, as a consequence of the reproducing formula and the Hahn-Banach theorem, the linear span of f g *Kz* : *z*∈ is dense in *Hp*ð Þ� ð Þ .

**Theorem 8.** *For each z*∈ *consider the linear functional γz* : *Hp*ð Þ� ð Þ! *defined*

ð Þ 1�j*z*j 1*=p*ð Þ*θ :*

*Consequently, the convergence in the Hp*ð Þ� ð Þ *-norm implies the uniform convergence*

The main tool for proving the previous theorem is following the version of a Forelli-Rudin inequality, adapted to the case of variable exponents. Here, the Log-

**Lemma 9.** *Suppose p* : ½ �! 0, 2*π* ½ Þ 1, ∞ *and q* : ½ �! 0, 2*π* ½ Þ 1, ∞ *are such that* 1

*q*ð Þ*θ* ¼ 1 *for every θ* ∈½ � 0, 2*π . Let* 1*=*2< *r*<1 *and z* ¼ ∣*z*∣*eiθ. Define the function φ* :

*φ*ðÞ¼ *t* ð Þ 1�j*z*j 1*=q*ð Þ*θ*

∣1 � ∣*z*∣*rei t*ð Þ �*θ* ∣

*:*

d*m*ð Þ*ζ :*

∥ *fr*∥*Lp*ð Þ� ð Þ < ∞*:*

ð Þ ,

ð Þ , we have the reproducing formula:

*p*ð Þ*θ* þ

that hold for all balls *B* ⊂ . Here ∣*B*∣ stands for the Lebesgue measure of *B*, *p*þð Þ¼ *B* ess sup*z* ∈*Bp z*ð Þ, and *p*�ð Þ¼ *B* ess inf *z*∈*Bp z*ð Þ. This will be a useful tool when studying complex function spaces since it allows to pass from an exponent that varies over a ball to a constant exponent at the center of such ball.

One interesting point of the theory of variable exponents is that, in general, the classical approach to Bergman spaces seems to fail in the variable framework. For example, in [19] it is shown that the Bergman projection is bounded from *Ap*ð Þ� ð Þ to *Ap*ð Þ� ð Þ . This is done by using the theory of Békollé-Bonami weights and a theorem that extends Rubio de Francia's extrapolation theory to variable Lebesgue spaces. This method is quite different from the usual ways of showing that the Bergman projection is bounded, which use either Schur's test or Calderón-Zygmund theory.

#### **3. Variable exponent Hardy spaces**

In this section we will give an introduction to the variable exponent Hardy spaces and the results and techniques that are usually found. We will follow the presentation as in [20].

**Definition 2.** For each *z* in the unit disk , the Poisson kernel *P z*ð Þ , *ζ* is defined as

$$P(z,\zeta) = \frac{1-\left|z\right|^2}{\left|z-\zeta\right|^2},$$

and the Poisson transform of a function *f* ∈*Lp*ð Þ� ð Þ is defined as

$$Pf(z) = \int\_{\mathbb{T}} P(z,\zeta)f(\zeta)m(\zeta),$$

where *m* denotes the normalized Lebesgue measure on . We will use the following result from [21]:

**Theorem 3.** *Suppose that p is* log *-Hölder continuous. For each* 0≤*r*<1*, and f* ∈*Lp*ð Þ� ð Þ *, the linear operator Pr* : *Lp*ð Þ� ð Þ! *Lp*ð Þ� ð Þ *defined as*

$$P\_{\mathcal{Y}}f(\zeta) = Pf(r\zeta).$$

is uniformly bounded on *Lp*ð Þ� ð Þ , ∥*Pr*ð Þ*f* ∥*Lp*ð Þ� ð Þ ≲∥*f* ∥*Lp*ð Þ� ð Þ , and for every *f* ∈*Lp*ð Þ� ð Þ ,

$$\|f - P\_{\mathfrak{F}}f\|\_{L^{p(\cdot)}(\mathbb{T})} \to \mathbf{0}, \quad \text{as } r \to \mathbf{1}^-.$$

We are now ready to define the harmonic Hardy spaces with variable exponents. Given *f* : ! and 0 ≤*r*<1, the *dilations f r* : ! of *f* are defined as *fr*ð Þ¼ *ζ f r*ð Þ*ζ* .

**Definition 4.** The *variable exponent harmonic Hardy space hp*ð Þ� ð Þ is defined as the space of harmonic functions *f* : ! such that

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

j j *B p*�ð Þ� *B p*þð Þ *B* ≤*C*,

that hold for all balls *B* ⊂ . Here ∣*B*∣ stands for the Lebesgue measure of *B*, *p*þð Þ¼ *B* ess sup*z* ∈*Bp z*ð Þ, and *p*�ð Þ¼ *B* ess inf *z*∈*Bp z*ð Þ. This will be a useful tool when studying complex function spaces since it allows to pass from an exponent

In this section we will give an introduction to the variable exponent Hardy spaces and the results and techniques that are usually found. We will follow the

**Definition 2.** For each *z* in the unit disk , the Poisson kernel *P z*ð Þ , *ζ* is

*P z*ð Þ¼ , *ζ* 1 � j j *z*

**Theorem 3.** *Suppose that p is* log *-Hölder continuous. For each* 0≤*r*<1*, and*

*Prf*ð Þ¼ *ζ Pf r*ð Þ*ζ*

∥*f* � *Prf* ∥*Lp*ð Þ� ð Þ ! 0, *as r* ! 1�*:*

exponents. Given *f* : ! and 0 ≤*r*<1, the *dilations f r* : ! of *f* are defined as

**Definition 4.** The *variable exponent harmonic Hardy space hp*ð Þ� ð Þ is defined as

is uniformly bounded on *Lp*ð Þ� ð Þ , ∥*Pr*ð Þ*f* ∥*Lp*ð Þ� ð Þ ≲∥*f* ∥*Lp*ð Þ� ð Þ , and for every

We are now ready to define the harmonic Hardy spaces with variable

and the Poisson transform of a function *f* ∈*Lp*ð Þ� ð Þ is defined as

ð

*Pf z*ð Þ¼

where *m* denotes the normalized Lebesgue measure on .

*f* ∈*Lp*ð Þ� ð Þ *, the linear operator Pr* : *Lp*ð Þ� ð Þ! *Lp*ð Þ� ð Þ *defined as*

We will use the following result from [21]:

the space of harmonic functions *f* : ! such that

2 j j *z* � *ζ* 2 ,

*P z*ð Þ , *ζ f*ð Þ*ζ m*ð Þ*ζ* ,

One interesting point of the theory of variable exponents is that, in general, the classical approach to Bergman spaces seems to fail in the variable framework. For example, in [19] it is shown that the Bergman projection is bounded from *Ap*ð Þ� ð Þ to *Ap*ð Þ� ð Þ . This is done by using the theory of Békollé-Bonami weights and a theorem that extends Rubio de Francia's extrapolation theory to variable Lebesgue spaces. This method is quite different from the usual ways of showing that the Bergman projection is bounded, which use either Schur's test or Calderón-

that varies over a ball to a constant exponent at the center of such ball.

Zygmund theory.

presentation as in [20].

defined as

*f* ∈*Lp*ð Þ� ð Þ ,

*fr*ð Þ¼ *ζ f r*ð Þ*ζ* .

**68**

**3. Variable exponent Hardy spaces**

*Advances in Complex Analysis and Applications*

$$\|f\|\_{h^{p(\cdot)}(\mathbb{D})} = \sup\_{0 \le r < 1} \|f\_r\|\_{L^{p(\cdot)}(\mathbb{T})} < \infty.$$

Notice that, since *Pf* is harmonic for *f* ∈*Lp*ð Þ� ð Þ , then Theorem 3 shows that the Poisson transform *P* : *Lp*ð Þ� ð Þ! *hp*ð Þ� ð Þ is bounded. Moreover, we have the following theorem analogous to the constant exponent context.

**Theorem 5.** *Suppose that p* : ½ �! 0, 2*π* ½ Þ 1, ∞ *is* log *-Hölder continuous. Then for every U* ∈ *hp*ð Þ� ð Þ *, there exists u* ∈*Lp*ð Þ� ð Þ *such that Pu* ¼ *U and moreover,* ∥*U*∥*hp*ð Þ� ð Þ � ∥*u*∥*Lp*ð Þ� ð Þ *.*

**Theorem 6.** *The following chain of inclusions holds*

$$h^{p^{+}}(\mathbb{D}) \subset h^{p(\cdot)}(\mathbb{D}) \subset h^{p^{-}}(\mathbb{D}),$$

and moreover, the inclusions are continuous.

**Definition 7.** The variable exponent Hardy space *Hp*ð Þ� ð Þ is defined as the space of analytic functions *f* : ! such that *f* ∈*hp*ð Þ� ð Þ .

In an analogous way to the classical setting *Hp*ð Þ� ð Þ can be identified with the subspace of functions in *Lp*ð Þ� ð Þ whose negative Fourier coefficients are zero, and as such, *Hp*ð Þ� ð Þ is a Banach space.

Recall that for all functions *f* ∈ *H*1 ð Þ , we have the reproducing formula:

$$f(\mathbf{z}) = \int\_{\mathbb{T}} \frac{f(\boldsymbol{\zeta})}{\mathbf{1} - \overline{\boldsymbol{\zeta}}\boldsymbol{z}} \, \mathrm{d}m(\boldsymbol{\zeta})\, .$$

For each *z*∈ , the functions *Kz* : ! defined as

$$K\_{\underline{x}}(w) = \frac{1}{1 - \overline{z}w},$$

are called *reproducing kernels*. They are bounded on and consequently belong to every space *Hp*ð Þ� ð Þ . Moreover, as a consequence of the reproducing formula and the Hahn-Banach theorem, the linear span of f g *Kz* : *z*∈ is dense in *Hp*ð Þ� ð Þ . Consequently, the set of polynomials is also dense in *Hp*ð Þ� ð Þ .

**Theorem 8.** *For each z*∈ *consider the linear functional γz* : *Hp*ð Þ� ð Þ! *defined as γz*ð Þ¼ *f f z*ð Þ*. Then γz is a bounded operator for every z* ¼ ∣*z*∣*eiθ* ∈ *and*

$$\|\gamma\_x\| \lesssim \frac{1}{\left(1 - |z|\right)^{1/p(\theta)}}.$$

*Consequently, the convergence in the Hp*ð Þ� ð Þ *-norm implies the uniform convergence on compact subsets of .*

The main tool for proving the previous theorem is following the version of a Forelli-Rudin inequality, adapted to the case of variable exponents. Here, the Log-Hölder continuity plays a key role.

**Lemma 9.** *Suppose p* : ½ �! 0, 2*π* ½ Þ 1, ∞ *and q* : ½ �! 0, 2*π* ½ Þ 1, ∞ *are such that* 1 *p*ð Þ*θ* þ 1 *q*ð Þ*θ* ¼ 1 *for every θ* ∈½ � 0, 2*π . Let* 1*=*2< *r*<1 *and z* ¼ ∣*z*∣*eiθ. Define the function φ* : ½ �! 0, 2*π* þ *as*

$$\wp(t) = \frac{(1 - |\mathbf{z}|)^{1/q(\theta)}}{|\mathbf{1} - |\mathbf{z}| \eta e^{i(t-\theta)} |}.$$

*Then if φ*ð Þ*t* >1*, it holds that*

$$
\varrho(t)^{p(t)} \lesssim \varrho(t)^{p(\theta)}.
$$

The additional term that appears in the previous inequality makes the boundedness of the domain a necessary condition to use such technique. One question to address in future investigations is how the situation is for Bergman spaces defined on unbounded domains. In such case, the error term accumulates despite the regu-

One technique developed to study variable exponent Bergman spaces is a combination of mollifiers and dilations. Such concept was developed for studying the density of the set of polynomials in Bergman spaces. In the classical case, this is a consequence of the fact that convergence in *Ap*ð Þ implies convergence on compact

*fr*ð Þ¼ *z f rz* ð Þ

One way of overcoming this problem in the case of variable exponent Lebesgue spaces is the use of a *mollification operator*. But in the case of the unit disk, changes need to be made. That is, when we had the idea to introduce a *mollified dilation*,

*f rw*ð Þ*ηr*ð Þ *z* � *w* d*A w*ð Þ*:*

2 *η*

0, if ∣*z*∣ ⩾1,

∥ *fr*∥*Ap*ð Þ� ð Þ ≲ ∥*f* ∥*Ap*ð Þ� ð Þ *:*

As a consequence of the previous theorem, we can approximate a function

2*rz* 1 � *r* � �

, if ∣*z*∣ <1,

2*r* . Hence, the sequence of Taylor

2 , 1 � �, define the *mollified dilation*

converge in norm to *f* for *r* ! 1�. This is a simple fact that depends on the dilation-invariant nature of Bergman Spaces, something that does not hold in the case of variable exponent Bergman spaces. This is one of the main difficulties for

larity condition on *p*ð Þ� , and consequently a different technique is needed.

subsets of and that for any *f* ∈ *Ap*ð Þ , its *radial dilations*

*fr*ð Þ*z* ≔

where *ρ* stands for the complex disk with radius *ρ*,

*η*ð Þ*z* ≔ exp

8 ><

>:

sup ½ ⩽*r*<1

*Moreover,* ∥ *f r* � *f* ∥*Ap*ð Þ� ð Þ ! 0 *as r* ! 1�*.*

*f* ∈ *Ap*ð Þ� ð Þ by a function *fr* which is analytic in 1þ*r*

ð

*ηr*ð Þ*z* ≔ 4*r*2

ð Þ 1 � *r*

1 j j *z* 2 � 1

Mollified dilations allow us to approximate analytic functions in the variable exponent Bergman spaces by functions that are analytic on a neighborhood of the

**Theorem 14.** *Let p*∈Plogð Þ *and let f* ∈ *Ap*ð Þ� ð Þ *. Then for* ½ ⩽*r*<1*,*

!

**4.1 Mollified dilations**

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

generalizing the classical theory.

2*r* ! of *f* as

*fr* : 1þ*r*

and

unit disk.

**71**

*fr* ∈ *Ap*ð Þ� ð Þ , *and*

given a function *f* ∈ *Ap*ð Þ� ð Þ and *r*∈ 1

#### **4. Variable exponent Bergman spaces**

The theory of Bergman spaces was introduced by S. Bergman in [22] and since the 1990s has gained a great deal of attention mainly due to some major breakthroughs at the time. For details on the theory of Bergman spaces, we refer to the books [5, 23].

**Definition 10.** Given a measurable function *p* ∈Pð Þ , we define the *variable exponent Bergman space Ap*ð Þ� ð Þ as the space of all analytic functions on that belong to the variable exponent Lebesgue space *Lp*ð Þ� ð Þ with respect to the area measure d*A* on the unit disk , i.e.,

$$\mathcal{A}^{p(\cdot)}(\mathbb{D}) = \left\{ f \text{ is an analytic function and} \int\_{\mathbb{D}} |f(z)|^{p(x)} \mathrm{d}A(z) < \infty \right\}.$$

One of the first results proven in [19] about this theory was the boundedness of the evaluation functionals that, as we saw before, will allow us to conclude that convergence in *Ap*ð Þ� ð Þ implies uniform convergence on compact subsets of and consequently *Lp*ð Þ� -limits of sequences in *Ap*ð Þ� ð Þ are analytic. Hence *Ap*ð Þ� ð Þ is a closed subspace of *Lp*ð Þ� ð Þ (and hence a Banach space). The first approach to such result used the following interpolation result based on Forelli-Rudin estimates:

**Lemma 11.** *Let p*∈Pð Þ *, then for every function f* ∈*Lp*ð Þ� ð Þ *we have*

$$\|f\|\_{L^{p(\cdot)}(\mathbb{D})} \lesssim \|f\|\_{L^{1}(\mathbb{D})}^{1/p^{+}} \|f\|\_{L^{\infty}(\mathbb{D})}^{1-1/p^{+}} \cdot \|$$

Using different techniques, the following sharp estimate was obtained in [24]: **Theorem 12.** *Let p*ð Þ� ∈Plogð Þ *. Then for every z*∈ *we have that*

$$\|\|\boldsymbol{\chi}\_{x}\|\|\_{p(\cdot)} \asymp \frac{1}{\left(1-|x|^{2}\right)^{2/p(x)}}.\tag{2}$$

The proof of this fact relies on the Log-Hölder condition and on the following version of a Jensen-type inequality for variable exponent spaces (see [17]).

**Lemma 13.** *Suppose that p*ð Þ� ∈Plogð Þ *and let*

$$\frac{1}{q(z,\boldsymbol{y})} \coloneqq \max\left\{ \frac{1}{p(\boldsymbol{z})} - \frac{1}{p(\boldsymbol{y})}, \mathbf{0} \right\}.$$

*Then for* 0<*γ* < 1*,*

$$\left(\int\_{B(\mathbf{x},r)} |f(\mathbf{y})| \mathbf{d} \mathbf{A}(\mathbf{y})\right)^{p(\mathbf{x})} \lesssim \int\_{B(\mathbf{x},r)} |f(\mathbf{y})|^{p(\mathbf{y})} \mathbf{d} \mathbf{A}(\mathbf{y}) + \int\_{B(\mathbf{x},r)} \mathbf{y}^{q(\mathbf{x},\mathbf{y})} \mathbf{d} \mathbf{A}(\mathbf{y}) \lesssim \frac{\varepsilon}{2}$$

for all *z*∈ *B x*ð Þ ,*r* provided that ∥*f* ∥*Lp*ð Þ� ð Þ ⩽1*:* The constant depends on *γ* and *p*ð Þ� .

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

The additional term that appears in the previous inequality makes the boundedness of the domain a necessary condition to use such technique. One question to address in future investigations is how the situation is for Bergman spaces defined on unbounded domains. In such case, the error term accumulates despite the regularity condition on *p*ð Þ� , and consequently a different technique is needed.

#### **4.1 Mollified dilations**

*Then if φ*ð Þ*t* >1*, it holds that*

*Advances in Complex Analysis and Applications*

measure d*A* on the unit disk , i.e.,

*Ap*ð Þ� ð Þ¼ *f* is an analytic function and

∥*f* ∥*Lp*ð Þ� ð Þ ⩽∥*f* ∥

**Theorem 12.** *Let p*ð Þ� ∈Plogð Þ *. Then for every z*∈ *we have that*

∥*γz*∥*p*ð Þ� ≍

**Lemma 13.** *Suppose that p*ð Þ� ∈Plogð Þ *and let*

j*f y*ð Þjd*A y*ð Þ !*p z*ð Þ

*Then for* 0<*γ* < 1*,*

**70**

⨏ *B x*ð Þ *;r* 1

*q z*ð Þ , *y* ≔ max

≲ ⨏ *B x*ð Þ *;r*

books [5, 23].

**4. Variable exponent Bergman spaces**

*φ*ð Þ*t p t*ð Þ≲*φ*ð Þ*t p*ð Þ*θ :*

The theory of Bergman spaces was introduced by S. Bergman in [22] and since the 1990s has gained a great deal of attention mainly due to some major breakthroughs at the time. For details on the theory of Bergman spaces, we refer to the

**Definition 10.** Given a measurable function *p* ∈Pð Þ , we define the *variable exponent Bergman space Ap*ð Þ� ð Þ as the space of all analytic functions on that belong to the variable exponent Lebesgue space *Lp*ð Þ� ð Þ with respect to the area

> ð

1�1*=p*þ *L*∞ð Þ *:*

2 � �2*=p z*ð Þ *:* (2)

� �

One of the first results proven in [19] about this theory was the boundedness of the evaluation functionals that, as we saw before, will allow us to conclude that convergence in *Ap*ð Þ� ð Þ implies uniform convergence on compact subsets of and consequently *Lp*ð Þ� -limits of sequences in *Ap*ð Þ� ð Þ are analytic. Hence *Ap*ð Þ� ð Þ is a closed subspace of *Lp*ð Þ� ð Þ (and hence a Banach space). The first approach to such result used the following interpolation result based on Forelli-Rudin estimates: **Lemma 11.** *Let p*∈Pð Þ *, then for every function f* ∈*Lp*ð Þ� ð Þ *we have*

> 1*=p*þ *L*1ð Þ ∥*f* ∥

Using different techniques, the following sharp estimate was obtained in [24]:

1 � j j *z*

The proof of this fact relies on the Log-Hölder condition and on the following

1 *p z*ð Þ � 1

for all *z*∈ *B x*ð Þ ,*r* provided that ∥*f* ∥*Lp*ð Þ� ð Þ ⩽1*:* The constant depends on *γ* and *p*ð Þ� .

*p y*ð Þ, 0

*:*

*B x*ð Þ *;r*

*γq z*ð Þ *;y* d*A y*ð Þ

� �

j j *f y*ð Þ *p y*ð Þd*A y*ð Þþ ⨏

version of a Jensen-type inequality for variable exponent spaces (see [17]).

1

j j *f z*ð Þ *p z*ð Þd*A z*ð Þ< ∞

*:*

One technique developed to study variable exponent Bergman spaces is a combination of mollifiers and dilations. Such concept was developed for studying the density of the set of polynomials in Bergman spaces. In the classical case, this is a consequence of the fact that convergence in *Ap*ð Þ implies convergence on compact subsets of and that for any *f* ∈ *Ap*ð Þ , its *radial dilations*

$$f\_r(\mathbf{z}) = f(\mathbf{z}\mathbf{z})$$

converge in norm to *f* for *r* ! 1�. This is a simple fact that depends on the dilation-invariant nature of Bergman Spaces, something that does not hold in the case of variable exponent Bergman spaces. This is one of the main difficulties for generalizing the classical theory.

One way of overcoming this problem in the case of variable exponent Lebesgue spaces is the use of a *mollification operator*. But in the case of the unit disk, changes need to be made. That is, when we had the idea to introduce a *mollified dilation*, given a function *f* ∈ *Ap*ð Þ� ð Þ and *r*∈ 1 2 , 1 � �, define the *mollified dilation fr* : 1þ*r* 2*r* ! of *f* as

$$f\_r(z) \coloneqq \int\_{\mathbb{D}} f(rw) \eta\_r(z - w) \mathrm{d}A(w).$$

where *ρ* stands for the complex disk with radius *ρ*,

$$\eta\_r(z) \coloneqq \frac{4r^2}{\left(1-r\right)^2} \eta\left(\frac{2rz}{1-r}\right),$$

and

$$\eta(z) \coloneqq \begin{cases} \exp\left(\frac{1}{\left|z\right|^2 - 1}\right), & \text{if} \quad |z| < 1, \\ 0, & \text{if} \quad |z| \geqslant 1, \end{cases}$$

Mollified dilations allow us to approximate analytic functions in the variable exponent Bergman spaces by functions that are analytic on a neighborhood of the unit disk.

**Theorem 14.** *Let p*∈Plogð Þ *and let f* ∈ *Ap*ð Þ� ð Þ *. Then for* ½ ⩽*r*<1*, fr* ∈ *Ap*ð Þ� ð Þ , *and*

$$\sup\_{\forall \boldsymbol{\varepsilon} \leqslant r < 1} \|f\_r\|\_{A^{p(\cdot)}(\mathbb{D})} \lesssim \|f\|\_{A^{p(\cdot)}(\mathbb{D})}.$$

*Moreover,* ∥ *f r* � *f* ∥*Ap*ð Þ� ð Þ ! 0 *as r* ! 1�*.*

As a consequence of the previous theorem, we can approximate a function *f* ∈ *Ap*ð Þ� ð Þ by a function *fr* which is analytic in 1þ*r* 2*r* . Hence, the sequence of Taylor polynomials associated with *fr* converges uniformly on to *fr*. Therefore, given *ε*>0, there exists 1*=*2≤ *r*< 1 such that ∥*f* � *fr*∥*Ap*ð Þ� ð Þ <*ε=*2. And there exists a polynomial *p* such that ∥*p* � *fr*∥*Ap*ð Þ� ð Þ <*ε=*2.

A sequence of points f g *zn* ⊂ is a zero set for the Bergman space *Ap*ð Þ if there

1 � *r* � � � � , *r* ! 1�*:*

> þ 1 2*π* ð2*π* 0

¼ *O nr*ð Þ1*=p* � �, *r* ! 1�*:*

Finding an extension of such result to variable exponents would include making

A related concept that we would like to address in variable exponent Bergman spaces is that of sampling sequences. A sequence of points f g *zk* ⊂ is sampling in

One difference that is expected to occur in the case of variable exponents is that sampling sequences will probably not be invariant under automorphism of the unit disk. This is true for a constant exponent, and it will be interesting to find counterexamples if the exponent varies. Sampling sequences are related to the concept of "frames" in Hilbert spaces and can be thought as sequences that contain great information of the space. It is expected that a sampling sequence is a somehow "big" and "spread out" subset of the unit disk. The notion is clearly related in *Ap* to the

> 1 � j j *zk* 2 � �2

*δzk*

j j *f z*ð Þ*k p* ≤*C*∥*f* ∥*p*

*p:*

This equation holds for every analytic function on . Consequently, if we want to find out whether a similar result about the growth of the zeroes holds in the variable exponent case, then it is necessary to study the radial growth of functions in *Ap*ð Þ� ð Þ . Such results, in the case of a constant exponent again, depend on the

Although the problem of completely describing the zero sets in *Ap*ð Þ is still open, some information is available. For example, if *n r*ð Þ denotes the number of zeroes of a function *f* ∈ *Ap*ð Þ having modulus smaller than *r*>0, then it is

1 � *r* log 1

This result depends on studying the radial growth of functions in the Bergman space and on Jensen's formula for the sequence f g *zk* of zeroes of *f* such

> log ∣*zk*∣ *r* � �

�∈f g *zn* .

log ∣*f reiθ* � �∣d*θ:*

exists *f* ∈ *Ap*ð Þ such that *f z*ð Þ¼ *n* 0 for all *n* and *f z*ð Þ 6¼ 0 if *z*

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

*n r*ð Þ¼ *O* 1

log ∣*f*ð Þ 0 ∣ ¼ X*n r*ð Þ

.

Y*n r*ð Þ

1 ∣*zk*∣

*k*¼1

*c*∥*f* ∥*p*

*p* ≤ X∞ *k*¼1

*k*¼1

Another result involving zero sets is that if *f*ð Þ 0 6¼ 0, then

sense of the right-hand side term that is not even clear in this context.

*Ap*ð Þ if there exists constants *c*> 0 and *C*> 0 such that for all *f* ∈ *Ap*ð Þ

*μ* ¼ X∞ *k*¼1

1 � j j *zk* 2 � �2

known that

that ∣*zk*Þ<*r*.

subharmonicity of j j *f p*

**5.2 Sampling sequences**

concept of Carleson measures.

**73**

If we define a discrete measure *μ* as

#### **4.2 Carleson measures**

Another classical problem that has a variable exponent equivalent consists on finding a geometrical characterization of Carleson measures in variable exponent Bergman spaces. Given a positive Borel measure *μ* defined on the unit disk , we say that *μ* is a *Carleson measure* for the variable exponent Bergman space *Ap*ð Þ� ð Þ if there exists a constant *C*>0 such that

$$\|f\|\_{L^{p(\cdot)}(\mathbb{D},\mu)} \leqslant C \|f\|\_{A^{p(\cdot)}(\mathbb{D})}.$$

In other words, *μ* is a Carleson measure on *Ap*ð Þ� ð Þ if *Ap*ð Þ� ð Þ is continuously embedded in *Lp*ð Þ� ð Þ , *μ* .

Lennart Carleson gave a geometric characterization of Carleson measures on Hardy spaces *Hp* and used this result in his proof of the Corona theorem and some interpolation problems (cf. [25]). After that, Carleson measures have gained interest, and a geometric characterization is usually important in spaces of analytic functions.

In the case of the classical Bergman spaces *Ap*ð Þ , Carleson measures are characterized in terms of the measure on pseudo-hyperbolic disks (see [26]). An interesting fact is the independence of the characterization on the exponent *p*. In the case of variable exponent Bergman spaces, the independence of *p*ð Þ� does not hold unless the *p*ð Þ� satisfies the Log-Hölder condition. The counterexample for the general case and the theorem for the restricted case were found in [24].

**Theorem 15.** *Let μ be a finite, positive Borel measure on and p*ð Þ� ∈Plogð Þ *. Then μ is a Carleson measure for Ap*ð Þ� ð Þ *if and only if there exists a constant C*> 0 *such that for every* 0< *r*<1 *and every a*∈ *, μ*ð Þ *Dr*ð Þ *a* ≤*C*∣*Dr*ð Þ *a* ∣*, where Dr*ð Þ *a denotes pseudohyperbolic disk with center at a*∈ *and radius r:*

$$D\_r(\mathfrak{a}) \coloneqq \left\{ z \in \mathbb{D} \, : \, \left| \frac{\mathfrak{a} - z}{1 - \overline{a}z} \right| < r \right\}.$$

The main tool for the proof of the previous theorem is Eq. (2) that implies the following estimate for the reproducing kernels.

**Theorem 16.** *Let p*ð Þ� ∈Plogð Þ *. Then for every z*∈ *we have that*

$$\|k\_x\|\_{\mathcal{A}^{p(\cdot)}(\mathbb{D})} \asymp \frac{1}{\left(1-\left|\boldsymbol{z}\right|^2\right)^{2\left(1-1/p(\boldsymbol{z})\right)}}\ .$$

#### **5. Open problems**

#### **5.1 Zero sets**

An interesting area will be to study some analytic properties of functions belonging to variable exponent Bergman spaces. One starting point will be to study the structure of zero sets in the space.

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

polynomials associated with *fr* converges uniformly on to *fr*. Therefore, given *ε*>0, there exists 1*=*2≤ *r*< 1 such that ∥*f* � *fr*∥*Ap*ð Þ� ð Þ <*ε=*2. And there exists a

Another classical problem that has a variable exponent equivalent consists on finding a geometrical characterization of Carleson measures in variable exponent Bergman spaces. Given a positive Borel measure *μ* defined on the unit disk , we say that *μ* is a *Carleson measure* for the variable exponent Bergman space *Ap*ð Þ� ð Þ if

∥*f* ∥*Lp*ð Þ� ð Þ ,*μ* ⩽*C*∥*f* ∥*Ap*ð Þ� ð Þ *:*

In other words, *μ* is a Carleson measure on *Ap*ð Þ� ð Þ if *Ap*ð Þ� ð Þ is continuously

Lennart Carleson gave a geometric characterization of Carleson measures on Hardy spaces *Hp* and used this result in his proof of the Corona theorem and some interpolation problems (cf. [25]). After that, Carleson measures have gained interest, and a geometric characterization is usually important in spaces of analytic

In the case of the classical Bergman spaces *Ap*ð Þ , Carleson measures are characterized in terms of the measure on pseudo-hyperbolic disks (see [26]). An interesting fact is the independence of the characterization on the exponent *p*. In the case of variable exponent Bergman spaces, the independence of *p*ð Þ� does not hold unless the *p*ð Þ� satisfies the Log-Hölder condition. The counterexample for the general case

**Theorem 15.** *Let μ be a finite, positive Borel measure on and p*ð Þ� ∈Plogð Þ *. Then μ is a Carleson measure for Ap*ð Þ� ð Þ *if and only if there exists a constant C*> 0 *such that for every* 0< *r*<1 *and every a*∈ *, μ*ð Þ *Dr*ð Þ *a* ≤*C*∣*Dr*ð Þ *a* ∣*, where Dr*ð Þ *a denotes pseudo-*

> � � �

The main tool for the proof of the previous theorem is Eq. (2) that implies the

1 � j j *z*

An interesting area will be to study some analytic properties of functions belonging to variable exponent Bergman spaces. One starting point will be to study

n o

1 � *az*

1

2 � �2 1ð Þ �1*=p z*ð Þ *:*

� � � <*r*

*:*

*Dr*ð Þ *a* ≔ *z*∈ : *a* � *z*

**Theorem 16.** *Let p*ð Þ� ∈Plogð Þ *. Then for every z*∈ *we have that*

∥*kz*∥*Ap*ð Þ� ð Þ ≍

and the theorem for the restricted case were found in [24].

*hyperbolic disk with center at a*∈ *and radius r:*

following estimate for the reproducing kernels.

**5. Open problems**

the structure of zero sets in the space.

**5.1 Zero sets**

**72**

polynomial *p* such that ∥*p* � *fr*∥*Ap*ð Þ� ð Þ <*ε=*2.

*Advances in Complex Analysis and Applications*

there exists a constant *C*>0 such that

**4.2 Carleson measures**

embedded in *Lp*ð Þ� ð Þ , *μ* .

functions.

A sequence of points f g *zn* ⊂ is a zero set for the Bergman space *Ap*ð Þ if there exists *f* ∈ *Ap*ð Þ such that *f z*ð Þ¼ *n* 0 for all *n* and *f z*ð Þ 6¼ 0 if *z* �∈f g *zn* .

Although the problem of completely describing the zero sets in *Ap*ð Þ is still open, some information is available. For example, if *n r*ð Þ denotes the number of zeroes of a function *f* ∈ *Ap*ð Þ having modulus smaller than *r*>0, then it is known that

$$m(r) = O\left(\frac{1}{1-r}\log\left(\frac{1}{1-r}\right)\right), \qquad r \to 1^-.$$

This result depends on studying the radial growth of functions in the Bergman space and on Jensen's formula for the sequence f g *zk* of zeroes of *f* such that ∣*zk*Þ<*r*.

$$\log|f(0)| = \sum\_{k=1}^{n(r)} \log\left(\frac{|x\_k|}{r}\right) + \frac{1}{2\pi} \int\_0^{2\pi} \log|f'(re^{i\theta})| \,\mathrm{d}\theta \,\mathrm{d}\theta$$

This equation holds for every analytic function on . Consequently, if we want to find out whether a similar result about the growth of the zeroes holds in the variable exponent case, then it is necessary to study the radial growth of functions in *Ap*ð Þ� ð Þ . Such results, in the case of a constant exponent again, depend on the subharmonicity of j j *f p* .

Another result involving zero sets is that if *f*ð Þ 0 6¼ 0, then

$$\prod\_{k=1}^{n(r)} \frac{1}{|z\_k|} = O\left(n(r)^{1/p}\right), \qquad r \to 1^{-}.$$

Finding an extension of such result to variable exponents would include making sense of the right-hand side term that is not even clear in this context.

#### **5.2 Sampling sequences**

A related concept that we would like to address in variable exponent Bergman spaces is that of sampling sequences. A sequence of points f g *zk* ⊂ is sampling in *Ap*ð Þ if there exists constants *c*> 0 and *C*> 0 such that for all *f* ∈ *Ap*ð Þ

$$c \| f \|\_{p}^{p} \le \sum\_{k=1}^{\infty} \left( 1 - |z\_k|^2 \right)^2 |f(z\_k)|^p \le C \| f \|\_{p}^{p}.$$

One difference that is expected to occur in the case of variable exponents is that sampling sequences will probably not be invariant under automorphism of the unit disk. This is true for a constant exponent, and it will be interesting to find counterexamples if the exponent varies. Sampling sequences are related to the concept of "frames" in Hilbert spaces and can be thought as sequences that contain great information of the space. It is expected that a sampling sequence is a somehow "big" and "spread out" subset of the unit disk. The notion is clearly related in *Ap* to the concept of Carleson measures.

If we define a discrete measure *μ* as

$$\mu = \sum\_{k=1}^{\infty} \left( \mathbf{1} - |\mathbf{z}\_k|^2 \right)^2 \delta\_{\mathbf{z}\_k}$$

where *δzk* is the Dirac measure with a point mass at *zk*, then the sequence f g *zk* is sampling if *μ* is a Carleson measure and satisfies a "reverse" Carleson condition. Then it becomes relevant to find a geometric characterization of the measures that satisfy a reverse Carleson condition (see [27–29]). The characterization depends on the concept of dominant sets and it is independent of *p*. The methods used by Luecking for Bergman spaces rely on a calculation of the norm of evaluation functionals and a submean value property. Different techniques must be developed for the variable exponent counterpart.

on *Ap*ð Þ� ð Þ . One approach to follow is to address the boundedness problems of linear operators acting on *Ap*ð Þ� ð Þ through the use Rubio de Francia extrapolation. Given an open set Ω ⊂ , we denote by F a family of pairs of non-negative,

ess sup*z* ∈ Ω

ð Ω

*Given p* ∈Pð Þ Ω *, if p*0 ≤ *p*� ≤*p*þ < ∞ *and the maximal operator is bounded on*

∥*F*∥*Lp*ð Þ� ð Þ Ω ≤*Cp*ð Þ� ∥*G*∥*Lp*ð Þ� ð Þ Ω *:*

This result allows, under certain conditions, to pass from the of studying operators defined on variable exponent spaces, to study operators defined on weighted constant exponent spaces, and therefore it is possible to use known results from the

Another type of spaces we are interested is the class of analytic Besov spaces *Bp*.

These are the spaces of all analytic functions in the unit disk such that

ð *f* 0 ð Þ*z* � � � �

the exponents that leave the space invariant under disk automorphisms.

functions in *Bp*. It can be seen (see, e.g., [33]) that *f* ∈*Bp* if and only if

j j *f z*ð Þ� *f w*ð Þ *p* j j 1 � *zw*

*p* 1 � j j *z* 2 � �*p*�2

One particular property of this space is that they are Möbius invariant in the sense that ∥*f*∘*φ*∥*Bp* ¼ ∥*f* ∥*Bp* for every automorphism of the unit disk *φ*. This is probably not true in general for a variable exponent version due to the problems with the change of variables formula. A first question to address is to characterize

One useful technique to study such spaces is a derivative-free characterization of

A similar result in the case of variable exponents could serve as a starting point

ð *f* 0 ð Þ*z* � � � �

For each *p* this is a normed space with norm

∥*f* ∥*Bp* ¼ ∣*f*ð Þ 0 ∣ þ

ð ð

for investigating other properties of the space.

**75**

weights *w*. That is, the non-negative functions *w* such that

*F x*ð Þ*p*0*w x*ð Þd*x*≤*C*0

*of functions* F *is such that for all w* ∈ *A*1*,*

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

> ð Ω

ð Þ Ω *, then*

Muckenhoupt weights theory.

**5.4 Analytic Besov spaces**

*L p*ð Þ� *=p* ð Þ0 0

measurable functions defined on Ω, and by *A*1 we denote the class of Muckenhoupt

where *M* denotes the Hardy-Littlewood maximal operator. Rubio de Francia extrapolation in the setting of variable exponent spaces can be stated as follows. **Proposition 17** (Thm. 5.24 in [1]). *Suppose that for some p*0 ≥ 1 *the family of pairs*

*Mw z*ð Þ *w z*ð Þ < ∞

*G x*ð Þ*p*0*w x*ð Þd*x*, ð Þ *F*, *G* ∈ F*:*

d*A z*ð Þ< ∞*:*

*:*

*p* 1 � j j *z* 2 � �*p*�2 d*A z*ð Þ � �1*=p*

4 d*A z*ð Þd*A w*ð Þ< ∞*:*

#### **5.3 Operators in variable exponent Bergman spaces**

The characterization of Carleson measures obtained in [24] opens the possibility of studying the boundedness and compactness of certain operators acting on variable exponent Bergman spaces.

Among those operators worth studying are multiplication operators. Those are formally defined on a functions space F as

$$\mathcal{M}\_f(\mathbf{g}) = f \mathbf{g}$$

where *g* ∈ F. One natural question is to find the set of symbols *f* that make the operator *Mf* map the space *Ap*ð Þ� ð Þ to itself.

Such question has been addressed in the case of weighted Bergman spaces in [23]. The proof relies on a geometric characterization of Carleson measures, a composition with a disk isomorphism, and a theorem of change of variables. Such tool is difficult to use in the setting of variable exponents since any change of variables will also affect the exponent and consequently the spaces are not necessarily invariant under composition with isomorphisms. This makes the situation different from the case of a constant exponent.

Other operators related to multiplication are Toeplitz operators. Those have already being studied in [30] for the case of weighted Bergman spaces with nonradial weights. Given a function *φ*∈ *L*1 ð Þ , the Toeplitz operator *Tφ* is defined on the set of polynomials as

$$T\_{\boldsymbol{\varrho}}p(\boldsymbol{z}) = \int\_{\mathbb{D}} p(\boldsymbol{w}) k\_{\boldsymbol{w}}(\boldsymbol{z}) \boldsymbol{\varrho}(\boldsymbol{w}) \mathrm{d}A(\boldsymbol{w}).$$

One natural question is to ask whether this operator can be extended boundedly to the space *Ap*ð Þ� ð Þ . This question is addressed in [31] were a characterization of the boundedness and compactness of *Tφ* is found in terms of Carleson measures and the Berezin transform associated with *Tφ*. This type of results are sometimes referred to as Axler-Zheng theorems, and similar questions looking to establish the relation between the Berezin transform of a finite product of Toeplitz operators with its compactness have been addressed by several authors; a survey of this type of results is given in [32]. It is then reasonable to ask weather an Axler-Zheng type result for products of Toeplitz operators holds in the context of variable exponents.

The composition operators are another type of operators to be studied. Given an analytic self-map *φ* of the unit disk, the composition operator *Cφ* can be formally defined as

$$\mathsf{C}\_{\mathfrak{q}}(f) = f \circ \mathfrak{q}.$$

A natural question is to find function-theoretic conditions on *φ* that guarantee the boundedness and/or compactness of its associated composition operator acting

where *δzk* is the Dirac measure with a point mass at *zk*, then the sequence f g *zk* is sampling if *μ* is a Carleson measure and satisfies a "reverse" Carleson condition. Then it becomes relevant to find a geometric characterization of the measures that satisfy a reverse Carleson condition (see [27–29]). The characterization depends on the concept of dominant sets and it is independent of *p*. The methods used by Luecking for Bergman spaces rely on a calculation of the norm of evaluation functionals and a submean value property. Different techniques must be developed for

The characterization of Carleson measures obtained in [24] opens the possibility of studying the boundedness and compactness of certain operators acting on vari-

Among those operators worth studying are multiplication operators. Those are

*Mf*ð Þ¼ *g fg*

where *g* ∈ F. One natural question is to find the set of symbols *f* that make the

Such question has been addressed in the case of weighted Bergman spaces in [23]. The proof relies on a geometric characterization of Carleson measures, a composition with a disk isomorphism, and a theorem of change of variables. Such tool is difficult to use in the setting of variable exponents since any change of variables will also affect the exponent and consequently the spaces are not necessarily invariant under composition with isomorphisms. This makes the situation

Other operators related to multiplication are Toeplitz operators. Those have already being studied in [30] for the case of weighted Bergman spaces with non-

*p w*ð Þ*kw*ð Þ*z φ*ð Þ *w* d*A w*ð Þ*:*

One natural question is to ask whether this operator can be extended boundedly to the space *Ap*ð Þ� ð Þ . This question is addressed in [31] were a characterization of the boundedness and compactness of *Tφ* is found in terms of Carleson measures and the Berezin transform associated with *Tφ*. This type of results are sometimes referred to as Axler-Zheng theorems, and similar questions looking to establish the relation between the Berezin transform of a finite product of Toeplitz operators with its compactness have been addressed by several authors; a survey of this type of results is given in [32]. It is then reasonable to ask weather an Axler-Zheng type result for products of Toeplitz operators holds in the context of variable exponents. The composition operators are another type of operators to be studied. Given an analytic self-map *φ* of the unit disk, the composition operator *Cφ* can be formally

*Cφ*ð Þ¼ *f f*∘*φ:*

A natural question is to find function-theoretic conditions on *φ* that guarantee the boundedness and/or compactness of its associated composition operator acting

ð Þ , the Toeplitz operator *Tφ* is defined on

the variable exponent counterpart.

*Advances in Complex Analysis and Applications*

able exponent Bergman spaces.

formally defined on a functions space F as

operator *Mf* map the space *Ap*ð Þ� ð Þ to itself.

different from the case of a constant exponent.

*Tφp z*ð Þ¼

ð

radial weights. Given a function *φ*∈ *L*1

the set of polynomials as

defined as

**74**

**5.3 Operators in variable exponent Bergman spaces**

on *Ap*ð Þ� ð Þ . One approach to follow is to address the boundedness problems of linear operators acting on *Ap*ð Þ� ð Þ through the use Rubio de Francia extrapolation. Given an open set Ω ⊂ , we denote by F a family of pairs of non-negative, measurable functions defined on Ω, and by *A*1 we denote the class of Muckenhoupt weights *w*. That is, the non-negative functions *w* such that

$$\text{ess } \sup\_{z \in \Omega} \frac{Mw(z)}{w(z)} < \infty$$

where *M* denotes the Hardy-Littlewood maximal operator. Rubio de Francia extrapolation in the setting of variable exponent spaces can be stated as follows.

**Proposition 17** (Thm. 5.24 in [1]). *Suppose that for some p*0 ≥ 1 *the family of pairs of functions* F *is such that for all w* ∈ *A*1*,*

$$\int\_{\mathfrak{U}} F(\mathfrak{x})^{p\_0} w(\mathfrak{x}) \mathrm{d}\mathfrak{x} \le \mathcal{C}\_0 \int\_{\mathfrak{U}} G(\mathfrak{x})^{p\_0} w(\mathfrak{x}) \mathrm{d}\mathfrak{x}, \quad (F, G) \in \mathcal{F}.$$

*Given p* ∈Pð Þ Ω *, if p*0 ≤ *p*� ≤*p*þ < ∞ *and the maximal operator is bounded on L p*ð Þ� *=p* ð Þ0 0 ð Þ Ω *, then*

$$\|F\|\_{L^{p(\cdot)}(\mathfrak{\Omega})} \le C\_{p(\cdot)} \|G\|\_{L^{p(\cdot)}(\mathfrak{\Omega})}.$$

This result allows, under certain conditions, to pass from the of studying operators defined on variable exponent spaces, to study operators defined on weighted constant exponent spaces, and therefore it is possible to use known results from the Muckenhoupt weights theory.

#### **5.4 Analytic Besov spaces**

Another type of spaces we are interested is the class of analytic Besov spaces *Bp*. These are the spaces of all analytic functions in the unit disk such that

$$\int\_{\mathbb{D}} \left| \left. f'(z) \right|^p \left( 1 - \left| z \right|^2 \right)^{p-2} \mathrm{d}A \left( z \right) < \infty. \right| $$

For each *p* this is a normed space with norm

$$\|\|f\|\|\_{B\_p} = |f(\mathbf{0})| + \left(\int\_{\mathbb{D}} |f'(z)|^p \left(\mathbf{1} - |z|^2\right)^{p-2} \mathbf{d}A(z)\right)^{1/p}.$$

One particular property of this space is that they are Möbius invariant in the sense that ∥*f*∘*φ*∥*Bp* ¼ ∥*f* ∥*Bp* for every automorphism of the unit disk *φ*. This is probably not true in general for a variable exponent version due to the problems with the change of variables formula. A first question to address is to characterize the exponents that leave the space invariant under disk automorphisms.

One useful technique to study such spaces is a derivative-free characterization of functions in *Bp*. It can be seen (see, e.g., [33]) that *f* ∈*Bp* if and only if

$$\int\_{\mathcal{D}} \int\_{\mathcal{D}} \frac{|f(z) - f(w)|^p}{|1 - z\overline{w}|^4} \mathbf{d}A(z) \mathbf{d}A(w) < \infty.$$

A similar result in the case of variable exponents could serve as a starting point for investigating other properties of the space.

*Advances in Complex Analysis and Applications*

**References**

2013

461-482

817-822

[1] Cruz-Uribe D, Fiorenza A. Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Basel: Birkhäuser;

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

> restoration, numerical methods and extensions. In: Proceedings of the 1997 IEEE International Conference on Image Processing, Volume III; 1997.

[11] Bollt EM, Chartrand R, Esedoglu S,

compromise between total variation and

[12] Chen Y, Guo W, Zeng Q, Liu Y. A

reconstruction of apparent diffusion coefficient profiles from diffusion weighted images. Inverse Problems and

Schultz P, Vixie KR. Graduated adaptive image denoising, local

isotropic diffusion. Advances in Computational Mathematics. 2009;**31**

nonstandard smoothing in

Imaging. 2008;**2**(2):205-224

Mathematical Analysis and

[13] Wunderli T. On time flows of minimizers of general convex functionals of linear growth with variable exponent in BV space and stability of pseudo-solutions. Journal of

Applications. 2010;**364**(2):5915-5998

[14] Harjulehto P, Hästoö P, Lê ÚV, Nuortio M. Overview of differential equations with non-standard growth. Nonlinear Analysis: Theory Methods & Applications. 2010;**72**(12):4551-4574

[15] Mingione G. Regularity of minima, an invitation to the dark side of the calculus of variations. Applications of Mathematics. 2006;**51**(4):355-426

[16] Cowen CC Jr, MacCluer BI. Composition Operators on Spaces of Analytic Functions (Studies in Advanced Mathematics). Boca Ratón,

[17] Diening L, Harjulehto P, Hästö P, Růžička M. Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics. Berlin:

FL: CRC Press; 1995

Springer-Verlag; 2011

pp. 384-387

(1-3):61-85

[2] Samko S. On a progress in the theory of Lebesgue spaces with variable exponent: Maximal and singular operators. Integral Transforms and Special Functions. 2005;**16**(5–6):

[3] Acerbi E, Mingione G. Regularity results for electrorheological fluids, the stationary case. Comptes rendus de l'Académie des Sciences. 2002;**334**(9):

[4] Acerbi E, Mingione G. Regularity results for stationary electrorheological fluids. Archive for Rational Mechanics and Analysis. 2002;**164**(3):213-259

[5] Růžička M. Electrorheological Fluids, Modeling and Mathematical Theory. Lecture Notes in Mathematics. Berlin:

mathematical and numerical analysis of electrorheological fluids. Applications of Mathematics. 2004;**49**(6):565-609

[7] Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM Journal on Applied Mathematics. 2006;**66**(4):

Springer-Verlag; 2000

1383-1406 (electronic)

[8] Aboulaich R, Boujena S,

El Guarmah E. Sur un modèle nonlinéaire pour le débruitage de l'image. Comptes Rendus de l'Académie des Sciences. 2007;**345**(8):425-429

[9] Aboulaich R, Meskine D, Souissi A. New diffusion models in image

processing. Computers & Mathematics with Applications. 2008;**56**(4):874-882

[10] Blomgren P, Chan T, Mulet P, Wong CK. Total variation image

**77**

[6] Růžička M. Modeling,

### **Author details**

Gerardo A. Chacón1 \* and Gerardo R. Chacón2

\*Address all correspondence to<EMAIL_ADDRESS>
© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

### **References**

[1] Cruz-Uribe D, Fiorenza A. Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Basel: Birkhäuser; 2013

[2] Samko S. On a progress in the theory of Lebesgue spaces with variable exponent: Maximal and singular operators. Integral Transforms and Special Functions. 2005;**16**(5–6): 461-482

[3] Acerbi E, Mingione G. Regularity results for electrorheological fluids, the stationary case. Comptes rendus de l'Académie des Sciences. 2002;**334**(9): 817-822

[4] Acerbi E, Mingione G. Regularity results for stationary electrorheological fluids. Archive for Rational Mechanics and Analysis. 2002;**164**(3):213-259

[5] Růžička M. Electrorheological Fluids, Modeling and Mathematical Theory. Lecture Notes in Mathematics. Berlin: Springer-Verlag; 2000

[6] Růžička M. Modeling, mathematical and numerical analysis of electrorheological fluids. Applications of Mathematics. 2004;**49**(6):565-609

[7] Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM Journal on Applied Mathematics. 2006;**66**(4): 1383-1406 (electronic)

[8] Aboulaich R, Boujena S, El Guarmah E. Sur un modèle nonlinéaire pour le débruitage de l'image. Comptes Rendus de l'Académie des Sciences. 2007;**345**(8):425-429

[9] Aboulaich R, Meskine D, Souissi A. New diffusion models in image processing. Computers & Mathematics with Applications. 2008;**56**(4):874-882

[10] Blomgren P, Chan T, Mulet P, Wong CK. Total variation image

restoration, numerical methods and extensions. In: Proceedings of the 1997 IEEE International Conference on Image Processing, Volume III; 1997. pp. 384-387

[11] Bollt EM, Chartrand R, Esedoglu S, Schultz P, Vixie KR. Graduated adaptive image denoising, local compromise between total variation and isotropic diffusion. Advances in Computational Mathematics. 2009;**31** (1-3):61-85

[12] Chen Y, Guo W, Zeng Q, Liu Y. A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images. Inverse Problems and Imaging. 2008;**2**(2):205-224

[13] Wunderli T. On time flows of minimizers of general convex functionals of linear growth with variable exponent in BV space and stability of pseudo-solutions. Journal of Mathematical Analysis and Applications. 2010;**364**(2):5915-5998

[14] Harjulehto P, Hästoö P, Lê ÚV, Nuortio M. Overview of differential equations with non-standard growth. Nonlinear Analysis: Theory Methods & Applications. 2010;**72**(12):4551-4574

[15] Mingione G. Regularity of minima, an invitation to the dark side of the calculus of variations. Applications of Mathematics. 2006;**51**(4):355-426

[16] Cowen CC Jr, MacCluer BI. Composition Operators on Spaces of Analytic Functions (Studies in Advanced Mathematics). Boca Ratón, FL: CRC Press; 1995

[17] Diening L, Harjulehto P, Hästö P, Růžička M. Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics. Berlin: Springer-Verlag; 2011

**Author details**

**76**

Gerardo A. Chacón1

\* and Gerardo R. Chacón2

\*Address all correspondence to<EMAIL_ADDRESS>
© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

1 Universidad Antonio Nariño, Bogotá, Colombia

*Advances in Complex Analysis and Applications*

2 Gallaudet University, Washington, D.C., USA

provided the original work is properly cited.

[18] Rafeiro H, Rojas E. Espacios de Lebesgue con exponente variable: Un espacio de Banach de funciones medibles (Spanish). Vol. 2014. Caracas: IVIC—Instituto Venezolano de Investigaciones Científicas. p. XI+136

[19] Chacón GR, Rafeiro H. Variable exponent Bergman spaces. Nonlinear Analysis: Theory Methods & Applications. 2014;**105**:41-49

[20] Chacón GA, Chacón GR. Analytic variable exponent Hardy spaces. Advances in Operator Theory. 2019;**4**: 738-749

[21] Sharapudinov II. Uniform boundedness in *Lp* (*p* ¼ *p x*ð Þ) of some families of convolution operators. Mathematical Notes. 1996;**59**:205-212

[22] Bergman S. The Kernel Function and Conformal Mapping. 2nd ed. Providence, R. I.: American Mathematical Society; 1970

[23] Zhao R. Pointwise multipliers from weighted Bergman spaces and Hardy spaces to weighted Bergman spaces. Annales Academiæ Scientiarum Fennicæ. 2004;**29**:139-150

[24] Chacón GR, Rafeiro H, Vallejo JC. Carleson measures for variable exponent Bergman spaces. Complex Analysis and Operator Theory. 2017;**11**: 1623-1638. DOI: 10.1007/s11785-016- 0573-0

[25] Garnett JW. Bounded Analytic Functions. Berlin: Springer-Verlag; 2007

[26] Duren P, Schuster A. Mathematical surveys and monographs, 100. In: Bergman Spaces. Providence, RI: American Mathematical Society; 2004

[27] Luecking D. Closed-range restriction operators on weighted Bergman spaces. Pacific Journal of Mathematics. 1984;**110**:145-160

[28] Luecking D. Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives. American Journal of Mathematics. 1985; **107**:85-111

**Chapter 6**

*Francisco Bulnes*

the reinterpretation of vector fields.

integral operators

**1. Introduction**

**79**

**Abstract**

Integral Geometry and

Riemannian Manifold

Cohomology in Field Theory on

The study of the relationships between the integration invariants and the different classes of operators, as well as of functions inside the context of the integral geometry, establishes diverse homologies in the dual space of the functions. This is given in the class of cohomology of the integral operators that give solution to certain class of differential equations in field theory inside a holomorphic context. By this way, using a cohomological theory of appropriate operators that establish equivalences among cycles and cocycles of closed submanifolds, line bundles and contours can be obtained by a cohomology of general integrals, useful in the evaluation and measurement of fields, particles, and physical interactions of diverse nature that occurs in the space-time geometry and phenomena. Some of the results applied through this study are the obtaining of solutions through orbital integrals for the tensor of curvature *Rμν*, of Einstein's equations, and using the imbedding of cycles in a complex Riemannian manifold through the duality: line bundles with cohomological contours and closed submanifolds with cohomological functional. Concrete results also are obtained in the determination of Cauchy type integral for

**Keywords:** complex Riemannian manifold, cocycles, cohomology, cohomology of

Obtaining an integral cohomology of general integral operators that determine complex analytic solutions through classes of cohomology born of the *∂*‐cohomology is necessary to use a holomorphic language with the purpose of obtaining the holomorphic forms that involve exact forms. In fact, this methodology is a way of so many perspectives that suggest the use of complex hyperholomorphic functions in approaching functions in complex analysis, although using fibrations on some quaternionic algebra. The holomorphic forms required in this language, are good to express the integral of complex vector fields as integral of line, which have more than enough lines and hyperplanes, respectively, in n and n, visualizing these

cycles, geometrical integration, integral curvature, integration invariants,

**AMS Classification:** 32A19 32C26 32C30 23C15 58 K70 14 J17

the Space-Time as Complex

[29] Mengestie T. Carleson type measures for Fock-Sobolev spaces. Complex Analysis and Operator Theory. 2014;**8**:1225-1256. DOI: 10.1007/s11785- 013-0321-7

[30] Chacón GR. Toeplitz operators on weighted Bergman spaces. Journal of Function Spaces and Applications. 2013; **2013**:5. Article ID: 753153. DOI: 10.1155/ 2013/753153

[31] Chacón GR, Rafeiro H. Toeplitz operators on variable exponent Bergman spaces. Mediterranean Journal of Mathematics. 2016;**13**:3525-3536. DOI: 10.1007/s00009-016-0701-0

[32] Cucković Z, Şahutoğlu S, Zheng A. Type theorem on a class of domains in *n*. Integral Equations and Operator Theory. 2013;**77**:397

[33] Zhu K. Operator Theory in Function Spaces, Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society; 2007

#### **Chapter 6**

[18] Rafeiro H, Rojas E. Espacios de Lebesgue con exponente variable: Un espacio de Banach de funciones

IVIC—Instituto Venezolano de Investigaciones Científicas. p. XI+136

Analysis: Theory Methods & Applications. 2014;**105**:41-49

[21] Sharapudinov II. Uniform

738-749

0573-0

**78**

medibles (Spanish). Vol. 2014. Caracas:

*Advances in Complex Analysis and Applications*

[28] Luecking D. Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives. American Journal of Mathematics. 1985;

[29] Mengestie T. Carleson type measures for Fock-Sobolev spaces. Complex Analysis and Operator Theory. 2014;**8**:1225-1256. DOI: 10.1007/s11785-

[30] Chacón GR. Toeplitz operators on weighted Bergman spaces. Journal of Function Spaces and Applications. 2013; **2013**:5. Article ID: 753153. DOI: 10.1155/

[31] Chacón GR, Rafeiro H. Toeplitz operators on variable exponent

Bergman spaces. Mediterranean Journal of Mathematics. 2016;**13**:3525-3536. DOI: 10.1007/s00009-016-0701-0

[32] Cucković Z, Şahutoğlu S, Zheng A. Type theorem on a class of domains in *n*. Integral Equations and Operator

[33] Zhu K. Operator Theory in Function Spaces, Mathematical Surveys and Monographs. Providence, RI: American

**107**:85-111

013-0321-7

2013/753153

Theory. 2013;**77**:397

Mathematical Society; 2007

[19] Chacón GR, Rafeiro H. Variable exponent Bergman spaces. Nonlinear

[20] Chacón GA, Chacón GR. Analytic variable exponent Hardy spaces. Advances in Operator Theory. 2019;**4**:

boundedness in *Lp* (*p* ¼ *p x*ð Þ) of some families of convolution operators. Mathematical Notes. 1996;**59**:205-212

[22] Bergman S. The Kernel Function and Conformal Mapping. 2nd ed. Providence, R. I.: American Mathematical Society; 1970

[23] Zhao R. Pointwise multipliers from weighted Bergman spaces and Hardy spaces to weighted Bergman spaces. Annales Academiæ Scientiarum Fennicæ. 2004;**29**:139-150

[24] Chacón GR, Rafeiro H, Vallejo JC. Carleson measures for variable exponent Bergman spaces. Complex Analysis and Operator Theory. 2017;**11**: 1623-1638. DOI: 10.1007/s11785-016-

[25] Garnett JW. Bounded Analytic Functions. Berlin: Springer-Verlag; 2007

[27] Luecking D. Closed-range restriction operators on weighted Bergman spaces. Pacific Journal of Mathematics. 1984;**110**:145-160

[26] Duren P, Schuster A. Mathematical surveys and monographs, 100. In: Bergman Spaces. Providence, RI: American Mathematical Society; 2004

## Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex Riemannian Manifold

*Francisco Bulnes*

### **Abstract**

The study of the relationships between the integration invariants and the different classes of operators, as well as of functions inside the context of the integral geometry, establishes diverse homologies in the dual space of the functions. This is given in the class of cohomology of the integral operators that give solution to certain class of differential equations in field theory inside a holomorphic context. By this way, using a cohomological theory of appropriate operators that establish equivalences among cycles and cocycles of closed submanifolds, line bundles and contours can be obtained by a cohomology of general integrals, useful in the evaluation and measurement of fields, particles, and physical interactions of diverse nature that occurs in the space-time geometry and phenomena. Some of the results applied through this study are the obtaining of solutions through orbital integrals for the tensor of curvature *Rμν*, of Einstein's equations, and using the imbedding of cycles in a complex Riemannian manifold through the duality: line bundles with cohomological contours and closed submanifolds with cohomological functional. Concrete results also are obtained in the determination of Cauchy type integral for the reinterpretation of vector fields.

**Keywords:** complex Riemannian manifold, cocycles, cohomology, cohomology of cycles, geometrical integration, integral curvature, integration invariants, integral operators

**AMS Classification:** 32A19 32C26 32C30 23C15 58 K70 14 J17

#### **1. Introduction**

Obtaining an integral cohomology of general integral operators that determine complex analytic solutions through classes of cohomology born of the *∂*‐cohomology is necessary to use a holomorphic language with the purpose of obtaining the holomorphic forms that involve exact forms. In fact, this methodology is a way of so many perspectives that suggest the use of complex hyperholomorphic functions in approaching functions in complex analysis, although using fibrations on some quaternionic algebra. The holomorphic forms required in this language, are good to express the integral of complex vector fields as integral of line, which have more than enough lines and hyperplanes, respectively, in n and n, visualizing these

fields like holomorphic sections of complex holomorphic bundles of fibrations X ! M.

But the *∂*‐cohomology exists naturally in coverings of Stein X ! M, like holomorphic forms. Then, the integral can be expressed on spaces M*δ*, and Δz, [1, 2], that are given as lines and hyperplanes of n and n, and that as such they are integral orbital of the complex manifolds M ¼ G*=*L and Δ ¼ Γ*=*Σ, belonging to a *∂*‐cohomology in holomorphic language.

The cohomologies of functionals and functions, respectively, that they can built through the complex cohomology of hyperspaces are generalizable for vector fields in the same sense of the coverings of Stein and therefore of the *∂*‐cohomology.

The following question arises, how to establish isomorphisms of cohomological classes for functions, functional, and vector fields inside the holomorphic context possible? How to determine a cohomological theory of integral operators that establish equivalences among these objects and the geometric objects of closed submanifolds, bundles of lines, and Feynman diagrams? How everything can decrease to a single cohomology of general integrals on contours or a cohomology of generalized functionals?

Before giving an answer to the previous questions, we give some preliminary definitions that we will use to fix concepts and outlines of the wanted general theory.

Let M, be a complex Riemannian manifold and be a sheaf of germs of holomorphic sections of a vector holomorphic sheaf.

**Definition 1.1.** We say that a spaceH•ð Þ M, J , is an integral cohomology (not in the sense of the set , but yes in the sense of the integrals of partial differential equations) of those *∂*‐equations, if this is a class of solutions or general integrals of these equations in M [1, 3].

**Definition 1.2.** An integral as generalized solution of a *∂*‐equation is a realization of an irreducible representation of a *∂*‐cohomology of complex closed submanifolds [2–4].

If the irreducible representations are unitary, then we have a complex L2 � cohomology or *∂*‐cohomology with coefficients in *L*2 . The corresponding integral operators to their integral cohomology are those of the complex Fourier analysis, which in the complex geometrical context (geometrical analysis) could be integrals constructed through integral transforms as the Hilbert transforms and other [3].

In the case of a real reductive Lie group, the generalized integrals come to be determined by their orbital integrals. Let G, be a real form of G, and P, their parabolic subgroup. The generalized integrals in G, are the integrals on open orbits of the generalized flag manifold G*=*P. For this way, if G ¼ U n, 1 ð Þ, and the generalized flag manifold is then n, the open orbit is the group of positive lines þ, which is an U n, 1 ð Þ�orbit in nð Þ . The integrals are of John type [3, 5, 6]:

$$\rho(\mathbb{P}) = \bigcap\_{\mathbb{L}^+ \subset \mathbb{P}^a(\mathbb{C})} \mathbf{f},\tag{1}$$

(see **Figure 1**). Then, it is possible to assign a vector bundle of lines with a unitary

*Electromagnetic waves in conformal actions of the group* SU 2, 2 ð Þ *on a two-dimensional flat model of the*

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex…*

The concepts of general integral and generalized integral are different because one refers to the whole class of cohomology of solutions of those *∂*‐equations about a complex analytic manifold, and the other refers to the classes of cohomology of solutions on cycles or cocycles of the complex Riemannian manifold [1, 2].

Another example in the recovery of a space of functions mainly the space M, is the recovery of real functions in the space n, through values of certain integral

∧ð Þ *ξ*, p ½ � p � ð Þ *ξ*, x �n

where the integral on p, is understood in terms of its regularization (role that carries out the Hilbert transform). The constant c depends on the dimension parity

To answer the first question, we need a structure of complexes that induce

the corresponding fibers X ! M and X ! Δ, of the double fibration [1].

**Definition 1.3.** A covering of Stein is a set of manifolds of Stein1 M*δ* and Δz, of

Let us consider the complexes given in Ref. [1], and let us consider the structure defined by a covering of Stein given by the set of open Mf g*δ* , and f g Δz , in the topology

1 A Stein manifold is an open orbit of a semi-simple Lie group G, in a generalized flag manifold whose nilpotent radical is opposite to the parabolic subgroup P, of G [12]. A definition of the Stein manifold that

Let be G ¼ G0, a real form of G, and *F*D ¼ G*=*H, an open orbit in the flag manifold *F* ¼ G*=*P ¼ G0*=*U*: F*D is Stein if H is compact (or equivalently, if *F*D is Hermitian symmetric). Likewise, a Stein

*τ*X ¼ f g ð Þ z, *ξ* ∈ M � ΞjX⊂ M � Ξð Þ ⇔M*δ* ∩ Ξz , (4)

dp

9 = ;

ωð Þ*ξ* (3)

operators. Such is the case of the formula tof xð Þ, recovered on n,

ð ∞

�∞ f

representation, where it can be classified.

*DOI: http://dx.doi.org/10.5772/intechopen.92969*

**Figure 1.**

*space-time [9–11].*

f xð Þ¼ c

isomorphisms in integral cohomology.

manifold is a Hermitian symmetric flag domain.

**81**

ð

8 < :

Γ

of the space n, where it was carrying the tomography [6].

uses the Hermitian structure of a complex holomorphic manifold is:

The general integral in this case is given by the twistor transform [7] on the corresponding homogeneous bundle of lines, that is to say:

$$\mathcal{T}: \mathbf{H}^1(\mathbb{P}^+, \mathbf{O}(-\mathbf{n} - \mathbf{2})) \to \mathbf{H}^1(\mathbb{P}^{+\*}, \mathbf{O}(\mathbf{n} - \mathbf{2})),\tag{2}$$

Using the twistor transform like intertwining operator of induced tempered representations on a *∂*‐cohomology, we have representations of SU 2, 2 ð Þ that are orbits of a fundamental unitary g, K ð Þ�module of the electrodynamics [8]

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex… DOI: http://dx.doi.org/10.5772/intechopen.92969*

#### **Figure 1.**

fields like holomorphic sections of complex holomorphic bundles of fibrations

But the *∂*‐cohomology exists naturally in coverings of Stein X ! M, like holomorphic forms. Then, the integral can be expressed on spaces M*δ*, and Δz, [1, 2], that are given as lines and hyperplanes of n and n, and that as such they are integral orbital of the complex manifolds M ¼ G*=*L and Δ ¼ Γ*=*Σ, belonging to a

The cohomologies of functionals and functions, respectively, that they can built through the complex cohomology of hyperspaces are generalizable for vector fields in the same sense of the coverings of Stein and therefore of the *∂*‐cohomology. The following question arises, how to establish isomorphisms of cohomological classes for functions, functional, and vector fields inside the holomorphic context possible? How to determine a cohomological theory of integral operators that establish equivalences among these objects and the geometric objects of closed submanifolds, bundles of lines, and Feynman diagrams? How everything can decrease to a single cohomology of general integrals on contours or a cohomology of

Before giving an answer to the previous questions, we give some preliminary definitions that we will use to fix concepts and outlines of the wanted general theory. Let M, be a complex Riemannian manifold and be a sheaf of germs of

**Definition 1.1.** We say that a spaceH•ð Þ M, J , is an integral cohomology (not in the sense of the set , but yes in the sense of the integrals of partial differential equations) of those *∂*‐equations, if this is a class of solutions or general integrals of

**Definition 1.2.** An integral as generalized solution of a *∂*‐equation is a realization of an irreducible representation of a *∂*‐cohomology of complex closed

If the irreducible representations are unitary, then we have a complex L2

operators to their integral cohomology are those of the complex Fourier analysis, which in the complex geometrical context (geometrical analysis) could be integrals constructed through integral transforms as the Hilbert transforms and other [3]. In the case of a real reductive Lie group, the generalized integrals come to be determined by their orbital integrals. Let G, be a real form of G, and P, their parabolic subgroup. The generalized integrals in G, are the integrals on open orbits

of the generalized flag manifold G*=*P. For this way, if G ¼ U n, 1 ð Þ, and the

which is an U n, 1 ð Þ�orbit in nð Þ . The integrals are of John type [3, 5, 6]:

*φ*ð Þ¼ P

corresponding homogeneous bundle of lines, that is to say:

generalized flag manifold is then n, the open orbit is the group of positive lines þ,

The general integral in this case is given by the twistor transform [7] on the

Using the twistor transform like intertwining operator of induced tempered representations on a *∂*‐cohomology, we have representations of SU 2, 2 ð Þ that are orbits of a fundamental unitary g, K ð Þ�module of the electrodynamics [8]

ð

Lþ ⊂nð Þ

T : H1 þ ð , Oð Þ �n � 2 Þ ! H1 þ ∗ ð Þ ,O nð Þ � 2 , (2)

�

. The corresponding integral

f, (1)

X ! M.

*∂*‐cohomology in holomorphic language.

*Advances in Complex Analysis and Applications*

holomorphic sections of a vector holomorphic sheaf.

cohomology or *∂*‐cohomology with coefficients in *L*2

generalized functionals?

these equations in M [1, 3].

submanifolds [2–4].

**80**

*Electromagnetic waves in conformal actions of the group* SU 2, 2 ð Þ *on a two-dimensional flat model of the space-time [9–11].*

(see **Figure 1**). Then, it is possible to assign a vector bundle of lines with a unitary representation, where it can be classified.

The concepts of general integral and generalized integral are different because one refers to the whole class of cohomology of solutions of those *∂*‐equations about a complex analytic manifold, and the other refers to the classes of cohomology of solutions on cycles or cocycles of the complex Riemannian manifold [1, 2].

Another example in the recovery of a space of functions mainly the space M, is the recovery of real functions in the space n, through values of certain integral operators. Such is the case of the formula tof xð Þ, recovered on n,

$$\mathbf{f}(\mathbf{x}) = \mathbf{c} \left\{ \left\{ \int\_{-\infty}^{\infty} \mathbf{f}^{\wedge}(\xi, \mathbf{p}) [\mathbf{p} - (\xi, \mathbf{x})]^{-\mathbf{n}} d\mathbf{p} \right\} \alpha(\xi) \tag{3}$$

where the integral on p, is understood in terms of its regularization (role that carries out the Hilbert transform). The constant c depends on the dimension parity of the space n, where it was carrying the tomography [6].

To answer the first question, we need a structure of complexes that induce isomorphisms in integral cohomology.

**Definition 1.3.** A covering of Stein is a set of manifolds of Stein1 M*δ* and Δz, of the corresponding fibers X ! M and X ! Δ, of the double fibration [1].

Let us consider the complexes given in Ref. [1], and let us consider the structure defined by a covering of Stein given by the set of open Mf g*δ* , and f g Δz , in the topology

$$\tau\_{\mathbf{X}} = \{ (\mathbf{z}, \boldsymbol{\xi}) \in \mathbf{M} \times \boldsymbol{\Xi} | \mathbf{X} \subset \mathbf{M} \times \boldsymbol{\Xi} (\Leftrightarrow \mathbf{M}\_{\delta} \cap \boldsymbol{\Xi}\_{\mathbf{z}}) \},\tag{4}$$

1 A Stein manifold is an open orbit of a semi-simple Lie group G, in a generalized flag manifold whose nilpotent radical is opposite to the parabolic subgroup P, of G [12]. A definition of the Stein manifold that uses the Hermitian structure of a complex holomorphic manifold is:

Let be G ¼ G0, a real form of G, and *F*D ¼ G*=*H, an open orbit in the flag manifold *F* ¼ G*=*P ¼ G0*=*U*: F*D is Stein if H is compact (or equivalently, if *F*D is Hermitian symmetric). Likewise, a Stein manifold is a Hermitian symmetric flag domain.

Then, a complex in X is the space such that Ωr h, for any complex Ωr , in a corresponding long succession is given as follows:

$$
\{\mathfrak{a}^\prime\} \in \mathfrak{b}^\prime\}\tag{5}
$$

For the theorem of Buchdall Eastwood [12], we have that the orbits generalized in X, give us a new cohomological class that is related to the previous for an integral

and such that H• Ξ, *τ*‐1 ð Þffi Jð Þ*ν T* kerf g *D*‐equations G*=*H , has more than enough.

In particular, kerf g *D*‐equations , takes place the correspondence with the cycles of H•ð Þ M, J . In fact, kerf g *D*‐equations , is similar to a compact number of components on which G, acts transitively, and these belong together to the cocycles of

But kerf g *D*‐equations only exists as integral of those *∂*‐equations in M (with M,

M ¼ ∪ *σ* ∈ΣV*σ*, and Ξ ¼ ∪ *γ* ∈ΣV*γ* , then for n‐dimensional planes of a Grassmann manifold, G1,n had that M*δ* ¼ ∪ M*π=*z, and Δz ¼ ∪ Δz*=π*, which defines cycles in

However, fixed G exists alone a finite number of flag manifolds of certain biholomorphism of this type. These are in bijective correspondence with the conjugated classes of parabolic subalgebras of g, and each flag manifold admits a finite

Then, kerf g *D*‐equations , is made of a finite number of G�orbits, all which are closed and (iii) == > (ii). Then since each one of these G�orbits exists like an K�orbit of the space of classes G*=*K, with Nijenhuis null curvature tensor, then each

Its integrals are orbital, and their extensions to M*δ*, and Δz, are generalized integrals (since they are integral of line along the fibers of M*δ*, and Δz, respectively)

2 **Theorem.** The K� invariance given for the G�structure SGð Þ M , of complex holomorphic M, is induced to each closed submanifold given for the flag manifolds of the corresponding vector holomorphic G� bundle. Furthermore, the given integral cohomology on such complex submanifolds is equivalent to the

3 **Theorem.** Let be M ¼ G*=*K, an internal symmetric simply connected Riemannian manifold and of

T ∗ ð Þ M ⊗ E � � RM

� , �

It consists of a finite number of connected components on each a G, that acts transitively. Further, any

The requirement of the transitive action of G, on the orbits is, for example, indispensable to the spatial isotropy hypothesis in the constructions of an integral cohomology to the space–time curvature, since the curvature integrals must be determined on G�invariant orbits and will be calculated for reduction of the

I ðÞ¼j 0g �

I ð Þj ∈End ∧2

flag submanifold is an K�orbit of the vector holomorphic G�bundle of the 2n� dimensional irreducible symmetrical Riemannian manifold J Mð Þ.

� . This g establishes a generalized structure of M, which

the theorem II.2 [1],3 we have that each canonical fibration of a flag manifold will

, the G�orbits are K�orbits in X. Then (i) == > (ii). Now for

, mþ ½ �⊂ m, for integrability). Then, ∀z∈ M

ð Þ Vw , ∀w ∈F. Then,

i

Jð Þ*ν* � �, (7)

H•ð Þ! M, Jð Þ*ν* H• Ξ, *τ*‐1

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex…*

give a G�orbit in *Z*, for some internal symmetrical G�space M.

� � exist z∈ F, such that TzF ¼ *σ*TwF∈P

operator *T*, defined for

By the theorem II.1 [1]2

H• Ξ, *τ*‐1 ð Þ Jð Þ*ν* .

integrable) if RM

and *σ* ∈Σ ffi P

compact type. Then

**83**

I ðÞ¼j 0 �

i Vz

number of canonical fibrations.

underlies in its composition (in fact mþ,

*DOI: http://dx.doi.org/10.5772/intechopen.92969*

Hn‐1 U, *ρ* ð Þ ‐1 O Vð Þ , with U ⊂ M. Then (ii) implies (iii).

for which it is continued. Then, (ii) implies (i).♦.

integral cohomology on submanifolds of a maximum complex torus.

Z ¼ RM

flag G�manifold is realized as a such orbit for some M.

corresponding holonomy group on K� invariant orbits.

(i.e., to say, all the subcomplexes Ωr h, of the complex Ωr ). Then, the integral operator cohomology H•ð Þ M, J , in a complex manifold M, is that whose complexes conform a holomorphic structure that induces (in the corresponding integral manifold) a generalized according structure of integral submanifolds.

The integral submanifolds represent solutions of those *∂*‐equations in cycles of M. The integral submanifolds are the corresponding cocycles of M, under the integral operators of H•ð Þ M, J .

For example, if we take the complex manifold *M*, like a manifold of rational curves Ez, of a twistor manifold I [where should understand each manifold I, as the manifold of integral submanifolds (locally)], then its structure comes from a structure projective of their line on Ez, guided according to the vectors in TzM.

These correspond to sections of a normal bundle N Ez, to the curve Ez (infinitesimal deformations to the curve), that is to say, these conform the holomorphic structure that will induce the corresponding structure (that is to say in the corresponding integral manifold). In this case, the generalized structure of integral submanifolds is the Vð Þ k ‐conformal integrable structure given by I. The integral cohomology in this case is given by the family of rational curves.

The twistor content in this case helps and is necessary to establish the deformation of the integral curves of the vector sheaf of lines O kð Þ. In such case, the integral cohomology is H•ð Þ¼ M, J H0ð Þ I,O kð Þ .

This example is interesting not only for the fact of the definition of the integral cohomology, which defines, for this way, a class of integrals for M, but also for the fact of satisfaction of the integrability condition for the equation of the tensor of Weyl Wij ¼ 0, where H0 Iþ ð Þ ,O kð Þ (respectively, H0ð Þ I,O kð Þ ) are the solutions or integral of Wþ ¼ 0 (respectively, W� ¼ 0 [13, 14].

#### **2. Duality: line bundles with cohomological contours and closed submanifolds with cohomological functional**

We consider the following result on integral cohomology for integral geometry. **Proposition 2.1.** In the integral cohomology H•ð Þ M, J , on complex manifolds, the following statements are equivalent:

$$\text{c. } \mathbf{M}\_{\delta} = \cup\_{\mathbf{M}} \underline{\mathbf{z}}/\underline{\mathbf{z}} \text{, and } \Delta\_{\mathbf{z}} = \cup\_{\Delta} \underline{\mathbf{z}}/\underline{\mathbf{z}} \text{, where } \mathbf{H}^{\star}(\mathbf{M}, \mathfrak{J}) = \mathbf{H}^{\mathbf{n} \cdot \mathbf{1}}(\mathbf{U}, \rho^{\cdot} \mathbf{O}(\mathbf{V})).$$

*Proof.* The integrals on the open G� orbits satisfy the G�invariant integration:

$$\int\_{\mathbf{G}/\mathcal{H}} \mathbf{f} \mathbf{d} \Phi(\boldsymbol{\phi}^{\mathbf{r}}) = \int\_{\mathbf{G}/\mathcal{H}} (\mathbf{f} \circ \boldsymbol{\Phi}^{\mathbf{r}}) \boldsymbol{\phi}^{\mathbf{r}},\tag{6}$$

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex… DOI: http://dx.doi.org/10.5772/intechopen.92969*

For the theorem of Buchdall Eastwood [12], we have that the orbits generalized in X, give us a new cohomological class that is related to the previous for an integral operator *T*, defined for

$$\mathbf{H}^\bullet(\mathsf{M}, \mathfrak{J}(\nu)) \to \mathbf{H}^\bullet\left(\Xi, \mathfrak{r}^{\cdot 1}\mathfrak{J}(\nu)\right),\tag{7}$$

and such that H• Ξ, *τ*‐1 ð Þffi Jð Þ*ν T* kerf g *D*‐equations G*=*H , has more than enough. By the theorem II.1 [1]2 , the G�orbits are K�orbits in X. Then (i) == > (ii). Now for the theorem II.2 [1],3 we have that each canonical fibration of a flag manifold will give a G�orbit in *Z*, for some internal symmetrical G�space M.

In particular, kerf g *D*‐equations , takes place the correspondence with the cycles of H•ð Þ M, J . In fact, kerf g *D*‐equations , is similar to a compact number of components on which G, acts transitively, and these belong together to the cocycles of H• Ξ, *τ*‐1 ð Þ Jð Þ*ν* .

But kerf g *D*‐equations only exists as integral of those *∂*‐equations in M (with M, integrable) if RM I ðÞ¼j 0 � � . This g establishes a generalized structure of M, which underlies in its composition (in fact mþ, , mþ ½ �⊂ m, for integrability). Then, ∀z∈ M and *σ* ∈Σ ffi P i Vz � � exist z∈ F, such that TzF ¼ *σ*TwF∈P i ð Þ Vw , ∀w ∈F. Then, M ¼ ∪ *σ* ∈ΣV*σ*, and Ξ ¼ ∪ *γ* ∈ΣV*γ* , then for n‐dimensional planes of a Grassmann manifold, G1,n had that M*δ* ¼ ∪ M*π=*z, and Δz ¼ ∪ Δz*=π*, which defines cycles in Hn‐1 U, *ρ* ð Þ ‐1 O Vð Þ , with U ⊂ M. Then (ii) implies (iii).

However, fixed G exists alone a finite number of flag manifolds of certain biholomorphism of this type. These are in bijective correspondence with the conjugated classes of parabolic subalgebras of g, and each flag manifold admits a finite number of canonical fibrations.

Then, kerf g *D*‐equations , is made of a finite number of G�orbits, all which are closed and (iii) == > (ii). Then since each one of these G�orbits exists like an K�orbit of the space of classes G*=*K, with Nijenhuis null curvature tensor, then each flag submanifold is an K�orbit of the vector holomorphic G�bundle of the 2n� dimensional irreducible symmetrical Riemannian manifold J Mð Þ.

Its integrals are orbital, and their extensions to M*δ*, and Δz, are generalized integrals (since they are integral of line along the fibers of M*δ*, and Δz, respectively) for which it is continued. Then, (ii) implies (i).♦.

$$Z = \{ \mathsf{R}\_{\mathcal{T}}^{\mathsf{M}}(\mathbf{j}) \in \mathrm{End}\left(\wedge^{2}\mathsf{T}\*(\mathsf{M})\otimes\_{\mathbb{R}}\mathbf{E}\right)|\mathsf{R}\_{\mathcal{T}}^{\mathsf{M}}(\mathbf{j}) = \mathbf{0} \},$$

Then, a complex in X is the space such that Ωr

corresponding long succession is given as follows:

(i.e., to say, all the subcomplexes Ωr

*Advances in Complex Analysis and Applications*

operators of H•ð Þ M, J .

h, for any complex Ωr

h, of the complex Ωr

operator cohomology H•ð Þ M, J , in a complex manifold M, is that whose complexes conform a holomorphic structure that induces (in the corresponding integral manifold) a generalized according structure of integral submanifolds.

The integral submanifolds represent solutions of those *∂*‐equations in cycles of M. The integral submanifolds are the corresponding cocycles of M, under the integral

For example, if we take the complex manifold *M*, like a manifold of rational curves Ez, of a twistor manifold I [where should understand each manifold I, as the manifold of integral submanifolds (locally)], then its structure comes from a structure projective of their line on Ez, guided according to the vectors in TzM. These correspond to sections of a normal bundle N Ez, to the curve Ez (infinitesimal deformations to the curve), that is to say, these conform the holomorphic structure that will induce the corresponding structure (that is to say in the

corresponding integral manifold). In this case, the generalized structure of integral submanifolds is the Vð Þ k ‐conformal integrable structure given by I. The integral

deformation of the integral curves of the vector sheaf of lines O kð Þ. In such case, the

This example is interesting not only for the fact of the definition of the integral cohomology, which defines, for this way, a class of integrals for M, but also for the fact of satisfaction of the integrability condition for the equation of the tensor of Weyl Wij ¼ 0, where H0 Iþ ð Þ ,O kð Þ (respectively, H0ð Þ I,O kð Þ ) are the solutions or

We consider the following result on integral cohomology for integral geometry. **Proposition 2.1.** In the integral cohomology H•ð Þ M, J , on complex manifolds,

a. The open M*δ*, and Δz, are G�orbits opened up in X, and their integrals are

b. Exists an integral operator *T*, such that H•ð Þffi M, J *T*kerf g *D*‐equations ,

c. M*δ* ¼ ∪ M*π=*z, and Δz ¼ ∪ Δz*=π*, where H•ð Þ¼ M, J Hn‐1 U, *ρ* ð Þ ‐1O Vð Þ .

fdΦ *ϕ*r ð Þ¼

*Proof.* The integrals on the open G� orbits satisfy the G�invariant integration:

ð

G*=*H

f ∘ Φ‐1 � �*ϕ*r

, (6)

The twistor content in this case helps and is necessary to establish the

**2. Duality: line bundles with cohomological contours and closed**

cohomology in this case is given by the family of rational curves.

integral cohomology is H•ð Þ¼ M, J H0ð Þ I,O kð Þ .

integral of Wþ ¼ 0 (respectively, W� ¼ 0 [13, 14].

the following statements are equivalent:

generalized integral for M,

**82**

**submanifolds with cohomological functional**

ð

G*=*H

, in a

). Then, the integral

ð5Þ

2 **Theorem.** The K� invariance given for the G�structure SGð Þ M , of complex holomorphic M, is induced to each closed submanifold given for the flag manifolds of the corresponding vector holomorphic G� bundle. Furthermore, the given integral cohomology on such complex submanifolds is equivalent to the integral cohomology on submanifolds of a maximum complex torus.

3 **Theorem.** Let be M ¼ G*=*K, an internal symmetric simply connected Riemannian manifold and of compact type. Then

It consists of a finite number of connected components on each a G, that acts transitively. Further, any flag G�manifold is realized as a such orbit for some M.

The requirement of the transitive action of G, on the orbits is, for example, indispensable to the spatial isotropy hypothesis in the constructions of an integral cohomology to the space–time curvature, since the curvature integrals must be determined on G�invariant orbits and will be calculated for reduction of the corresponding holonomy group on K� invariant orbits.

**Proposition 2.2.** The nð Þ� � 1 *∂*� complex cohomology with coefficients in a complex holomorphic bundle of M, is a cohomology of hyperlines and hyperplanes4 .

Their demonstration is a simple consequence of the digression in part II of Ref. [1], on some basic integral *∂*�cohomologies on n�dimensional complex spaces

(see **Figure 2**), of this same philosophical dissertation of integral operators

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex…*

cohomology of contours (cohomological functional).

connecting map in the succession of Mayer-Vietoris:

with rule of correspondence for a complex coordinates system z1, z2, … , zn,

2 � �n‐1ð

*D*

the evaluation of f ∈ H Dð Þ, in the complex hyperplane *π*ð Þz , of D, with rule of correspondence:

∧ � � ⊗ z þ f

followed residual *∂*‐cohomology and the complex Radon transform on subvarieties of 1 [16].

where in particular the exterior algebra ∧0,qð Þ T ∗ ð Þ M ⊗ V , is generated for elements *π*ð Þz ∧*∂*R fð Þð ÞÞ z . Then

∧ ∈L Dð Þ, we have that:

∧ð Þz � � ¼ *∂* f

∧ *ζ*1, *ζ*2, … , *ζ*n ð Þ¼ i

*∂* f

example with cohomology space H3*υ*ð Þ M � Sing M, .

We define the following concept.

*DOI: http://dx.doi.org/10.5772/intechopen.92969*

**Proposition 2.3.** The integrals of contour are generalized function in a

is an integral cohomology of the form H•ð Þ M � Sing M, , where M, can be

understood as a twistor space corresponding to M. (see **Figure 3**). For example, this class belongs to the Feynman integrals.

manifolds X, and Y, respectively, *α*∈ Hpð Þ X, S , and *β* ∈ Hqð Þ Y, S .

*∂* ∗ : Hpþqð Þ! X∩ Y, S ⊗ T Hpþqþ1

**Definition 2.1.** Cohomological function of a cohomology H• M � SingM, Ωr ð Þ, 5

We consider p and differential q forms of the cohomologies on the complex

We consider their cup product given for *α* ∪ *β* ∈ Hpþqð Þ X∩Y, S ⊗ T , and the

R : H Dð Þ! L Dð Þ,

We consider *∂*, complex scalar mapping defined to Dolbeault cohomology. Let D ∗ ¼ Lð Þ H Dð Þ, , be the set of

evf : Lð Þ! H Dð Þ, ,

*π*ð Þz ð Þ¼ f <*π*ð Þz , f >,

R*∂*ð Þf ⊗ evf ¼ <*π*ð Þz f, *∂*ð Þ RF >,

5 Here, M, is the product of all the twistor spaces and Sing M, is the union of all the subspaces on which the Zð Þ *α*W*α* ‐1

, are singular factors. Its differential form is integrated over a contour, which can be traditional contour, for

Therefore p ¼ 0. Then R*∂*ð Þf ⊗ evf ∈ A0,qð Þ V . Then H0,q n ð Þ¼ *=*D, V H0,qð Þ D1, V , which is a Dolbeault cohomology. Of this form, it can be established that through the hyperspaces geometry can be determined a certain class of holomorphic complex functions using an appropriate *∂*‐ cohomology. Also Gindikin generalizes this idea using the ð Þ n‐1‐q ‐*∂*‐cohomology determining the Radon transform on hyperplanes defined as linearly concave domains of dimension q, first to the real case and after to the complex case [16]. For example, a good modern research to the respect is the

f zð Þ 1, z2, … , zn *δ* s‐ *ζ*1, *ζ*2, … , *ζ*n ð Þ ½ � ð Þ, zð Þ 1, z2, … , zn •

<dz1, dz2, … , dzn, dz1, dz2, … , dznÞ,

∧*∂*ð Þ¼ z *∂*ð Þz ⊗ R fð Þ ∈ Ap,q‐1

ð Þ V ,

,

ð Þ X∪Y, S ⊗ T , (8)

published in 2007.

f

hyperplanes corresponding to D. Let

Due to that *∂* R fð Þ¼ R*∂*ð Þf , ∀f

∀f ∈ H Dð Þ.

and Z*α* ð Þ A*α* ‐1

**85**

Let be M ffi n, and we consider concave linearly domains D, in n (or so better in n [16]). D, has structure of complex vector space. Let be D1 ¼ *ξ*n*=*D, a holomorphic convex linear domain conformed for holomorphic hyperplanes *π*ið Þ D , i ¼ 1, 2, 3, … , in D1. Let be H1 ð Þ D , the complex holomorphic functions space defined on D1. Let D ! M, be a fiber vector bundle seated in the complex holomorphic manifold M. Let Ap,qð Þ D , be the p, q ð Þ‐forms space on M, with values in D (that is to say, the global sections space of the fiber tangent bundle ∧p,qT ∗ ð Þ M ⊗ D). Of this way, the bi-graded algebra is the space

$$\mathcal{A}(\mathbf{D}) = \oplus\_{\mathbf{n}+\mathbf{m}-\mathbf{p}} \mathbf{A}^{\mathbf{n},\mathbf{m}}(\mathbf{D}),$$

Let *∂*‐be the scalar operator on the complex manifold M, with values on the vector bundle of global sections E ∧p,q ð Þ T ∗ ð Þ M ⊗ D , that is to say, the differential operator

$$\overline{\mathcal{O}} : \mathcal{E}(\wedge^{\mathbb{P},\mathbb{q}} \mathcal{T} \* (\mathbb{M}) \otimes \mathcal{D} \mid) \to \mathcal{E}\left(\wedge^{\mathbb{P},\mathbb{q}+1} \mathcal{T} \* (\mathbb{M}) \otimes \mathcal{D} \mid\right),$$

The operator *∂*, complies certain anticommutative properties [1]. Now, we consider the Radon transform on the complex holomorphic functions H Dð Þ, and its analogous for the complex holomorphic functionals H Dð Þ ∗ , through the corresponding diagram:

$$\begin{array}{l} \mathrm{D} \xrightarrow{\mathrm{f}} \mathrm{H}(\mathrm{D}) \xrightarrow{\mathrm{R}\_{7}} \mathrm{L}(\mathrm{D})\\ \downarrow \mathrm{F} \text{unctional} \downarrow \mathrm{F} \text{unctional} \downarrow \mathrm{F} \text{unctional} \downarrow,\\ \mathrm{D} \ast \xrightarrow{\wp} \mathrm{H}(\mathrm{D} \ast) \xrightarrow{\mathrm{R}\_{7}\ast} \mathrm{L}(\mathrm{D} \ast) \end{array}$$

and we consider an *∂*‐operator in D. The before diagram represents a first cohomological advance on the relation between functionals of the nð Þ� � 1 dimensional *∂*‐ cohomology with coefficients in a complex vector bundle Ωn(holomorphic complex bundle) and the integration of the cohomology on hyperplanes in D, which an integral geometry is equivalent to consider an adequate Radon transform in D. Likewise, and using the satellites *∂* R, and *∂*R�1 , of the before diagram and composing the diagram with the correspondences to R, on D, and D ∗ , we can have:

$$
\overline{\partial}\,\mathsf{R}(\mathsf{f}) = \mathsf{R}\overline{\partial}(\mathsf{f}),
$$

The details of the demonstration of this identity can be seen in Ref. [1].

Due to that R, is injective, this is equivalent to have the exact succession of Radon transform images:

$$\mathbf{0} \to \mathsf{R}\_{\mathsf{D}} \mathbf{A}^{0,0}(\mathbf{L}(\mathbf{D})^{\prime}) \to \mathbf{R} \mathbf{A}^{0,1}(\mathsf{H}(\mathbf{D})^{\prime}) \to \mathbf{R} \ast \mathbf{A}^{0,1}(\mathsf{L}(\mathbf{D})^{\prime}) \to \mathbf{0},$$

or in equivalent way, for the exact succession:

$$\mathbf{0} \to \mathbf{R}\_0^{-1} \mathbf{B}^{0,0} (\mathbf{L}(\mathbf{D})^{\ast}) \to \mathbf{R}^{-1} \mathbf{B}^{0,1} (\mathbf{H}(\mathbf{D})^{\ast}) \to \mathbf{R} \ast^{-1} \mathbf{B}^{0,1} (\mathbf{L}(\mathbf{D})^{\prime}) \to \mathbf{0},$$

Here the 00 and 0 denote the projectivizing of spaces for R. Then the Radon transform can be generalized to the *∂*‐

cohomology classes on the complex spaces D, respectively D1, in n, as the mapping:

R*∂* : *∂*‐cohomology of dimension n ! *∂*‐cohomology of dimension nð Þ ‐1 ,

whose restriction to a domain D, is the mapping:

$$
\mathsf{R}\_{\mathsf{\eth}}|\_{\mathsf{D}}: \mathsf{H}(\mathsf{D}) \to \mathsf{L}(\mathsf{D}),
$$

which satisfies the diagram of correspondences for functionals of a nð Þ ‐1 ‐dimensional cohomology with coefficients in a holomorphic vector bundle E ! M.

The natural question arises, what relation there is between the two different corresponding objects to functionals, that is to say, cohomology and functions?

The relation is an integral relation of certain *∂*‐cohomology, which comes defined for the Radon transform of the Dolbeault cohomology R*∂*.

The Radon transform can be viewed as the mapping of cohomological spaces:

$$\mathsf{R\_{\mathfrak{F}}} \colon \mathsf{H}^{\mathsf{p},\mathsf{n}}(\mathsf{D},\mathsf{V}) \to \mathsf{H}^{\mathsf{p},\mathsf{n}-1}(\mathsf{D},\mathsf{V}),$$

Therefore, it is enough to demonstrate that R*∂* Hp,n ð Þ ð Þ D, V , is the q*th*‐projection nð Þ ‐1 ‐dimensional of Hp,nð Þ D, V , in H0,qð Þ¼ D, V H0,nð Þ D, V . Remember that the Radon transform in the complex context D, is the continuous and analytic mapping [17]:

4 We can have a little digression with certain details on the complex Radon transform using submanifolds in the space 1 , to the *∂*� cohomology. Let be M, a complex holomorphic manifold (or complex Riemannian manifold [15]). We consider its corresponding reductive homogeneous space determined for the flag manifold *F* ¼ G*=*P, with P, a parabolic subgroup of G. We consider the open orbit given for the Stein manifold *F*D ¼ G*=*H (as was defined in the footnote 1) with H, a compact subgroup of the real form G0, of G.

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex… DOI: http://dx.doi.org/10.5772/intechopen.92969*

Their demonstration is a simple consequence of the digression in part II of Ref. [1], on some basic integral *∂*�cohomologies on n�dimensional complex spaces (see **Figure 2**), of this same philosophical dissertation of integral operators published in 2007.

**Proposition 2.3.** The integrals of contour are generalized function in a cohomology of contours (cohomological functional).

We define the following concept.

**Definition 2.1.** Cohomological function of a cohomology H• M � SingM, Ωr ð Þ, 5 is an integral cohomology of the form H•ð Þ M � Sing M, , where M, can be understood as a twistor space corresponding to M. (see **Figure 3**).

For example, this class belongs to the Feynman integrals.

We consider p and differential q forms of the cohomologies on the complex manifolds X, and Y, respectively, *α*∈ Hpð Þ X, S , and *β* ∈ Hqð Þ Y, S .

We consider their cup product given for *α* ∪ *β* ∈ Hpþqð Þ X∩Y, S ⊗ T , and the connecting map in the succession of Mayer-Vietoris:

$$\partial \ast : \mathbf{H}^{p+q}(\mathbf{X} \cap \mathbf{Y}, \mathbf{S} \otimes \mathbf{T}) \to \mathbf{H}^{p+q+1}(\mathbf{X} \cup \mathbf{Y}, \mathbf{S} \otimes \mathbf{T}),\tag{8}$$

$$
\mathsf{R}: \mathsf{H}(\mathsf{D}) \to \mathsf{L}(\mathsf{D}),
$$

with rule of correspondence for a complex coordinates system z1, z2, … , zn,

$$\begin{split} \mathbf{f}^{\wedge}(\zeta\_{1},\zeta\_{2},\ldots,\zeta\_{n}) &= \left(\frac{\mathbf{i}}{2}\right)^{\bullet \cdot 1} \int\_{D} \mathbf{f}(\mathbf{z}\_{1},\mathbf{z}\_{2},\ldots,\mathbf{z}\_{n}) \delta(\mathbf{s}\cdot[(\zeta\_{1},\zeta\_{2},\ldots,\zeta\_{n}),(\mathbf{z}\_{1},\mathbf{z}\_{2},\ldots,\mathbf{z}\_{n})]) \bullet \\ & \qquad \qquad \qquad \qquad \qquad < \mathrm{d}\mathbf{z}\_{1}, \mathrm{d}\mathbf{z}\_{2}, \ldots,\mathbf{d}\mathbf{z}\_{n}, \mathrm{d}\overline{\mathbf{z}}\_{1}, \operatorname{d}\overline{\mathbf{z}}\_{2}, \ldots,\mathbf{d}\overline{\mathbf{z}}\_{n}), \end{split}$$

∀f ∈ H Dð Þ.

**Proposition 2.2.** The nð Þ� � 1 *∂*� complex cohomology with coefficients in a complex holomorphic bundle of M, is a cohomology of hyperlines and hyperplanes4

4 We can have a little digression with certain details on the complex Radon transform using submanifolds in the space

Let be M ffi n, and we consider concave linearly domains D, in n (or so better in n [16]). D, has structure of complex vector space. Let be D1 ¼ *ξ*n*=*D, a holomorphic convex linear domain conformed for holomorphic hyperplanes

values in D (that is to say, the global sections space of the fiber tangent bundle ∧p,qT ∗ ð Þ M ⊗ D).

fiber vector bundle seated in the complex holomorphic manifold M. Let Ap,qð Þ D , be the p, q ð Þ‐forms space on M, with

Að Þ¼ D ⊕nþm¼pAn,mð Þ D ,

Let *∂*‐be the scalar operator on the complex manifold M, with values on the vector bundle of global sections

The operator *∂*, complies certain anticommutative properties [1]. Now, we consider the Radon transform on the complex holomorphic functions H Dð Þ, and its analogous for the complex holomorphic functionals H Dð Þ ∗ , through the

> L Dð Þ ↓Functional↓Functional↓Functional ,

L Dð Þ ∗ and we consider an *∂*‐operator in D. The before diagram represents a first cohomological advance on the relation between functionals of the nð Þ� � 1 dimensional *∂*‐ cohomology with coefficients in a complex vector bundle Ωn(holomorphic complex bundle) and the integration of the cohomology on hyperplanes in D, which an integral geometry is equivalent to

*∂* R fð Þ¼ R*∂*ð Þf ,

0 ! R0A0,0ð Þ! L Dð Þ″ RA0,1ð Þ! H Dð Þ″ R ∗ A0,1 L Dð Þ0 ! 0,

R*∂*jD :H Dð Þ! L Dð Þ, which satisfies the diagram of correspondences for functionals of a nð Þ ‐1 ‐dimensional cohomology with coefficients in a

The natural question arises, what relation there is between the two different corresponding objects to functionals, that is

ð Þ D, V ,

The relation is an integral relation of certain *∂*‐cohomology, which comes defined for the Radon transform of the

R*∂* : Hp,nð Þ! D, V Hp,n‐1

Therefore, it is enough to demonstrate that R*∂* Hp,n ð Þ ð Þ D, V , is the q*th*‐projection nð Þ ‐1 ‐dimensional of Hp,nð Þ D, V , in H0,qð Þ¼ D, V H0,nð Þ D, V . Remember that the Radon transform in the complex context D, is the continuous and analytic

Here the 00 and 0 denote the projectivizing of spaces for R. Then the Radon transform can be generalized to the *∂*‐

B0,1ð Þ! H Dð Þ″ R ∗ �1

B0,1 L Dð Þ0 ! 0,

*∂* : E ∧p,q ð T ∗ ð Þ M ⊗ D Þ!E ∧p,qþ1

*φ* H Dð Þ! ∗ R*∂* ∗

Due to that R, is injective, this is equivalent to have the exact succession of Radon transform images:

D ! f H Dð Þ!R*∂*

D ∗ !

consider an adequate Radon transform in D. Likewise, and using the satellites *∂* R, and *∂*R�1

composing the diagram with the correspondences to R, on D, and D ∗ , we can have:

0 B0,0ð Þ! L Dð Þ″ R‐1

cohomology classes on the complex spaces D, respectively D1, in n, as the mapping:

R*∂* : *∂*‐cohomology of dimension n ! *∂*‐cohomology of dimension nð Þ ‐1 ,

The Radon transform can be viewed as the mapping of cohomological spaces:

The details of the demonstration of this identity can be seen in Ref. [1].

or in equivalent way, for the exact succession: 0 ! R‐1

whose restriction to a domain D, is the mapping:

holomorphic vector bundle E ! M.