**Author details**

Abdolreza Jamilian1\*, Alireza Darnahal1 and Letizia Perillo2

\*Address all correspondence to<EMAIL_ADDRESS>
1 Tehran Dental Branch, Department of Orthodontics, Islamic Azad University, Tehran, Iran

2 Departments of Orthodontics, Second University of Naples, Naples, Italy


We notice that (*M*.1) and (*M*.3) are usual conditions of logarithmic convexity and non-quasianalyticity, respectively, and when *σ* = 1 and *τ* > 0 the conditions ( -*M*.2) and ( -*M*.2) become the standard conditions of stability under differential and ultradifferential operators, (*M*.2) and (*M*.2), respectively (see [5]). In the sequel we consider the sequence *Mτ*,*σ p* = *pτpσ* , *p* ∈ **N**, which fulfills the above mentioned conditions (see Lemma 2.2 in [1]). This particular choice slightly simplifies our exposition. Clearly, by choosing *σ* = 1 and *τ* > 1 we recover the well known Gevrey sequence *p*! *τ*.

Recall [4], the associated function related to the sequence *pτpσ* is defined by

$$T\_{\tau,\sigma,h}(k) = \sup\_{p \in \mathbb{N}} \ln \frac{h^{p^{\mathcal{F}}} k^p}{p^{\tau p^{\mathcal{F}}}}, \quad k > 0. \tag{1}$$

For *h* = *σ* = 1 and *τ* > 1, *Tτ*,1,1(*k*) is the associated function of the Gevrey sequence *p*! *τ*.

In the next lemma we derive the precise asymptotic behavior of the function *Tτ*,*σ*,*h* associated with the sequence *pτpσ* . This in turn highlights the essential difference between *Tτ*,*σ*,*h* and the associated functions determined by Gevrey type sequences.

We first introduce some notation. The Lambert *W* function is defined as the inverse function of *zez*, *z* ∈ **C**, wherefrom

$$\mathbf{x} = \mathcal{W}(\mathbf{x})e^{\mathcal{W}(x)}, \quad \mathbf{x} \ge 0.$$

We denote its principal (real) branch by *W*(*x*), *x* ≥ 0 (see [24,25]). It is a continuous, increasing and concave function on [0, ∞), *W*(0) = 0, *W*(*e*) = 1, and *W*(*x*) > 0, *x* > 0. It can be shown that *W* can be represented in the form of the absolutely convergent series

$$\mathcal{W}(\mathbf{x}) = \ln \mathbf{x} - \ln(\ln \mathbf{x}) + \sum\_{k=0}^{\infty} \sum\_{m=1}^{\infty} c\_{km} \frac{(\ln(\ln \mathbf{x}))^m}{(\ln \mathbf{x})^{k+m}}, \quad \mathbf{x} \ge \mathbf{x}\_0 > \mathbf{c}\_1$$

with suitable constants *ckm* and *x*0. Thus the following estimates hold:

$$
\ln x - \ln(\ln x) \le W(x) \le \ln x - \frac{1}{2}\ln(\ln x), \quad x \ge \varepsilon,\tag{2}
$$

with the equality in (2) if and only if *x* = *e*.

For given *σ* > 1, *τ*, *h* > 0, let

$$\mathcal{R}(h,k) := h^{-\frac{\sigma-1}{\tau}} e^{\frac{\sigma-1}{\sigma}} \frac{\sigma-1}{\tau \sigma} \ln k = h^{-\frac{\sigma}{\tau \sigma'}} e^{\frac{1}{\sigma'}} \frac{1}{\tau \sigma'} \ln k, \quad k > c\_{\tau}$$

where

$$\frac{1}{\sigma} + \frac{1}{\sigma'} = 1, \quad \text{i.e.} \quad \sigma' = \frac{\sigma}{\sigma - 1}.$$

**Lemma 1.** *Let h* > 0*, and let Tτ*,*σ*,*h be given by* (1)*. Then there exist constants B*1, *B*2, *b*1, *b*2 > 0 *such that*

$$\begin{aligned} \left( \,\_{B\_1} k \right)^{b\_1 \left( \frac{\ln k}{\ln(\ln k)} \right)^{\frac{1}{\sigma - 1}}} & \leq \exp \left\{ T\_{\tau, \sigma, h}(k) \right\} \leq B\_2 \, k^{\frac{b\_2 \left( \frac{\ln k}{\ln(\ln k)} \right)^{\frac{1}{\sigma - 1}}}} \quad \text{,} \quad k > \sigma. \end{aligned}$$

*More precisely, if*

$$c\_1 = \left(\frac{\sigma - 1}{\tau \sigma}\right)^{\frac{1}{\sigma - 1}}, \quad \text{and} \quad c\_2 = h^{-\frac{\sigma - 1}{\tau}} \sigma^{\frac{\sigma - 1}{\sigma}} \frac{\sigma - 1}{\tau \sigma}.$$

*then there exist constants A*1, *A*2 > 0 *such that*

$$\begin{aligned} A\_1 \, k^{\frac{1}{2}\frac{c-1}{\delta^\*}c\_1 \left(\frac{\ln k}{\ln(c\_2 \ln k)}\right)^{\frac{1}{\delta^\*-1}}} & \leq \exp\{T\_{\mathsf{r}, \mathsf{r}, h}(k)\} \leq A\_2 \, k^{c\_1 \left(\frac{\ln k}{\ln(c\_2 \ln k)}\right)^{\frac{1}{\delta^\*-1}}}, \quad k > c. \end{aligned}$$

**Proof.** Lemma 1 can be proved by following the arguments used in the proof of Theorem 2.1 in [4]. There it is shown that for given *h* > 0, *τ* > 0 and *σ* > 1 the following inequalities hold:

$$\begin{aligned} \left\| A\_{\tau,\sigma,h} \exp \left\{ \left( \frac{\ln k}{\left( 2^{\tau-1} \tau W(\mathcal{R}(h,k)) \right)^{\frac{1}{\tau}} \sigma'} \right)^{\sigma'} \right\} \right\| & \leq c^{T\_{\tau,\sigma,h}(k)} \\ & \leq A\_{\tau,\sigma,h} \exp \left\{ \left( \frac{\ln k}{\left( \tau \sigma' \mathcal{W}(\mathcal{R}(h,k)) \right)^{\frac{1}{\tau}}} \right)^{\sigma'} \right\}, \quad k > e, \end{aligned}$$

for some *Aτ*,*σ*,*h*, *A*˜ *τ*,*σ*,*h* > 0. Moreover, in the view of (2), it follows that

$$\begin{aligned} W^{-\frac{\sigma'}{\delta'}} (\mathcal{R}(h,k)) \left(\ln k\right)^{\sigma'} &\asymp \left(\frac{\ln k}{\ln \left(\mathcal{C}\_h \ln k\right)}\right)^{\frac{\sigma'}{\delta'}} \ln k, \quad k \to \infty, \\\\ \eta^{-\frac{\sigma-1}{\delta}} \varepsilon^{\frac{\sigma-1}{\delta'}} \frac{\sigma-1}{\cdots} &= h^{-\frac{\sigma'}{\pi \sigma'}} \varepsilon^{\frac{1}{\sigma'}} (\tau \sigma')^{-1}. \end{aligned}$$

with *Ch* := *h*− *σ*−1 *τ e τσ* = *h*− *σ e* (*τσ* Details are left for the reader.

We define (following the classical approach [5]):

$$T^\*\_{\tau,r,h}(k) = \sup\_{p \in \mathbb{N}} \ln \frac{h^{p^{r^\*}} k^p}{p^{p(\tau p^{r^\*-1} - 1)}}, \quad k > 0. \tag{3}$$

It turns out that *T*∗ *τ*,*σ*,*h*(*k*) enjoys the same asymptotic behavior as *Tτ*,*σ*,*h*; cf. Lemma A1 (a) in Appendix A. This is another difference between our approach and the classical ultradistribution theory, where *T*∗ plays an important role.

Next we recall the definition of spaces E*τ*,*σ*(*U*) and D*τ*,*σ*(*U*), where *U* is an open set in **R***d* ([1]).

Let *K* ⊂⊂ **R***d* be a regular compact set. Then, E*τ*,*σ*,*h*(*K*) is the Banach space of functions *φ* ∈ *C*∞(*K*) such that

$$\|\|\phi\|\|\_{\mathcal{E}\_{\mathbf{r},\mathbf{r},h}(K)} = \sup\_{a \in \mathbb{N}^d} \sup\_{\mathbf{x} \in K} \frac{|\partial^a \phi(\mathbf{x})|}{h^{|a|^r} |a|^{\tau |a|^r}} < \infty. \tag{4}$$

We have

$$\mathcal{E}\_{\mathsf{T}\_1,\mathsf{T}\_1,\mathsf{h}\_1}(K) \hookrightarrow \mathcal{E}\_{\mathsf{T}\_2,\mathsf{t}\mathsf{T}\_2,\mathsf{h}\_2}(K), \quad 0 < \mathsf{h}\_1 < \mathsf{h}\_2, \ 0 < \mathsf{\tau}\_1 < \mathsf{\tau}\_2, \ 1 < \sigma\_1 < \sigma\_2.$$

where → denotes the strict and dense inclusion.

The set of functions from E*τ*,*σ*,*h*(*K*) supported by *K* is denoted by D*K τ*,*σ*,*h* . Next,

$$\mathcal{E}\_{\{\tau,\sigma\}}(\mathsf{II}) = \varprojlim\_{K \subset \subset \mathsf{U}} \varinjlim\_{h \to \infty} \mathcal{E}\_{\tau,\sigma,h}(K),\tag{5}$$

$$\mathcal{E}\_{\left(\tau,\sigma\right)}\left(\mathcal{U}\right) = \varprojlim\_{K \subseteq \subset \mathcal{U}} \varprojlim\_{h \to 0} \mathcal{E}\_{\tau,\sigma,h}\left(K\right),\tag{6}$$

$$\mathcal{D}\_{\{\tau,\tau\}}(\mathcal{U}) = \varinjlim\_{K \subset \subset \mathcal{U}} \mathcal{D}\_{\{\tau,\tau\}}^{K} = \varinjlim\_{K \subset \subset \mathcal{U}} (\varinjlim\_{h \to \infty} \mathcal{D}\_{\tau,\sigma,h}^{K}) \, , \tag{7}$$

$$\mathcal{D}\_{(\mathsf{r},\mathsf{r})}(\mathsf{U}) = \varinjlim\_{K \subset \subset \mathsf{U}} \mathcal{D}\_{(\mathsf{r},\mathsf{r})}^{K} = \varinjlim\_{K \subset \subset \mathsf{U}} \underbrace{\left(\lim\_{h \to 0} \mathcal{D}\_{\mathsf{r},\mathsf{r},h}^{K}\right)}\_{\text{KL} \subset \mathbb{U}} \cdot \mathbf{1}\_{\mathsf{F}} \,\tag{8}$$

Spaces in (5) and (7) are called Roumieu type spaces, and (6) and (8) are Beurling type spaces. Note that all the spaces of ultradifferentiable functions defined by Gevrey type sequences are contained in the corresponding spaces defined above.

For the corresponding spaces of ultradistributions we have:

$$\mathcal{D}'\_{\{\tau,\sigma\}}(\mathcal{U}) = \varprojlim\_{K \subset \subset \mathcal{U}} \varprojlim\_{h \to 0} (\mathcal{D}^{K}\_{\tau,\sigma,h})', \quad \mathcal{D}'\_{\{\tau,\sigma\}}(\mathcal{U}) = \varprojlim\_{K \subset \subset \mathcal{U}} \varprojlim\_{h \to \infty} (\mathcal{D}^{K}\_{\tau,\sigma,h})'.$$

Topological properties of all those spaces are the same as in the case of Beurling and Roumieu type spaces given in [5].

We will use abbreviated notation *τ*, *σ* for {*τ*, *σ*} or (*τ*, *σ*). Clearly,

$$\mathcal{D}'(\mathcal{U}) \hookrightarrow \mathcal{D}'\_{\tau,\sigma}(\mathcal{U}) \hookrightarrow \varprojlim\_{t \to 1} \mathcal{D}'\_t(\mathcal{U})\_{\prime t}$$

where D *t*(*U*) = D *t*,1(*U*) denotes the space of Gevrey ultradistributions with index *t* > 1. More precisely, if (for *σ* > 1) we put

$$\mathcal{D}^{(\sigma)}(\mathcal{U}) = \varprojlim\_{\tau \to 0} \mathcal{D}\_{\tau,\sigma}(\mathcal{U}), \quad \text{and} \quad \mathcal{D}^{\{\sigma\}}(\mathcal{U}) = \varprojlim\_{\tau \to \infty} \mathcal{D}\_{\tau,\sigma}(\mathcal{U}),$$

then

$$\mathcal{D}'(\mathcal{U}) \hookrightarrow \mathcal{D}'^{\{\sigma\}}(\mathcal{U}) \hookrightarrow \mathcal{D}'^{\{\sigma\}}(\mathcal{U}) \hookrightarrow \varprojlim\_{t \to 1} \mathcal{D}'\_t(\mathcal{U})\_t$$

where D (*σ*)(*U*) and D {*σ*}(*U*) are dual spaces of D(*σ*)(*U*) and D{*σ*}(*U*), respectively.

Thus we are dealing with intermediate spaces between the space of Schwartz distributions and spaces of Gevrey ultradistributions. In the next section we show the boundary value result in the given framework. This, however, asks for the use of new techniques.

## **3. Main Result**

The condition (*M*.2) (also known as the stability under the ultradifferentiable operators), essential for the boundary value theorems in the framework of ultradistribution

spaces [5,13], is in our approach replaced by the condition ( -*M*.2). We note that in [19] a more general condition than (*M*.2) is considered. In the case of the sequence *Mτ*,*σ p* = *pτpσ* , *p* ∈ **N**, the asymptotic behaviour given in Lemma 1 is essentially used to prove our main result as follows.

**Theorem 1.** *Let σ* > 1*, U be an open set in* **R***d,* Γ *an open cone in* **R***d and γ* > 0*. Assume that F*(*z*)*, z* ∈ *Z is an analytic function, where*

$$Z = \{ z \in \mathbb{C}^d \mid \text{Re}\, z \in \mathcal{U}, \text{Im}\, z \in \Gamma, |\text{Im}\, z| < \gamma \},$$

*and such that*

$$|F(z)| \le A|y|^{-H\left(\frac{\ln(1/|y|)}{\ln(\ln(1/|y|))}\right)^{\frac{1}{s-1}}}, \quad z = x + iy \in Z\_{\prime\prime}$$

*for some A*, *H* > 0 *(resp. for every H* > 0 *there exists A* > 0*). Then*

$$F(\mathbf{x} + i\mathbf{y}) \to F(\mathbf{x} + i\mathbf{0}), \quad \mathbf{y} \to \mathbf{0}, \mathbf{y} \in \Gamma,\tag{9}$$

*in* D (*σ*)(*U*) *(resp.* D {*σ*}(*U*)*). More precisely, if*

$$|F(z)| \le A \exp\{T\_{(2^r - 1)\tau, \sigma, H}(1/|y|) \} \quad z = x + iy \in Z,\tag{10}$$

*for some A*, *H* > 0 *(resp. for every H* > 0 *there exists A* > 0*) then* (9) *holds in* D (*τ*/2*σ*−1,*σ*)(*U*) *(resp.* D {*τ*/2*σ*−1,*σ*}(*U*)*).*

**Proof.** Let *K* ⊂⊂ *U* and *ϕ* ∈ D*K τ*/2*σ*−1,*σ*. Moreover, let *κ* ∈ D*τ*/2*σ*−1,*σ*(**R***d*) be such that supp *κ* ⊆ *B*(0, 2), *κ* = 1 on *B*(0, 1).

In the sequel we denote *mp* = *pτ*((2*p*)*σ*−1−1) , *p* ∈ **N**. Clearly, *mp* is an increasing sequence and *mp* → ∞ as *p* → ∞.

Fix *h* > 0, and let

$$\kappa\_{\mathfrak{a}}(y) = \kappa(4lm\_{|\mathfrak{a}|}y), \quad \mathfrak{a} \in \mathbf{N}^d, \ y \in \mathbf{R}^d.$$

Note that

$$\text{supp}\,\kappa\_{\mathfrak{k}} \subseteq \{ y \in \mathbf{R}^d \mid |y| \le 1/(2\hbar m\_{|a|}) \},\tag{11}$$

and for *j* = 1, . . . , *d*,

$$\text{supp}\,\partial\_{\mathcal{Y}/\mathfrak{A}} \subseteq \{ y \in \mathbb{R}^d \, | \, 1/(4lm\_{|\mathfrak{a}|}) \le |y| \le 1/(2lm\_{|\mathfrak{a}|}) \}, \quad \mathfrak{a} \in \mathbb{N}^d. \tag{12}$$

Let

$$\Phi(z) = \sum\_{a \in \mathbf{N}^d} \frac{\partial^a \varphi(\mathbf{x})}{|a|^{\mathsf{T}[a]}} (iy)^a \kappa\_a(y), \quad z = \mathbf{x} + iy \in \mathbf{C}^d. \tag{13}$$

Clearly, Φ is a smooth function in R2*d* and Φ(*x*) = *ϕ*(*x*) for *x* ∈ *K*. Fix *Y* = (*Y*1,...,*Yd*) ∈ Γ, *Y* = 0, |*Y*| < *γ*, and set

$$Z\_Y = \{ \mathbf{x} + it\mathbf{Y} \mid \mathbf{x} \in \mathbb{K}, t \in (0, 1] \}. \tag{14}$$

In order to use Stoke's formula (see [13]) we need to estimate Φ and its derivatives on *ZY*. To that end we had to adjust the standard technique in a nontrivial manner.

Let us show that there exists *Ah* > 0 such that

$$|\Phi(z)| \le A\_h \|\|\varphi\|\|\_{\mathcal{E}\_{\mathbf{r}/2^{\mathfrak{s}t-1}, \mathfrak{a}, \mathbf{h}} \prime} \quad \mathbf{h} > \mathbf{0}, \; z \in Z\_Y. \tag{15}$$

Note that (11) implies

$$|tY|^{|a|}|\kappa\_a(tY)| \le \frac{1}{(2\lim\_{|a|})^{|a|}} = \frac{|a|^{\tau|a|}}{(2h)^{|a|}|a|^{2^{r-1}\tau|a|^{\tau}}}, \quad t \in (0,1], a \in \mathbb{N}^d.$$

and therefore we obtain

$$\begin{split} |\Phi(\boldsymbol{z})| \leq \sum\_{\boldsymbol{a} \in \mathbf{N}^{d}} \frac{|\partial^{\boldsymbol{a}}\varrho(\boldsymbol{x})|}{|\boldsymbol{a}|^{\tau|\boldsymbol{a}|}} |tY|^{\boldsymbol{a}} |\kappa\_{\boldsymbol{a}}(tY)| \leq \sum\_{\boldsymbol{a} \in \mathbf{N}^{d}} \frac{|\partial^{\boldsymbol{a}}\varrho(\boldsymbol{x})|}{(2\hbar)^{|\boldsymbol{a}|} |\boldsymbol{a}|^{2^{\sigma-1}\tau|\boldsymbol{a}|^{\sigma}}} \\ \leq ||\varrho||\_{\mathcal{E}\_{\boldsymbol{r}/2^{\sigma-1},\boldsymbol{r},\boldsymbol{h}}} \sum\_{\boldsymbol{a} \in \mathbf{N}^{d}} \frac{\hbar^{|\boldsymbol{a}|^{\sigma}} |\boldsymbol{a}|^{(\tau/2^{\sigma-1})|\boldsymbol{a}|^{\sigma}}}{(2\hbar)^{|\boldsymbol{a}|} |\boldsymbol{a}|^{2^{\sigma'-1}\tau|\boldsymbol{a}|^{\sigma}}} = A\_{\boldsymbol{h}} ||\varrho||\_{\mathcal{E}\_{\boldsymbol{r}/2^{\sigma-1},\boldsymbol{r},\boldsymbol{h}}}. \end{split}$$

where *Ah* = ∑ *α*∈**N***d h*|*α*| *σ*−|*α*| 2|*α*| |*α*|*τ*0|*α*|*σ* < ∞ for *τ*0 = *τ*(2*σ*−1 − 1 2*σ*−1 ) > 0. Hence (15) follows.

Next we estimate *∂zj* Φ(*z*), *j* ∈ {1, ... , *d*}, when *z* ∈ *ZY*. More precisely, we show that for a given *h* > 0, there exists *Bh* > 0 such that

$$|\partial\_{\overline{z\_{j}}}\Phi(z)| \le B\_{h} \|\varphi\|\_{\mathcal{E}\_{\mathbf{r}/2^{r}-1}\mathbf{r},h} \exp\{-T\_{(2^{r}-1)\mathbf{r},\sigma}(1/|tY|)\}, \quad z \in Z\_{\mathbf{Y}}.\tag{16}$$

By (11) and (12) it is sufficient to prove (16) for

$$1/(4\text{l}m\_{|a|}) \le |tY| \le 1/(2\text{l}m\_{|a|}), \quad 0 < t \le 1, a \in \mathbb{N}^d.\tag{17}$$

Note that for *z* ∈ *ZY* we have

$$\begin{split} \partial\_{\Xi\_{l}^{k}}\Phi(z) &= \frac{1}{2} \Big( \sum\_{\mathbf{a}\in\mathbf{N}^{d}} \frac{\partial^{\mathbf{a}+\varepsilon\_{j}}\varrho(\mathbf{x})}{|\mathbf{a}|^{\mathsf{T}[\mathbf{a}]}} i^{|\mathbf{a}|}(tY)^{\mathbf{a}}\kappa\_{\mathbf{a}}(tY) \\ &+ \sum\_{\mathbf{a}\in\mathbf{N}^{d}} \frac{\partial^{\mathbf{a}}\varrho(\mathbf{x})}{|\mathbf{a}|^{\mathsf{T}[\mathbf{a}]}} a\_{j} i^{|\mathbf{a}|+1} t^{|\mathbf{a}|} Y^{\mathbf{a}-\varepsilon\_{j}} \kappa\_{\mathbf{a}}(tY) \\ &+ \sum\_{\mathbf{a}\in\mathbf{N}^{d}} \frac{\partial^{\mathbf{a}}\varrho(\mathbf{x})}{|\mathbf{a}|^{\mathsf{T}[\mathbf{a}]}} i^{|\mathbf{a}|+1} t^{|\mathbf{a}|+1} Y^{\mathbf{a}} 4im\_{|\mathbf{a}|} \cdot (\mathfrak{d}\_{y\_{j}}\mathbf{x}) (4im\_{|\mathbf{a}|} tY) \Big) = \frac{1}{2} (\mathcal{S}\_{1} + \mathcal{S}\_{2} + \mathcal{S}\_{3})(\mathbf{z}). \end{split}$$

We will show that there exists a constant *Bh* > 0 such that

$$\mathbb{E}\exp\{T\_{(2^{\sigma}-1)\tau,\sigma,h}(1/|tY|)\}|\mathcal{S}\_1(z)| \le B\_h ||\varrho||\_{\mathcal{E}\_{\tau/2^{\sigma}-1,\sigma,h'}} \qquad z \in Z\_Y.$$

The estimates for *S*2 and *S*3 can be obtained in a similar way. Let *Ch* = *C* max{*h*, *h*2*σ*−1 } where *C* > 0 is the constant from ( -*M*.2) . Using

$$(p+1)^{\sigma} \le 2^{\sigma-1}(p^{\sigma}+1), \quad p \in \mathbb{N}\_{\prime}$$

we obtain

$$\begin{split} \frac{h^{|\beta|^{\tau}}}{|tY|^{|\beta|}|\beta|^{(2^{\prime}-1)\tau|\beta|^{\prime}}}|S\_{1}(z)| &\leq \mathcal{C}\_{h} \|\,\varphi\|\!\!|\_{\mathcal{E}\_{\boldsymbol{\tau}/2^{\prime\prime}-1,\rho,\boldsymbol{h}}} \\ &\left(\sum\_{\begin{subarray}{c}\boldsymbol{a}\in\mathbf{N}^{d}\\ |\boldsymbol{a}|\leq|\boldsymbol{\beta}|\end{subarray}} + \sum\_{\begin{subarray}{c}\boldsymbol{a}\in\mathbf{N}^{d}\\ |\boldsymbol{a}|\leq|\boldsymbol{\beta}|\end{subarray}} \right) \frac{h^{|\boldsymbol{\beta}|^{\sigma}}\mathcal{C}\_{h}^{|\boldsymbol{a}|^{\sigma}}|\boldsymbol{a}|^{(\tau/2^{\prime\prime}-1)|\boldsymbol{a}|^{\sigma}}}{|\boldsymbol{\beta}|^{(2^{\prime\prime}-1)\tau|\beta|^{\prime}}|\boldsymbol{a}|^{\tau|\boldsymbol{a}|}} |tY|^{|\boldsymbol{a}|^{-|\boldsymbol{\beta}|}|\boldsymbol{\beta}|}|\kappa\_{\boldsymbol{a}}(tY)| \\ &= \mathcal{C}\_{h} ||\,\varphi\|\_{\mathcal{E}\_{\boldsymbol{\tau}/2^{\prime\prime}-1,\rho,\boldsymbol{h}}}(I\_{1,\boldsymbol{\beta}}+I\_{2,\boldsymbol{\beta}}), \quad \boldsymbol{\beta}\in\mathbb{N}^{d}, z\in\boldsymbol{Z}\_{Y}. \end{split}$$

It remains to show that sup*β*∈**N***d I*1,*β* and sup*β*∈**N***d I*2,*β* are finite. First we estimate *I*1,*β*. Note that for |*α*|≤|*β*|, the left-hand side in (17) implies

$$\begin{split} |tY|^{|a|-|\beta|} |\kappa\_a(tY)| &\leq (4\hbar m\_{|a|})^{|\beta|-|a|} \leq \frac{(4\hbar)^{|\beta|} m\_{|\beta|}^{|\beta|}}{h^{|a|} m\_{|a|}^{|a|}} \\ &\leq \frac{(4\hbar)^{|\beta|} |a|^{\tau|a|} |\beta|^{2^{\tau-1}\tau|\beta|^{\sigma}}}{h^{|a|} |a|^{2^{\tau-1}\tau|a|^{\sigma}}}, \quad t \in (0,1], a, \beta \in \mathbb{N}^d. \end{split} \tag{18}$$

Again, when *τ*0 = *τ*(2*σ*−1 − 1 2*σ*−1 ), by (18) we have

$$I\_{1,\beta} \leq \frac{(4h)^{|\beta|} h^{|\beta|^{r}}}{|\beta|^{(2^{r}-1-1)\tau|\beta|^{r}}} \sum\_{a \in \mathbf{N}^{d}} \frac{\mathsf{C}\_{h}^{|a|^{r}}}{h^{|a|} |a|^{\tau\_{0}|a|^{r}}} = \mathsf{C}\_{h}' \frac{(4h)^{|\beta|} h^{|\beta|^{r}}}{|\beta|^{(2^{r-1}-1)\tau|\beta|^{r}}}, \ \beta \in \mathbf{N}^{d}.$$

Hence, we conclude sup*β*∈**N***d I*1,*β* ≤ *C h* exp{*T*(2*σ*−1−1)*τ*,*σ*,*h*(4*h*)} < ∞. To estimate *I*2,*β* we first note that for |*α*| > |*β*| the right-hand side in (17) implies

$$|tY|^{|a-\beta|}|\kappa\_a(tY)| \le (1/(2\text{Im}\_{|a|}))^{|a-\beta|} \le 1/((2\text{h})^{|a-\beta|}m^{|a-\beta|}\_{|a-\beta|})$$

$$\le \frac{|a|^{\tau|a|}}{(2h)^{|a-\beta|}|a-\beta|^{2^{\tau-1}\tau|a-\beta|^{\tau}}}, \quad t \in (0,1], \ \kappa, \beta \in \mathbb{N}^d. \tag{19}$$

Set *C h* = *C* max{*Ch*, *C*2*σ*−1 *h* }. Using ( -*M*.2), (19) and (A4), for *β* ∈ **N***d*, we have

$$\begin{split} I\_{2,\delta} \leq & \sum\_{\substack{a \in \mathbb{N}^d \\ |\alpha| \geq |\beta|}} \frac{h^{|\beta|^{\sigma}} \mathsf{C}\_h^{|a|^{\sigma}} |a|^{(\pi/2^{\sigma-1})|a|^{\sigma}}}{|\beta|^{(2^{\sigma}-1)\tau|\beta|^{\sigma}} (2h)^{|\alpha-\beta|} |a-\beta|^{2^{\sigma-1}\tau|a-\beta|^{\sigma}}} \\ & \leq \mathbb{C} \sum\_{\substack{a \in \mathbb{N}^d \\ |\alpha| > |\beta|}} \frac{h^{|\beta|^{\sigma}} (\mathsf{C}\_h^{\prime\prime})^{|a-\beta|^{\sigma}} (\mathsf{C}\_h^{\prime\prime})^{|\beta|^{\sigma}} |a-\beta|^{\tau|a-\beta|^{\sigma}} |\beta|^{\tau|\beta|^{\sigma}}}{|\beta|^{(2^{\sigma}-1)\tau|\beta|^{\sigma}} (2h)^{|\alpha-\beta|} |a-\beta|^{2^{\sigma-1}\tau|a-\beta|^{\sigma}}} \\ & \leq \frac{(\mathsf{C}\_h^{\prime\prime} h)^{|\beta|^{\sigma}}}{|\beta|^{\tau(2^{\sigma}-2)|\beta|^{\sigma}}} \mathbb{C} \sum\_{\delta \in \mathbb{N}^d} \frac{(\mathsf{C}\_h^{\prime\prime})^{|\delta|^{\sigma}}}{(2h)^{|\delta|} |\delta|^{(2^{\sigma-1}-1)\tau|\delta|^{\sigma}}} = \mathsf{C}\_h^{\prime\prime\prime} \frac{(\mathsf{C}\_h^{\prime\prime} h)^{|\beta|^{\sigma}}}{|\beta|^{\tau(2^{\sigma}-2)|\beta|^{\sigma}}}. \end{split}$$

In particular, sup*β*∈**N***d I*2,*β* ≤ *C h* exp{*T*(2*σ*−2)*τ*,*σ*,*C h h*(1)} < ∞. Now, Stoke's formula gives

$$\begin{aligned} \langle F(\mathbf{x} + i0), \boldsymbol{\varrho}(\mathbf{x}) \rangle &= \int\_{K} F(\mathbf{x} + iY) \boldsymbol{\Phi}(\mathbf{x} + iY) d\mathbf{x} \\ &+ 2i \sum\_{j=1}^{d} Y\_{j} \int\_{0}^{1} \int\_{K} \partial\_{\Xi\_{j}^{\ast}} \boldsymbol{\Phi}(\mathbf{x} + itY) F(\mathbf{x} + itY) dt d\mathbf{x}, \end{aligned} \tag{20}$$

and we have used the assumptions in Theorem 1, and inequalities (15) and (16). Note that for *H* = *h*, (10) and (15) imply that there exists *Ah* > 0 such that

$$\begin{split} |F(\mathbf{x} + iY)\Phi(\mathbf{x} + iY)| &\leq A\_{\hbar} \|\varphi\|\_{\mathcal{E}\_{\mathbf{r}/2^{\sigma}-1, \mathbf{r}/\hbar}} \exp\{T\_{(2^{\sigma}-1)\mathbf{r}, \sigma, \hbar}(1/|Y|)\} \\ &= A\_{\hbar}^{\prime} \|\varphi\|\_{\mathcal{E}\_{\mathbf{r}/2^{\sigma}-1, \mathbf{r}/\hbar}} \quad \mathbf{x} \in \mathsf{K}\_{\prime} \quad \text{(21)} \end{split} \tag{21}$$

where *A h* = *Ah* exp{*T*(2*σ*−1)*τ*,*σ*,*h*(1/|*Y*|)}.

Moreover, (10) and (16) imply that there exists *Bh* > 0 such that

$$|\partial\_{\Xi\_{\vec{j}}}\Phi(z)F(z)| \le B\_h \|\|\varphi\|\|\_{\mathcal{E}\_{r/2^{r-1},r,h'}} \quad 1 \le j \le d, \ z \in Z\_Y. \tag{22}$$

Now (20)–(22) imply

$$|\langle F(\mathbf{x} + i\mathbf{0}), \boldsymbol{\varrho}(\mathbf{x})\rangle| \le B\_{\mathbf{h}}' ||\boldsymbol{\varrho}||\_{\mathcal{E}\_{\mathbf{r}/2^{\mathbf{r}} - \mathbf{1}\_{\boldsymbol{\varrho}^\*}, \mathbf{h}}} $$

for suitable constant *B h* > 0. This completes the proof of the second part of theorem, and the first part follows immediately.

**Remark 1.** *In order to show that any ultradistribution f is locally (on a bounded open set U) the sum of boundary values of analytic functions defined in the corresponding cone domains* Γ*j, j* = 1, ... , *k, one should proceed as in the classical theory. We multiply f by a cutoff test function κU equal to* 1 *over U, and obtain f*0 = *f κU equals f on U*. *Then we divide* R*n* \ {0} *into regular non overlapping cones* Γ*j*0*, j* = 1, . . . , *k, dual cones of* Γ*j, and define*

$$F\_j(z) = \langle f\_0(t), \int\_{\Gamma\_{j0}} \exp\{2\pi i (z - t)\eta\} d\eta \rangle, \quad z \in \mathbb{R}^n + i\Gamma\_j, \ j = 1, \dots, k.$$

*Now one can get the growth conditions for Fj*, *j* = 1, . . . , *k*, *and show that*

$$f \mathfrak{o} = \sum\_{j=1}^{k} \lim\_{\substack{y \to 0, y \in \Gamma\_j}} F\_j(x + iy), \quad x \in \mathbb{R}^n.$$

*The details will be given in a separate contribution where we will consider Lp versions of new ultradistributions spaces similar to the corresponding ones in [11].*

## **4. Wave Front Sets**

In this section we analyze wave front sets WF*τ*,*σ*(*u*) related to the classes E*τ*,*σ*(*U*) introduced in Section 2. We refer to [2–4,10,20] for properties of WF*τ*,*σ*(*u*) when *u* is a Schwartz distribution.

We begin with the definition.

**Definition 1.** *Let τ* > 0*, σ* > 1*, U open set in* **R***d and* (*x*0, *ξ*0) ∈ *U* × **R***d*\{0}*. Then for u* ∈ D {*τ*,*σ*}(*U*) *(respectively* D (*τ*,*σ*) (*U*)*),* (*x*0, *ξ*0) ∈ WF{*τ*,*σ*}(*u*) *(resp.* (*x*0, *ξ*0) ∈ WF(*τ*,*σ*)(*u*)*) if and only if there exists a conic neighborhood* Γ *of ξ*0*; a compact neighborhood K of x*0*; and φ* ∈ D{*τ*,*σ*}(*U*) *(respectively φ* ∈ D(*τ*,*σ*)(*U*)*) such that* supp *φ* ⊆ *K, φ* = 1 *on some neighborhood of x*0 *and*

$$|\bar{\phi}\bar{u}(\xi)| \le A \exp\{-T\_{\tau,\sigma,h}(|\xi|) \}, \quad \xi \in \Gamma,$$

*for some A*, *h* > 0 *(resp. for any h* > 0 *there exists A* > 0*).*

We will write WF*τ*,*σ*(*u*) for WF(*τ*,*σ*)(*u*) or WF{*τ*,*σ*}(*u*).

**Remark 2.** *Note that* WF*τ*,1(*u*) = WF*τ*(*u*)*, τ* > 1*, are Gevrey wave front sets investigated in [12].*

Moreover (cf. [3]), for 0 < *τ*1 < *τ*2 and *σ* > 1 we have

$$\text{WF}(u) \subset \text{WF}\_{\mathsf{T}\downarrow\mathcal{F}}(u) \subset \text{WF}\_{\mathsf{T}\downarrow\mathcal{F}}(u) \subset \bigcap\_{t>1} \text{WF}\_t(u) \subset \text{WF}\_A(u), \quad u \in \mathcal{D}'(\mathcal{U})\_\ast$$

where WF*A* denotes the analytic wave front set.

Let

$$\text{WF}^{\{\sigma\}}(\mathfrak{u}) = \bigcap\_{\tau > 0} \text{WF}\_{\tau, \sigma}(\mathfrak{u}), \quad \mathfrak{u} \in \mathcal{D}^{'\{\sigma\}}(\mathcal{U}),$$

and

$$\mathrm{WF}^{(\sigma)}(\mu) = \bigcup\_{\tau > 0} \mathrm{WF}\_{\tau,\sigma}(\mu), \quad \mu \in \mathcal{D}^{'(\sigma)}(\mathcal{U}).$$

For such wave front sets we have the following corollary which is an immediate consequence of Lemma 1.

**Corollary 1.** *Let u* ∈ D {*σ*}(*U*) *(resp.* D (*σ*)(*U*)*), σ* > 1*. Then* (*x*0, *ξ*0) ∈ WF{*σ*}(*u*) *(resp.* (*x*0, *ξ*0) ∈ WF(*σ*) (*u*)*) if and only if there exists a conic neighborhood* Γ *of ξ*0*; a compact neighborhood K of x*0*; and φ* ∈ D{*σ*}(*U*) *(resp φ* ∈ D(*σ*)(*U*)*) such that* supp *φ* ⊆ *K, φ* = 1 *on some neighborhood of x*0 *and*

$$|\widehat{\phi\mu}(\xi)| \le A|\xi|^{-H\left(\frac{\ln|\xi|}{\ln(\ln|\xi|)}\right)^{\frac{1}{\sigma-1}}}, \quad \xi \in \Gamma,$$

*for some A*, *H* > 0 *(resp. for any H* > 0 *there exists A* > 0*).*

We write *u*(*x*) = *F*(*x* + *i*Γ 0) if *u*(*x*) is obtained as boundary value of the analytic function *F* as *y* → 0 in Γ. Recall (cf. [9])

$$\Gamma\_0 = \{ \emptyset \in \mathbf{R}^d \mid y \cdot \emptyset \ge 0 \text{ for all } y \in \Gamma \}$$

denotes the dual cone of Γ.

To conclude the paper we prove the following theorem.

**Theorem 2.** *Let the assumptions of Theorem 1 hold, and let u*(*x*) = *F*(*x* + *i*Γ 0) ∈ D (*σ*)(*U*) *(resp.* D {*σ*}(*U*)*). Then*

$$\mathsf{WF}^{(\sigma)}(\mathsf{u}) \subseteq \mathsf{U} \times \Gamma\_{0\prime} \quad (resp. \ \mathsf{WF}^{\{\sigma\}}(\mathsf{u}) \subseteq \mathsf{U} \times \Gamma\_{0}).$$

*More precisely, if u*(*x*) = *F*(*x* + *i*Γ 0) ∈ D {*τ*/2*σ*−1,*σ*}(*U*) *(resp.* D (*τ*/2*σ*−1,*σ*) (*U*)*) then*

$$\text{WF}\_{\{(2^{\boldsymbol{\tau}}-1)\boldsymbol{\tau},\boldsymbol{\tau}\}}(\boldsymbol{\mu}) \subseteq \boldsymbol{\mathcal{U}} \times \Gamma\_{0\prime} \quad (resp. \quad \text{WF}\_{\{(2^{\boldsymbol{\tau}}-1)\boldsymbol{\tau},\boldsymbol{\tau}\}} \subseteq \boldsymbol{\mathcal{U}} \times \Gamma\_{0}).$$

**Proof.** Fix *x*0 ∈ *U* and *ξ*0 ∈ Γ0\{0}. Then there exists *Y* = (*Y*1, ... ,*Yd*) ∈ Γ, |*Y*| < *γ*, such that *Y* · *ξ*0 < 0. Moreover, there exists conical neighborhood *V* of *ξ*0 and constant *γ*1 > 0 such that *Y* · *ξ* ≤ −*γ*1|*ξ*|, for all *ξ* ∈ *V*. To see that, note that there exists *Br*(*ξ*0) such that *Y* · *ξ* < 0 for all *ξ* ∈ *Br*(*ξ*0). The assertion follows for *V* = {*sξ* |*s* > 0, *ξ* ∈ *Br*(*ξ*0)} and *γ*1 = inf *ξ*∈*V*, |*ξ*|=1 (−*Y*) · *ξ*.

Let *τ* > 0 and *τ*0 = (2*σ* − 1)*τ*. If *u*(*x*) = *F*(*x* + *i*Γ 0) ∈ D *τ*/2*σ*−1,*σ*(*U*) as in Theorem 1, then

$$|F(z)| \le A \exp\{T\_{\pi\_0, \sigma, h\_1}(1/|y|)\}, \quad z = x + iy \in Z,\tag{23}$$

for suitable constants *A*, *h*1 > 0.

Choose *ϕ* ∈ D*K τ*/4*σ*−1,*σ* such that *ϕ* = 1 in a neighborhood of *x*0 and let *ZY* be as in (14). Then there exists Φ (see (13)) such that

$$|\Phi(z)| \le A\_1, \quad \text{and} \quad |\partial\_{\overline{z}\_j^\*} \Phi(z)| \le A\_2 \exp\{-T\_{\tau\_0/2^{r-1}, r, l\_2} (1/|tY|)\},\tag{24}$$

*z* ∈ *ZY*, 1 ≤ *j* ≤ *d*, for suitable constants *A*1, *A*2, *h*2 > 0. Note that formula (20) implies

$$\begin{split} \widehat{(\!(\!qu\!u)}\!) \_{\ell}^{\mathbb{F}} (\!(\!z\!) ) &= \langle u(\!(\!x\!)e^{-i\, \!x\cdot \frac{\pi}{\sqrt{}}}\!) \!y\!(\!x\!) \rangle = \int\_{K} F(\!(\!x+i\, \!Y\!)e^{-i(\!(\!x+i\, \!Y)\cdot \frac{\pi}{\sqrt{}}\!)} \Phi(\!(x+i\, \!Y\!) ) dx \\ &+ 2i \sum\_{j=1}^{d} \mathcal{Y}\_{j} \int\_{0}^{1} \int\_{K} \partial\_{\Xi\_{j}^{\mathbb{F}}} \Phi(\!(x+it\, Y\!)F(x+it\,Y) \!e^{-i(\!(x+it\, Y\!) ) \cdot \frac{\pi}{\sqrt{}}} dt dx, \quad \!\!(\!z\!) \in V. \end{split}$$

Using (23) and (24) we have

$$|F(\mathbf{x} + iY)\Phi(\mathbf{x} + iY)e^{-i(\mathbf{x} + iY)\tilde{\xi}}| \le B|e^{-\gamma\_1|\tilde{\xi}|}, \quad \mathbf{x} \in K, \tilde{\xi} \in V,\tag{26}$$

for some *B* > 0.

Moreover, for *z* ∈ *ZY* and *ξ* ∈ *V* we have

$$\begin{split} |F(z)\partial\_{\overline{z\_{j}}}\Phi(z)e^{-i\overline{z}\cdot\overline{z}}| \\ \leq & \mathbb{C}\exp\{T\_{\overline{\eta},\sigma,h\_{1}}(1/|tY|) - T\_{\overline{\eta},/2^{r-1},\sigma\,h\_{2}}(1/|tY|) - t\gamma\_{1}|\xi|\} \\ \leq & \mathbb{C}\_{1}\exp\{-T\_{\overline{\eta},\sigma,\zeta\_{h\_{1},h\_{2}}}(1/(t\gamma\_{1})) - t\gamma\_{1}|\xi|\} \leq \mathbb{C}\_{2}\exp\{-T\_{\overline{\eta},\sigma,\zeta\_{h\_{1},h\_{2}}}(|\xi|) \}, \end{split} \tag{27}$$

for suitable constants *C*1, *C*2, *ch*1,*h*2 , *c h*1,*h*2 > 0, where we have used inequalities (A2) and (A3). Finally, using (25)–(27) we obtain

$$|\widehat{(\![\![\!qu]\!] }\!)| \leq B\_1(e^{-\gamma\_1|\xi|} + \exp\{-T\_{\pi\_0\sigma,\varepsilon'\_{h\_1,b\_2}}(|\xi| \!)\}) \leq B\_2 \exp\{-T\_{\pi\_0\sigma,\varepsilon'\_{h\_1,b\_2}}(|\xi| \!)\},$$

for *ξ* ∈ *V* and for suitable constant *B*2 > 0. This completes the proof.

### **5. Conclusions**

Various classes of (ultra)distributions are commonly introduced as topological duals of suitable test function spaces, or as equivalence classes of certain Cauchy sequences of smooth functions. Another approach is given through their representations as finite sums of boundary values of analytic functions. We refer to [11] for a history, motivation and a detailed study of the subject. In this paper, we give a characterization of analytic functions whose boundary values are elements of intermediate spaces between the spaces of Schwartz distributions and any space of Gevrey ultradistributions. Test function spaces for such spaces of ultradistributions are related to the so-called extended Gervey regularity studied by the authors in [1–4,10,20]. We note that the extended Gevrey classes are different from any Carleman class which appears to be essential for many calculations, leading to the use of novel tools and techniques. In particular, we have used asymptotic properties of the Lambert *W* function in order to describe appropriate logarithmic growth rate in our calculations. This tool appears in the theory of new ultradistribution spaces for the first time (see also [4]). Since we relaxed the condition of stability under ultradifferentiable operators, to prove our main result, Theorem 1, we had to preform nontrivial changes in proofs of related results of classical theory (cf. [13]). This methodology could be used in other situations as well. For example, in future research we will consider the Paley–Wiener theorem for the new spaces of ultradistributions. This, in turn, will be used to prove the structure theorems in terms of boundary values of analytic functions; cf. Remark 1. We will also study other classes of two parameter sequences *Mτ*,*σ p* , apart from *pτpσ* , *p* ∈ **N**, *τ* > 0, *σ* > 1. This will be done in the spirit of Komastu [5].

**Author Contributions:** Individual contributions of the authors were equally distributed. Conceptualization, methodology, writing—original draft preparation and writing—review and editing, S.P., N.T. and F.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Ministry of Education, Science and Technological Development, Republic of Serbia, project numbers 451-03-68/2020-14/200125 and 451-03-68/2020-14/200156; the Serbian Academy of Sciences and Arts, project F10; and MNRVOID of the Republic of Srpska, project 19.032/961103/19.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the writing of the manuscript, or in the decision to publish the results.

#### **Appendix A**

In the following Lemma we study *Tτ*,*σ*,*h*(*k*) in some detail.

**Lemma A1.** *Let h* > 0*, and Tτ*,*σ*,*h be given by* (1)*, and let T*∗ *τ*,*σ*,*h be given by* (3)*. Then*

(*a*) *If h*1 < *h*2 *then Tτ*,*σ*,*h*1 (*k*) < *Tτ*,*σ*,*h*2 (*k*)*, k* > 0*. Moreover, for any h* > 0 *there exists H* > *h such that*

$$T\_{\tau,\sigma,h}(k) \le T\_{\tau,\sigma,h}^\*(k) \le T\_{\tau,\sigma,H}(k), \quad k > 0. \tag{A1}$$

(*b*) *For h*1, *h*2 > 0 *there exists C*, *ch*1,*h*2 > 0 *such that*

$$T\_{\tau,\sigma,h\_1}(k) + T\_{\tau,\sigma,h\_2}(k) \le T\_{\tau/2^{\varepsilon-1},\sigma,c\_{h\_1,h\_2}}(k) + \ln \mathbb{C} \quad k > 0,\tag{A2}$$

(*c*) *For every h* > 0 *there exists H* > 0 *such that*

$$T\_{\tau,\sigma,H}(l) \le T\_{\tau,\sigma,h}(1/k) + kl, \quad k, l > 0. \tag{A3}$$

**Proof.** (a) Notice that for arbitrary *h* > 0,

$$\ln \frac{h^{p^{\tau^r}} k^p}{p^{\tau p^{\tau}}} \le \ln \frac{p^p h^{p^{\tau^r}} k^p}{p^{\tau p^{\tau}}} \le \ln \frac{(Ch)^{p^{\tau^r}} k^p}{p^{\tau p^{\tau}}}, \quad k > 0,$$

where for the second inequality we use that for every *σ* > 1 there exists *C* > 1 such that *pp* ≤ *Cpσ* , *p* ∈ **N** (see the proof of Proposition 2.1. in [1]). Now (A1) follows by putting *H* = *Ch*.

(b) Let *h*1, *h*2 > 0. We will use the following simple inequality

$$p^{\sigma} + q^{\sigma} \le (p+q)^{\sigma} \le 2^{\sigma-1}(p^{\sigma} + q^{\sigma}), \quad p, q \in \mathbb{N}.\tag{A4}$$

Since, *hpσ* 1 *h qσ* 2 ≤ (*h*1 + *h*2)*pσ*+*qσ* we conclude that

$$h\_1^{p^{r^r}} h\_2^{q^{r^r}} \le (h\_1 + h\_2)^{(p+q)^{r^r}} \quad \text{where} \quad h\_1 + h\_2 \ge 1$$

and

$$h\_1^{p^{r^r}} h\_2^{q^{r^r}} \le (h\_1 + h\_2)^{(1/2^{r-1})(p+q)^{r^r}} \quad \text{when} \quad 0 < h\_1 + h\_2 < 1.$$

Hence, there exists 0 < *cσ* ≤ 1 such that

$$\ln \frac{h\_1^{p^{\mathcal{F}}} k^p}{p^{\tau p^{\mathcal{F}}}} + \ln \frac{h\_2^{\tau^{\mathcal{F}}} k^q}{q^{\tau q^{\mathcal{F}}}} \le \ln \frac{(\mathcal{C}(h\_1 + h\_2)^{c\_{\mathcal{F}}})^{(p+q)^{\mathcal{F}}} k^{p+q}}{(p+q)^{(\tau/2^{\mathcal{F}-1})(p+q)^{\mathcal{F}}}} + \ln \mathcal{C}, \quad p, q \in \mathbb{N}\_{\text{max}}$$

where *C* > 0 is constant appearing in ( -*M*.2). Now (A2) follows after taking supremums over *p*, *q* ∈ **N**.

(c) Recall (see [6]) that there exists *A* > 0 such that *kl* = sup *p*∈**N** ln *Apkplp pp* . Note that for every *σ* > 1 there exists 0 < *C* < 1 such that 1 *pp* ≥ *Cpσ* , *p* ∈ **N**.

Then for arbitrary *h* > 0 we have

$$T\_{\mathsf{r},\mathsf{r},h}(1/k) + kl = \sup\_{p,q \in \mathsf{N}} \ln \frac{h^{p^{\mathsf{r}}}}{k^p p^{\mathsf{r}p^{\mathsf{r}}}} \frac{A^q k^q l^q}{q^q} \ge \sup\_{p,q \in \mathsf{N}, p=q} \ln \frac{(A^\prime Ch)^{p^{\mathsf{r}}} l^p}{p^{\mathsf{r}p^{\mathsf{r}}}} = T\_{\mathsf{r},\mathsf{r},H}(l), \quad k, l > 0, \mathsf{r}$$

where *A* = min{1, *A*}. This proves (A3).

Finally, we discuss certain stability and embedding properties of E*τ*,*σ*(*U*) given by (5) and (6). Analogous considerations hold when the spaces D*τ*,*σ*(*U*) from (7) and (8) are considered instead.

Let *aα* ∈ E(*τ*,*σ*)(*U*) (resp. *aα* ∈ E{*τ*,*σ*}(*U*)), where *U* is an open set in **R***d*. Then we say that

$$P(\mathbf{x}, \partial) = \sum\_{|\alpha|=0}^{\infty} a\_{\alpha}(\mathbf{x}) \partial^{\alpha}$$

is an ultradifferential operator of class (*τ*, *σ*) (resp. {*τ*, *σ*}) on *U* if for every *K* ⊂⊂ *U* there exists constant *L* > 0 such that for any *h* > 0 there exists *A* > 0 (resp. for every *K* ⊂⊂ *U* there exists *h* > 0 such that for any *L* > 0 there exists *A* > 0) such that

$$\sup\_{x \in K} |\partial^{\beta} a\_{\alpha}(x)| \le A h^{|\beta|^{r'}} |\beta|^{\pi |\beta|^{r'}} \frac{L^{|\alpha|^{r}}}{|\alpha|^{\pi 2^{r-1}|\alpha|^{r'}}}, \quad \alpha, \beta \in \mathbf{N}^{d}.$$

We refer to [2] for the proof of the following continuity and embedding properties.

## **Proposition A1.**

(*a*) *Let P*(*x*, *∂*) *be an ultradifferential operator of class* (*τ*, *σ*) *(resp.* {*τ*, *σ*}*). Then P*(*x*, *∂*)*:* E*τ*,*σ*(*U*) −→ E*τ*2*σ*−1,*σ*(*U*) *is a continuous linear mapping; the same holds for*

$$P(\mathfrak{x}, \mathfrak{d}): \quad \varinjlim\_{\tau \to \infty} \mathcal{E}\_{\tau, \sigma}(\mathcal{U}) \longrightarrow \varinjlim\_{\tau \to \infty} \mathcal{E}\_{\tau, \sigma}(\mathcal{U}).$$

(*b*) *Let σ*1 ≥ 1*. Then for every σ*2 > *σ*1

$$\varinjlim\_{\tau \to \infty} \mathcal{E}\_{\tau, \sigma\_1}(\mathcal{U}) \hookrightarrow \varprojlim\_{\tau \to 0^+} \mathcal{E}\_{\tau, \sigma\_2}(\mathcal{U}).$$

(*c*) *If* 0 < *τ*1 < *τ*2*, then*

$$\mathcal{E}\_{\{\tau\_1,\sigma\}}(\mathsf{U}) \hookrightarrow \mathcal{E}\_{\{\tau\_2,\sigma\}}(\mathsf{U}) \hookrightarrow \mathcal{E}\_{\{\tau\_2,\sigma\}}(\mathsf{U}), \quad \sigma > 1,$$

*and*

$$\varinjlim\_{\tau \to \infty} \mathcal{E}\_{\{\tau, \sigma\}}(\mathcal{U}) = \varinjlim\_{\tau \to \infty} \mathcal{E}\_{(\tau, \sigma)}(\mathcal{U}),$$

$$\varinjlim\_{\tau \to 0^{+}} \mathcal{E}\_{\{\tau, \sigma\}}(\mathcal{U}) = \varprojlim\_{\tau \to 0^{+}} \mathcal{E}\_{(\tau, \sigma)}(\mathcal{U}), \quad \sigma > 1.$$

*Consequently we obtain that*

$$\varinjlim\_{t \to \infty} \mathcal{E}\_t(\mathsf{U}) \hookrightarrow \mathcal{E}\_{\tau, \sigma}(\mathsf{U}) \hookrightarrow \mathcal{C}^{\infty}(\mathsf{U}), \quad \tau > 0, \ \sigma > 1, \epsilon$$

*where* E*t*(*U*) *is Gevrey space with index t* > 1*.*

## **References**

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www.mdpi.com ISBN 978-3-0365-3172-4


Abstract

Web services clustering is the task of extracting and selecting the features from a collection of Web services and forming groups of closely related services. The implementation of novel and efficient algorithms for Web services clustering is relevant for the organization of service repositories on the Web. Counting with well-organized collections of Web services promotes the efficiency of Web service discovery, search, selection, substitution, and invocation. In recent years, methods inspired by nature using biological analogies have been adapted for clustering problems, among which genetic algorithms, evolutionary strategies, and algorithms that imitate the behavior of some animal species have been implemented. Computation inspired by nature aims at imitating the steps that nature has developed and adapting them to find a solution of a given problem. In this chapter, we investigate how biologically inspired clustering methods can be applied to clustering Web services and present a hybrid approach for Web services clustering using the Artificial Bee Colony (ABC) algorithm, K-means, and Consensus. This hybrid algorithm was implemented, and a series of experiments were conducted using three collections of Web services. Results of the experiments show that the solution approach is adequate and efficient to carry out the clustering of very large collections of Web services.

Keywords: artificial bee colony, K-means, Consensus, hybrid algorithms, Web services clustering, semantic similarity measures


**How Touchless was modified?**

We added the calibration application to interact with touchless SDK for better performance. The main purpose is to calibrate the resolution of the screen and the virtual display. The application is mainly to improve the accuracy of the markers interactions.

Figure 10 depicts the main classes of ATB. The first set of Classes includes the following classes: ActivateSystem, Marker, and CalibrationPoints. In the ActivateSystem class, the user (teacher or admin) can startSystem and EndSystem. While in the Marker class, the teacher can DefineMarker or add NewMarker. Moreover, the teacher or admin can setCalibrationPoints in the class CalibrationPoints. However, the class CalibrationPoints is aggregated to the class Marker; that is, if the user didn't define a Marker, he or she can't setCalibrationPoints. Also, the class Marker is aggregated to the class ActivateSystem. The teacher can't defineMarker or add newMarker without Activating the system first.

The second set of Classes includes three classes which are: Subject, Lesson, and Test. In the Subject class the user (Autistic User) can chooseSubject or close the Application. While in the Lesson class the user can start NewLesson, OpenLesson (open a previous lesson), SaveLesson, and CloseLesson. In the Test class, the user can start NewTest, RepeatTest (repeat a previous test), and check the TestScore. Also, the class Test is aggregated to the class Lesson, and the class Lesson is aggregated to the class Subject. That is, the user can't take a test if there is no lesson, and the user can't take a lesson if there is no subject.

#### **4.5 Using ATB**

Figure 11 depicts the sequence diagram of calibration process. The user "StartSystem()" to start the ATB System. After that, the admin should select "defineMarker" to define a new Marker. After defining a Marker, the admin or the teacher should carefully set the calibration points through selecting "SetCalibrationPoints()". This will set the dimensions of the whiteboard. Doing this process will convert the whiteboard to a smart interactive board. As a result, the user or the admin can interact with the whiteboard through hand gestures. So, he or she can run the Educational Applications to "chooseSubject()".

Fig. 10. Class diagram of ATB.

28 E-Learning – Organizational Infrastructure and Tools for Specific Areas

ATB mainly consists of three layers (Figure 9). The layers are: interface layer, tracking layer

The interface Layer consists of the PC desktop or the educational application. The user shall interact directly with the PC through the hand gestures to form different marker events (i.e.

This layer is the middle layer. It consists of the projector and the webcam. It helps convert the whiteboard to an electronic touch-screen, where any program can be controlled through hand gestures. The webcam will track the hand gestures through detecting pre-defined

This is the bottom layer, which consists of the Calibration Applications and Microsoft

 Calibration Application: It is software to measure the dimension of the PC's screen and represent it on the whiteboard through the projector. The main idea is to calibrate the

 Modified Touchless SDK: In this application we modified the click action to accomodate the interaction method. The code was modified so that the click event is generated if the user holds his hands/markers over an object (on the wall or screen) for around 2

We added the calibration application to interact with touchless SDK for better performance. The main purpose is to calibrate the resolution of the screen and the virtual display. The

Figure 10 depicts the main classes of ATB. The first set of Classes includes the following classes: ActivateSystem, Marker, and CalibrationPoints. In the ActivateSystem class, the user (teacher or admin) can startSystem and EndSystem. While in the Marker class, the teacher can DefineMarker or add NewMarker. Moreover, the teacher or admin can setCalibrationPoints in the class CalibrationPoints. However, the class CalibrationPoints is aggregated to the class Marker; that is, if the user didn't define a Marker, he or she can't setCalibrationPoints. Also, the class Marker is aggregated to the class ActivateSystem. The

The second set of Classes includes three classes which are: Subject, Lesson, and Test. In the Subject class the user (Autistic User) can chooseSubject or close the Application. While in the Lesson class the user can start NewLesson, OpenLesson (open a previous lesson), SaveLesson, and CloseLesson. In the Test class, the user can start NewTest, RepeatTest (repeat a previous test), and check the TestScore. Also, the class Test is aggregated to the class Lesson, and the class Lesson is aggregated to the class Subject. That is, the user can't

Figure 11 depicts the sequence diagram of calibration process. The user "StartSystem()" to start the ATB System. After that, the admin should select "defineMarker" to define a new Marker. After defining a Marker, the admin or the teacher should carefully set the

resolution of the pc screen to the dimensions of the white board.

seconds. The advantage of this method is that it is fairly simple.

application is mainly to improve the accuracy of the markers interactions.

teacher can't defineMarker or add newMarker without Activating the system first.

take a test if there is no lesson, and the user can't take a lesson if there is no subject.

**4.4 ATB design**

and software layer. **a. Interface Layer**

click, drag…etc.) **b. Tracking Layer**

**c. Software layer**

**How Touchless was modified?**

Touch-less SDK.

**4.5 Using ATB**

colors.

Fig. 11. Sequence Diagram of using ATB.

Towards Economical E-Learning Educational Environments for Physically Challenged Students 31

Fig. 13. Using ATB.

Fig. 14. Matching animals and voices game.

ATB has been tested by some young students and teachers. The initial feedback is positive. We are in the process of getting permission to deploy the system in real schools for autistic children to have concrete surveys and statistics regarding how ATB affected their education

**4.6 Assessments**


**Author details**

Michael Langemeier Purdue University, West Lafayette, Indiana, United States

\*Address all correspondence to<EMAIL_ADDRESS>
© 2023 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.


**1. Introduction**

The pelvic floor consists of muscles and connective tissue. In the past, the components' relative contribution to the structural support of the pelvic floor and its functions has been a subject of controversy (Corton 2009). With increasing age women can develop vaginal and pelvic organ prolapse as well as symptoms such as stress urinary incontinence, voiding dysfunction, urgency and frequency and nocturia, and may also develop fecal incontinence, obstructive defecation and pelvic pain (Petros 2010). All of these symptoms can be associated - to a greater or lesser extent - with pelvic floor defects.

What events are responsible for these defects? One theory says that an important cause of prolapse and pelvic floor dysfunction is likely to be partial denervation (Swash et al 1985, Smith et al. 1989). But Pierce et al. (2008) demonstrated in nulliparous monkeys that bilateral transection of the levator ani nerve resulted in atrophy of denervated levator ani muscles but not in failure of pelvic support. This indicates that connective tissue components could compensate for weakened pelvic floor muscles. According to South et al. (2009), in up to 30 percent of all vaginal childbirths, pelvic floor muscles are partially denervated. However, such functions are known to recover and reinnervate often within months (Snooks et al 1984, Lin et al. 2010) .

In a direct test of the question, "connective tissue or muscle damage?", Petros et al 2008 performed a blinded prospective study with muscle biopsies of m.pubococcygeus taken at the same time as a midurethral sling operation for urinary stress incontinence (USI) was done, an operation which works by creating an artificial collagenous neoligament (Petros PE, Ulmsten U, Papadimitriou 1990). Out of 39 patients with histological evidence of muscle damage, 33 (85%) were cured immediately after surgery, indicating that connective tissue, not muscle damage was most likely the major cause of the USI.

Further, the muscle itself can change. It is known that the number and density of urethral striated muscle fibers declines with age (Huisman 1983, Perucchini et al. 2002), an idea that has been confirmed in studies about the vastus lateralis muscle (Lexell et al. 1988). Muscle

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The Role of Altered Connective Tissue in the Causation of Pelvic Floor Symptoms 5

process occurs in women with descending or prolapsing vaginal wall and pelvic organs. The muscles which attaches directly or indirectly to the vagina or the pelvic organs change their length and their direction of action. This alters muscle force and function according to the relation shown in Figure 1. After re-positioning of the prolapsed organs, the muscle can reach its normal length and function. Hence, atrophy of muscle by immobilization (Hvid et

A prime example of this principle is restoration of urethral closure by a midurethral sling which restores the integrity of the pubourethral ligament. In the original description of the "tension-free" sling (Ulmsten et al 1996), the operation was performed under local anesthesia and the tape was lifted upwards while the patient was coughing, until the urine

Fig. 1. Relationship of maximal muscle force to muscle (sarcomere)length (modified after Gordon 1966). Maximal muscle strength is exerted over a very short length (between red lines). Contractile strength falls rapidly with muscle lengthening and shortening, for

In many studies morphology of pelvic floor muscles has been explained with only few limited reference to muscle action. There is no doubt that the pelvic floor muscles and ligaments have immense importance for stress incontinence, micturition and anorectal functions. It was P. Petros who explained the directional muscle forces (Petros and Ulmsten

From a functional and clinical aspect, it is important to consider 4 major muscle groups of the pelvic floor which are able to move the vaginal wall and pelvic floor organs (Petros

1993, Petros and Ulmsten 1997) and their significant role in pelvic floor dysfunctions.

al. 2011) can be avoided at least with some patients.

example, due to lax connective tissue attachments.

**3. Pelvic floor muscles and their functions (figure 2)**

leakage ceased.

2010):

avulsions have been reported at the pelvic floor (Dietz and Lanzarone 2005, Dietz et al. 2007), but it is more likely that the insertion areas of muscles are dislocated by connective tissue alterations than muscle tears (Petros 2008).

From a mechanical point of view, the pelvic floor is composed of both muscles and connective tissue. The muscles are the active components that are – through their contractions - responsible for all functions of the pelvic floor. The connective tissues, with their elastic and collagen fibres and their extracellular matrices, provide structural support for the vagina and other organs such as uterus, urethra, bladder and rectum (Abramowitch 2009). It has been shown, that connective tissue changes occur during pregnancy (Rechberger et al. 1988, Harkness 1959). Weakening of collagen cross bonding (Rechberger et al. 1988) added to dilatation of the vaginal canal at childbirth can lead to overdistension or rupture of connective tissue. Extracellular matrix proteases contribute to progression of pelvic organ prolapse in mice and humans (Budatha et al. 2011, Connell 2011). The first vaginal birth is especially associated with the development of a prolapse, whereas additional vaginal births do not show significant increases in the odds of prolapse (Quiroz et al. 2011). Aging is characterized by a loss of collagen, degeneration of the elastic fibre network and a loss of hydration as a result of imbalance between biosynthesis and degradation (Uitto und Bernstein 1998, Campisi 1998)

In addition to that, there is a significant variability of tissue due to inborn variations (Dietz et al 2004) and collagen-associated disorders (Lammers et al. 2011, Campeau et al. 2011).

Surgical procedures can reduce structural support of the organs, especially those which cut or displace the uterosacral and cardinal ligaments during hysterectomy or which partially resect vaginal tissue or perineal body during colporrhaphy.

Petros and Ulmsten (1993) stated that looseness or laxity of the vagina and its supporting ligaments can cause stress incontinence as well as urge. Since then the theory has been expanded to include other symptoms such as pelvic pain, voiding dysfunction and more recently, fecal incontinence and constipation (Petros & Swash 2008). In order to fix such loose ligaments Petros et al. (1990) have introduced alloplastic material for planned formation of an artificial neo-ligament. From this rather basic research, new surgical techniques have been developed, such as tapes for midurethral slings (TVT, TOT) and for repair of other pelvic floor ligaments (Petros and Ulmsten 1990, 1993). The new developments and the recent focus on connective tissue are important, not least because looseness of tissues can be repaired surgically..