# A duality between
Lorentzian Twisted Spectral Triples
and Spectral Triples

G<EMAIL_ADDRESS>
DIMA, Universita di Genova

###### Abstract

We present a duality between spectral triple and a Lorentzian version of
twisted spectral triples, offering an original approach to Lorentzian geometry
within the context of noncommutative geometry. This duality is used to define
a spectral triple for which the fermionic action is Lorentz invariant,
providing a new candidate for the almost-commutative manifold’s spectral
triple of the noncommutative standard model of particles physics.

##  1 Introduction

One of the cornerstones of the road towards Noncommutative Geometry (NCG) is
the duality between topological spaces $\mathcal{M}$ and the commutative
algebras of smooth functions $C^{\infty}(\mathcal{M})$ pointed out by the
Gelfand-Naimark theorem. The key idea that emerged was that we can describe
“spaces” starting from such algebras. The algebra becomes the basic structure,
and fundamental frameworks such as topology, Riemannian geometry and
differential calculus have been successfully translated within this new
perspective. An important result known as the Connes reconstruction theorem
(see [7]) states that taking $C^{\infty}(\mathcal{M})$,
$L^{2}(\mathcal{M},\mathcal{S})$ the Hilbert space of square integrable
sections of spinor bundle over $\mathcal{M}$, $D$ the Dirac operator
associated with the Levi-Civita connection, $J$ a given anti-unitary operator
and $\Gamma$ a grading, then spectral triples given by
$(C^{\infty}(\mathcal{M}),L^{2}(\mathcal{M},\mathcal{S}),D,J,\Gamma)$ are in
one-to-one correspondence with Riemannian spin manifolds, which are
fundamental objects to understand Gravitation. Beyond the conceptual interest
of this approach, an added value is that this reformulation allowed to extend
our geometrical conception, by considering what this structure becomes when
moving to a more general NonCommutative (NC) algebra $\mathcal{A}$.

A significant achievement of NCG has been the re-expression of Euclidean
Gravitation and Yang-Mills theories within a unified geometric framework in
which gauge and Higgs fields emerges as fluctuations of the Dirac operator.
This is called the Noncommutative Standard Model of Particle Physics (NCSMPP),
we refer to [6] for a detailed explanation of the model. However, this
approach faces a fundamental problem, the model is inherently Euclidean. This
is because Connes reconstruction theorem [7] works only for positive definite
signature. We therefore lose the characteristic causal structure of spaces
with Lorentz signature, thus missing out on physics.

This missing point is at the origin of several attempts to find a Lorentzian
version of the framework of spectral triples, such as [1, 12, 17, 18, 2, 10].
A common feature shared by many of these approaches is the replacement of the
Hilbert space by a Krein space, turning the inner product
$\langle\,.\,,\,.\,\rangle$ into an indefinite inner product
$\langle\,.\,,\,.\,\rangle_{\gamma}\vcentcolon=\langle\,.\,,\,\gamma.\,\rangle$
with $\gamma^{2}={\text{\char 49\relax}}$ by definition. In these approach, a
particular choice is to take $\gamma=\gamma^{0}$, the first gamma matrix,
giving the inner product that we use in quantum field theory, to get Lorentz
invariant scalars such as the Dirac action. Krein’s spaces appear for the
first time in the context of twisted spectral triple in [10] as a natural new
feature of this twisted version of spectral triples (presented in section 3),
connecting the Krein product with the twist $\rho(a)=\gamma a\gamma^{\dagger}$
which appear inside the twisted commutator
$[D,a]_{\rho}\vcentcolon=Da-\rho(a)D$.

This article proposes a new perspective on the signature problem. This is
based on the following two considerations:

  1. 1.

If we consider the usual Lorentzian Dirac operator $D^{L\,}$, the physical
Dirac action is given by
$\mathcal{S}_{D}=\langle\psi,D^{L\,}\psi\rangle_{\gamma^{0}}$. A good approach
for the NCSMPP must naturally produce such an invariant. Interestingly, the
operator defined by $D\vcentcolon=\gamma^{0}D^{L\,}$ is a self-adjoint Dirac
operator, so that we also have $\mathcal{S}_{D}=\langle\psi,D\psi\rangle$.

  2. 2.

Connes reconstruction theorem gives strong reasons to believe into the axioms
of spectral triples. In particular, an important requirement is that the Dirac
operator must be self-adjoint.

The approach proposed in this article is an attempt to reconcile these two
considerations, by setting up (in section 4) a duality between spectral
triples and “Lorentzian twisted spectral triple”, as defined in section 5.

A first insight into the idea is given by the following consideration. Taking
$(\langle\,.\,,\,.\,\rangle,\,D\vcentcolon=\gamma^{0}D^{L\,},\,\delta(\,.\,)=[D,\,.\,])$
in a usual spectral triple $(\mathcal{A},\mathcal{H},D)$, and
$(\langle\,.\,,\,.\,\rangle_{\gamma^{0}},\,D^{L\,},\,\delta_{\rho}(\,.\,)=[D^{L\,},\,.\,]_{\rho})$
with $\rho(A)={\gamma^{0}}A{\gamma^{0}}^{\dagger}$ for any
$A\in\mathcal{B}(\mathcal{H})$ and
$[D^{L\,},a]_{\rho}=D^{L\,}a-\rho(a)D^{L\,}$ in a Lorentzian version (in which
the Dirac operator is the Lorentzian one) of the corresponding twisted
spectral triple, then we have that $\forall a\in\mathcal{A}$ and
$\psi\in\mathcal{H}$:

$\langle\psi,D^{L\,}\psi\rangle_{\gamma^{0}}=\langle\psi,D\psi\rangle\qquad\text{and}\qquad\langle\psi,[D^{L\,},a]_{\rho}\psi\rangle_{\gamma^{0}}=\langle\psi,[D,a]\psi\rangle$

which defines the first elements of the duality between spectral triples and
Lorentzian twisted spectral triples by means of this equality.

The aim of this article is to show that this duality extends until the level
of the fluctuations of the Dirac operator so that the fermionic and spectral
actions in this two context became also equals. Then, stating that this is a
fundamental duality between Lorentzian twisted spectral triples and spectral
triples, we present in section 5 the defining axioms of the $\gamma$-twisted
spectral triple dual to a spectral triple. The particular example of the
$\gamma^{0}$-twisted spectral triple for the 4-dimensional spin manifold with
Dirac operator $D^{L\,}$ permit to define the so called Lorentzian twisted
spectral triple. Then, starting from this particular case, we determinate the
corresponding spectral triple, leading to the definition of the so called
Lorentzian twisted spectral triple in definition 5.2 with its dual spectral
triple. Interestingly, this spectral triple does not correspond to the one of
the actual NCSMPP, since the Dirac operator is given by
$D\vcentcolon=\gamma^{0}D^{L\,}$. An interesting point is that such a Dirac
operator has already appeared in the context of NCSMPP (see [3]) as an attempt
to solve the fermion doubling problem. We hope that the approach presented
here will provide a solution to the signature problem and to the fermion
doubling problem, naturally providing $\mathcal{S}_{D}$ as a fermionic action.

##  2 Spectral triples

We remind some important properties of spectral triples, in particular
concerning the way to generate the fluctuations of the Dirac operator together
with the associated spectral invariants.

######  Definition 2.1 (Spectral triple).

A Spectral triple $(\mathcal{A},\mathcal{H},D)$ is the data of an involutive
unital algebra $\mathcal{A}$ represented by bounded operators on a Hilbert
space $\mathcal{H}$, and of a self-adjoint operator $D$ acting on
$\mathcal{H}$ such that the resolvent $(i+D^{2})^{-1}$ is compact and that for
any $a\in\mathcal{A}$, $[D,a]$ is a bounded operator.

A Real and Even spectral triple $(\mathcal{A},\mathcal{H},D,\Gamma,J)$ is
defined by the enlargement of the spectral triple by two operators $\Gamma$
and $J$ such that $J$ is a anti-unitary operator and $\Gamma$ a
$\mathbb{Z}_{2}$-grading on $\mathcal{H}$ satisfying $\Gamma^{2}={\text{\char
49\relax}}$ and $\Gamma^{\dagger}=\Gamma$ so that

$J^{2}=\epsilon,\qquad JD=\epsilon^{\prime}DJ,\qquad
J\Gamma=\epsilon^{\prime\prime}\Gamma J,\qquad\Gamma
D+D\Gamma=0,\quad\text{and}\quad\Gamma\pi(a)=\pi(a)\Gamma$

$\forall a\in\mathcal{A}$, with
$\epsilon,\epsilon^{\prime},\epsilon^{\prime\prime}=\pm 1$.

In addition, the two following conditions must be satisfied:

  * •

The commutant property: $[a,Jb^{\dagger}J^{-1}]=0$.

  * •

The first-order condition $[[D,a],Jb^{\dagger}J^{-1}]=0$.

The derivative is given by $\delta(.)=[D,\,.\,]$ so that the
$\mathcal{A}$-bimodule of Connes’s differential one-forms is given by
$\Omega_{D}^{1}(\mathcal{A})\vcentcolon=\left\\{\sum_{k}a_{k}[D,b_{k}]:a_{k},b_{k}\in\mathcal{A}\right\\}$.
In this way a one form $A\in\Omega_{D}^{1}(\mathcal{A})$ gives rise to a
fluctuated Dirac operator defined by the following:

$D_{A}\vcentcolon=D+A+\epsilon^{\prime}JAJ^{-1}.$

Taking $u\in\mathcal{U}(\mathcal{A})$, we can define the adjoint unitary
$U=uJuJ^{-1}$ so that the inner fluctuation of $D_{A}$ became equivalent to a
gauge transformation of the one-form $A$:

$(D_{A})^{u}=UD_{A}U^{\dagger}=D_{A^{u}}\qquad\text{with}\qquad
A^{u}\vcentcolon=uAu^{\dagger}+u\text{{d}}_{U}u^{\dagger}.$

Taking this, a way to obtain a spectral invariant is given by the spectral and
fermionic actions. The spectral action is given by the one introduced in [4]:

$\displaystyle\mathcal{S}[D_{A}]$
$\displaystyle\vcentcolon=\operatorname{Tr}f(D_{A}D_{A}^{\dagger}/\Lambda^{2})$
(2.1)

with $f:\mathbb{R}^{+}\,\to\,\mathbb{R}^{+}$ a positive and even function that
decay at $\pm\infty$, $\Lambda\in\mathbb{R}^{+}$ being a cutoff parameter, see
[5].

A fermionic action is defined by:

$\mathcal{S}_{f}(D_{A},\psi)\vcentcolon=\langle\psi,D_{A}\psi\rangle\qquad\quad\text{with}\qquad\quad\psi\in\mathcal{H}.$

##  3 Twisted Spectral Triples

###  3.1 Introduction to Twisted Spectral Triples

First introduced by A. Connes and H. Moscovici in [8], a twisted spectral
triple is obtained by considering a spectral triple
$(\mathcal{A},\mathcal{H},D)$ together with a regular automorphism
$\rho\in\operatorname{Aut}(\mathcal{A})$ (i.e. satisfying
$\rho(a^{\dagger})=(\rho^{-1}(a))^{\dagger}\,\forall a\in\mathcal{A}$) called
twist, using the twisted commutator $[D,a]_{\rho}:=Da-\rho(a)D$ instead of the
$[D,a]$ and requiring only the twisted one to be bounded. The twisted spectral
triple is then given by the set $(\mathcal{A},\mathcal{H},D,\rho)$. We can
then define the twisted derivation $\delta_{\rho}(a):=[D,a]_{\rho}$ so that
the twisted Leibniz rule
$\delta_{\rho}(ab)=\rho(a)\delta_{\rho}(b)+\delta_{\rho}(a)b$ is verified
$\forall a,b\in\mathcal{A}$.

Twisted spectral triples were then implemented in the context of NCGFT by A.
Devastato and P. Martinetti in [9] then developed in [13] and [14] by G. Landi
and P. Martinetti. This approach was first motivated with hope that it can
generate the missing scalar field necessary for the prediction of the Higgs
mass, see [9], [11], and then, surprisingly that it can be connected with a
pseudo-Riemannian geometry (see [10]), and generate a Lorentz invariant
fermionic action [15].

A novelty in this approach is the replacement of the inner product
$\langle\,.\,,\,.\,\rangle$ by the $\rho$-product
$\langle\,.\,,\,.\,\rangle_{\rho}$ which by definition satisfies

$\langle\psi,A\psi^{\prime}\rangle_{\rho}=\langle\rho(A)^{\dagger}\psi,\psi^{\prime}\rangle_{\rho}\qquad\forall\psi,\psi^{\prime}\in\mathcal{H}\qquad\text{and}\qquad
A\in\mathcal{B}(\mathcal{H}).$

The associated adjoint will be called $\rho$-adjoint, it is defined by
$(\,.\,)^{+}:=\rho(\,.\,)^{\dagger}$, an operator
$A\in\mathcal{B}(\mathcal{H})$ is called “$\rho$-self-adjoint” if $A^{+}=A$.
We restrict to twist which respect $\rho^{2}={\text{\char 49\relax}}$ in the
following, so that $(\,.\,)^{+}$ is an involution and
$\rho(A^{\dagger})=\rho(A)^{\dagger}$. A $\rho$-unitary is an operator
$U_{\rho}$ which is unitary for the $\rho$-product i.e.
$U_{\rho}U_{\rho}^{+}=U_{\rho}^{+}U_{\rho}={\text{\char 49\relax}}$. If
$\rho\in\operatorname{Inn}(\mathcal{B}(\mathcal{H}))$, there is a unitary
operator $\gamma$ acting on $\mathcal{H}$ such that $\rho(A)=\gamma
A\gamma^{\dagger}$, in this case, the $\rho$-product became specified by
$\gamma$ so that:

$\langle\,.\,,\,.\,\rangle_{\rho}:=\langle\,.\,,\gamma.\,\rangle.$

Since different $\gamma$ can generate the twist $\rho$, a more convenient
notation for the corresponding inner product will be in terms of the chosen
twist $\gamma$ so that we will use the specific notation222Specific to the
context where $\rho\in\operatorname{Inn}(\mathcal{B}(\mathcal{H}))$. in terms
of the $\gamma$-product
$\langle\,.\,,\,.\,\rangle_{\gamma}\vcentcolon=\langle\,.\,,\,\gamma.\,\rangle$.
In the same way, the associated $\gamma$-adjoint is given by the notation
$(\,.\,)^{{\dagger_{\gamma}}}\vcentcolon=\rho(\,.\,)^{\dagger}$. A
$\gamma$-unitary operator is then an operator $\tilde{U}_{\,\gamma}$ which
satisfy
$\tilde{U}_{\,\gamma}\tilde{U}_{\,\gamma}^{\dagger_{\gamma}}=\tilde{U}_{\,\gamma}^{\dagger_{\gamma}}\tilde{U}_{\,\gamma}={\text{\char
49\relax}}$ and the set of all $\gamma$-unitaries is called
$\mathcal{U}_{\gamma}(\mathcal{B}(\mathcal{H}))$.

The axioms of twisted spectral triples are the same as for spectral triples to
the exception of the introduction of a fourth characteristic number
$\epsilon^{\prime\prime\prime}$ such that $\,\gamma
J=\epsilon^{\prime\prime\prime}J\gamma$, and the replacement of the first
order condition by the twisted first order condition
$[[D,a]_{\rho},b^{\circ}]_{\rho^{\circ}}=0\,,\,\forall
a\in\mathcal{A},\,\,b^{\circ}\in\mathcal{A}^{\circ}$, with
$\mathcal{A}^{\circ}$ the opposite algebra, and the opposite twist
$\rho^{\circ}$ acting on $\mathcal{A}^{\circ}$ defined by
$\rho^{\circ}(a^{\circ}):=(\rho^{-1}(a))^{\circ}$ (see [13]). Twisted 1-forms
are then given by elements $A_{\rho}=\sum_{k}a_{k}[D,b_{k}]_{\rho}$ with
$a_{k},b_{k}\in\mathcal{A}$ so that twisted-fluctuations of the Dirac operator
will take the form

$D_{A_{\rho}}:=D+A_{\rho}+\epsilon^{\prime}JA_{\rho}J^{-1}.$

As shown in [16], there are two ways to generate such twisted-fluctuations:

  1. 1.

$D_{A_{\rho}}=UDU^{\dagger_{\gamma}}$ with $U=uJuJ^{-1}$ and $u$ a unitary
operator in $\mathcal{A}$.

  2. 2.

$D_{A_{\rho}}=U_{\gamma}DU_{\gamma}^{\dagger}$ with
$U_{\gamma}=u_{\gamma}Ju_{\gamma}J^{-1}$ and $u_{\gamma}$ a $\gamma$-unitary
operator in $\mathcal{A}$.

In this context, following [15], the spectral action is the same as for
spectral triples, taking $D_{A_{\rho}}$ for the Dirac operator. The fermionic
action became a $\gamma$-twisted fermionic action, defined by:

$\mathcal{S}_{f}^{\gamma}(D_{A_{\rho}},\psi)\vcentcolon=\langle\psi,D_{A_{\rho}}\psi\rangle_{\gamma}\qquad\quad\text{with}\qquad\quad\psi\in\mathcal{H}.$

###  3.2 The Minimal Twist of the Manifold

An important way to twist a spectral triple, called “minimal twist” is defined
by leaving $D$ and $\mathcal{H}$ unchanged while transforming the algebra into
$\mathcal{A}\otimes\mathcal{B}$ with $\mathcal{B}$ a unital involutive
algebra. Let’s consider a manifold of dimension $2m$ with $m$ an integer. As
shown in [13], the minimal (non-trivial) twist is given by
$(C^{\infty}(\mathcal{M})\otimes\mathbb{C}^{2},L^{2}(\mathcal{M},\mathcal{S}),D,J,\Gamma,\rho)$.
Using the projections $p_{\pm}:=({\text{\char 49\relax}}\pm\Gamma)/2$ so that
$\mathcal{H}=p_{+}\mathcal{H}\oplus
p_{-}\mathcal{H}:=\mathcal{H}_{+}\oplus\mathcal{H}_{-}$, all elements
$a=(f,g)\in C^{\infty}(\mathcal{M})\otimes\mathbb{C}^{2}$ will be represented
accordingly as

$\pi(f,g)\vcentcolon=p_{+}\pi_{0}(f)+p_{-}\pi_{0}(g)=\begin{pmatrix}\pi_{+}(f)&0\\\
0&\pi_{-}(g)\\\ \end{pmatrix}$

with $\pi_{0}$ the representation used for $C^{\infty}(\mathcal{M})$ as
defined in [13]. The action of $\rho$ on $a$ is given by $\rho(f,g)=(g,f)$,
this is called “twist by grading”. As shown in [13], such twists are the only
ways to minimally twist an even-dimensional spin manifold, making this model
be a canonical way to twist the spectral triple of the Riemannian manifold.

As shown in [16] such a twist can be generated by a unitary action
$\rho(a)=\gamma a\gamma^{\dagger}$ where $\gamma$ can be any one of the
Euclidean gamma matrices $\gamma^{\mu}$ so that $\gamma^{2}={\text{\char
49\relax}}$, implying that $\langle\,.\,,\,.\,\rangle_{\gamma}$ is a Krein
product.

######  Remark 3.1.

$\mathcal{U}_{\gamma}(\mathcal{B}(\mathcal{H}))$ is the natural symmetry group
associated with the scalars defined by $\langle\psi,\psi\rangle_{\gamma}$ with
$\psi\in\mathcal{H}$ since
$\langle\tilde{U}_{\,\gamma}\psi,\tilde{U}_{\,\gamma}\psi\rangle_{\gamma}=\langle\psi,\tilde{U}_{\,\gamma}^{\dagger_{\gamma}}\tilde{U}_{\,\gamma}\psi\rangle_{\gamma}=\langle\psi,\psi\rangle_{\gamma}$.

######  Remark 3.2.

The Lorentzian Dirac operator $D^{L\,}$ of quantum field theory corresponds to
a $\gamma$-self-adjoint operator for $\gamma=\gamma^{0}$. As shown in [16],
the Lorentz group is a subgroup of
$\mathcal{U}_{\gamma^{0}}(\mathcal{B}(\mathcal{H}))$. Additionally, the
evaluation of $D^{L\,}$ by the $\gamma$-product gives the usual Dirac action
of the standard model of particles physics:

$\displaystyle\mathcal{S}_{D}=\langle\psi,D^{L\,}\psi\rangle_{\gamma^{0}}.$

The novelty of the approach is to consider a $\gamma$-self-adjoint Dirac
operator $D^{\,\gamma}$ inside the minimal twist (by grading) of the even-
dimensional spin manifold.

######  Definition 3.3 ($\gamma$-twisted spectral triple).

A $\gamma$-twisted spectral triple is a twisted spectral triple for which the
Dirac operator is a $\gamma$-self-adjoint operator.

##  4 The Duality

Let’s consider a manifold of dimension $2m$ with $m$ an integer. The algebra
is $\mathcal{A}=C^{\infty}(\mathcal{M})\otimes\mathbb{C}^{2}$ and the
representation is given by the grading $\Gamma$ according to its chiral
eigenspace, as in subsection 3.2. This section aims at presenting the duality
at the origin of this article. This duality works at the level of the spectral
triples of the two-sheeted manifold. This is a duality between the spectral
triple
$\mathcal{ST}(\mathcal{A})=(\mathcal{A},L^{2}(\mathcal{M},\mathcal{S}),\langle\,.\,,\,.\,\rangle,D,J,\Gamma)$
and the $\gamma$-twisted spectral triple
$\mathcal{TST}_{\gamma}(\mathcal{A})=(\mathcal{A},L^{2}(\mathcal{M},\mathcal{S}),\langle\,.\,,\,.\,\rangle_{\gamma},D^{\,\gamma},J,\Gamma,\rho(.)=\gamma(.)\gamma^{\dagger})$
where $D^{\,\gamma}$ is a $\gamma$-self-adjoint operator and $\gamma$ one of
the Euclidean gamma matrices implementing the twist $\rho$.

######  Proposition 4.1.

$\gamma$-self-adjoint Dirac operators $D^{\,\gamma}$ and self-adjoint Dirac
operators $D$ are in one-to-one correspondence through the relation
$D^{\,\gamma}=\gamma D$.

###### Proof

If $D$ is self-adjoint then $D^{\,\gamma}=\gamma D$ is $\gamma$-self-adjoint:

$(D^{\,\gamma})^{\dagger_{\gamma}}=(\gamma
D^{\,\gamma}\gamma^{\dagger})^{\dagger}=(\gamma\gamma
D\gamma^{\dagger})^{\dagger}=\gamma D=D^{\,\gamma}.$

If $D^{\,\gamma}$ is $\gamma$-self-adjoint then $D=\gamma D^{\,\gamma}$ is
self-adjoint:

$D^{\dagger}=(\gamma
D^{\,\gamma})^{\dagger}=(D^{\,\gamma})^{\dagger}\gamma^{\dagger}=\gamma
D^{\,\gamma}=D.$

######  Proposition 4.2 (Duality at the level of the Dirac operators.).

There is a duality between twisted Dirac operators $D^{\,\gamma}$ and Dirac
operators $D\vcentcolon=\gamma D^{\,\gamma}$ when evaluated on their
respective inner products.

###### Proof

$\displaystyle\langle\psi,D^{\,\gamma}\psi\rangle_{\gamma}=\langle\psi,\gamma\gamma
D\psi\rangle=\langle\psi,D\psi\rangle.$

######  Remark 4.3.

Taking $D=\gamma D^{\,\gamma}$, we have that
$\mathcal{TST}_{\gamma={\text{\char
49\relax}}}(\mathcal{A})=\mathcal{ST}(\mathcal{A})$.

######  Proposition 4.4 (Duality at the level of derivations.).

There is a duality between twisted derivations evaluated by
$\langle\,.\,,\,.\,\rangle_{\gamma}$ and derivations evaluated by
$\langle\,.\,,\,.\,\rangle$.

###### Proof

Since $D=\gamma D^{\,\gamma}$ implies $[D^{\,\gamma},a]_{\rho}=\gamma[D,a]$,
we have that

$\displaystyle\langle\psi,[D^{\,\gamma},a]_{\rho}\psi\rangle_{\gamma}=\langle\psi,[D,a]\psi\rangle$

$\forall a\in\mathcal{A}$ and $\psi\in\mathcal{H}$.

######  Proposition 4.5 (Duality at the level of one forms.).

There is a duality between twisted one-forms $A_{\rho}$ evaluate by
$\langle\,.\,,\,.\,\rangle_{\gamma}$ and one-forms $A$ evaluated by
$\langle\,.\,,\,.\,\rangle$.

###### Proof

Noting that $A_{\rho}=\rho(a)[D^{\,\gamma},b]_{\rho}=\gamma a[D,b]=\gamma A$,
then

$\displaystyle\langle\psi,\rho(a)[D^{\,\gamma},b]_{\rho}\psi\rangle_{\gamma}=\langle\psi,a[D,b]\psi\rangle$

$\forall a,b\in\mathcal{A}$ and $\psi\in\mathcal{H}$.

######  Proposition 4.6 (Duality at the level of first-order conditions.).

The first-order condition $[[D,a],b^{\circ}]=0$ implies the twisted first-
order condition.

###### Proof

By a direct computation we have that:

$\displaystyle[[D^{\,\gamma},a]_{\rho},b^{\circ}]_{\rho^{\circ}}=\gamma[[D,a],b^{\circ}]=0.$

Concerning the inner fluctuations of the Dirac operator, we first have to
determinate the form the twisted fluctuations of $D^{\,\gamma}$ takes.

######  Proposition 4.7.

Taking the Dirac operator $D^{\,\gamma}$, twisted fluctuations takes the form

$D^{\,\gamma}_{A_{\rho}}=D^{\,\gamma}+A_{\rho}+\epsilon^{\prime}\epsilon^{\prime\prime\prime}JA_{\rho}J^{-1}.$

###### Proof

Taking $U=uJuJ^{-1}=uv$, with $u$ a unitary operator and noting that
$JD=\epsilon^{\prime}DJ$ implies
$JD^{\,\gamma}=\epsilon^{\prime}\epsilon^{\prime\prime\prime}D^{\,\gamma}J$,
then we have

$\displaystyle UD^{\,\gamma}U^{\dagger_{\gamma}}$
$\displaystyle=uvD^{\,\gamma}v^{\dagger_{\gamma}}u^{\dagger_{\gamma}}=v(uD^{\,\gamma}u^{\dagger_{\gamma}})v^{\dagger_{\gamma}}=v(D^{\,\gamma}+u[D^{\,\gamma},u^{\dagger_{\gamma}}]_{\rho})v^{\dagger_{\gamma}}$
$\displaystyle=vD^{\,\gamma}v^{\dagger_{\gamma}}+vuv^{\dagger}[D^{\,\gamma},u^{\dagger_{\gamma}}]_{\rho}\qquad\text{(Twisted
first-order condition.)}$
$\displaystyle=vD^{\,\gamma}v^{\dagger_{\gamma}}+vv^{\dagger}u[D^{\,\gamma},u^{\dagger_{\gamma}}]_{\rho}\qquad\text{(Zero-
order condition.)}$
$\displaystyle=vD^{\,\gamma}v^{\dagger_{\gamma}}+u[D^{\,\gamma},u^{\dagger_{\gamma}}]_{\rho}\qquad\text{($v$
is unitary.)}$
$\displaystyle=JuJ^{-1}D^{\,\gamma}Ju^{\dagger_{\gamma}}J^{-1}+u[D^{\,\gamma},u^{\dagger_{\gamma}}]_{\rho}$
$\displaystyle=\epsilon^{\prime}\epsilon^{\prime\prime\prime}JuJ^{-1}JD^{\,\gamma}u^{\dagger_{\gamma}}J^{-1}+u[D^{\,\gamma},u^{\dagger_{\gamma}}]_{\rho}\qquad\text{(via
$JD^{\,\gamma}=\epsilon^{\prime}\epsilon^{\prime\prime\prime}D^{\,\gamma}J$)}$
$\displaystyle=\epsilon^{\prime}\epsilon^{\prime\prime\prime}J(D^{\,\gamma}+u[D^{\,\gamma},u^{\dagger_{\gamma}}]_{\rho})J^{-1}+u[D^{\,\gamma},u^{\dagger_{\gamma}}]_{\rho}$
$\displaystyle=D^{\,\gamma}+u[D^{\,\gamma},u^{\dagger_{\gamma}}]_{\rho}+\epsilon^{\prime}\epsilon^{\prime\prime\prime}Ju[D^{\,\gamma},u^{\dagger_{\gamma}}]_{\rho}J^{-1}.$

In this way, twisted fluctuations takes the form
$D^{\,\gamma}_{A_{\rho}}=D^{\,\gamma}+A_{\rho}+\epsilon^{\prime}\epsilon^{\prime\prime\prime}JA_{\rho}J^{-1}$.

######  Proposition 4.8 (Duality at the level of the fluctuations of the Dirac
operators).

There is a duality between twisted fluctuations
$D^{\,\gamma}_{A_{\rho}}=D^{\,\gamma}+A_{\rho}+\epsilon^{\prime}\epsilon^{\prime\prime\prime}JA_{\rho}J^{-1}$
and untwisted fluctuation $D_{A}=D+A+\epsilon^{\prime}JAJ^{-1}$ given by their
evaluations on their respective inner products.

###### Proof

We must show that $\forall\psi\in\mathcal{H}$ we have

$\langle\psi,D^{\,\gamma}_{A_{\rho}}\psi\rangle_{\gamma}=\langle\psi,D_{A}\psi\rangle$

when taking $A_{\rho}=\rho(a)[D^{\,\gamma},b]_{\rho}$ and $A=a[D,b]$. Thanks
to proposition 4.2 and 4.5 it remains to prove the equivalence at the level of
$\epsilon^{\prime}\epsilon^{\prime\prime\prime}JA_{\rho}J^{-1}$ and
$\epsilon^{\prime}JAJ^{-1}$:

$\displaystyle\langle\psi,\epsilon^{\prime}\epsilon^{\prime\prime\prime}JA_{\rho}J^{-1}\psi\rangle_{\gamma}=\langle\psi,\epsilon^{\prime}\epsilon^{\prime\prime\prime}J\gamma
AJ^{-1}\psi\rangle_{\gamma}=\langle\psi,\epsilon^{\prime}\gamma
JAJ^{-1}\psi\rangle_{\gamma}=\langle\psi,\epsilon^{\prime}JAJ^{-1}\psi\rangle$

Taking an element $a\in\mathcal{A}$, we have by definition
$\rho(Ad(a))\vcentcolon=Ad(\rho(a))=\rho(a)J\rho(a)J^{-1}=\rho(a)\rho^{\circ}(JaJ^{-1})$.
As mentioned in section 3, there are two ways to generate twisted
fluctuations:

  1. 1.

$D^{\,\gamma}_{A_{\rho}}=UD^{\,\gamma}U^{\dagger_{\gamma}}$ with
$U=Ad(u)=uJuJ^{-1}$ and $u$ a unitary operator in $\mathcal{A}$.

  2. 2.

$D^{\,\gamma}_{A_{\rho}}=U_{\gamma}D^{\,\gamma}U_{\gamma}^{\dagger}$ with
$U_{\gamma}=Ad(u_{\gamma})=u_{\gamma}Ju_{\gamma}J^{-1}$ and $u_{\gamma}$ a
$\gamma$-unitary operator in $\mathcal{A}$.

Concerning the generation of usual fluctuations, we have that:

  1. 1.

$D_{A}=UDU^{\dagger}$ with $U=Ad(u)=uJuJ^{-1}$ and $u$ a unitary operator in
$\mathcal{A}$.

  2. 2.

$D_{A}=U_{\gamma}DU_{\gamma}^{\dagger_{\gamma}}$ with
$U_{\gamma}=Ad(u_{\gamma})=u_{\gamma}Ju_{\gamma}J^{-1}$ and $u_{\gamma}$ a
$\gamma$-unitary operator in $\mathcal{A}$.

######  Proposition 4.9 (Duality at the level of inner fluctuations).

There is a duality between the inner actions that generate twisted
fluctuations and the ones that generate usual fluctuations, given by
$UD^{\,\gamma}U^{\dagger_{\gamma}}=\gamma VDV^{\dagger}$ with $U=uJuJ^{-1}$
and $V=\rho(U)=\rho(u)J\rho(u)J^{-1}$ another unitary or by the relation
$U_{\gamma}D^{\,\gamma}U_{\gamma}^{\dagger}=\gamma
V_{\gamma}DV_{\gamma}^{\dagger_{\gamma}}$ with
$U_{\gamma}=u_{\gamma}Ju_{\gamma}J^{-1}$ and
$V_{\gamma}=\rho(U_{\gamma})=\rho(u_{\gamma})J\rho(u_{\gamma})J^{-1}$ another
$\gamma$-unitary operator.

###### Proof

Using that $a\gamma^{\mu}=\gamma^{\mu}\rho(a)$ for any $a\in\mathcal{A}$ (see
[13] for the proof), so that
$Ad(a)\gamma=\gamma\rho(a)J\rho(a)J^{-1}=\gamma\rho(Ad(a))$, then the first
way to generate twisted fluctuation is equivalent to a generation of usual
fluctuation in the duality:

$\displaystyle
D^{\,\gamma}_{A_{\rho}}=UD^{\,\gamma}U^{\dagger_{\gamma}}=U\gamma
DU^{\dagger_{\gamma}}=\gamma\rho(U)DU^{\dagger_{\gamma}}=\gamma
VDV^{\dagger}\vcentcolon=\gamma D_{A}$

Where $V\vcentcolon=\rho(U)$ is also an unitary operator since
$V=\rho(u)J\rho(u)J^{-1}$ with $\rho(u)\in\mathcal{U}(\mathcal{A})$. For the
second kind of twisted fluctuations:

$\displaystyle
D^{\,\gamma}_{A_{\rho}}=U_{\gamma}D^{\,\gamma}U_{\gamma}^{\dagger}=U_{\gamma}\gamma
DU_{\gamma}^{\dagger}=\gamma\rho(U_{\gamma})DU_{\gamma}^{\dagger}=\gamma
V_{\gamma}DV_{\gamma}^{\dagger_{\gamma}}\vcentcolon=\gamma D_{A}$

with $V_{\gamma}\vcentcolon=\rho(U_{\gamma})$ being also a $\gamma$-unitary
operator since $\rho(u_{\gamma})\in\mathcal{U}_{\gamma}(\mathcal{A})$.

######  Proposition 4.10 (Duality at the level of fermionic actions).

The usual fermionic action evaluated on $D_{A}$ and the $\gamma$-twisted
fermionic action evaluated on $D^{\,\gamma}_{A_{\rho}}=\gamma D_{A}$ are
equals.

###### Proof

A direct computation gives

$\mathcal{S}_{f}^{\gamma}(D^{\,\gamma}_{A_{\rho}},\psi)=\langle\psi,D^{\,\gamma}_{A_{\rho}}\psi\rangle_{\gamma}=\langle\psi,\gamma\gamma
D_{A}\psi\rangle=\mathcal{S}_{f}(D_{A},\psi)$

######  Proposition 4.11 (Duality at the level of spectral actions).

The evaluation of the spectral action is the same in both context.

###### Proof

As $D^{\,\gamma}_{A_{\rho}}=\gamma D_{A}$, we have that

$\displaystyle\mathcal{S}[D^{\,\gamma}_{A_{\rho}}]=\operatorname{Tr}f(D^{\,\gamma}_{A_{\rho}}(D^{\,\gamma}_{A_{\rho}})^{\dagger}/\Lambda^{2})=\operatorname{Tr}f(\gamma
D_{A}D_{A}^{\dagger}\gamma^{\dagger}/\Lambda^{2})=\operatorname{Tr}f(D_{A}D_{A}^{\dagger}/\Lambda^{2})=\mathcal{S}[D_{A}]$

##  5 Implications of the Duality

Starting from the axioms of spectral triples, as given in section 2, and using
the duality pointed in section 4, we deduce the axioms of a $\gamma$-twisted
spectral triple dual to a usual spectral triple.

######  Proposition 5.1.

A $\gamma$-twisted spectral triple with Dirac operator $D^{\,\gamma}=\gamma D$
is dual to a spectral triple (as defined in section 2) with Dirac operator $D$
iff $JD^{\,\gamma}=\epsilon_{\gamma}^{\prime}D^{\,\gamma}J$ with
$\epsilon_{\gamma}^{\prime}=\epsilon^{\prime}\epsilon^{\prime\prime\prime}$
and $\Gamma D^{\,\gamma}=D^{\,\gamma}\Gamma$.

###### Proof

The relation $JD=\epsilon^{\prime}DJ$ in the usual spectral triple implies
$JD^{\,\gamma}=\epsilon^{\prime}\epsilon^{\prime\prime\prime}D^{\,\gamma}J$
and since $\gamma^{\mu}\Gamma=-\Gamma\gamma^{\mu}$, using $D\Gamma=-\Gamma D$
we get directly that $\Gamma D^{\,\gamma}=D^{\,\gamma}\Gamma$.

######  Definition 5.2 (Lorentzian twisted spectral triple).

We define Lorentzian twisted spectral triple as a $\gamma$-twisted spectral
triple which is dual to a spectral triple.

An important example of $\gamma$-twisted spectral triple is given by the
$4$-dimensional manifold, taking $\gamma=\gamma^{0}$ as in remark 3.2, with
$D^{L\,}$ for the $\gamma^{0}$-self-adjoint Dirac operator. This
$\gamma$-twisted spectral triple is considered as a good candidate for a
Lorentzian version of space-time’s spectral triples for the following reasons:

  * •

It endows the Lorentzian Dirac operator $D^{L\,}$ within a meaningful context
given by the $\gamma^{0}$-product and a duality with usual spectral triple.

  * •

The evaluation of $\mathcal{S}_{f}^{\gamma^{0}}(D^{L\,},\psi)$ gives the
physical Dirac action $\mathcal{S}_{D}$.

  * •

$\mathcal{S}_{D}$ is invariant under the symmetry group associated with
$\mathcal{U}_{\gamma^{0}}$, which in particular contain the Lorentz group as a
sub-group (see [16]).

Now taking the Lorentzian Dirac operator $D^{L\,}$ within the corresponding
$\gamma$-twisted spectral triple, an important question is to determine the
corresponding spectral triple.

The Lorentzian gamma matrices are defined according to the Euclidean ones by
$\gamma^{0}_{L}\vcentcolon=\gamma^{0}$ and
$\gamma^{i}_{L}\vcentcolon=i\gamma^{i}$ for $i=1,2,3$. We then obtain the dual
spectral triple, containing the following Dirac operator

$D\vcentcolon=\gamma^{0}D^{L\,}=\gamma^{0}\gamma_{L}^{\mu}\nabla_{\mu}^{S}$

Such a Dirac operator $D$ as also been used in the name of Krein-shifted Dirac
operator (see [3]) as an attempt to solve the fermion doubling problem,
another major weak point of the NCSMPP.

##  6 Conclusion

The particularity of the minimal twist by grading of the manifold is that it
allow the algebra to act differently on each chirality eigenspace and to
change the action of the commutator according to this splitting. The present
duality permit to obtain a different presentation of the usual spectral triple
in which the Dirac operator is the Lorentzian one while giving the same
spectral invariants. We hope that this unexpected duality will offer a
concrete path toward the definition of a Lorentzian approach within NCG and
also to modify the actual NCSMPP by using the dual spectral triple into the
almost commutative manifold, to obtain the good Lorentz invariant fermionic
action and solve the fermion doubling problem. It remains to investigate
further the implications of this duality, to study the metric properties of
$D=\gamma^{0}D^{L\,}$, the associated distance, the causal structure and also
to generalize this result to more general spectral triples. This will be done
in the next article.

## References

  * Barrett [2007] John W Barrett.  Lorentzian version of the noncommutative geometry of the standard model of particle physics.  _Journal of mathematical physics_ , 48(1), 2007.
  * Bochniak and Sitarz [2018] Arkadiusz Bochniak and Andrzej Sitarz.  Finite pseudo-riemannian spectral triples and the standard model.  _Physical Review D_ , 97(11):115029, 2018.
  * Bochniak and Sitarz [2020] Arkadiusz Bochniak and Andrzej Sitarz.  Spectral geometry for the standard model without fermion doubling.  _Physical Review D_ , 101(7):075038, 2020.
  * Chamseddine and Connes [1997] Ali H Chamseddine and Alain Connes.  The spectral action principle.  _Communications in Mathematical Physics_ , 186(3):731–750, 1997.
  * Chamseddine and Connes [2010] Ali H Chamseddine and Alain Connes.  Noncommutative geometry as a framework for unification of all fundamental interactions including gravity. part i.  _Fortschritte der Physik_ , 58(6):553–600, 2010\.
  * Connes [2006] Alain Connes.  Noncommutative geometry and the standard model with neutrino mixing.  _Journal of High Energy Physics_ , 2006(11):081, 2006.
  * Connes [2013] Alain Connes.  On the spectral characterization of manifolds.  _Journal of Noncommutative Geometry_ , 7(1):1–82, 2013.
  * Connes and Moscovici [2006] Alain Connes and Henri Moscovici.  Type iii and spectral triples.  _arXiv preprint math/0609703_ , 2006.
  * Devastato and Martinetti [2017] Agostino Devastato and Pierre Martinetti.  Twisted spectral triple for the standard model and spontaneous breaking of the grand symmetry.  _Mathematical Physics, Analysis and Geometry_ , 20:1–43, 2017.
  * Devastato et al. [2018] Agostino Devastato, Shane Farnsworth, Fedele Lizzi, and Pierre Martinetti.  Lorentz signature and twisted spectral triples.  _Journal of High Energy Physics_ , 2018(3):1–21, 2018.
  * Filaci et al. [2021] Manuele Filaci, Pierre Martinetti, and Simone Pesco.  Minimal twist for the standard model in noncommutative geometry: the field content.  _Physical Review D_ , 104(2):025011, 2021.
  * Franco [2014] Nicolas Franco.  Temporal lorentzian spectral triples.  _Reviews in Mathematical Physics_ , 26(08):1430007, 2014.
  * Landi and Martinetti [2016] Giovanni Landi and Pierre Martinetti.  On twisting real spectral triples by algebra automorphisms.  _Letters in Mathematical Physics_ , 106(11):1499–1530, 2016.
  * Landi and Martinetti [2018] Giovanni Landi and Pierre Martinetti.  Gauge transformations for twisted spectral triples.  _Letters in Mathematical Physics_ , 108:2589–2626, 2018\.
  * Martinetti and Singh [2022] Pierre Martinetti and Devashish Singh.  Lorentzian fermionic action by twisting euclidean spectral triples.  _Journal of Noncommutative Geometry_ , 16(2):513–559, 2022.
  * Martinetti et al. [2024] Pierre Martinetti, Gaston Nieuviarts, and Ruben Zeitoun.  Torsion from twisted spectral triples.  _preprint arXiv:2401.07848_ , 2024.
  * Paschke and Sitarz [2006] Mario Paschke and Andrzej Sitarz.  Equivariant lorentzian spectral triples.  _arXiv preprint math-ph/0611029_ , 2006.
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On a General Concept of a Hausdorff-Type Operator

A. R. Mirotin

<EMAIL_ADDRESS>
Abstract. A unified approach to the concept of a Hausdorff operator is
proposed in such a way that a number of classical and new operators feet into
the given definition. Conditions are given for the boundedness of the
operators under consideration in $L^{p}$ and in the atomic Hardy space
$H^{1}$, and their regularity property is investigated. Examples are
considered. The author hopes that this approach will allow one to unify the
study of a lot of extensions and analogs of the classical Hausdorff operator.

2020 Mathematics Subject Classification: Primary 47B38; Secondary 47B15, 46E30

Key words and phrases. Hausdorff operator, topological group, discrete Hilbert
transform, Hilbert transform, Cauchy transform, Hardy space, space of
homogeneous type, regularity.

##  1\.  Introduction

In resent two decades different notions of a Hausdorff operator have been
suggested (see, e. g., [18, 19, 5, 17, 24, 38, 39, 22, 27, 28] and
bibliography therein). In our opinion, the unified approach to this notion may
be as follows.

Let $\mathfrak{S}$ be a set which is an object of some category. In particular
one can assume that $\mathfrak{S}$ is endowed with some mathematical structure
(algebraical, topological, analytical, algebraical-topological, order,
measure, etc.) in a sense of N. Bourbaki. Let $\mathrm{Aut}(\mathfrak{S})$
stands for the set of all automorphisms of $\mathfrak{S}$ in this category,
and $(\Omega,\mu)$ denotes some measure space. Finally, let
$A:\Omega\to\mathrm{Aut}(\mathfrak{S})$ be some measurable map (in a sense
which will be specified in each concrete situation, see, e.g., definitions
4.4, 5.7, and 5.11 below) defined a. e. $[\mu]$, and $\Phi$ a $\mu$-measurable
function.

######  Definition 1.1.

A Hausdorff operator acts on a functions $f:\mathfrak{S}\to X$ (here $X$ is
some topological vector space) by the rule

$(\mathcal{H}_{\Phi,A}f)(x)=\int_{\Omega}\Phi(u)f(A(u)(x))d\mu(u)$ (1.1)

provided the integral converge in a suitable sense.

The author hopes that this approach will allow one to unify the study of a lot
of extensions and analogs of the classical Hausdorff operator.

The next examples show that a number of classical and new operators feet into
Definition 1.1.

##  2\. Special cases

######  Example 1.

(Hausdorff operator over a matrix algebra.) Let
$\mathfrak{S}=\mathrm{Mat}_{n}(\mathfrak{k})$ be the algebra of square
matrices $M=(m_{ij})$ of order $n$ over a field $\mathfrak{k}$. Every
$M\in\mathrm{Mat}_{n}(\mathfrak{k})$ has the form $M=(m_{1},\dots,m_{n})$,
where $m_{j}$ stands for the $j$th column of $M$. For each permutation
$\sigma\in\mathbf{S}_{n}$ ($\mathbf{S}_{n}$ denotes the symmetric group of
order $n$) we denote by $A(\sigma)$ the bijection of
$\mathrm{Mat}_{n}(\mathfrak{k})$ (automorphism in the category of sets) such
that $A(\sigma)(M)=(m_{\sigma(1)},\dots,m_{\sigma(n)})$. We equip the set
$\Omega=\mathbf{S}_{n}$ with the counting measure. Let
$\Phi(\sigma)=\mathrm{sgn}(\sigma)$, where $\mathrm{sgn}(\sigma):=1$ if
$\sigma$ is even, and $\mathrm{sgn}(\sigma):=-1$ otherwise. Then a Hausdorff
operator in the sense of Definition 1.1 acts on the function
$f:\mathrm{Mat}_{n}(\mathfrak{k})\to X$ as

$(\mathcal{H}_{\Phi,A}f)(M)=\sum_{\sigma\in\mathbf{S}_{n}}\mathrm{sgn}(\sigma)f(A(\sigma)(M)).$

In particular, if we take $\mathfrak{k}=X=\mathbb{C}$,
$f_{0}(M)=\prod_{i=1}^{n}m_{ii}$, then

$\displaystyle(\mathcal{H}_{\Phi,A}f_{0})(M)$ $\displaystyle=$
$\displaystyle\sum_{\sigma\in\mathbf{S}_{n}}\mathrm{sgn}(\sigma)f_{0}(A(\sigma)(M))$
$\displaystyle=$
$\displaystyle\sum_{\sigma\in\mathbf{S}_{n}}\mathrm{sgn}(\sigma)f_{0}(A(\sigma)(m_{1},\dots,m_{n})).$

Since the right-hand side here is an alternate multilinear form (as a function
of column vectors $m_{1},\dots,m_{n}$), we have

$(\mathcal{H}_{\Phi,A}f_{0})(M)=\mathrm{det}(M)$

(see also [1, p. 202]).

One can take also as $\mathfrak{S}$ any subset of
$\mathrm{Mat}_{n}(\mathfrak{k})$ which is invariant with respect to some
family of automorphisms $(A(\sigma))_{\sigma\in\Sigma}$.

######  Example 2.

(The Discrete Hilbert transform.) Let $\mathfrak{S}=\mathbb{Z}$ be the ring of
integers with its natural order, $\Omega=\mathbb{Z}$ endowed with a discrete
measure $\mu(\\{k\\})=p_{k}$, and $A(u)(k)=k-u$ ($k,u\in\mathbb{Z}$) an order
preserving bijections of $\mathbb{Z}$ (automorphisms in the category of
linearly ordered sets). Let

$\Phi(u)=\begin{cases}\frac{2}{\pi u},u\mbox{ odd}\\\ 0,u\mbox{
even}.\end{cases}$

In this case, (1.1) takes the form

$(Hf)(k)=\sum_{u\in\mathbb{Z}}\Phi(u)f(k-u)p_{u}=\begin{cases}\frac{2}{\pi}\sum\limits_{n\
\mathrm{odd}}\frac{f(n)}{k-n}p_{k-n},k\mbox{ even}\\\
\frac{2}{\pi}\sum\limits_{n\ \mathrm{even}}\frac{f(n)}{k-n}p_{k-n},k\mbox{
odd}.\end{cases}$

For $p_{k}\equiv 1$ this is the discrete Hilbert transform of a function
$f:\mathbb{Z}\to X$ (for the case $X=\mathbb{C}$ see [15]).

######  Example 3.

(The Hilbert transform.) Let $\mathfrak{S}=\Omega$ be the real line
$\mathbb{R}$ with Euclidean metric and Lebesgue measure, $A(u)(x)=x-u$
($x,u\in\mathbb{R}$) a distance preserving bijections of $\mathbb{R}$, and
$\Phi(u)=\frac{1}{\pi u}$, the Cauchi kernel. In this case, (1.1) takes the
form

$(\mathrm{H}f)(x)=\frac{1}{\pi}\mbox{p.v.}\int_{-\infty}^{\infty}\frac{f(x-u)}{u}du,$

the Hilbert transform of a measurable function $f:\mathbb{R}\to X$.

Calderón-Zygmund operators can be considered in a similar manner.

The previous example can be generalized in the following way (see [37, 34,
35]).

######  Example 4.

(The Hilbert transform along curves.) Let $\mathfrak{S}=\mathbb{R}^{n}$ with
Euclidean metric, $\Omega=\mathbb{R}$ with Lebesgue measure, and
$A(u)(x)=x-\gamma(u)$ ($x\in\mathbb{R}^{n}$) a distance preserving bijections
of $\mathbb{R}^{n}$ where $\gamma:\mathbb{R}\to\mathbb{R}^{n}$ is a suitable
function (say polynomial) satisfying $\gamma(0)=0$. Then the singular integral
operator

$(\mathrm{H}f)(x):=\int_{-\infty}^{\infty}\Phi(u)f(x-\gamma(u))du,$

where $\Phi$ is a Calderón-Zygmund kernel, is of the form (1.1).

From now on we shall assume that the integral in (1.1) exists in the sense of
Lebesgue.

######  Example 5.

(A Cauchi transform over a circular manifold.) Let $\mathfrak{S}$ be a complex
submanifold of $\mathbb{C}^{n}$ with automorphisms (biholomorphic mappings)
$A(u)(z)=(u_{1}z_{1},\dots,u_{n}z_{n})$, where
$u=(u_{1},\dots,u_{n})\in\mathbb{T}^{n}$ (e. g., let $\mathfrak{S}$ be a
Reinchart domain in $\mathbb{C}^{n}$, or the torus $\mathbb{T}^{n}$). Let also
$\Omega=\mathbb{T}^{n}$ endowed with the Lebesgue measure, and
$\Phi(u)=\frac{1}{(2\pi\imath)^{n}}\frac{1}{(u_{1}-1)\dots(u_{n}-1)}$. In this
case, (1.1) turns to the operator

$(\mathcal{C}f)(z)=\frac{1}{(2\pi\imath)^{n}}\int_{\mathbb{T}^{n}}\frac{f(u_{1}z_{1},\dots,u_{n}z_{n})}{(u_{1}-1)\dots(u_{n}-1)}du_{1}\dots
du_{n},\ z\in\mathfrak{S}.$

On can call this operator a Cauchi transform of a measurable function $f$ on
$\mathfrak{S}$. Indeed, if $\mathfrak{S}=\mathbb{T}^{n}$, then putting
$u_{j}=\zeta_{j}/z_{j}$ for $j=1,\dots,n$ we get

$(\mathcal{C}f)(z)=\frac{1}{(2\pi\imath)^{n}}\int_{\mathbb{T}^{n}}\frac{f(\zeta_{1},\dots,\zeta_{n})}{(\zeta_{1}-z_{1})\dots(\zeta_{n}-z_{n})}d\zeta_{1}\dots
d\zeta_{n},\ z\in\mathbb{T}^{n}$

the Cauchi transform of a measurable function $f$ on $\mathbb{T}^{n}$.

######  Example 6.

(A convolution with a measure.) Let $\mathfrak{S}=G=\Omega$ be a
multiplicative group equipped with some (left) invariant metric,
$\mathrm{Aut}(\mathfrak{S})=\mathrm{Iso}(G)$ the set of isometries of $G$
(automorphisms in the category of metric spaces), $A(u)(x)=u^{-1}x$ ($u\in
G$), $\Phi(u)=1$. In this case, (1.1) turns into a convolution operator
$f\mapsto f\ast\mu$ on $G$.

######  Example 7.

(Hausdorff operator over a topological group.) Let $\mathfrak{S}=G$ be a
topological group, $\mathrm{Aut}(\mathfrak{S})=\mathrm{Aut}(G)$ the group of
all topological automorphisms of $G$. In this case, we have got a general
definition of a Hausdorff operator on topological groups. This example
contains several known definitions of Hausdorff operators for classical groups
(see [24], [25], and examples therein).

######  Example 8.

(Discrete Hausdorff operator over the Euclidean space.) Let
$\mathfrak{S}=\mathbb{R}^{d}$ be a $d$-dimensional Euclidean space considered
as an additive topological group. Then the group
$\mathrm{Aut}(\mathbb{R}^{d})$ of all topological automorphisms of
$\mathbb{R}^{d}$ can be identified with the general linear group
$\mathrm{GL}(d,\mathbb{R})$. Let $\Omega=\mathbb{Z}$ be endowed with the
counting measure. In this case, (1.1) turns into a so-called discrete
Hausdorff operator

$(\mathcal{H}_{\Phi,A}f)(x)=\sum_{k\in\mathbb{Z}}\Phi(k)f(A(k)x),$

where $A(k)\in\mathrm{GL}(d,\mathbb{R})$, $x\in\mathbb{R}^{d}$ is a column
vector. For the spectral theory of such operators see [32, 33].

######  Example 9.

(Hausdorff operator over a homogeneous space.) Let $\mathfrak{S}=G/K$ be a
homogeneous space of a locally compact group $G$, $K$ a compact subgroup of
$G$. In this case, $\mathrm{Aut}(\mathfrak{S})$ can be identified with the
group $\mathrm{Aut}_{K}(G)$ of all topological automorphisms of $G$ which map
$K$ onto itself (see [26], [27]) and examples therein).

######  Example 10.

(Hausdorff operator over a double coset space of a topological group.) Let
$\mathfrak{S}=G//K$ be a double coset space of a locally compact group $G$,
$K$ a compact subgroup of $G$. In this case again $\mathrm{Aut}(\mathfrak{S})$
can be identified with the group $\mathrm{Aut}_{K}(G)$ of all topological
automorphisms of $G$ which map $K$ onto itself (see [28] for details).

######  Example 11.

(Hausdorff operator over the unit disc.) Let $\mathfrak{S}=\mathbb{D}$ be the
unite disc in the complex plane with its natural analytic structure,
$\mathrm{Aut}(\mathfrak{S})=\mathrm{Aut}_{0}(\mathbb{D})$ the group of all
involutive Mobius automorphisms of $\mathbb{D}$,
$A(u)(z)=\frac{u-z}{1-\bar{u}z}$, $\Omega=\mathbb{D}$. In this case, (1.1)
turns to a so called Hausdorff-Zhu operator (see [2, 22, 23, 16, 8]).

A similar construction works if $\mathfrak{S}$ is the unit ball in
$\mathbb{C}^{n}$.

######  Example 12.

(Hausdorff operator over the upper half-plane.) Let $\mathbb{C}^{+}$ be the
upper half-plane with its natural analytic structure,
$\mathfrak{S}=(\mathbb{C}^{+})^{n}$ , $\Omega=(0,\infty)^{n}$, and
$A(u)(z)=(\frac{z_{1}}{u_{1}},\dots,\frac{z_{n}}{u_{n}})$ ($u\in\Omega$) a
biholomorphic map of $(\mathbb{C}^{+})^{n}$. In the case $n=1$, (1.1) turns to
a so called Hausdorff operator over the upper half-plane (see [39], [38], [2]
and the bibliography therein).

##  3\. The regularity property of general Hausdorff operators

To examine the regularity property of the transformation
$\mathcal{H}_{\Phi,A}$ we need the following definition.

######  Definition 3.2.

Let $\mathcal{B}$ be a filter base on the set $\mathfrak{S}$. We say that a
family $(A(u))_{u\in\Omega}$ of automorphisms of $\mathfrak{S}$ agrees with
the filter base $\mathcal{B}$ if $A(u)^{-1}(B)$ belongs to $\mathcal{B}$ for
each $B\in\mathcal{B}$ and every $u\in\Omega$.

The next proposition is a wide generalization of the classical result of
Gerabedyan and Rogosinskii (see [9]).

######  Proposition 3.3.

Suppose that the conditions of Definition 1.1 are fulfilled and a filter base
$\mathcal{B}$ on $\mathfrak{S}$ is countable. Let a family
$(A(u))_{u\in\Omega}$ of automorphisms of $\mathfrak{S}$ agrees with
$\mathcal{B}$. In order that the transformation $\mathcal{H}_{\Phi,A}$ should
be regular, i. e. that for every bounded function $f$ on $\mathfrak{S}$ the
equality $\lim_{x,\mathcal{B}}f(x)=l$ should imply
$\lim_{x,\mathcal{B}}(\mathcal{H}_{\Phi,A}f)(x)=l$ it is necessary and
sufficient that

$\int_{\Omega}\Phi(u)d\mu(u)=1.$ (3.2)

###### Proof.

If $f(x)=1$ then $(\mathcal{H}_{\Phi,A}f)(x)=\int_{\Omega}\Phi(u)d\mu(u).$
Thus, $\int_{\Omega}\Phi(u)d\mu(u)=1$ is a necessary condition.

To prove the sufficiency, let $\int_{\Omega}\Phi(u)d\mu(u)=1$, and
$\lim_{x,\mathcal{B}}f(x)=l$. Then

$\lim_{x,\mathcal{B}}f(A(u)(x))=l$ (3.3)

for all $u\in\Omega$. Indeed, for every $\varepsilon>0$ there exists such
$B_{\varepsilon}\in\mathcal{B}$ that $|f(y)-l|<\varepsilon$ for all $y\in
B_{\varepsilon}$. It follows that $|f(A(u)(x))-l|<\varepsilon$ for all $x\in
A(u)^{-1}(B_{\varepsilon})$, as well. By the Definition 3.2 we have
$A(u)^{-1}(B_{\varepsilon})\in\mathcal{B}$ for each $u\in\Omega$ and (3.3)
follows. Now by the Lebesgue Theorem (one can apply the Lebesgue Theorem,
since the filter base $\mathcal{B}$ is countable) one has

$\lim_{x,\mathcal{B}}(\mathcal{H}_{\Phi,A}f)(x)=\int_{\Omega}\Phi(u)ld\mu(u)=l.$

∎

######  Example 13.

Let in the Example 7 $G$ be sigma-compact and $\mathcal{B}$ be the set of all
complements $G\setminus K$ of compact subsets $K\subset G$. In this case, all
the conditions of Definition 3.2 are fulfilled for every topological
automorphosm $A$ of $G$, and $\lim_{x,\mathcal{B}}f(x)=\lim_{x\to\infty}f(x)$.
Thus the Proposition 3.3 implies that under the condition (3.2) one has
$\lim_{x\to\infty}(\mathcal{H}_{\Phi,A}f)(x)=l$ whenever
$\lim_{x\to\infty}f(x)=l$ for a bounded measurable function $f$ on $G$.

######  Example 14.

Let in the Example 5 $\mathcal{B}$ be the set of all balls
$B_{k}:=\\{|z|<1/k\\}$, $k\in\mathbb{N}$ in $\mathbb{C}^{n}$ and
$\mathfrak{S}=B_{1}$. Then the Proposition 3.3 implies that under the
condition (3.2) one has $\lim_{z\to 0}(\mathcal{C}f)(z)=l$ whenever
$\lim_{z\to 0}f(z)=l$ for a bounded measurable function $f$ on $B_{1}$.

##  4\. $L^{p}$ boundedness of general Hausdorff-type operators

To formulate a result on the $L^{p}$ boundedness of the operator (1.1) we need
the following notion.

######  Definition 4.4.

Let the set $\mathfrak{S}$ be equipped with some sigma-finite positive measure
$\nu$. We say that the family $(A(u))_{u\in\Omega}$ of authomorphisms of
$\mathfrak{S}$ agrees with the measure $\nu$ if for each $\nu$-measurable set
$E\subset\mathfrak{S}$ of finite measure and for every $u\in\Omega$ the set
$A(u)^{-1}(E)$ is $\nu$-measurable and

$\nu(A(u)^{-1}(E))=m(A(u))^{-1}\nu(E)$

for some positive $\mu$-measurable function $u\mapsto m(A(u))$.

######  Proposition 4.5.

Let the set $\mathfrak{S}$ be equipped with some sigma-finite positive measure
$\nu$, the family $(A(u))_{u\in\Omega}$ agrees with $\nu$, and $1\leq
p\leq\infty$. If

$\|\Phi\|_{A,p}:=\int_{\Omega}|\Phi(u)|m(A(u))^{-1/p}d\mu(u)<\infty$

(here $\|\Phi\|_{A,\infty}:=\|\Phi\|_{L^{1}(\mu)}$), then the operator
$\mathcal{H}_{\Phi,A}$ is bounded in $L^{p}(\nu)$ and its norm does not exceed
$\|\Phi\|_{A,p}$.

###### Proof.

Using Minkowskii integral inequality we have for $1<p<\infty$ and $f\in
L^{p}(\nu)$

$\displaystyle\|\mathcal{H}_{\Phi,A}f\|_{L^{p}(\nu)}$ $\displaystyle=$
$\displaystyle\left(\int_{\mathfrak{S}}\left|\int_{\Omega}\Phi(u)f(A(u)(x))d\mu(u)\right|^{p}d\nu(x)\right)^{1/p}$
$\displaystyle\leq$
$\displaystyle\int_{\Omega}\left(\int_{\mathfrak{S}}|\Phi(u)|^{p}|f(A(u)(x))|^{p}d\nu(x)\right)^{1/p}d\mu(u)$
$\displaystyle=$
$\displaystyle\int_{\Omega}|\Phi(u)|\left(\int_{\mathfrak{S}}|f(A(u)(x))|^{p}d\nu(x)\right)^{1/p}d\mu(u).$

Since by the Definition 4.4

$\int_{\mathfrak{S}}|f(A(u)(x))|^{p}d\nu(x)=m(A(u))^{-1}\int_{\mathfrak{S}}|f(x)|^{p}d\nu(x)$
(4.4)

(it suffices to verify the last equality for $f=\chi_{E}$, the indicator of a
$\nu$-measurable set $E\subset\mathfrak{S}$ of finite measure), we have

$\displaystyle\|\mathcal{H}_{\Phi,A}f\|_{L^{p}(\nu)}$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}|\Phi(u)|m(A(u))^{-1/p}d\mu(u)\left(\int_{\mathfrak{S}}|f(x)|^{p}d\nu(x)\right)^{1/p}$
$\displaystyle=$ $\displaystyle\|\Phi\|_{A,p}\|f\|_{L^{p}(\nu)}.$

For $p=1$ the statement of the proposition follows from Fubini Theorem and for
$p=\infty$ it is obvious. ∎

######  Example 15.

Let in the Example 7 $G$ be locally compact, $\nu$ the Haar measure of $G$,
and the family $\mathrm{Aut}(G)$ of all topological automorphisms of a locally
compact group $G$ is equipped with its natural topology (see, e. g., [11,
(26.1)]). Assume that the map $u\mapsto A(u)$ from $\Omega$ to
$\mathrm{Aut}(G)$ is measurable with respect to the measure $\mu$ in $\Omega$
and the Borel structure in $\mathrm{Aut}(G)$. In this case all the conditions
of Definition 4.4 are fulfilled for $(A(u))_{u\in\Omega}$. Indeed, we have
$m(A(u))={\rm mod}(A(u))$, the modulus of $A(u)$, and the map $A\mapsto{\rm
mod}(A)$ from $\mathrm{Aut}(G)$ to $(0,\infty)$ is continuous (see [11,
(26.21)]). Thus, the family $(A(u))_{u\in\Omega}$ agrees with the measure
$\nu$ and the Proposition 4.5 is applicable.

##  5\. $H^{1}$ boundedness of general Hausdorff-type operators

In this section we shall be working in the following setting. We assume that
$\mathfrak{S}$ is at quasi-metric space with quasi-metric $\rho$ and positive
regular Borel measure $\nu$. Moreover, the following doubling condition holds:

There exists of a constant $C$ such that

$\nu(B(x,2r))\leq C\nu(B(x,r))$

for each $x\in\mathfrak{S}$ and $r>0$.

(Here and below $B(x,r)$ denotes a quasi-ball with respect to $\rho$ with a
center $x$ and radius $r>0$).

In this case, the triple $(\mathfrak{S},\rho,\nu)$ is called a quasi-metric
measure space of homogeneous type [7].

The doubling constant  is the smallest constant $C\geq 1$ for which the
doubling inequality holds. We denote this constant by $C_{\nu}.$ Then for each
$x\in\mathfrak{S},k\geq 1$ and $r>0$

$\nu(B(x,kr))\leq C_{\nu}k^{s}\nu(B(x,r)),$ (5.5)

where $s=\log_{2}C_{\nu}$ (see, e.g., [10, p. 76]). The number $s$ sometimes
takes the role of a “dimension” for a doubling quasi-metric measure space.

######  Definition 5.6.

Let $(\Omega,\mu)$ be a measure space. We say that a family of automorphisms
$(A(u))_{u\in\Omega}$ of a quasi-metric space $(\mathfrak{S},\rho)$ agrees
with the quasi-metric if there exists a $\mu$-measurable function $k(u)$ which
depends on $u\in\Omega$ only, such that for every $x\in\mathfrak{S}$, for
every $u\in\Omega$, and for every $r>0$

$A(u)^{-1}(B(x,r))\subseteq B(x^{\prime},k(u)r)$ (5.6)

for some point $x^{\prime}=x^{\prime}(x,u,r)\in\mathfrak{S}$111In fact, $k(u)$
depends on $A(u)$..

Remark 1. Let $\Omega$ be a $\sigma$-compact quasi-metric space with Radon
measure $\mu$. If $\mathfrak{S}=G$ is a (finite dimensional real or complex)
connected Lie group with left invariant Riemann metric $\rho$, then every
automorphism $A\in\mathrm{Aut}(G)$ is Lipschitz and $\mathrm{Aut}(G)$ agrees
with $\rho$ by [27, Lemma 2.6].

######  Definition 5.7.

Let $(\Omega,\mu)$ be a measure space. We say that a family of automorphisms
$(A(u))_{u\in\Omega}$ is $\mu$-$\nu$ measurable if for every
$x\in\mathfrak{S}$ the map $u\mapsto A(u)(x)$ from $(\Omega,\mu)$ to
$(\mathfrak{S},\nu)$ is measurable.

Recall [7] that a $\nu$-measurable function $a$ on $\mathfrak{S}$ is an
$(1,q)$-atom ($q\in(1,\infty]$) if

(i) the support of $a$ is contained in a ball $B(x,r)$;

(ii) $\|a\|_{\infty}\leq\frac{1}{\nu(B(x,r))}$ if $q=\infty$, and

$\|a\|_{q}\leq\nu(B(x,r))^{\frac{1}{q}-1}$ if $q\in(1,\infty)$222
$\|\cdot\|_{q}$ denotes the $L^{q}$ norm.;

(iii) $\int_{G}a(x)d\nu(x)=0$.

In case $\nu(\mathfrak{S})<\infty$ we shall assume $\nu(\mathfrak{S})=1$; in
this case the constant function having value $1$ is also considered to be an
atom.

From now on by atom we mean an $(1,q)$-atom.

######  Definition 5.8.

[7, p. 592] Let $q\in(1,\infty]$. We define the Hardy space
$H^{1,q}(\mathfrak{S})$ as a space of such functions $f$ on $\mathfrak{S}$
that $f$ admits an atomic decomposition of the form

$f=\sum_{j=1}^{\infty}\alpha_{j}a_{j},$ (5.7)

where $a_{j}$ are $(1,q)$-atoms, and $\sum_{j=1}^{\infty}|\alpha_{j}|<\infty$
(the sums (5.7) are convergent in the $L^{1}$ norm). 333It is known that
$H^{1,q}$ does not depend on $q\in(1,\infty]$ [7, Theorem A, p. 592]. We write
$H^{1,q}$ instead of $H^{1}$ in order to stress the fact that we use the norm
$\|\cdot\|_{H^{1,q}}$ described below. In this case,

$\|f\|_{H^{1,q}(\mathfrak{S})}:=\inf\sum_{j=1}^{\infty}|\alpha_{j}|,$

and infimum is taken over all decompositions above of $f$.

Since $\|a\|_{L^{1}}\leq 1$, one has
$\|f\|_{L^{1}(\mathfrak{S})}\leq\|f\|_{H^{1,q}(\mathfrak{S})}$ for a function
$f$ in $H^{1,q}(\mathfrak{S})$, in particular $H^{1,q}(\mathfrak{S})\subset
L^{1}(\mathfrak{S})$.

For the proof of the $H^{1}$ boundedness of $\mathcal{H}_{\Phi,A}$ we shall
use the following lemma.

######  Lemma 5.9.

[24, Lemma 2] Let $(\mathfrak{S};\nu)$ be a measure space,
$\mathcal{F}(\mathfrak{S})$ some Banach space of $\nu$-measurable functions on
$\mathfrak{S}$, $(\Omega,\mu)$ a $\sigma$-compact quasi-metric space with
positive Radon measure $\mu$, and $F(u,x)$ a function on
$\Omega\times\mathfrak{S}$. Assume that

(a) the convergence of a sequence in norm in $\mathcal{F}(\mathfrak{S})$
yields the convergence of some subsequence to the same function for $\nu$-a.
e. $x\in\mathfrak{S}$;

(b) $F(u,\cdot)\in\mathcal{F}(\mathfrak{S})$ for $\mu$-a. e. $u\in\Omega$;

(c) the map $u\mapsto F(u,\cdot):\Omega\to\mathcal{F}(\mathfrak{S})$ is
Bochner integrable with respect to $\mu$.

Then for $\nu$-a. e. $x\in\mathfrak{S}$ one has

$\left((B)\int_{\Omega}F(u,\cdot)d\mu(u)\right)(x)=\int_{\Omega}F(u,x)d\mu(u).$

In the following we put

$N(\Phi,A,q)=C_{\nu}^{1-\frac{1}{q}}\int_{\Omega}|\Phi(u)|k(u)^{s\left(1-\frac{1}{q}\right)}m(A(u))^{-\frac{1}{q}}d\mu(u).$

######  Theorem 5.10.

Let $\Omega$ be a $\sigma$-compact quasi-metric space with positive Radon
measure $\mu$ and let $(\mathfrak{S},\rho,\nu)$ be a quasi-metric measure
space of homogeneous type such that the space $H^{1,q}(\mathfrak{S})$ is
separable ($q\in(1,\infty]$). If a $\mu$-$\nu$\- measurable family of
automorphisms $(A(u))_{u\in\Omega}$ of $\mathfrak{S}$ agrees with the quasi-
metric $\rho$ and with the measure $\nu$, and $N(\Phi,A,q)<\infty$, then a
Hausdorff operator $\mathcal{H}_{\Phi,A}$ is bounded in
$H^{1,q}(\mathfrak{S})$ and its norm does not exceed $N(\Phi,A,q)$.

###### Proof.

We use the approach from [24]. First we are going to show that the conditions
of Lemma 5.9 are fulfilled with
$\mathcal{F}(\mathfrak{S})=H^{1,q}(\mathfrak{S})$ and
$F(u,x)=\Phi(u)f(A(u)(x))$ where $f\in H^{1,q}(\mathfrak{S})$.

Let $1<q<\infty$. Since for a function $f\in H^{1,q}(\mathfrak{S})$ one has
$\|f\|_{L^{1}(\mathfrak{S})}\leq\|f\|_{H^{1,q}(\mathfrak{S})}$, the condition
(a) of the Lemma follows from the well known theorem of F. Riesz.

To verify conditions (b) and (c), consider a function $f\in
H^{1,q}(\mathfrak{S})$ with an atomic representation (5.7). Then

$f\circ A(u)=\sum_{j=1}^{\infty}\alpha_{j}a_{j}\circ A(u),$ (5.8)

for all $u\in\Omega$. We claim that a function

$a^{\prime}_{j,u}:=C_{\nu}^{\frac{1}{q}-1}k(u)^{s(\frac{1}{q}-1)}m(A(u))^{\frac{1}{q}}a_{j}\circ
A(u)$

is an atom, as well.

Indeed, if an atom $a_{j}$ is supported in a ball $B(x_{j},r_{j})$ then
$a^{\prime}_{j,u}$ is supported in $A(u)^{-1}(B(x_{j},r_{j}))\subset
B(x^{\prime}_{j},k(u)r_{j})$ by (5.6).

Next, since the property (ii) holds for $a_{j}$ , we have by (4.4)

$\displaystyle\|a^{\prime}_{j,u}\|_{q}$ $\displaystyle=$ $\displaystyle
C_{\nu}^{\frac{1}{q}-1}k(u)^{s(\frac{1}{q}-1)}m(A(u))^{\frac{1}{q}}\|a_{j}\circ
A(u)\|_{q}$ $\displaystyle=$ $\displaystyle
C_{\nu}^{\frac{1}{q}-1}k(u)^{s(\frac{1}{q}-1)}m(A(u))^{\frac{1}{q}}\left(\int_{\mathfrak{S}}|a_{j}(A(u)(x))|^{q}d\nu(x)\right)^{\frac{1}{q}}$
$\displaystyle=$ $\displaystyle
C_{\nu}^{\frac{1}{q}-1}k(u)^{s(\frac{1}{q}-1)}\|a_{j}\|_{q}$
$\displaystyle\leq$ $\displaystyle
C_{\nu}^{\frac{1}{q}-1}k(u)^{s(\frac{1}{q}-1)}\nu(B(x_{j},r_{j}))^{\frac{1}{q}-1}.$

On the other hand, the doubling condition (5.5) yields

$\nu(B(x^{\prime}_{j},k(u)r_{j}))\leq C_{\nu}k(u)^{s}\nu(B(x_{j},r_{j}))$

and therefore

$\nu(B(x_{j},r_{j}))^{\frac{1}{q}-1}\leq(C_{\nu}k(u)^{s})^{1-\frac{1}{q}}\nu(B(x^{\prime}_{j},k(u)r_{j}))^{\frac{1}{q}-1}.$

Now (5) implies that

$\|a^{\prime}_{j,u}\|_{q}\leq\nu(B(x^{\prime}_{j},k(u)r_{j}))^{\frac{1}{q}-1},$

i.e., (ii) holds for $a^{\prime}_{j,u}$.

Finally, the cancellation condition (iii) for $a^{\prime}_{j,u}$ follows from
(4.4) and the corresponding condition for $a_{j}$.

Further, since for all $u\in\Omega$

$a_{j}\circ
A(u)=C_{\nu}^{1-\frac{1}{q}}k(u)^{s(1-\frac{1}{q})}m(A(u))^{-\frac{1}{q}}a^{\prime}_{j,u},$

formula (5.8) reeds as

$f\circ
A(u)=\sum_{j=1}^{\infty}\left(\alpha_{j}C_{\nu}^{1-\frac{1}{q}}k(u)^{s(1-\frac{1}{q})}m(A(u))^{-\frac{1}{q}}\right)a^{\prime}_{j,u}.$

It follows that $f\circ A(u)\in H^{1,q}(\mathfrak{S})$ (and therefore the
condition (b) holds) and

$\|f\circ
A(u)\|_{H^{1,q}}\leq\left(C_{\nu}^{1-\frac{1}{q}}k(u)^{s(1-\frac{1}{q})}m(A(u))^{-\frac{1}{q}}\right)\|f\|_{H^{1,q}}.$
(5.10)

The condition (c) holds, too. Indeed, since $H^{1,q}(\mathfrak{S})$ is
separable, to verify that the $H^{1,q}(\mathfrak{S})$-valued function
$u\mapsto f\circ A(u)$ is strongly $\mu$-measurable it suffices to prove that
it is weakly $\mu$-measurable. To this end, in view of (5.7), it suffices to
consider the case where $f=a$ is an atom. Let $l^{*}$ be a linear continuous
functional on $H^{1,q}(\mathfrak{S})$. Then [7, Theorem B] there is such a
function $l\in BMO(\mathfrak{S})$ that

$l^{*}(a\circ A(u))=\int_{\mathfrak{S}}l(x)a(A(u)(x))d\nu(x).$

The map $u\mapsto l^{*}(a\circ A(u))$ is $\mu$-measurable, if the map
$\phi(u):=a\circ A(u)(x)$ is $\mu$-measurable for each $x$. To verify the last
property one can assume that $a$ is real-valued. Let
$E_{c}=\\{y\in\mathfrak{S}:a(y)<c\\}$ ($c\in\mathbb{R}$). Then $E_{c}$ is
$\nu$-measurable and so the set
$\phi^{-1}((-\infty,c))=\\{u\in\Omega:A(u)(x)\in E_{c}\\}$ is $\mu$-measurable
by Definition 5.7.

Now the inequality (5.10) and the condition $N(\Phi,A,q)<\infty$ imply that
the function $u\mapsto\|F(u,\cdot)\||_{H^{1,q}}$ is Lebesgue $\mu$-integrable
and (c) from the Lemma 5.9 holds.

Thus, by Lemma 5.9,

$\mathcal{H}_{\Phi,A}f=\int_{\Omega}\Phi(u)f\circ A(u)d\mu(u)$

(the Bochner integral), and therefore

$\displaystyle\|\mathcal{H}_{\Phi,A}f\|_{H^{1,q}}$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}|\Phi(u)|\|f\circ A(u)\|_{H^{1,q}}d\mu(u)$
$\displaystyle\leq$ $\displaystyle N(\Phi,A,q)\|f\|_{H^{1,q}}.$

The case $q=\infty$ can be treated in a similar manner.

The proof is complete.

∎

Remark 2. The proof of Theorem 5.10 shows that the condition that the space
$H^{1,q}(\mathfrak{S})$ is separable can be replaced by the condition that for
every fixed $f\in H^{1,q}(\mathfrak{S})$ the range of the map $u\mapsto f\circ
A(u)$, $\Omega\to H^{1,q}(\mathfrak{S})$ is almost separable. Since $\Omega$
is separable, $f\circ A(\Omega)$ is separable if this map is measurable [13,
Lemma 1.1.12].

If $\Omega$ is countable it is obvious that the subspace $f\circ A(\Omega)$ of
$H^{1,q}(\mathfrak{S})$ is separable.

In the following we shall assume that the family $\mathrm{Aut}(G)$ of all
topological automorphisms of a locally compact group $G$ is equipped with its
natural (Braconnier) topology (see, e. g., [11, (26.1)], [12, Section III.3]).

######  Definition 5.11.

Let $(\Omega,\mu)$ be a $\sigma$-compact quasi-metric space with positive
Radon measure $\mu$. A family of topological automorphisms
$(A(u))_{u\in\Omega}$ of a locally compact group $G$ is called measurable if
the map $u\mapsto A(u)$ is measurable with respect to the measure $\mu$ and
the Borel structure in $\mathrm{Aut}(G)$. 444Here we concretisize the notion
of the measurable family of topological automorphisms of a locally compact
group from [24, 27, 26, 28, 25].

######  Corollary 5.12.

(cf. [24], [27]). Let $(\Omega,\mu)$ be a $\sigma$-compact quasi-metric space
with positive Radon measure $\mu$. Let $\mathfrak{S}=G$ be a locally compact
group with Haar measure $\nu$, and the topology of $G$ is generated by a
quasi-metric $\rho$. Assume that $(G,\rho,\nu)$ is a space of homogeneous
type, and the space $H^{1,q}(G)$ is separable ($q\in(1,\infty]$). If a
measurable family of topological automorphisms $(A(u))_{u\in\Omega}$ of $G$
agrees with the quasi-metric $\rho$ and $N(\Phi,A,q)<\infty$, then a Hausdorff
operator $\mathcal{H}_{\Phi,A}$ is bounded in $H^{1,q}(G)$ and its norm does
not exceed $N(\Phi,A,q)$.

###### Proof.

The only conditions of Theorem 5.10 we need to verify are that the family
$(A(u))_{u\in\Omega}$ agrees with the measure $\nu$ and that it is
$\mu$-$\nu$-measurable.

For the proof of the first property note that in our case we have
$m(A(u))=\mathrm{mod}(A(u))$, and the map $u\mapsto\mathrm{mod}(A(u))$ is
$\mu$-measurable, since the family $(A(u))_{u\in\Omega}$ is measurable, and
the map $A\mapsto\mathrm{mod}A$ from $\mathrm{Aut}(G)$ to $(0,\infty)$ is
continuous (see [11, (26.21)]).

Finally, since for each $x\in G$ the map $\mathrm{Aut}(G)\to G$ sending $A$
onto $A(x)$ is continuous [12, Proposition III.3.1, p. 40], and the family
$(A(u))_{u\in\Omega}$ is measurable, it is $\mu$-$\nu$-measurable. ∎

Remark 3. As in Theorem 5.10 in Corollary 5.12 the condition that the space
$H^{1,q}(G)$ is separable can be replaced by the condition that for every
fixed $f\in H^{1,q}(G)$ the range of the map $u\mapsto f\circ A(u)$,
$\Omega\to H^{1,q}(G)$ is almost separable. Since $\Omega$ is separable, and
the map $u\mapsto A(u)$ is measurable, Lemma 1.1.12 from [13] shows that
$f\circ A(\Omega)$ is separable if the map $A\mapsto f\circ A$,
$\mathrm{Aut}(G)\to H^{1,q}(G)$ is measurable.

Also it is obvious that $f\circ A(\Omega)$ is separable if $\Omega$ is
countable.

Conjecture. Let $G$ be a locally compact group with Haar measure $\nu$, and
the topology of $G$ is generated by a quasi-metric $\rho$. If $(G,\rho,\nu)$
is a space of homogeneous type, then the map $A\mapsto f\circ A$,
$\mathrm{Aut}(G)\to H^{1,q}(G)$ is continuous.

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Department of Mathematics and Programming Technologies, Francisk Skorina Gomel
State University, Gomel, 246019, Belarus $\&$ Regional Mathematical Center,
Southern Federal University, Rostov-on-Don, 344090, Russia.