incollectioninproceedings

THE REAL AND COMPLEX TECHNIQUES
IN HARMONIC ANALYSIS
FROM THE POINT OF VIEW OF COVARIANT TRANSFORM

Vladimir V. Kisil

_Dedicated to Prof. Sergei V. Rogosin on the occasion of his 60th birthday_

Key words: wavelet, coherent state, covariant transform, reconstruction
formula, the affine group, $ax+b$-group, square integrable representations,
admissible vectors, Hardy space, fiducial operator, approximation of the
identity, atom, nucleus, atomic decomposition, Cauchy integral, Poisson
integral, Hardy–Littlewood maximal function, grand maximal function, vertical
maximal function, non-tangential maximal function, intertwining operator,
Cauchy-Riemann operator, Laplace operator, singular integral operator, SIO,
Hilbert transform, boundary behaviour, Carleson measure, Littlewood–Paley
theory

AMS Mathematics Subject Classification: Primary 42-02; Secondary 42A20, 42B20,
42B25, 42B35, 42C40, 43A50, 43A80.

Abstract. This paper reviews complex and real techniques in harmonic analysis.
We describe the common source of both approaches rooted in the covariant
transform generated by the affine group.

###### Contents

  1. 1 Introduction
  2. 2 Two approaches to harmonic analysis
  3. 3 Affine group and its representations
  4. 4 Covariant transform
  5. 5 The contravariant transform
  6. 6 Intertwining properties of covariant transforms
  7. 7 Composing the covariant and the contravariant transforms
  8. 8 Transported norms
  9. 9 Conclusion

##  1 Introduction

There are two main approaches in harmonic analysis on the real line. The real
variables technique uses various maximal functions, dyadic cubes and,
occasionally, the Poisson integral [Stein93]. The complex variable technique
is based on the Cauchy integral and fine properties of analytic functions
[Nikolski02a, Nikolski02b].

Both methods seem to have clear advantages. The real variable technique:

  1. i.

does not require an introduction of the imaginary unit for a study of real-
valued harmonic functions of a real variable (Occam’s Razor: among competing
hypotheses, the one with the fewest assumptions should be selected);

  2. ii.

allows a straightforward generalization to several real variables.

By contrast, access to the beauty and power of analytic functions (e.g.,
Möbius transformations, factorisation of zeroes, etc. [Koosis98a]) is the main
reason to use the complex variable technique. A posteriori, a multidimensional
analytic version was also discovered [McIntosh95a], it is based on the
monogenic Clifford-valued functions [BraDelSom82].

Therefore, propensity for either techniques becomes a personal choice of a
researcher. Some of them prefer the real variable method, explicitly cleaning
out any reference to analytic or harmonic functions [Stein93]*Ch. III, p. 88.
Others, e.g. [Krantz09a, CoifmanJonesSemmes89], happily combine the both
techniques. However, the reasons for switching between two methds at
particular places may look mysterious.

The purpose of the present paper is to revise the origins of the real and
complex variable techniques. Thereafter, we describe the common group-
theoretical root of both. Such a unification deepens our understanding of both
methods and illuminates their interaction.

######  Remark 1.1.

In this paper, we consider only examples which are supported by the affine
group $\mathrm{Aff}$ of the real line. In the essence, $\mathrm{Aff}$ is the
semidirect product of the group of dilations acting on the group of
translations. Thus, our consideration can be generalized to the semidirect
product of dilations and homogeneous (nilpotent) Lie groups, cf. [FollStein82,
Kisil12b]. Other important extensions are the group
$\mathrm{SL}_{2}(\mathbb{R}{})$ and associated hypercomplex algebras, see
Rems. 3.5, 4.10 and [Kisil05a, Kisil12a, Kisil11c]. However, we do not aim
here to a high level of generality, it can be developed in subsequent works
once the fundamental issues are sufficiently clarified.

##  2 Two approaches to harmonic analysis

As a starting point of our discussion, we provide a schematic outline of
complex and real variables techniques in the one-dimensional harmonic
analysis. The application of complex analysis may be summarised in the
following sequence of principal steps:

Integral transforms.

For a function $f\in L_{p}{}(\mathbb{R}{})$, we apply the Cauchy or Poisson
integral transforms:

$\displaystyle[\mathcal{C}f](x+\mathrm{i}y)$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\int_{\mathbb{R}{}}\frac{f(t)}{t-(x+\mathrm{i}y)}\,dt\,,$
(2.1) $\displaystyle[\mathcal{P}f](x,y)$
$\displaystyle=\frac{1}{\pi}\int_{\mathbb{R}{}}\frac{y}{(t-x)^{2}+y^{2}}\,f(t)\,dt\,.$
(2.2)

An equivalent transformation on the unit circle replaces the Fourier series
$\sum_{k}c_{k}e^{\mathrm{i}kt}$ by the Taylor series
$\sum_{k=0}^{\infty}c_{k}z^{k}$ in the complex variable $z=re^{\mathrm{i}t}$,
$0\leq r<1$. It is used for the Abel summation of trigonometric series
[Zygmund02]*§ III.6.

Domains.

Above integrals (2.1)–(2.2) map the domain of functions from the real line to
the upper half-plane, which can be conveniently identified with the set of
complex numbers having a positive imaginary part. The larger domain allows us
to inspect functions in greater details.

Differential operators.

The image of integrals (2.1) and (2.2) consists of functions,belonging to the
kernel of the Cauchy–Riemann operator $\partial_{\bar{z}}$ and Laplace
operator $\Delta$ respectively, i.e.:

$\partial_{\bar{z}}=\frac{\partial}{\partial
x}+\mathrm{i}\frac{\partial}{\partial
y}\,,\qquad\Delta=\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\,.$ (2.3)

Such functions have numerous nice properties in the upper half-plane, e.g.
they are infinitely differentiable, which make their study interesting and
fruitful.

Boundary values and SIO.

To describe properties of the initial function $f$ on the real line we
consider the boundary values of $[\mathcal{C}f](x+\mathrm{i}y)$ or
$[\mathcal{P}f](x,y)$, i.e. their limits as $y\rightarrow 0$ in some sense.
The Sokhotsky–Plemelj formula provides the boundary value of the Cauchy
integral [MityushevRogosin00a]*(2.6.6):

$[\mathcal{C}f](x,0)=\frac{1}{2}f(x)+\frac{1}{2\pi\mathrm{i}}\int_{\mathbb{R}{}}\frac{f(t)}{t-x}\,dt.$
(2.4)

The last term is a singular integral operator defined through the principal
value in the Cauchy sense:

$\frac{1}{2\pi\mathrm{i}}\int_{\mathbb{R}{}}\frac{f(t)}{t-x}\,dt=\lim_{\varepsilon\rightarrow
0}\frac{1}{2\pi\mathrm{i}}\int\limits_{-\infty}^{x-\varepsilon}+\int\limits_{x+\varepsilon}^{\infty}\frac{f(t)}{t-x}\,dt\,.$
(2.5)

For the Abel summation the boundary values are replaced by the limit as
$r\rightarrow 1^{-}$ in the series
$\sum_{k=0}^{\infty}c_{k}(re^{\mathrm{i}t})^{k}$.

Hardy space.

Sokhotsky–Plemelj formula (2.4) shows, that the boundary value
$[\mathcal{C}f](x,0)$ may be different from $f(x)$. The vector space of
functions $f(x)$ such that $[\mathcal{C}f](x,0)=f(x)$ is called the Hardy
space on the real line [Nikolski02a]*A.6.3.

Summing up this scheme: we replace a function (distribution) on the real line
by a nicer (analytic or harmonic) function on a larger domain—the upper half-
plane. Then, we trace down properties of the extensions to its boundary values
and, eventually, to the initial function.

The real variable approach does not have a clearly designated path in the
above sense. Rather, it looks like a collection of interrelated tools, which
are efficient for various purposes. To highlight similarity and differences
between real and complex analysis, we line up the elements of the real
variable technique in the following way:

Hardy–Littlewood maximal function

is, probably, the most important component [Koosis98a]*§ VIII.B.1
[Stein93]*Ch. 2 [Garnett07a]*§ I.4 [Burenkov12a] of this technique. The
maximal function $f^{M}$ is defined on the real line by the identity:

$f^{M}(t)=\sup_{a>0}\left\\{\frac{1}{2a}\int\limits^{t+a}_{t-a}\left|f\left(x\right)\right|\,dx\right\\}.$
(2.6)

Domain

is not apparently changed, the maximal function $f^{M}$ is again defined on
the real line. However, an efficient treatment of the maximal functions
requires consideration of tents [Stein93]*§ II.2, which are parametrised by
their vertices, i.e. points $(a,b)$, $a>0$, of the upper half-plane. In other
words, we repeatedly need values of all integrals
$\frac{1}{2a}\int\limits^{t+a}_{t-a}\left|f\left(x\right)\right|\,dx$, rather
than the single value of the supremum over $a$.

Littlewood–Paley theory

[CoifmanJonesSemmes89]*§ 3 and associated dyadic squares technique
[Garnett07a]*Ch. VII, Thm. 1.1 [Stein93]*§ IV.3 as well as stopping time
argument [Garnett07a]*Ch. VI, Lem. 2.2 are based on bisection of a function’s
domain into two equal parts.

SIO

is a natural class of bounded linear operators in $L_{p}{}(\mathbb{R}{})$.
Moreover, maximal operator $M:f\rightarrow f^{M}$ (2.6) and singular integrals
are intimately related [Stein93]*Ch. I.

Hardy space

can be defined in several equivalent ways from previous notions. For example,
it is the class of such functions that their image under maximal operator
(2.6) or singular integral (2.5) belongs to $L_{p}{}(\mathbb{R}{})$
[Stein93]*Ch. III.

The following discussion will line up real variable objects along the same
axis as complex variables. We will summarize this in Table 1.

##  3 Affine group and its representations

It is hard to present harmonic analysis and wavelets without touching the
affine group one way or another. Unfortunately, many sources only mention the
group and do not use it explicitly. On the other hand, it is equally difficult
to speak about the affine group without a reference to results in harmonic
analysis: two theories are intimately intertwined. In this section we collect
fundamentals of the affine group and its representations, which are not yet a
standard background of an analyst.

Let $G=\mathrm{Aff}$ be the $ax+b$ (or the _affine_) group [AliAntGaz00]*§
8.2, which is represented (as a topological set) by the upper half-plane
$\\{(a,b)\,\mid\,a\in\mathbb{R}_{+}{},\ b\in\mathbb{R}{}\\}$. The group law
is:

$(a,b)\cdot(a^{\prime},b^{\prime})=(aa^{\prime},ab^{\prime}+b).$ (3.1)

As any other group, $\mathrm{Aff}$ has the _left regular representation_ by
shifts on functions $\mathrm{Aff}\rightarrow\mathbb{C}{}$:

$\Lambda(a,b):f(a^{\prime},b^{\prime})\mapsto
f_{(a,b)}(a^{\prime},b^{\prime})=f\left(\frac{a^{\prime}}{a},\frac{b^{\prime}-b}{a}\right).$
(3.2)

A left invariant measure on $\mathrm{Aff}$ is $dg=a^{-2}\,da\,db$, $g=(a,b)$.
By the definition, the left regular representation (3.2) acts by unitary
operators on $L_{2}{}(\mathrm{Aff},dg)$. The group is not unimodular and a
right invariant measure is $a^{-1}\,da\,db$.

There are two important subgroups of the $ax+b$ group:

$A=\\{(a,0)\in\mathrm{Aff}\,\mid\,a\in\mathbb{R}_{+}{}\\}\quad\text{ and
}\quad N=\\{(1,b)\in\mathrm{Aff}\,\mid\,b\in\mathbb{R}{}\\}.$ (3.3)

An isometric representation of $\mathrm{Aff}$ on $L_{p}{}(\mathbb{R}{})$ is
given by the formula:

$[{\rho_{p}}(a,b)\,f](x)=a^{-\frac{1}{p}}\,f\left(\frac{x-b}{a}\right).$ (3.4)

Here, we identify the real line with the subgroup $N$ or, even more
accurately, with the homogeneous space $\mathrm{Aff}/N$ [ElmabrokHutnik12a]*§
2. This representation is known as _quasi-regular_ for its similarity with
(3.2). The action of the subgroup $N$ in (3.4) reduces to shifts, the subgroup
$A$ acts by dilations.

######  Remark 3.1.

The $ax+b$ group definitely escapes Occam’s Razor in harmonic analysis, cf.
the arguments against the imaginary unit in the Introduction. Indeed, shifts
are required to define convolutions on $\mathbb{R}^{n}{}$, and an
_approximation of the identity_ [Stein93]*§ I.6.1 is a convolution with the
dilated kernel. The same scaled convolutions define the fundamental _maximal
functions_ , see [Stein93]*§ III.1.2 cf. Example 7.6 below. Thus, we can avoid
usage of the upper half-plane $\mathbb{C}_{+}{}$, but the same set will anyway
re-invent itself in the form of the $ax+b$ group.

The representation (3.4) in $L_{2}{}(\mathbb{R}{})$ is reducible and the space
can be split into irreducible subspaces. Following the philosophy presented in
the Introduction to the paper [Kisil09e]*§ 1 we give the following

######  Definition 3.2.

For a representation ${\rho}$ of a group $G$ in a space $V$, a _generalized
Hardy space_ $H{}$ is an ${\rho}$-irreducible (or ${\rho}$-primary, as
discussed in Section 7) subspace of $V$.

######  Example 3.3.

Let $G=\mathrm{Aff}$ and the representation ${\rho_{p}}$ be defined in
$V=L_{p}{}(\mathbb{R}{})$ by (3.4). Then the classical Hardy spaces
$H_{p}{}(\mathbb{R}{})$ are ${\rho_{p}}$-irreducible, thus are covered by the
above definition.

Some ambiguity in picking the Hardy space out of all (well, two, as we will
see below) irreducible components is resolved by the traditional preference.

######  Remark 3.4.

We have defined the Hardy space completely in terms of representation theory
of $ax+b$ group. The traditional descriptions, via the Fourier transform or
analytic extensions, will be corollaries in our approach, see Prop. 3.6 and
Example 6.9.

######  Remark 3.5.

It is an interesting and important observation, that the Hardy space in
$L_{p}{}(\mathbb{R}{})$ is invariant under the action of a larger group
$\mathrm{SL}_{2}(\mathbb{R}{})$, the group of $2\times 2$ matrices with real
entries and determinant equal to $1$, the group operation coincides with the
multiplication of matrices. The $ax+b$ group is isomorphic to the subgroup of
the upper-triangular matrices in $\mathrm{SL}_{2}(\mathbb{R}{})$. The group
$\mathrm{SL}_{2}(\mathbb{R}{})$ has an isometric representation in
$L_{p}{}(\mathbb{R}{})$:

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}:\ f(x)\ \mapsto\
\frac{1}{\left|a-cx\right|^{\frac{2}{p}}}\,f\left(\frac{dx-b}{a-cx}\right),$
(3.5)

which produces quasi-regular representation (3.4) by the restriction to upper-
triangular matrices. The Hardy space $H_{p}{}(\mathbb{R}{})$ is invariant
under the above action as well. Thus, $\mathrm{SL}_{2}(\mathbb{R}{})$ produces
a refined version in comparison with the harmonic analysis of the $ax+b$ group
considered in this paper. Moreover, as representations of the $ax+b$ group are
connected with complex numbers, the structure of
$\mathrm{SL}_{2}(\mathbb{R}{})$ links all three types of hypercomplex numbers
[Kisil05a] [Kisil12a]*§ 3.3.4 [Kisil11c]*§ 3, see also Rem. 4.10.

To clarify a decomposition of $L_{p}{}(\mathbb{R}{})$ into irreducible
subspaces of representation (3.4) we need another realization of this
representation. It is called _co-adjoint_ and is related to the _orbit method_
of Kirillov [Kirillov04a]*§ 4.1.4 [Folland95]*§ 6.7.1. Again, this isometric
representation can be defined on $L_{p}{}(\mathbb{R}{})$ by the formula:

$[{\hat{\rho}_{p}}(a,b)\,f](\lambda)=a^{\frac{1}{p}}\,\mathrm{e}^{-2\pi\mathrm{i}b\lambda}f(a\lambda).$
(3.6)

Since $a>0$, there is an obvious decomposition into invariant subspaces of
${\hat{\rho}_{p}}$:

$L_{p}{}(\mathbb{R}{})=L_{p}{}(-\infty,0)\oplus L_{p}{}(0,\infty).$ (3.7)

It is possible to demonstrate, that these components are irreducible. This
decomposition has a spatial nature, i.e., the subspaces have disjoint
supports. Each half-line can be identified with the subgroup $A$ or with the
homogeneous space $\mathrm{Aff}/N$.

The restrictions ${\hat{\rho}^{+}_{p}}$ and ${\hat{\rho}^{-}_{p}}$ of the co-
adjoint representation ${\hat{\rho}_{p}}$ to invariant subspaces (3.7) for
$p=2$ are not unitary equivalent. Any irreducible unitary representation of
$\mathrm{Aff}$ is unitary equivalent either to ${\hat{\rho}^{+}_{2}}$ or
${\hat{\rho}^{-}_{2}}$. Although there is no intertwining operator between
${\hat{\rho}^{+}_{p}}$ and ${\hat{\rho}^{-}_{p}}$, the map:

${J}:\ L_{p}{}(\mathbb{R}{})\rightarrow L_{p}{}(\mathbb{R}{}):\
{f}(\lambda)\mapsto{f}(-\lambda),$ (3.8)

has the property

${\hat{\rho}^{-}_{p}}(a,-b)\circ{J}={J}\circ{\hat{\rho}^{+}_{p}}(a,b)$ (3.9)

which corresponds to the outer automorphism $(a,b)\mapsto(a,-b)$ of
$\mathrm{Aff}$.

As was already mentioned, for the Hilbert space $L_{2}{}(\mathbb{R}{})$,
representations (3.4) and (3.6) are unitary equivalent, i.e., there is a
unitary intertwining operator between them. We may guess its nature as
follows. The eigenfunctions of the operators ${\rho_{2}}(1,b)$ are
$\mathrm{e}^{2\pi\mathrm{i}\omega x}$ and the eigenfunctions of
${\hat{\rho}_{2}}(1,b)$ are $\delta(\lambda-\omega)$. Both sets form
“continuous bases” of $L_{2}{}(\mathbb{R}{})$ and the unitary operator which
maps one to another is the Fourier transform:

$\mathcal{F}:f(x)\mapsto\hat{f}(\lambda)=\int_{\mathbb{R}{}}e^{-2\pi\mathrm{i}\lambda
x}\,f(x)\,dx.$ (3.10)

Although, the above arguments were informal, the intertwining property
$\mathcal{F}{\rho_{2}}(a,b)={\hat{\rho}_{2}}(1,b)\mathcal{F}$ can be directly
verified by the appropriate change of variables in the Fourier transform.
Thus, cf. [Nikolski02a]*Lem. A.6.2.2:

######  Proposition 3.6.

The Fourier transform maps irreducible invariant subspaces $H_{2}{}$ and
$H_{2}^{\perp}{}$ of (3.4) to irreducible invariant subspaces
$L_{2}{}(0,\infty)=\mathcal{F}(H_{2}{})$ and
$L_{2}{}(-\infty,0)=\mathcal{F}(H_{2}^{\perp}{})$ of co-adjoint representation
(3.6). In particular, $L_{2}{}(\mathbb{R}{})=H_{2}{}\oplus H_{2}^{\perp}{}$.

Reflection $J$ (3.8) anticommutes with the Fourier transform:
$\mathcal{F}J=-J\mathcal{F}$. Thus, $J$ also interchange the irreducible
components ${\rho^{+}_{p}}$ and ${\rho^{-}_{p}}$ of quasi-regular
representation (3.4) according to (3.9).

Summing up, the unique rôle of the Fourier transform in harmonic analysis is
based on the following facts from the representation theory. The Fourier
transform

  * •

intertwines shifts in quasi-regular representation (3.4) to operators of
multiplication in co-adjoint representation (3.6);

  * •

intertwines dilations in (3.4) to dilations in (3.6);

  * •

maps the decomposition $L_{2}{}(\mathbb{R}{})=H_{2}{}\oplus H_{2}^{\perp}{}$
into spatially separated spaces with disjoint supports;

  * •

anticommutes with $J$, which interchanges ${\rho^{+}_{2}}$ and
${\rho^{-}_{2}}$.

Armed with this knowledge we are ready to proceed to harmonic analysis.

##  4 Covariant transform

We make an extension of the wavelet construction defined in terms of group
representations. See [Kirillov76] for a background in the representation
theory, however, the only treated case in this paper is the $ax+b$ group.

######  Definition 4.1.

[Kisil09d, Kisil11c] Let ${\rho}$ be a representation of a group $G$ in a
space $V$ and $F$ be an operator acting from $V$ to a space $U$. We define a
_covariant transform_ $\mathcal{W}_{F}^{{\rho}}$ acting from $V$ to the space
$L{}(G,U)$ of $U$-valued functions on $G$ by the formula:

$\mathcal{W}_{F}^{{\rho}}:v\mapsto\hat{v}(g)=F({\rho}(g^{-1})v),\qquad v\in
V,\ g\in G.$ (4.1)

The operator $F$ will be called a _fiducial operator_ in this context (cf. the
fiducial vector in [KlaSkag85]).

We may drop the sup/subscripts from $\mathcal{W}_{F}^{{\rho}}$ if the
functional $F$ and/or the representation ${\rho}$ are clear from the context.

######  Remark 4.2.

We do not require that the fiducial operator $F$ be linear. Sometimes the
positive homogeneity, i.e. $F(tv)=tF(v)$ for $t>0$, alone can be already
sufficient, see Example 4.7.

######  Remark 4.3.

It looks like the usefulness of the covariant transform is in the reverse
proportion to the dimension of the space $U$. The covariant transform encodes
properties of $v$ in a function $\mathcal{W}_{F}^{{\rho}}v$ on $G$, which is a
scalar-valued function if $\dim U=1$. However, such a simplicity is not always
possible. Moreover, the paper [Kisil12b] gives an important example of a
covariant transform which provides a simplification even in the case $\dim
U=\dim V$.

We start the list of examples with the classical case of the group-theoretical
wavelet transform.

######  Example 4.4.

[Perelomov86, FeichGroech89a, Kisil98a, AliAntGaz00, KlaSkag85,
FeichGroech89a] Let $V$ be a Hilbert space with an inner product
$\left\langle\cdot,\cdot\right\rangle$ and ${\rho}$ be a unitary
representation of a group $G$ in the space $V$. Let
$F:V\rightarrow\mathbb{C}{}$ be the functional $v\mapsto\left\langle
v,v_{0}\right\rangle$ defined by a vector $v_{0}\in V$. The vector $v_{0}$ is
often called the _mother wavelet_ in areas related to signal processing, the
_vacuum state_ in the quantum framework, etc.

In this set-up, transformation (4.1) is the well-known expression for a
_wavelet transform_ [AliAntGaz00]*(7.48) (or _representation coefficients_):

$\mathcal{W}:v\mapsto\tilde{v}(g)=\left\langle{\rho}(g^{-1})v,v_{0}\right\rangle=\left\langle
v,{\rho}(g)v_{0}\right\rangle,\qquad v\in V,\ g\in G.$ (4.2)

The family of the vectors $v_{g}={\rho}(g)v_{0}$ is called _wavelets_ or
_coherent states_. The image of (4.2) consists of scalar valued functions on
$G$.

This scheme is typically carried out for a square integrable representation
${\rho}$ with $v_{0}$ being an admissible vector [Perelomov86, FeichGroech89a,
AliAntGaz00, Fuhr05a, ChristensenOlafsson09a, DufloMoore], i.e. satisfying the
condition:

$0<\left\|\tilde{v}_{0}\right\|^{2}=\int_{G}\left|\left\langle
v_{0},{\rho_{2}}(g)v_{0}\right\rangle\right|^{2}\,dg<\infty.$ (4.3)

In this case the wavelet (covariant) transform is a map into the square
integrable functions [DufloMoore] with respect to the left Haar measure on
$G$. The map becomes an isometry if $v_{0}$ is properly scaled. Moreover, we
are able to recover the input $v$ from its wavelet transform through the
reconstruction formula, which requires an admissible vector as well, see
Example 5.3 below. The most popularized case of the above scheme is provided
by the affine group.

######  Example 4.5.

For the $ax+b$ group, representation (3.4) is square integrable for $p=2$. Any
function $v_{0}$, such that its Fourier transform $\hat{v}_{0}(\lambda)$
satisfies

$\int\limits_{0}^{\infty}\frac{\left|\hat{v}_{0}(\lambda)\right|^{2}}{\lambda}\,d\lambda<\infty,$
(4.4)

is admissible in the sense of (4.3) [AliAntGaz00]*§ 12.2. The _continuous
wavelet transform_ is generated by representation (3.4) acting on an
admissible vector $v_{0}$ in expression (4.2). The image of a function from
$L_{2}{}(\mathbb{R}{})$ is a function on the upper half-plane square
integrable with respect to the measure $a^{-2}\,da\,db$. There are many
examples [AliAntGaz00]*§ 12.2 of useful admissible vectors, say, the _Mexican
hat_ wavelet: $(1-x^{2})e^{-x^{2}/2}$. For sufficiently regular $\hat{v}_{0}$
admissibility (4.4) of $v_{0}$ follows by a weaker condition

$\int_{\mathbb{R}{}}v_{0}(x)\,dx=0.$ (4.5)

We dedicate Section 8 to isometric properties of this transform.

However, square integrable representations and admissible vectors do not cover
all interesting cases.

######  Example 4.6.

For the above $G=\mathrm{Aff}$ and representation (3.4), we consider the
operators $F_{\pm}:L_{p}{}(\mathbb{R}{})\rightarrow\mathbb{C}{}$ defined by:

$F_{\pm}(f)=\frac{1}{\pi\mathrm{i}}\int_{\mathbb{R}{}}\frac{f(x)\,dx}{\mathrm{i}\mp
x}.$ (4.6)

In $L_{2}{}(\mathbb{R}{})$ we note that $F_{+}(f)=\left\langle
f,c\right\rangle$, where $c(x)=\frac{1}{\pi\mathrm{i}}\frac{1}{\mathrm{i}+x}$.
Computing the Fourier transform
$\hat{c}(\lambda)=\chi_{(0,+\infty)}(\lambda)\,e^{-\lambda}$, we see that
$\bar{c}\in H_{2}{}(\mathbb{R}{})$. Moreover, $\hat{c}$ does not satisfy
admissibility condition (4.4) for representation (3.4).

Then, covariant transform (4.1) is Cauchy integral (2.1) from
$L_{p}{}(\mathbb{R}{})$ to the space of functions $\tilde{f}(a,b)$ such that
$a^{-\frac{1}{p}}\tilde{f}(a,b)$ is in the Hardy space on the upper/lower
half-plane $H_{p}{}(\mathbb{R}^{2}_{\pm}{})$ [Nikolski02a]*§ A.6.3. Due to
inadmissibility of $c(x)$, the complex analysis become decoupled from the
traditional wavelet theory.

Many important objects in harmonic analysis are generated by inadmissible
mother wavelets like (4.6). For example, the functionals
$P=\frac{1}{2}(F_{+}+F_{-})$ and $Q=\frac{1}{2\mathrm{i}}(F_{+}-F_{-})$ are
defined by kernels:

$\displaystyle p(x)$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\left(\frac{1}{\mathrm{i}-x}-\frac{1}{\mathrm{i}+x}\right)=\frac{1}{\pi}\frac{1}{1+x^{2}},$
(4.7) $\displaystyle q(x)$
$\displaystyle=-\frac{1}{2\pi}\left(\frac{1}{\mathrm{i}-x}-\frac{1}{\mathrm{i}+x}\right)=-\frac{1}{\pi}\frac{x}{1+x^{2}}$
(4.8)

which are _Poisson kernel_ (2.2) and the _conjugate Poisson kernel_
[Grafakos08]*§ 4.1 [Garnett07a]*§ III.1 [Koosis98a]*Ch. 5 [Nikolski02a]*§
A.5.3, respectively. Another interesting non-admissible vector is the
_Gaussian_ $e^{-x^{2}}$.

######  Example 4.7.

A step in a different direction is a consideration of non-linear operators.
Take again the $ax+b$ group and its representation (3.4). We define $F$ to be
a homogeneous (but non-linear) functional $V\rightarrow\mathbb{R}_{+}{}$:

$F_{m}(f)=\frac{1}{2}\int\limits_{-1}^{1}\left|f(x)\right|\,dx.$ (4.9)

Covariant transform (4.1) becomes:

$[\mathcal{W}^{m}_{p}f](a,b)=F({\rho_{p}}({\textstyle\frac{1}{a},-\frac{1}{b}})f)=\frac{1}{2}\int\limits_{-1}^{1}\left|a^{\frac{1}{p}}f\left(ax+b\right)\right|\,dx=\frac{a^{\frac{1}{q}}}{2}\int\limits^{b+a}_{b-a}\left|f\left(x\right)\right|\,dx,$
(4.10)

where $\frac{1}{p}+\frac{1}{q}=1$, as usual. We will see its connections with
the Hardy–Littlewood maximal functions in Example 7.6.

Since linearity has clear advantages, we may prefer to reformulate the last
example using linear covariant transforms. The idea is similar to the
representation of a convex function as an envelope of linear ones, cf.
[Garnett07a]*Ch. I, Lem. 6.1. To this end, we take a collection $\mathbf{F}$
of linear fiducial functionals and, for a given function $f$, consider the set
of all covariant transforms $\mathcal{W}_{F}f$, $F\in\mathbf{F}$.

######  Example 4.8.

Let us return to the setup of the previous Example for $G=\mathrm{Aff}$ and
its representation (3.4). Consider the unit ball $B$ in $L_{\infty}{}[-1,1]$.
Then, any $\omega\in B$ defines a bounded linear functional $F_{\omega}$ on
$L_{1}{}(\mathbb{R}{})$:

$F_{\omega}(f)=\frac{1}{2}\int\limits_{-1}^{1}f(x)\,\omega(x)\,dx=\frac{1}{2}\int_{\mathbb{R}{}}f(x)\,\omega(x)\,dx.$
(4.11)

Of course, $\sup_{\omega\in B}F_{\omega}(f)=F_{m}(f)$ with $F_{m}$ from (4.9)
and for all $f\in L_{1}{}(\mathbb{R}{})$. Then, for the non-linear covariant
transform (4.10) we have the following expression in terms of the linear
covariant transforms generated by $F_{\omega}$:

$[\mathcal{W}^{m}_{1}f](a,b)=\sup_{\omega\in
B}\,[\mathcal{W}^{\omega}_{1}f](a,b).$ (4.12)

The presence of suprimum is the price to pay for such a “linearization”.

######  Remark 4.9.

The above construction is not much different to the _grand maximal function_
[Stein93]*§ III.1.2. Although, it may look like a generalisation of covariant
transform, grand maximal function can be realised as a particular case of
Defn. 4.1. Indeed, let $M(V)$ be a subgroup of the group of all invertible
isometries of a metric space $V$. If ${\rho}$ represents a group $G$ by
isometries of $V$ then we can consider the group $\tilde{G}$ generated by all
finite products of $M(V)$ and ${\rho}(g)$, $g\in G$ with the straightforward
action ${\tilde{\rho}}$ on $V$. The grand maximal functions is produced by the
covariant transform for the representation ${\tilde{\rho}}$ of $\tilde{G}$.

######  Remark 4.10.

It is instructive to compare action (3.5) of the large
$\mathrm{SL}_{2}(\mathbb{R}{})$ group on the mother wavelet
$\frac{1}{x+\mathrm{i}}$ for the Cauchy integral and the principal case
$\omega(x)=\chi_{[-1,1]}(x)$ (the characteristic function of $[-1,1]$) for
functional (4.11). The wavelet $\frac{1}{x+\mathrm{i}}$ is an eigenvector for
all matrices $\begin{pmatrix}\cos t&\sin t\\\ -\sin t&\cos t\end{pmatrix}$,
which form the one-parameter compact subgroup
$K\subset\mathrm{SL}_{2}(\mathbb{R}{})$. The respective covariant transform
(i.e., the Cauchy integral) maps functions to the homogeneous space
$\mathrm{SL}_{2}(\mathbb{R}{})/K$, which is the upper half-plane with the
Möbius (linear-fractional) transformations of complex numbers [Kisil05a]
[Kisil12a]*§ 3.3.4 [Kisil11c]*§ 3. By contrast, the mother wavelet
$\chi_{[-1,1]}$ is an eigenvector for all matrices $\begin{pmatrix}\cosh
t&\sinh t\\\ \sinh t&\cosh t\end{pmatrix}$, which form the one-parameter
subgroup $A\in\mathrm{SL}_{2}(\mathbb{R}{})$. The covariant transform (i.e.,
the averaging) maps functions to the homogeneous space
$\mathrm{SL}_{2}(\mathbb{R}{})/A$, which can be identified with a set of
double numbers with corresponding Möbius transformations [Kisil05a]
[Kisil12a]*§ 3.3.4 [Kisil11c]*§ 3. Conformal geometry of double numbers is
suitable for real variables technique, in particular, tents [Stein93]*§ II.2
make a Möbius-invariant family.

##  5 The contravariant transform

Define the left action $\Lambda$ of a group $G$ on a space of functions over
$G$ by:

$\Lambda(g):f(h)\mapsto f(g^{-1}h).$ (5.1)

For example, in the case of the affine group it is (3.2). An object invariant
under the left action $\Lambda$ is called _left invariant_. In particular, let
$L$ and $L^{\prime}$ be two left invariant spaces of functions on $G$. We say
that a pairing $\left\langle\cdot,\cdot\right\rangle:L\times
L^{\prime}\rightarrow\mathbb{C}{}$ is _left invariant_ if

$\left\langle\Lambda(g)f,\Lambda(g)f^{\prime}\right\rangle=\left\langle
f,f^{\prime}\right\rangle,\quad\textrm{ for all }\quad f\in L,\ f^{\prime}\in
L^{\prime},\ g\in G.$ (5.2)

######  Remark 5.1.

  1. i.

We do not require the pairing to be linear in general, in some cases it is
sufficient to have only homogeneity, see Example 5.5.

  2. ii.

If the pairing is invariant on space $L\times L^{\prime}$ it is not
necessarily invariant (or even defined) on large spaces of functions.

  3. iii.

In some cases, an invariant pairing on $G$ can be obtained from an _invariant
functional_ $l$ by the formula $\left\langle
f_{1},f_{2}\right\rangle=l(f_{1}{f}_{2})$.

For a representation ${\rho}$ of $G$ in $V$ and $w_{0}\in V$, we construct a
function $w(g)={\rho}(g)w_{0}$ on $G$. We assume that the pairing can be
extended in its second component to this $V$-valued functions. For example,
such an extension can be defined in the weak sense.

######  Definition 5.2.

[Kisil09d, Kisil11c] Let $\left\langle\cdot,\cdot\right\rangle$ be a left
invariant pairing on $L\times L^{\prime}$ as above, let ${\rho}$ be a
representation of $G$ in a space $V$, we define the function
$w(g)={\rho}(g)w_{0}$ for $w_{0}\in V$ such that $w(g)\in L^{\prime}$ in a
suitable sense. The _contravariant transform_ $\mathcal{M}_{w_{0}}^{{\rho}}$
is a map $L\rightarrow V$ defined by the pairing:

$\mathcal{M}_{w_{0}}^{{\rho}}:f\mapsto\left\langle
f,w\right\rangle,\qquad\text{ where }f\in L.$ (5.3)

We can drop out sup/subscripts in $\mathcal{M}_{w_{0}}^{{\rho}}$ as we did for
$\mathcal{W}_{F}^{{\rho}}$.

######  Example 5.3 (Haar paring).

The most used example of an invariant pairing on $L_{2}{}(G,d\mu)\times
L_{2}{}(G,d\mu)$ is the integration with respect to the Haar measure:

$\left\langle f_{1},f_{2}\right\rangle=\int_{G}f_{1}(g){f}_{2}(g)\,dg.$ (5.4)

If ${\rho}$ is a square integrable representation of $G$ and $w_{0}$ is an
admissible vector, see Example 4.4, then this pairing can be extended to
$w(g)={\rho}(g)w_{0}$. The contravariant transform is known in this setup as
the _reconstruction formula_ , cf. [AliAntGaz00]*(8.19):

$\mathcal{M}_{w_{0}}f=\int_{G}f(g)\,w(g)\,dg,\qquad\text{ where
}w(g)={\rho}(g)w_{0}.$ (5.5)

It is possible to use different admissible vectors $v_{0}$ and $w_{0}$ for
wavelet transform (4.2) and reconstruction formula (5.5), respectively, cf.
Example 7.4.

Let either

  * •

${\rho}$ be not a square integrable representation (even modulo a subgroup);
_or_

  * •

$w_{0}$ be an inadmissible vector of a square integrable representation
${\rho}$.

A suitable invariant pairing in this case is not associated with integration
over the Haar measure on $G$. In this case we speak about a _Hardy pairing_.
The following example explains the name.

######  Example 5.4 (Hardy pairing).

Let $G$ be the $ax+b$ group and its representation ${\rho}$ (3.4) in Example
4.5. An invariant pairing on $G$, which is not generated by the Haar measure
$a^{-2}da\,db$, is:

$\left\langle f_{1},f_{2}\right\rangle_{H}=\lim_{a\rightarrow
0}\int\limits_{-\infty}^{\infty}f_{1}(a,b)\,{f}_{2}(a,b)\,\frac{db}{a}.$ (5.6)

For this pairing, we can consider functions
$\frac{1}{\pi\mathrm{i}}\frac{1}{x+\mathrm{i}}$ or $e^{-x^{2}}$, which are not
admissible vectors in the sense of square integrable representations. For
example, for $v_{0}=\frac{1}{\pi\mathrm{i}}\frac{1}{x+\mathrm{i}}$ we obtain:

$[\mathcal{M}f](x)=\lim_{a\rightarrow
0}\int\limits_{-\infty}^{\infty}f(a,b)\,\frac{a^{-\frac{1}{p}}}{\pi\mathrm{i}(x+\mathrm{i}a-b)}\,db=-\lim_{a\rightarrow
0}\frac{a^{-\frac{1}{p}}}{\pi\mathrm{i}}\int\limits_{-\infty}^{\infty}\frac{f(a,b)\,db}{b-(x+\mathrm{i}a)}.$

In other words, it expresses the boundary values at $a=0$ of the Cauchy
integral $[-\mathcal{C}f](x+\mathrm{i}a)$.

Here is an important example of non-linear pairing.

######  Example 5.5.

Let $G=\mathrm{Aff}$ and an invariant homogeneous functional on $G$ be given
by the $L_{\infty}{}$-version of Haar functional (5.4):

$\left\langle f_{1},f_{2}\right\rangle_{\infty}=\sup_{g\in
G}\left|f_{1}(g){f}_{2}(g)\right|.$ (5.7)

Define the following two functions on $\mathbb{R}{}$:

${v}^{+}_{0}(t)=\left\\{\begin{array}[]{ll}1,&\text{ if }t=0;\\\ 0,&\text{ if
}t\neq 0,\end{array}\right.\quad\text{ and }\quad
v_{0}^{*}(t)=\left\\{\begin{array}[]{ll}1,&\text{ if }\left|t\right|\leq 1;\\\
0,&\text{ if }\left|t\right|>1.\end{array}\right.$ (5.8)

The respective contravariant transforms are generated by representation
${\rho_{\infty}}$ (3.4) are:

$\displaystyle[\mathcal{M}_{{v}^{+}_{0}}f](t)$ $\displaystyle=$ $\displaystyle
f^{+}(t)=\left\langle
f(a,b),{\rho_{\infty}}(a,b){v}_{0}^{+}(t)\right\rangle_{\infty}=\sup_{a}\left|f(a,t)\right|,$
(5.9) $\displaystyle{}[\mathcal{M}_{v^{*}_{0}}f](t)$ $\displaystyle=$
$\displaystyle f^{*}(t)=\left\langle
f(a,b),{\rho_{\infty}}(a,b)v_{0}^{*}(t)\right\rangle_{\infty}=\sup_{a>\left|b-t\right|}\left|f(a,b)\right|.$
(5.10)

Transforms (5.9) and (5.10) are the _vertical_ and _non-tangential maximal
functions_ [Koosis98a]*§ VIII.C.2, respectively.

######  Example 5.6.

Consider again $G=\mathrm{Aff}$ equipped now with an invariant linear
functional, which is a Hardy-type modification (cf. (5.6)) of
$L_{\infty}{}$-functional (5.7):

$\left\langle f_{1},f_{2}\right\rangle_{\stackrel{{\scriptstyle
H}}{{\infty}}}={\varlimsup_{a\rightarrow
0}}\,\sup_{b\in\mathbb{R}{}}(f_{1}(a,b){f}_{2}(a,b)),$ (5.11)

where $\varlimsup$ is the upper limit. Then, the covariant transform
$\mathcal{M}^{H}$ for this pairing for functions $v^{+}$ and $v^{*}$ (5.8)
becomes:

$\displaystyle[\mathcal{M}_{{v}^{+}_{0}}^{H}f](t)$ $\displaystyle=$
$\displaystyle\left\langle
f(a,b),{\rho_{\infty}}(a,b){v}_{0}^{+}(t)\right\rangle_{\stackrel{{\scriptstyle
H}}{{\infty}}}=\varlimsup_{a\rightarrow 0}f(a,t),$ (5.12)
$\displaystyle[\mathcal{M}_{v^{*}_{0}}^{H}f](t)$ $\displaystyle=$
$\displaystyle\left\langle
f(a,b),{\rho_{\infty}}(a,b)v_{0}^{*}(t)\right\rangle_{\stackrel{{\scriptstyle
H}}{{\infty}}}=\varlimsup_{\begin{subarray}{c}a\rightarrow 0\\\
\left|b-t\right|<a\end{subarray}}f(a,b).$ (5.13)

They are the _normal_ and _non-tangential_ upper limits from the upper-half
plane to the real line, respectively.

Note the obvious inequality $\left\langle
f_{1},f_{2}\right\rangle_{\infty}\geq\left\langle
f_{1},f_{2}\right\rangle_{\stackrel{{\scriptstyle H}}{{\infty}}}$ between
pairings (5.7) and (5.11), which produces the corresponding relation between
respective contravariant transforms.

There is an explicit duality between the covariant transform and the
contravariant transform. Discussion of the grand maximal function in the Rem.
4.9 shows usefulness of the covariant transform over a family of fiducial
functionals. Thus, we shall not be surprised by the contravariant transform
over a family of reconstructing vectors as well.

######  Definition 5.7.

Let $w:\mathrm{Aff}\rightarrow L_{1}{}(\mathbb{R}{})$ be a function. We define
a new function ${\rho_{1}}w$ on $\mathrm{Aff}$ with values in
$L_{1}{}(\mathbb{R}{})$ via the point-wise action
$[{\rho_{1}}w](g)={\rho_{1}}(g)w(g)$ of ${\rho_{\infty}}$ (3.4). If
$\sup_{g}\left\|w(g)\right\|_{1}<\infty$, then, for $f\in
L_{1}{}(\mathrm{Aff})$, we define the _extended contravariant transform_ by:

$[\mathcal{M}_{w}f](x)=\int_{\mathrm{Aff}}f(g)\,[{\rho_{1}}w](g)\,dg.$ (5.14)

Note, that (5.14) reduces to the contravariant transform (5.5) if we start
from the constant function $w(g)=w_{0}$.

######  Definition 5.8.

We call a function $r$ on $\mathbb{R}{}$ a _nucleus_ if:

  1. i.

$r$ is supported in $[-1,1]$,

  2. ii.

$\left|r\right|<\frac{1}{2}$ almost everywhere, and

  3. iii.

$\int_{\mathbb{R}{}}r(x)\,dx=0$, cf. (4.5).

Clearly, for a nucleus $r$, the function $s={\rho_{1}}(a,b)r$ has the
following properties:

  1. i.

$s$ is supported in a ball centred at $b$ and radius $a$,

  2. ii.

$\left|s\right|<\frac{1}{2a}$ almost everywhere, and

  3. iii.

$\int_{\mathbb{R}{}}s(x)\,dx=0$.

In other words, $s={\rho_{1}}(a,b)r$ is an _atom_ , cf. [Stein93]*§ III.2.2
and any atom may be obtained in this way from some nucleus and certain
$(a,b)\in\mathrm{Aff}$.

######  Example 5.9.

Let $f(g)=\sum_{j}\lambda_{j}\delta_{g_{j}}(g)$ with
$\sum_{j}\left|\lambda_{j}\right|<\infty$ be a countable sum of point masses
on $\mathrm{Aff}$. If all values of $w(g_{j})$ are nucleuses, then (5.14)
becomes:

$[\mathcal{M}_{w}f](x)=\int_{\mathrm{Aff}}f(g)\,[{\rho_{1}}w](g)\,dg=\sum_{j}\lambda_{j}s_{j},$
(5.15)

where $s_{j}={\rho_{1}}(g_{j})w(g_{j})$ are atoms. The right-hand side of
(5.15) is known as an _atomic decomposition_ of a function
$h(x)=[\mathcal{M}_{w}f](x)$, see [Stein93]*§ III.2.2.

##  6 Intertwining properties of covariant transforms

The covariant transform has obtained its name because of the following
property.

######  Theorem 6.1.

[Kisil09d, Kisil11c] Covariant transform (4.1) intertwines ${\rho}$ and the
left regular representation $\Lambda$ (5.1) on $L{}(G,U)$:

$\mathcal{W}{\rho}(g)=\Lambda(g)\mathcal{W}.$ (6.1)

######  Corollary 6.2.

The image space $\mathcal{W}(V)$ is invariant under the left shifts on $G$.

The covariant transform is also a natural source of _relative convolutions_
[Kisil94e, Kisil13a], which are operators $A_{k}=\int_{G}k(g){\rho}(g)\,dg$
obtained by integration a representation ${\rho}$ of a group $G$ with a
suitable kernel $k$ on $G$. In particular, inverse wavelet transform
$\mathcal{M}_{w_{0}}f$ (5.5) can be defined from the relative convolution
$A_{f}$ as well: $\mathcal{M}_{w_{0}}f=A_{f}w_{0}$.

######  Corollary 6.3.

Covariant transform (4.1) intertwines the operator of convolution $K$ (with
kernel $k$) and the operator of relative convolution $A_{k}$, i.e.
$K\mathcal{W}=\mathcal{W}A_{k}$.

If the invariant pairing is defined by integration with respect to the Haar
measure, cf. Example 5.3, then we can show an intertwining property for the
contravariant transform as well.

######  Proposition 6.4.

[Kisil98a]*Prop. 2.9 Inverse wavelet transform $\mathcal{M}_{w_{0}}$ (5.5)
intertwines left regular representation $\Lambda$ (5.1) on $L_{2}{}(G)$ and
${\rho}$:

$\mathcal{M}_{w_{0}}\Lambda(g)={\rho}(g)\mathcal{M}_{w_{0}}.$ (6.2)

######  Corollary 6.5.

The image $\mathcal{M}_{w_{0}}(L{}(G))\subset V$ of a left invariant space
$L{}(G)$ under the inverse wavelet transform $\mathcal{M}_{w_{0}}$ is
invariant under the representation ${\rho}$.

######  Remark 6.6.

It is an important observation, that the above intertwining property is also
true for some contravariant transforms which are not based on pairing (5.4).
For example, in the case of the affine group all pairings (5.6), (5.11) and
(non-linear!) (5.7) satisfy to (6.2) for the respective representation
${\rho_{p}}$ (3.4).

There is also a simple connection between a covariant transform and right
shifts.

######  Proposition 6.7.

[Kisil10c, Kisil11c] Let $G$ be a Lie group and ${\rho}$ be a representation
of $G$ in a space $V$. Let $[\mathcal{W}f](g)=F({\rho}(g^{-1})f)$ be a
covariant transform defined by a fiducial operator $F:V\rightarrow U$. Then
the right shift $[\mathcal{W}f](gg^{\prime})$ by $g^{\prime}$ is the covariant
transform $[\mathcal{W^{\prime}}f](g)=F^{\prime}({\rho}(g^{-1})f)]$ defined by
the fiducial operator $F^{\prime}=F\circ{\rho}(g^{-1})$.

In other words the covariant transform intertwines right shifts
$R(g):f(h)\mapsto f(hg)$ on the group $G$ with the associated action

${\rho_{B}}(g):F\mapsto F\circ{\rho}(g^{-1})$ (6.3)

on fiducial operators:

$R(g)\circ\mathcal{W}_{F}=\mathcal{W}_{{\rho_{B}}(g)F},\qquad g\in G.$ (6.4)

Although the above result is obvious, its infinitesimal version has
interesting consequences. Let $G$ be a Lie group with a Lie algebra
$\mathfrak{g}$ and ${\rho}$ be a smooth representation of $G$. We denote by
$d{\rho_{B}}$ the derived representation of the associated representation
${\rho_{B}}$ (6.3) on fiducial operators.

######  Corollary 6.8.

[Kisil10c, Kisil11c] Let a fiducial operator $F$ be a null-solution, i.e.
$AF=0$, for the operator $A=\sum_{j}a_{j}d{\rho^{X_{j}}_{B}}$, where
$X_{j}\in\mathfrak{g}$ and $a_{j}$ are constants. Then the covariant transform
$[\mathcal{W}_{F}f](g)=F({\rho}(g^{-1})f)$ for any $f$ satisfies

$D(\mathcal{W}_{F}f)=0,\qquad\text{where}\quad
D=\sum_{j}\bar{a}_{j}\mathfrak{L}^{X_{j}}.$

Here, $\mathfrak{L}^{X_{j}}$ are the left invariant fields (Lie derivatives)
on $G$ corresponding to $X_{j}$.

######  Example 6.9.

Consider representation ${\rho}$ (3.4) of the $ax+b$ group with the $p=1$. Let
$\mathsf{A}$ and $\mathsf{N}$ be the basis of $\mathfrak{g}$ generating one-
parameter subgroups $A$ and $N$ (3.3), respectively. Then, the derived
representations are:

$[d{\rho^{\mathsf{A}}}f](x)=-f(x)-xf^{\prime}(x),\qquad[d{\rho^{\mathsf{N}}}f](x)=-f^{\prime}(x).$

The corresponding left invariant vector fields on $ax+b$ group are:

$\mathfrak{L}^{\mathsf{A}}=a\partial_{a},\qquad\mathfrak{L}^{\mathsf{N}}=a\partial_{b}.$

The mother wavelet $\frac{1}{x+\mathrm{i}}$ in (4.6) is a null solution of the
operator

$-d{\rho^{\mathsf{A}}}-\mathrm{i}d{\rho^{\mathsf{N}}}=I+(x+\mathrm{i})\frac{d}{dx}.$
(6.5)

Therefore, the image of the covariant transform with fiducial operator $F_{+}$
(4.6) consists of the null solutions to the operator
$-\mathfrak{L}^{\mathsf{A}}+\mathrm{i}\mathfrak{L}^{\mathsf{N}}=\mathrm{i}a(\partial_{b}+\mathrm{i}\partial_{a})$,
that is in the essence Cauchy–Riemann operator $\partial_{\bar{z}}$ (2.3) in
the upper half-plane.

######  Example 6.10.

In the above setting, the function $p(x)=\frac{1}{\pi}\frac{1}{x^{2}+1}$ (4.7)
is a null solution of the operator:

$(d{\rho^{\mathsf{A}}})^{2}-d{\rho^{\mathsf{A}}}+(d{\rho^{\mathsf{N}}})^{2}=2I+4x\frac{d}{dx}+(1+x^{2})\frac{d^{2}}{dx^{2}}.$

The covariant transform with the mother wavelet $p(x)$ is the Poisson
integral, its values are null solutions to the operator
$(\mathfrak{L}^{\mathsf{A}})^{2}-\mathfrak{L}^{\mathsf{A}}+(\mathfrak{L}^{\mathsf{N}})^{2}=a^{2}(\partial_{b}^{2}+\partial_{a}^{2})$,
which is Laplace operator $\Delta$ (2.3).

######  Example 6.11.

Fiducial functional $F_{m}$ (4.9) is a null solution of the following
functional equation:

$\textstyle
F_{m}-F_{m}\circ{\rho_{\infty}}(\frac{1}{2},\frac{1}{2})-F_{m}\circ{\rho_{\infty}}(\frac{1}{2},-\frac{1}{2})=0.$

Consequently, the image of wavelet transform $\mathcal{W}^{m}_{p}$ (4.10)
consists of functions which solve the equation:

$\textstyle(I-R(\frac{1}{2},\frac{1}{2})-R(\frac{1}{2},-\frac{1}{2}))f=0\quad\text{
or }\quad
f(a,b)=f(\frac{1}{2}a,b+\frac{1}{2}a)+f(\frac{1}{2}a,b-\frac{1}{2}a).$

The last relation is the key to the stopping time argument [Garnett07a]*Ch.
VI, Lem. 2.2 and the dyadic squares technique, see for example [Stein93]*§
IV.3, [Garnett07a]*Ch. VII, Thm. 1.1 or the picture on the front cover of the
latter book.

The moral of the above Examples 6.9–6.11 is: there is a significant freedom in
choice of covariant transforms. However, some fiducial functionals have
special properties, which suggest the suitable technique (e.g., analytic,
harmonic, dyadic, etc.) following from this choice.

##  7 Composing the covariant and the contravariant transforms

From Props. 6.1, 6.4 and Rem. 6.6 we deduce the following

######  Corollary 7.1.

The composition $\mathcal{M}_{w}\circ\mathcal{W}_{F}$ of a covariant
$\mathcal{M}_{w}$ and contravariant $\mathcal{W}_{F}$ transforms is a map
$V\rightarrow V$, which commutes with ${\rho}$, i.e., intertwines ${\rho}$
with itself.

In particular for the affine group and representation (3.4),
$\mathcal{M}_{w}\circ\mathcal{W}_{F}$ commutes with shifts and dilations of
the real line.

Since the image space of $\mathcal{M}_{w}\circ\mathcal{W}_{F}$ is an
$\mathrm{Aff}$-invariant space, we shall be interested in the smallest
building blocks with the same property. For the Hilbert spaces, any group
invariant subspace $V$ can be decomposed into a direct integral $V=\oplus\int
V_{\mu}\,d\mu$ of _irreducible_ subspaces $V_{\mu}$, i.e. $V_{\mu}$ does not
have any non-trivial invariant subspace [Kirillov76]*§ 8.4. For
representations in Banach spaces complete reducibility may not occur and we
shall look for _primary_ subspace, i.e. space which is not a direct sum of two
invariant subspaces [Kirillov76]*§ 8.3. We already identified such subspaces
as generalized Hardy spaces in Defn. 3.2. They are also related to covariant
functional calculus [Kisil02a] [Kisil11c]*§ 6.

For irreducible Hardy spaces, we can use the following general principle,
which has several different formulations, cf. [Kirillov76]*Thm. 8.2.1:

######  Lemma 7.2 (Schur).

[AliAntGaz00]*Lem. 4.3.1 Let ${\rho}$ be a continuous unitary irreducible
representation of $G$ on the Hilbert space $H$. If a bounded operator
$T:H\rightarrow T$ commutes with ${\rho}(g)$, for all $g\in G$, then $T=kI$,
for some $\lambda\in\mathbb{C}{}$.

######  Remark 7.3.

A revision of proofs of the Schur’s Lemma, even in different formulations,
show that the result is related to the existence of joint invariant subspaces
for all operators ${\rho}(g)$, $g\in G$.

In the case of classical wavelets, the relation between wavelet transform
(4.2) and inverse wavelet transform (5.5) is suggested by their names.

######  Example 7.4.

For an irreducible square integrable representation and admissible vectors
$v_{0}$ and $w_{0}$, there is the relation [AliAntGaz00]*(8.52):

$\mathcal{M}_{w_{0}}\mathcal{W}_{v_{0}}=kI,$ (7.1)

as an immediate consequence from the Schur’s lemma. Furthermore, square
integrability condition (4.3) ensures that $k\neq 0$. The exact value of the
constant $k$ depends on $v_{0}$, $w_{0}$ and the Duflo–Moore operator
[DufloMoore] [AliAntGaz00]*§ 8.2.

It is of interest here, that two different vectors can be used as analysing
vector in (4.2) and for the reconstructing formula (5.5). Even a greater
variety can be achieved if we use additional fiducial operators and invariant
pairings.

For the affine group, recall the decomposition from Prop. 3.6 into invariant
subspaces $L_{2}{}(\mathbb{R}{})=H_{2}{}\oplus H_{2}^{\perp}{}$ and the fact,
that the restrictions ${\rho^{+}_{2}}$ and ${\rho^{-}_{2}}$ of ${\rho_{2}}$
(3.4) on $H_{2}{}$ and $H_{2}^{\perp}{}$ are not unitary equivalent. Then,
Schur’s lemma implies:

######  Corollary 7.5.

Any bounded linear operator $T:L_{2}{}(\mathbb{R}{})\rightarrow
L_{2}{}(\mathbb{R}{})$ commuting with ${\rho_{2}}$ has the form
$k_{1}I_{H_{2}{}}\oplus k_{2}I_{H_{2}^{\perp}{}}$ for some constants $k_{1}$,
$k_{2}\in\mathbb{C}{}$. Consequently, the Fourier transform maps $T$ to the
operator of multiplication by
$k_{1}\chi_{(0,+\infty)}+k_{2}\chi_{(-\infty,0)}$.

Of course, Corollary 7.5 is applicable to the composition of covariant and
contravariant transforms. In particular, the constants $k_{1}$ and $k_{2}$ may
have zero values: for example, the zero value occurs for $\mathcal{W}$ (4.2)
with an admissible vector $v_{0}$ and non-tangential limit
$\mathcal{M}_{v^{*}_{0}}^{H}$ (5.13)—because a square integrable function
$f(a,b)$ on $\mathrm{Aff}$ vanishes for $a\rightarrow 0$.

######  Example 7.6.

The composition of contravariant transform $\mathcal{M}_{v^{*}_{0}}$ (5.10)
with covariant transform $\mathcal{W}_{\infty}$ (4.10) is:

$\displaystyle[\mathcal{M}_{v^{*}_{0}}\mathcal{W}_{\infty}f](t)$
$\displaystyle=$
$\displaystyle\sup_{a>\left|b-t\right|}\left\\{\frac{1}{2a}\int\limits^{b+a}_{b-a}\left|f\left(x\right)\right|\,dx\right\\}$
$\displaystyle=$
$\displaystyle\sup_{b_{1}<t<b_{2}}\left\\{\frac{1}{b_{2}-b_{1}}\int\limits^{b_{2}}_{b_{1}}\left|f\left(x\right)\right|\,dx\right\\}.$

Thus, $\mathcal{M}_{v^{*}_{0}}\mathcal{W}_{\infty}f$ coincides with
_Hardy–Littlewood maximal function_ $f^{M}$ (2.6), which contains important
information on the original function $f$ [Koosis98a]*§ VIII.B.1. Combining
Props. 6.1 and 6.4 (through Rem. 6.6), we deduce that the operator $M:f\mapsto
f^{M}$ commutes with ${\rho_{p}}$: ${\rho_{p}}M=M{\rho_{p}}$. Yet, $M$ is non-
linear and Cor. 7.5 is not applicable in this case.

######  Example 7.7.

Let the mother wavelet $v_{0}(x)=\delta(x)$ be the Dirac delta function, then
the wavelet transform $\mathcal{W}_{\delta}$ generated by ${\rho_{\infty}}$
(3.4) on $C{}(\mathbb{R}{})$ is $[\mathcal{W}_{\delta}f](a,b)=f(b)$. Take the
reconstruction vector $w_{0}(t)=(1-\chi_{[-1,1]}(t))/t/\pi$ and consider the
respective inverse wavelet transform $\mathcal{M}_{w_{0}}$ produced by Hardy
pairing (5.6). Then, the composition of both maps is:

$\displaystyle[\mathcal{M}_{w_{0}}\circ\mathcal{W}_{\delta}f](t)$
$\displaystyle=$ $\displaystyle\lim_{a\rightarrow
0}\,\frac{1}{\pi}\\!\int\limits_{-\infty}^{\infty}f(b)\,{\rho_{\infty}}(a,b)w_{0}(t)\,\frac{db}{a}$
(7.3) $\displaystyle=$ $\displaystyle\lim_{a\rightarrow
0}\,\frac{1}{\pi}\\!\int\limits_{-\infty}^{\infty}f(b)\,\frac{1-\chi_{[-a,a]}(t-b)}{t-b}\,{db}$
$\displaystyle=$ $\displaystyle\lim_{a\rightarrow
0}\,\frac{1}{\pi}\\!\int\limits_{\left|b\right|>a}\frac{f(b)}{t-b}\,{db}.$

The last expression is the _Hilbert transform_
$\mathcal{H}=\mathcal{M}_{w_{0}}\circ\mathcal{W}_{\delta}$, which is an
example of a _singular integral operator_ (SIO) [Stein93]*§ I.5
[MityushevRogosin00a]*§ 2.6 defined through the principal value (2.5) (in the
sense of Cauchy). By Cor. 7.5 we know that $\mathcal{H}=k_{1}I_{H_{2}{}}\oplus
k_{2}I_{H_{2}^{\perp}{}}$ for some constants $k_{1}$, $k_{2}\in\mathbb{C}{}$.
Furthermore, we can directly check that $\mathcal{H}J=-J\mathcal{H}$, for the
reflection $J$ from (3.8), thus $k_{1}=-k_{2}$. An evaluation of $\mathcal{H}$
on a simple function from $H_{2}{}$ (say, the Cauchy kernel
$\frac{1}{x+\mathrm{i}}$) gives the value of the constant $k_{1}=-\mathrm{i}$.
Thus,
$\mathcal{H}=(-\mathrm{i}I_{H_{2}{}})\oplus(\mathrm{i}I_{H_{2}^{\perp}{}})$.

In fact, the previous reasons imply the following

######  Proposition 7.8.

[Stein70a]*§ III.1.1 Any bounded linear operator on $L_{2}{}(\mathbb{R}{})$
commuting with quasi-regular representation ${\rho_{2}}$ (3.4) and
anticommuting with reflection $J$ (3.8) is a constant multiple of Hilbert
transform (7.3).

######  Example 7.9.

Consider the covariant transform $\mathcal{W}_{q}$ defined by the inadmissible
wavelet $q(t)$ (4.8), the conjugated Poisson kernel. Its composition with the
contravariant transform $\mathcal{M}_{{v}^{+}_{0}}^{H}$ (5.12) is

$[\mathcal{M}_{{v}^{+}_{0}}^{H}\circ\mathcal{W}_{q}f](t)=\varlimsup_{a\rightarrow
0}\frac{1}{\pi}\int_{\mathbb{R}{}}\frac{f(x)\,(t-x)}{(t-x)^{2}+a^{2}}\,dx$
(7.4)

We can see that this composition satisfies to Prop. 7.8, the constant factor
can again be evaluated from the Cauchy kernel $f(x)=\frac{1}{x+\mathrm{i}}$
and is equal to $1$. Of course, this is a classical result [Grafakos08]*Thm.
4.1.5 in harmonic analysis that (7.4) provides an alternative expression for
Hilbert transform (7.3).

######  Example 7.10.

Let $\mathcal{W}$ be a covariant transfrom generated either by the functional
$F_{\pm}$ (4.6) (i.e. the Cauchy integral) or $\frac{1}{2}(F_{+}-F_{-})$ (i.e.
the Poisson integral) from the Example 4.6. Then, for contravariant transform
$\mathcal{M}_{v^{+}_{0}}^{H}$ (5.9) the composition
$\mathcal{M}_{v^{+}_{0}}^{H}\mathcal{W}$ becomes the normal boundary value of
the Cauchy/Poisson integral, respectively. The similar composition
$\mathcal{M}_{v^{*}_{0}}^{H}\mathcal{W}$ for reconstructing vector $v^{*}_{0}$
(5.8) turns to be the non-tangential limit of the Cauchy/Poisson integrals.

The maximal function and SIO are often treated as elementary building blocks
of harmonic analysis. In particular, it is common to define the Hardy space as
a closed subspace of $L_{p}{}(\mathbb{R}{})$ which is mapped to
$L_{p}{}(\mathbb{R}{})$ by either the maximal operator (7.6) or by the SIO
(7.3) [Stein93]*§ III.1.2 and § III.4.3 [DziubanskiPreisner10a]. From this
perspective, the coincidence of both characterizations seems to be non-
trivial. On the contrast, we presented both the maximal operator and SIO as
compositions of certain co- and contravariant transforms. Thus, these
operators act between certain $\mathrm{Aff}$-invariant subspaces, which we
associated with generalized Hardy spaces in Defn. 3.2. For the right choice of
fiducial functionals, the coincidence of the respective invariant subspaces is
quite natural.

The potential of the group-theoretical approach is not limited to the Hilbert
space $L_{2}{}(\mathbb{R}{})$. One of possibilities is to look for a suitable
modification of Schur’s Lemma 7.2, say, to Banach spaces. However, we can
proceed with the affine group without such a generalisation. Here is an
illustration to a classical question of harmonic analysis: to identify the
class of functions on the real line such that
$\mathcal{M}_{v^{*}_{0}}^{H}\mathcal{W}$ becomes the identity operator on it.

######  Proposition 7.11.

Let $B{}$ be the space of bounded uniformly continuous functions on the real
line. Let $F:B{}\rightarrow\mathbb{R}{}$ be a fiducial functional such that:

$\lim_{a\rightarrow 0}F({\rho_{\infty}}(1/a,0)f)=0,\quad\text{ for all }f\in
B{}\text{ such that }f(0)=0$ (7.5)

and $F({\rho_{\infty}}(1,b)f)$ is a continuous function of $b\in\mathbb{R}{}$
for a given $f\in B{}$.

Then, $\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}$ is a constant multiple
of the identity operator on $B{}$.

###### Proof.

First of all we note that $\mathcal{M}_{v^{+}_{0}}^{H}\mathcal{W}_{F}$ is a
bounded operator on $B{}$. Let $v^{*}_{(a,b)}={\rho_{\infty}}(a,b)v^{*}$.
Obviously, $v^{*}_{(a,b)}(0)=v^{*}(-\frac{b}{a})$ is an eigenfunction for
operators $\Lambda(a^{\prime},0)$, $a^{\prime}\in\mathbb{R}_{+}{}$ of the left
regular representation of $\mathrm{Aff}$:

$\Lambda(a^{\prime},0)v^{*}_{(a,b)}(0)=v^{*}_{(a,b)}(0).$ (7.6)

This and the left invariance of pairing (5.2) imply that
$\mathcal{M}_{v^{*}_{0}}^{H}\circ\Lambda(1/a,0)=\mathcal{M}_{v^{*}_{0}}^{H}$
for any $(a,0)\in\mathrm{Aff}$. Then, applying intertwining properties (6.1)
we obtain that

$\displaystyle[\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}f](0)$
$\displaystyle=$
$\displaystyle[\mathcal{M}_{v^{*}_{0}}^{H}\circ\Lambda(1/a,0)\circ\mathcal{W}_{F}f](0)$
$\displaystyle=$
$\displaystyle[\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}\circ{\rho_{\infty}}(1/a,0)f](0).$

Using the limit $a\rightarrow 0$ (7.5) and the continuity of
$F\circ{\rho_{\infty}}(1,b)$ we conclude that the linear functional
$l:f\mapsto[\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}f](0)$ vanishes for
any $f\in\mathbb{B}{}$ such that $f(0)=0$. Take a function $f_{1}\in B{}$ such
that $f_{1}(0)=1$ and define $c=l(f_{1})$. From linearity of $l$, for any
$f\in B{}$ we have:

$l(f)=l(f-f(0)f_{1}+f(0)f_{1})=l(f-f(0)f_{1})+f(0)l(f_{1})=cf(0).$

Furthermore, using intertwining properties (6.1) and (6.2):

$\displaystyle[\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}f](t)$
$\displaystyle=$
$\displaystyle[{\rho_{\infty}}(1,-t)\circ\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}f](0)$
$\displaystyle=$
$\displaystyle[\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}\circ{\rho_{\infty}}(1,-t)f](0)$
$\displaystyle=$ $\displaystyle l({\rho_{\infty}}(1,-t)f)$ $\displaystyle=$
$\displaystyle c[{\rho_{\infty}}(1,-t)f](0)$ $\displaystyle=$ $\displaystyle
cf(t).$

This completes the proof. ∎

To get the classical statement we need the following lemma.

######  Lemma 7.12.

For $w(t)\in L_{1}{}(\mathbb{R}{})$, define the fiducial functional on $B{}$:

$F(f)=\int_{\mathbb{R}{}}f(t)\,w(t)\,dt.$ (7.7)

Then $F$ satisfies the conditions (and thus the conclusions) of Prop. 7.11.

###### Proof.

Let $f$ be a continuous bounded function such that $f(0)=0$. For
$\varepsilon>0$ chose

  * •

$\delta>0$ such that $\left|f(t)\right|<\varepsilon$ for all
$\left|t\right|<\delta$;

  * •

$M>0$ such that $\int_{\left|t\right|>M}\left|w(t)\right|\,dt<\varepsilon$.

Then, for $a<\delta/M$, we have the estimation:

$\displaystyle\left|F({\rho_{\infty}}(1/a,0)f)\right|$ $\displaystyle=$
$\displaystyle\left|\int_{\mathbb{R}{}}f\left(at\right)\,w(t)\,dt\right|$
$\displaystyle\leq$
$\displaystyle\left|\int_{\left|t\right|<M}f\left(at\right)\,w(t)\,dt\right|+\left|\int_{\left|t\right|>M}f\left(at\right)\,w(t)\,dt\right|$
$\displaystyle\leq$
$\displaystyle\varepsilon(\left\|w\right\|_{1}+\left\|f\right\|_{\infty}).$

Finally, for a uniformly continuous function $g$ for $\varepsilon>0$ there is
$\delta>0$ such that $\left|g(t+b)-g(t)\right|<\varepsilon$ for all $b<\delta$
and $t\in\mathbb{R}{}$. Then

$\left|F({\rho_{\infty}}(1,b)g)-F(g)\right|=\left|\int_{\mathbb{R}{}}(g(t+b)-g(t))\,w(t)\,dt\right|\leq\varepsilon\left\|w\right\|_{1}.$

This proves the continuity of $F({\rho_{\infty}}(1,b)g)$ at $b=0$ and, by the
group property, at any other point as well. ∎

######  Remark 7.13.

A direct evaluation shows, that the constant $c=l(f_{1})$ from the proof of
Prop. 7.11 for fiducial functional (7.7) is equal to
$c=\int_{\mathbb{R}{}}w(t)\,dt$. Of course, for non-trivial boundary values we
need $c\neq 0$. On the other hand, admissibility condition (4.5) requires
$c=0$. Moreover, the classical harmonic analysis and the traditional wavelet
construction are two “orthogonal” parts of the same covariant transform theory
in the following sense. We can present a rather general bounded function
$w=w_{a}+w_{p}$ as a sum of an admissible mother wavelet $w_{a}$ and a
suitable multiple $w_{p}$ of the Poisson kernel. An extension of this
technique to unbounded functions leads to _Calderón–Zygmund decomposition_
[Stein93]*§ I.4.

The table integral $\int_{\mathbb{R}{}}\frac{dx}{x^{2}+1}=\pi$ tells that the
“wavelet” $p(t)=\frac{1}{\pi}\frac{1}{1+t^{2}}$ (4.7) is in
$L_{1}{}(\mathbb{R}{})$ with $c=1$, the corresponding wavelet transform is the
Poisson integral. Its boundary behaviour from Prop. 7.11 is the classical
result, cf. [Garnett07a]*Ch. I, Cor. 3.2. The comparison of our arguments with
the traditional proofs, e.g. in [Garnett07a], does not reveal any significant
distinctions. We simply made an explicit usage of the relevant group
structure, which is implicitly employed in traditional texts anyway, cf.
[Burenkov12a]. Further demonstrations of this type can be found in
[Albargi13a, ElmabrokHutnik12a].

##  8 Transported norms

If the functional $F$ and the representation ${\rho}$ in (4.1) are both
linear, then the resulting covariant transform is a linear map. If
$\mathcal{W}_{F}$ is injective, e.g. due to irreducibility of ${\rho}$, then
$\mathcal{W}_{F}$ transports a norm $\left\|\cdot\right\|$ defined on $V$ to a
norm $\left\|\cdot\right\|_{F}$ defined on the image space $\mathcal{W}_{F}V$
by the simple rule:

$\left\|u\right\|_{F}:=\left\|v\right\|,\quad\text{ where the unique }v\in
V\text{ is defined by }u=\mathcal{W}_{F}v.$ (8.1)

By the very definition, we have the following

######  Proposition 8.1.

  1. i.

$\mathcal{W}_{F}$ is an isometry
$(V,\left\|\cdot\right\|)\rightarrow(\mathcal{W}_{F}V,\left\|\cdot\right\|_{F})$.

  2. ii.

If the representation ${\rho}$ acts on $(V,\left\|\cdot\right\|)$ by
isometries then $\left\|\cdot\right\|_{F}$ is left invariant.

A touch of non-triviality occurs if the transported norm can be naturally
expressed in the original terms of $G$.

######  Example 8.2.

It is common to consider a unitary square integrable representation ${\rho}$
and an admissible mother wavelet $f\in V$. In this case, wavelet transform
(4.2) becomes an isometry to square integrable functions on $G$ with respect
to a Haar measure [AliAntGaz00]*Thm. 8.1.3. In particular, for the affine
group and setup of Example 4.5, the wavelet transform with an admissible
vector is a multiple of an isometry map from $L_{2}{}(\mathbb{R}{})$ to the
functions on the upper half-plane, i.e., the $ax+b$ group, which are square
integrable with respect to the Haar measure $a^{-2}\,da\,db$.

A reader expects that there are other interesting examples of the transported
norms, which are not connected to the Haar integration.

######  Example 8.3.

In the setup of Example 4.6, consider the space $L_{p}{}(\mathbb{R}{})$ with
representation (3.4) of $\mathrm{Aff}$ and Poisson kernel $p(t)$ (4.7) as an
inadmissible mother wavelet. The norm transported by $\mathcal{W}_{P}$ to the
image space on $\mathrm{Aff}$ is [Nikolski02a]*§ A.6.3:

$\left\|u\right\|_{p}=\sup_{a>0}\left(\int\limits_{-\infty}^{\infty}\left|u(a,b)\right|^{p}\,\frac{db}{a}\right)^{\frac{1}{p}}.$
(8.2)

In the theory of Hardy spaces, the $L_{p}{}$-norm on the real line and
transported norm (8.2) are naturally intertwined, cf. [Nikolski02a]*Thm.
A.3.4.1(iii), and are used interchangeably.

The second possibility to transport a norm from $V$ to a function space on $G$
uses an contravariant transform $\mathcal{M}_{v}$:

$\left\|u\right\|_{v}:=\left\|\mathcal{M}_{v}u\right\|.$ (8.3)

######  Proposition 8.4.

  1. i.

The contravariant transform $\mathcal{M}_{v}$ is an isometry
$(L,\left\|\cdot\right\|_{v})\rightarrow(V,\left\|\cdot\right\|)$.

  2. ii.

If the composition $\mathcal{M}_{v}\circ\mathcal{W}_{F}=cI$ is a multiple of
the identity on $V$ then transported norms $\left\|\cdot\right\|_{v}$ (8.3)
and $\left\|\cdot\right\|_{F}$ (8.1) differ only by a constant multiplier.

The above result is well-known for traditional wavelets.

######  Example 8.5.

In the setup of Example 7.4, for a square integrable representation and two
admissible mother wavelets $v_{0}$ and $w_{0}$ we know that
$\mathcal{M}_{w_{0}}\mathcal{W}_{v_{0}}=kI$ (7.1), thus transported norms
(8.1) and (8.3) differ by a constant multiplier. Thus, norm (8.3) is also
provided by the integration with respect to the Haar measure on $G$.

In the theory of Hardy spaces the result is also classical.

######  Example 8.6.

For the fiducial functional $F$ with property (7.5) and the contravariant
transform $\mathcal{M}_{v^{*}_{0}}^{H}$ (5.13), Prop. 7.11 implies
$\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}=cI$. Thus, the norm
transported to $\mathrm{Aff}$ by $\mathcal{M}_{v^{*}_{0}}^{H}$ from
$L_{p}{}(\mathbb{R}{})$ up to factor coincides with (8.2). In other words, the
transition to the boundary limit on the Hardy space is an isometric operator.
This is again a classical result of the harmonic analysis, cf.
[Nikolski02a]*Thm. A.3.4.1(ii).

The co- and contravariant transforms can be used to transport norms in the
opposite direction: from a classical space of functions on $G$ to a
representation space $V$.

######  Example 8.7.

Let $V$ be the space of $\sigma$-finite signed measures of a bounded variation
on the upper half-plane. Let the $ax+b$ group acts on $V$ by the
representation adjoint to
$[{\rho_{1}}(a,b)f](x,y)=a^{-1}f(\frac{x-b}{a},\frac{y}{a})$ on
$L_{2}{}(\mathbb{R}^{2}_{+}{})$, cf. (3.2). If the mother wavelet $v_{0}$ is
the indicator function of the square $\\{0<x<1,0<y<1\\}$, then the covariant
transform of a measure $\mu$ is $\tilde{\mu}(a,b)=a^{-1}\mu(Q_{a,b})$, where
$Q_{a,b}$ is the square $\\{b<x<b+a,0<y<a\\}$. If we request that
$\tilde{\mu}(a,b)$ is a bounded function on the affine group, then $\mu$ is a
Carleson measure [Garnett07a]*§ I.5. A norm transported from
$L_{\infty}{}(\mathrm{Aff})$ to the appropriate subset of $V$ becomes the
Carleson norm of measures. Indicator function of a tent taken as a mother
wavelet will lead to an equivalent definition.

It was already mentioned in Rem. 4.9 and Example 5.9 that we may be interested
to mix several different covariant and contravariant transforms. This motivate
the following statement.

######  Proposition 8.8.

Let $(V,\left\|\cdot\right\|)$ be a normed space and ${\rho}$ be a continuous
representation of a topological locally compact group $G$ on $V$. Let two
fiducial operators $F_{1}$ and $F_{2}$ define the respective covariant
transforms $\mathcal{W}_{1}$ and $\mathcal{W}_{2}$ to the same image space
$W=\mathcal{W}_{1}V=\mathcal{W}_{2}V$. Assume, there exists an contravariant
transform $\mathcal{M}:W\rightarrow V$ such that
$\mathcal{M}\circ\mathcal{W}_{1}=c_{1}I$ and
$\mathcal{M}\circ\mathcal{W}_{2}=c_{2}I$. Define by
$\left\|\cdot\right\|_{\mathcal{M}}$ the norm on $U$ transpordef from $V$ by
$\mathcal{M}$. Then

$\left\|\mathcal{W}_{1}v_{1}+\mathcal{W}_{2}v_{2}\right\|_{\mathcal{M}}=\left\|c_{1}v_{1}+c_{2}v_{2}\right\|,\quad\text{
for any }v_{1},v_{2}\in V.$ (8.4)

###### Proof.

Indeed:

$\begin{split}\left\|\mathcal{W}_{1}v_{1}+\mathcal{W}_{2}v_{2}\right\|_{\mathcal{M}}&=\left\|\mathcal{M}\circ\mathcal{W}_{1}v_{1}+\mathcal{M}\circ\mathcal{W}_{2}v_{2}\right\|\\\
&=\left\|c_{1}v_{1}+c_{2}v_{2}\right\|,\end{split}$

by the definition of transported norm (8.3) and the assumptions
$\mathcal{M}\circ\mathcal{W}_{i}=c_{i}I$. ∎

Although the above result is simple, it does have important consequences.

######  Corollary 8.9 (Orthogonality Relation).

Let ${\rho}$ be a square integrable representation of a group $G$ in a Hilbert
space $V$. Then, for any two admissible mother wavelets $f$ and $f^{\prime}$
there exists a constant $c$ such that:

$\int_{G}\left\langle v,{\rho}(g)f\right\rangle\,\overline{\left\langle
v^{\prime},{\rho}(g)f^{\prime}\right\rangle}\,dg=c\,\left\langle
v,v^{\prime}\right\rangle\quad\text{ for any }v_{1},v_{2}\in V.$ (8.5)

Moreover, the constant $c=c(f^{\prime},f)$ is a sesquilinear form of vectors
$f^{\prime}$ and $f$.

###### Proof.

We can derive (8.5) from (8.4) as follows. Let $\mathcal{M}_{f}$ be the
inverse wavelet transform (5.5) defined by the admissible vector $f$, then
$\mathcal{M}_{f}\circ\mathcal{W}_{f}=I$ on $V$ providing the right scaling of
$f$. Furthermore, $\mathcal{M}_{f}\circ\mathcal{W}_{f^{\prime}}=\bar{c}I$ by
(7.1) for some complex constant $c$. Thus, by (8.4):

$\left\|\mathcal{W}_{f}v+\mathcal{W}_{f^{\prime}}v^{\prime}\right\|_{\mathcal{M}}=\left\|v+\bar{c}v^{\prime}\right\|.$

Now, through the polarisation identity [KirGvi82]*Problem 476 we get the
equality (8.5) of inner products. ∎

The above result is known as the _orthogonality relation_ in the theory of
wavelets, for some further properties of the constant $c$ see
[AliAntGaz00]*Thm. 8.2.1.

Here is an application of Prop. 8.8 to harmonic analysis, cf.
[Grafakos08]*Thm. 4.1.7:

######  Corollary 8.10.

The covariant transform $\mathcal{W}_{q}$ with conjugate Poisson kernel $q$
(4.8) is a bounded map from $(L_{2}{}(\mathbb{R}{}),\left\|\cdot\right\|)$ to
$(L{}(\mathrm{Aff}),\left\|\cdot\right\|_{2})$ with norm
$\left\|\cdot\right\|_{2}$ (8.2). Moreover:

$\left\|\mathcal{W}_{q}f\right\|_{2}=\left\|f\right\|,\qquad\text{ for all
}f\in L_{2}{}(\mathbb{R}{}).$

###### Proof.

As we establish in Example 7.9 for contravariant transform
$\mathcal{M}_{{v}^{+}_{0}}^{H}$ (5.12),
$\mathcal{M}_{{v}^{+}_{0}}^{H}\circ\mathcal{W}_{q}=-\mathrm{i}I$ and
$\mathrm{i}I$ on $H_{2}{}$ and $H_{2}^{\perp}{}$, respectively. Take the
unique presentation $f=u+u^{\perp}$, for $u\in H_{2}{}$ and $u^{\perp}\in
H_{2}^{\perp}{}$. Then, by (8.4)

$\left\|\mathcal{W}_{q}f\right\|_{2}=\left\|-\mathrm{i}u+\mathrm{i}u^{\perp}\right\|=\left\|u+u^{\perp}\right\|=\left\|f\right\|.$

This completes the proof. ∎

##  9 Conclusion

We demonstrated that both, real and complex, techniques in harmonic analysis
have the same group-theoretical origin. Moreover, they are complemented by the
wavelet construction. Therefore, there is no any confrontation between these
approaches and they can be lined up as in Table 1. In other words, the binary
opposition of the real and complex methods resolves via Kant’s triad thesis-
antithesis-synthesis: complex-real-covariant.

## Acknowledgements

I am grateful to A. Albargi for careful reading of the paper and useful
comments. Prof. D. Yakubovich pointed out some recent publications in the
field. The anonymous referee made many useful suggestions which were
incorporated into the paper with gratitude.

Covariant scheme |  Complex variable |  Real variable
---|---|---
Covariant transform is $\mathcal{W}_{F}^{{\rho}}:v\mapsto\hat{v}(g)=F({\rho}(g^{-1})v)$. In particular, the wavelet transform for the mother wavelet $v_{0}$ is $\tilde{v}(g)=\left\langle v,{\rho}(g)v_{0}\right\rangle$. |  The Cauchy integral is generated by the mother wavelet $\frac{1}{2\pi\mathrm{i}}\frac{1}{x+\mathrm{i}}$. The Poisson integral is generated by the mother wavelet $\frac{1}{\pi}\frac{1}{x^{2}+1}$ |  The averaging operator $\tilde{f}(b)=\frac{1}{2a}\int\limits_{b-a}^{b+a}f(t)\,dt$ is defined by the mother wavelet $\chi_{[-1,1]}(t)$, to average the modulus of $f(t)$ we use all elements of the unit ball in $L_{\infty}{}[-1,1]$.
The covariant transform maps vectors to functions on $G$ or, in the induced case, to functions on the homogeneous space $G/H$. |  Functions are mapped from the real line to the upper half-plane parametrised by either the $ax+b$-group or the homogeneous space $\mathrm{SL}_{2}(\mathbb{R}{})/K$. |  Functions are mapped from the real line to the upper half-plane parametrised by either the $ax+b$-group or the homogeneous space $\mathrm{SL}_{2}(\mathbb{R}{})/A$.
Annihilating action on the mother wavelet produces functional relation on the image of the covariant transform |  The operator $-d{\rho^{\mathsf{A}}}-\mathrm{i}d{\rho^{\mathsf{N}}}=I+(x+\mathrm{i})\frac{d}{dx}$ annihilates the mother wavelet $\frac{1}{2\pi\mathrm{i}}\frac{1}{x+\mathrm{i}}$, thus the image of wavelet transform is in the kernel of the Cauchy–Riemann operator $-\mathfrak{L}^{\mathsf{A}}+\mathrm{i}\mathfrak{L}^{\mathsf{N}}=\mathrm{i}a(\partial_{b}+\mathrm{i}\partial_{a})$. Similarly, for the Laplace operator. |  The mother wavelet $v_{0}=\chi_{[-1,1]}$ satisfies the equality $\chi_{[-1,1]}=\chi_{[-1,0]}+\chi_{[0,1]}$, where both terms are again scaled and shifted $v_{0}$. The image of the wavelet transform is suitable for the stopping time argument and the dyadic squares technique.
An invariant pairing $\left\langle\cdot,\cdot\right\rangle$ generates the contravariant transform $[\mathcal{M}_{w_{0}}^{{\rho}}f]\left\langle f(g),{\rho}(g)w_{0}\right\rangle$ for |  The contravariant transform with the invariant Hardy pairing on the $ax+b$ group produces boundary values of functions on the real line. |  The covariant transform with the invariant $\sup$ pairing produces the vertical and non-tangential maximal functions.
The composition $\mathcal{M}_{v}\circ\mathcal{W}_{F}$ of the covariant and contravariant transforms is a multiple of the identity on irreducible components. |  SIO is a composition of the Cauchy integral and its boundary value. |  The Hardy–Littlewood maximal function is the composition of the averaging operator and the contravariant transform from the invariant $\sup$ pairing.
The Hardy space is an invariant subspace of the group representation. |  The Hardy space consists of the limiting values of the Cauchy integral. SIO is bounded on this space. |  The Hardy–Littlewood maximal operator is bounded on the Hardy space $H_{p}{}$ .
Table 1: The correspondence between different elements of harmonic analysis.

## References

Vladimir V. Kisil

School of Mathematics

University of Leeds

Leeds LS2 9JT

UK

E-mail<EMAIL_ADDRESS>
Web: http://www.maths.leeds.ac.uk/~kisilv/

Received: 12.09.2013

## Index

  * Abel summation item Integral transforms.
  * admissible
    * mother wavelet Example 8.2
  * admissible wavelet §4, Example 5.3
  * affine
    * group, see $ax+b$ group
  * approximation
    * identity, of the Remark 3.1
  * atom §5
  * atomic
    * decomposition Example 5.9
  * $ax+b$ group §3, Example 4.7, Example 5.4, Example 6.9
    * invariant measure §3
    * left regular representation §3
    * representation
      * co-adjoint §3
      * quasi-regular §3
  * calculus
    * functional
      * covariant §7
  * Calderón–Zygmund
    * decomposition Remark 7.13
  * Carleson
    * measure Example 8.7
    * norm Example 8.7
  * Cauchy
    * integral item Integral transforms., Example 4.6, Table 1
  * Cauchy–Riemann operator Example 6.9, Table 1
  * co-adjoint
    * representation of $ax+b$ group §3
  * complex numbers Remark 4.10
  * continuous
    * wavelet transform Example 4.5
  * contravariant
    * transform Definition 5.2, Table 1
  * convolution
    * relative §6
  * covariant
    * functional calculus §7
    * transform Definition 4.1, Table 1
      * inverse, see contravariant transform
  * decomposition
    * atomic Example 5.9
    * Calderón–Zygmund Remark 7.13
  * double numbers Remark 4.10
  * dyadic
    * squares item Littlewood–Paley theory, Example 6.11, Table 1
  * fiducial operator Definition 4.1
  * formula
    * reconstruction Example 5.3
    * Sokhotsky–Plemelj item Boundary values and SIO.
  * functional
    * calculus
      * covariant §7
    * invariant item iii
  * Gaussian Example 4.6
  * grand maximal function Remark 4.9
  * group
    * affine, see $ax+b$ group
    * $ax+b$ §3, Example 4.7, Example 5.4, Example 6.9
      * invariant measure §3
    * $\mathrm{SL}_{2}(\mathbb{R}{})$ Remark 1.1, Remark 3.5, Remark 4.10, Table 1
  * Haar measure, see invariant measure
  * Hardy
    * pairing §5
    * space item Hardy space., Example 3.3, Example 4.6, §7
      * generalized Definition 3.2, §7, §7
  * Hardy–Littlewood
    * maximal functions item Hardy–Littlewood maximal function, Example 4.7, Example 7.6, Table 1
  * Hilbert
    * transform Example 7.7, Example 7.9
  * hypercomplex numbers Remark 3.5
  * identity
    * approximation of the Remark 3.1
  * integral
    * Cauchy item Integral transforms., Example 4.6, Table 1
    * Poisson, see Poisson kernel
  * intertwining operator Theorem 6.1, Proposition 6.7, Example 7.6
  * invariant §5
    * functional item iii
    * measure §3, §4, Example 5.3, Example 5.4
    * pairing §5
    * subspace Remark 7.3
  * inverse
    * covariant transform, see contravariant transform
  * irreducible
    * representation Definition 3.2, §7
  * kernel
    * Poisson item Integral transforms., Example 4.6, Table 1
      * conjugated Example 4.6, Example 7.9
  * Laplace
    * operator Example 6.10
  * left regular representation
    * $ax+b$ group §3
  * Littlewood–Paley theory item Littlewood–Paley theory
  * maximal function
    * grand Remark 4.9
  * maximal functions Remark 3.1
    * Hardy–Littlewood item Hardy–Littlewood maximal function, Example 4.7, Example 7.6, Table 1
    * non-tangential Example 5.5, Table 1
    * vertical Example 5.5, Table 1
  * measure
    * Carleson Example 8.7
    * Haar, see invariant measure
    * invariant §3, §4, Example 5.3, Example 5.4
  * method
    * orbit §3
  * Mexican hat
    * wavelet Example 4.5
  * mother wavelet Example 4.4
    * admissible Example 8.2
  * non-tangential
    * maximal functions Example 5.5, Table 1
  * norm
    * Carleson Example 8.7
  * nucleus Definition 5.8
  * numbers
    * complex Remark 4.10
    * double Remark 4.10
    * hypercomplex Remark 3.5
  * operator
    * Cauchy–Riemann Example 6.9, Table 1
    * fiducial Definition 4.1
    * integral
      * singular item Boundary values and SIO., Example 7.7, Table 1
    * intertwining Theorem 6.1, Proposition 6.7, Example 7.6
    * Laplace Example 6.10
  * orbit
    * method §3
  * orthogonality
    * relation §8
  * pairing
    * Hardy §5
    * invariant §5
  * Poisson kernel item Integral transforms., Example 4.6, Table 1
    * conjugated Example 4.6, Example 7.9
  * primary
    * representation Definition 3.2
  * principal
    * value item Boundary values and SIO., Example 7.7
  * quasi-regular
    * representation of $ax+b$ group §3
  * reconstruction formula Example 5.3
  * relation
    * orthogonality §8
  * relative
    * convolution §6
  * representation
    * $ax+b$ group
      * co-adjoint §3
      * quasi-regular §3
    * coefficients, see wavelet transform
    * irreducible Definition 3.2, §7
    * left regular
      * $ax+b$ group §3
    * primary Definition 3.2
    * square integrable §4, Example 5.3, Example 8.2
  * singular
    * integral operator item Boundary values and SIO., Example 7.7, Table 1
  * seesingular integral operator Example 7.7
  * $\mathrm{SL}_{2}(\mathbb{R}{})$ Remark 1.1, Remark 3.5, Remark 4.10, Table 1
  * Sokhotsky–Plemelj formula item Boundary values and SIO.
  * space
    * Hardy item Hardy space., Example 3.3, Example 4.6, §7
      * generalized Definition 3.2, §7, §7
  * square integrable
    * representation §4, Example 5.3, Example 8.2
  * squares
    * dyadic item Littlewood–Paley theory, Example 6.11, Table 1
  * state
    * vacuum, see mother wavelet
  * stopping time argument Example 6.11, Table 1
  * subspace
    * invariant Remark 7.3
  * summation
    * Abel item Integral transforms.
  * tent item Domain, Remark 4.10, Example 8.7
  * theory
    * Littlewood–Paley item Littlewood–Paley theory
  * transform
    * contravariant Definition 5.2, Table 1
    * covariant Definition 4.1, Table 1
      * inverse, see contravariant transform
    * Hilbert Example 7.7, Example 7.9
    * wavelet Example 4.4, Table 1
      * continuous Example 4.5
  * vacuum state, see mother wavelet
  * value
    * principal item Boundary values and SIO., Example 7.7
  * vertical
    * maximal functions Example 5.5, Table 1
  * wavelet §4, Example 4.4
    * admissible §4, Example 5.3
    * Mexican hat Example 4.5
    * mother Example 4.4
      * admissible Example 8.2
    * transform Example 4.4, Table 1
      * continuous Example 4.5