# Functional calculi for sectorial operators and related function theory

Charles Batty  St. John’s College
University of Oxford
Oxford OX1 3JP, UK<EMAIL_ADDRESS>,  Alexander Gomilko  Faculty
of Mathematics and Computer Science
Nicolas Copernicus University
Chopin Street 12/18
87-100 Toruń, Poland<EMAIL_ADDRESS>and  Yuri Tomilov  Institute of
Mathematics
Polish Academy of Sciences
Śniadeckich 8
00-656 Warsaw, Poland<EMAIL_ADDRESS>
###### Abstract.

We construct two bounded functional calculi for sectorial operators on Banach
spaces, which enhance the functional calculus for analytic Besov functions, by
extending the class of functions, generalizing and sharpening estimates, and
adapting the calculus to the angle of sectoriality. The calculi are based on
appropriate reproducing formulas, they are compatible with standard functional
calculi and they admit appropriate convergence lemmas and spectral mapping
theorems. To achieve this, we develop the theory of associated function spaces
in ways which are interesting and significant. As consequences of our calculi,
we derive several well-known operator norm-estimates and provide
generalizations of some of them.

###### Key words and phrases:

Functional calculus, sectorial operator, bounded holomorphic semigroup,
Hardy–Sobolev space

###### 2010 Mathematics Subject Classification:

Primary 47A60, Secondary 30H25 46E15 47D03

This work was partially supported financially by a Leverhulme Trust Visiting
Research Professorship and an NCN grant UMO-2017/27/B/ST1/00078, and
inspirationally by the ambience of the Lamb & Flag, Oxford.

Dedicated to Vladimir Müller on the occasion of his 70th birthday

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 The Banach spaces $\mathcal{D}_{s}$ and their reproducing formulas
  4. 4 Hardy-Sobolev algebras on sectors
  5. 5 Dense sets in $\mathcal{D}_{s}$ and $\mathcal{H}_{\psi}$
  6. 6 Convergence Lemmas
  7. 7 The $\mathcal{D}$-calculus and its compatibility
  8. 8 The calculus on Hardy-Sobolev algebras
  9. 9 Convergence Lemmas and Spectral Mapping Theorems
  10. 10 Some applications to norm-estimates
  11. 11 Appendix: Shifts on $\mathcal{D}_{s}$ and $\mathcal{H}_{\psi}$
  12. 12 Appendix: The $\mathcal{D}$-calculus vs the HP- and the $\mathcal{B}$-calculi

##  1\. Introduction

The theory of functional calculi forms a basis for the study of sectorial
operators and semigroup generators. In particular, there were two functional
calculi used extensively in the research on operator semigroups and sectorial
operators through the past fifty years. One of them, the Hille-Phillips (HP)
functional calculus for semigroup generators, probably stemmed from the
foundational monograph [36], and it became an indispensable part of semigroup
theory. The systematic approach to the other one, the holomorphic functional
calculus for sectoral operators, was initiated by McIntosh and his
collaborators in the 1980s. While the two calculi appeared to be very useful
in applications, the operator norm-estimates within them are often
problematic. The estimates within the HP-calculus are direct but rather crude,
and the task of getting bounds within the holomorphic functional calculus is a
priori cumbersome since the calculus is not, in general, a bounded Banach
algebra homomorphism.

To circumvent those problems, a number of additional tools and methods
appeared in the literature. In particular, the advanced notions and techniques
related to bounded $H^{\infty}$-calculus, $R$-boundedness, Fourier multipliers
and transference were developed in depth, and one may consult [37], [40] and
[34] for many of these function-theoretical developments. Moreover, various
implications of positivity of functions and their derivatives (completely
monotone, Bernstein, $\mathcal{NP}_{+}$-functions) were adjusted to the
operator-theoretical set-up. For clarification of the role of positivity, see
[50], [30] and [5], for example.

Recently, in [6], a new functional calculus was constructed, the so-called
$\mathcal{B}$-calculus. First of all, the $\mathcal{B}$-calculus offers a
simple and efficient route to operator norm-estimates for functions of
semigroup generators, thus unifying a number of estimates in the literature
and leading to new ones. No supplementary arguments are required and the
estimates underline the strength of the calculi. Moreover, the
$\mathcal{B}$-calculus possesses all attributes of classical functional
calculi, see [6]. When combined properly they lead to new spectral mapping
theorems and generalizations of fundamentals of semigroup theory, see [7]. To
put our results into a proper context and to use some of the
$\mathcal{B}$-calculus properties in the sequel, we briefly recall the set-up
for the $\mathcal{B}$-calculus, see [6] for more details.

Let $\mathcal{B}$ be the algebra of holomorphic functions on the right half-
plane $\mathbb{C}_{+}$ such that

(1.1)
$\|f\|_{\mathcal{B}_{0}}:=\int_{0}^{\infty}\sup_{\beta\in\mathbb{R}}|f^{\prime}(\alpha+i\beta)|\,d\alpha<\infty.$

These functions are sometimes called analytic Besov functions, and they have
been considered in some detail in [6]. . In particular, every
$f\in\mathcal{B}$ belongs to $H^{\infty}(\mathbb{C}_{+})\cap
C(\overline{\mathbb{C}}_{+})$, and $\mathcal{B}$ is a Banach algebra with the
norm

(1.2) $\|f\|_{\mathcal{B}}:=\|f\|_{\infty}+\|f\|_{\mathcal{B}_{0}}.$

In the setting of power bounded operators on Hilbert spaces, the unit disc
counterpart of $\mathcal{B}$ was employed for the study of functional calculi
in [46].

Let $A$ be a densely defined closed operator on a Banach space $X$ such that
$\sigma(A)\subseteq\overline{\mathbb{C}}_{+}$, and

(1.3)
$\sup_{\alpha>0}\alpha\int_{\mathbb{R}}|\langle(\alpha+i\beta+A)^{-2}x,x^{*}\rangle|\,d\beta<\infty$

for all $x\in X$ and $x^{*}\in X^{*}$. This class of operators includes two
substantial subclasses, namely the (negative) generators of bounded
$C_{0}$-semigroups on Hilbert spaces $X$ and the generators of (sectorially)
bounded holomorphic $C_{0}$-semigroups on Banach spaces $X$. On the other
hand, every operator in the class is the negative generator of a bounded
$C_{0}$-semigroup.

Some results involving functional calculus of operators in $\mathcal{B}$ were
obtained in [35], [51], [52] and [55], but each of them applied only to one of
the two subclasses of operators, and only [51] and [52] considered all
functions in $\mathcal{B}$ applied to generators of bounded holomorphic
semigroups. In [6] we introduced a bounded functional calculus for all
operators satisfying (1.3), and we extended the theory in [7]. For
$f\in\mathcal{B}$, set

(1.4) $\displaystyle\langle f(A)x,x^{*}\rangle$
$\displaystyle=f(\infty)\langle
x,x^{*}\rangle-\frac{2}{\pi}\int_{0}^{\infty}\alpha\int_{\mathbb{R}}{f^{\prime}(\alpha+i\beta)}\langle(\alpha-i\beta+A)^{-2}x,x^{*}\rangle\,d\beta\,d\alpha$

for all $x\in X$ and $x^{*}\in X^{*}$, where
$f(\infty)=\lim_{\operatorname{Re}z\to\infty}f(z)$. Using (1.3) and the
definition of $\mathcal{B}$ (and the Closed Graph Theorem), it is easy to show
that $f(A)$ is a bounded linear mapping from $X$ to $X^{**}$, and that the
linear mapping

$\widetilde{\Phi}_{A}:\mathcal{B}\to\mathcal{L}(X,X^{**}),\qquad f\mapsto
f(A),$

is bounded.

It was discovered in [6] that much more is true. If $A$ belongs to any of the
classes of semigroup generators mentioned above, then the formula (1.4)
defines a bounded algebra homomorphism

$\Phi_{A}:\mathcal{B}\to L(X),\qquad\Phi_{A}(f):=f(A).$

It is natural to call the homomorphism $\Phi_{A}$ the ($\mathcal{B}$-)calculus
of $A$. It was proved in [6] that $\Phi_{A}$ possesses a number of useful
properties. In particular, it admits the spectral inclusion (spectral mapping,
in the case of bounded holomorphic semigroups) theorem and a Convergence Lemma
of appropriate form. The utility of the $\mathcal{B}$-calculus depends on the
facts that it (strictly) extends the Hille-Phillips (HP-) calculus and it is
compatible with the holomorphic functional calculi for sectorial and half-
plane type operators.

Moreover, the $\mathcal{B}$-calculus $\Phi_{A}$ is the only functional
calculus that one can define for $A$ satisfying (1.3) and for functions in
$\mathcal{B}$. Indeed, let $A$ be an operator on $X$ with dense domain, and
assume that $\sigma(A)\subseteq\overline{\mathbb{C}}_{+}$. A _(bounded)
$\mathcal{B}$-calculus_ for $A$ is, by definition, a bounded algebra
homomorphism $\Phi:\mathcal{B}\to L(X)$ such that
$\Phi((z+\cdot)^{-1})=(z+A)^{-1}$ for all $z\in\mathbb{C}_{+}$. As shown in
[7], if $A$ admits a $\mathcal{B}$-calculus, then the resolvent assumption
(1.3) holds, and the calculus is $\Phi_{A}$.

While the $\mathcal{B}$-calculus is optimal and unique for generators of
Hilbert space semigroups, the situation is far from being so for generators of
bounded holomorphic semigroups on Banach spaces (as this paper will, in
particular, show). Thus using the $\mathcal{B}$-calculus ideology as a guiding
principle, it is natural to try to extend it beyond the Besov algebra
$\mathcal{B}$ keeping all of its useful properties such as availability of
good norm-estimates, spectral mapping theorems, Convergence Lemmas,
compatibility with the other calculi, etc. Moreover, it is desirable to cover
all sectorial operators regardless their sectoriality angle.

In this paper we will construct some functional calculi encompassing wider
classes of functions (including some with singularities on $i\mathbb{R}$) and
providing finer estimates for all negative generators of (sectorially) bounded
holomorphic semigroups, and eventually for all sectorial operators. Functional
calculi for generators of some classes of bounded holomorphic semigroups were
constructed in [24], [25], [26], [27], and [39]. However, most of the results
in those papers concern sectorial operators of angle zero, and the approaches
there are based on fine estimates for the corresponding semigroups rather than
fine analytic properties of resolvents.

Let $A$ be a densely defined sectorial operator of sectorial angle
$\theta_{A}\in[0,\pi)$ on a Banach space $X$. It is well known that $-A$ is
the generator of a (sectorially) bounded holomorphic $C_{0}$-semigroup on $X$
if and only if and only if $A$ is sectorial and $\theta_{A}<\pi/2$ (we may
write $A\in\operatorname{Sect}(\pi/2)$ for this class). In this paper we
address the question whether the $\mathcal{B}$-calculus for $A$ can be
extended to more functions. Since the resolvent of $A$ satisfies the estimate

$M_{\psi}(A):=\sup_{z\in\Sigma_{\pi-\psi}}\|z(z+A)^{-1}\|<\infty,\qquad
z\in\Sigma_{\pi-\psi}.$

for all $\psi\in(\theta_{A},\pi)$, a direct way to define an appropriate
function algebra would be to introduce a Banach space of functions $f$ which
are holomorphic on sectors
$\Sigma_{\psi}:=\\{z\in\mathbb{C}:|\arg(z)|<\psi\\}$ such that

(1.5)
$\|f\|_{\psi}:=\int_{\partial\Sigma_{\psi}}\frac{|f(z)|}{|z|}|dz|<\infty.$

In order to apply this to all $A\in\operatorname{Sect}(\pi/2-)$, $f$ should be
holomorphic on $\mathbb{C}_{+}$ and the assumption (1.5) should hold for all
$\psi\in(0,\pi/2)$, and in order to provide an estimate which is uniform in
$\theta$ it is desirable to have $\sup_{\psi\in(0,\pi/2)}\|f\|_{\psi}<\infty$.
To our knowledge, no spaces of this type have been studied systematically in
the literature, although they appear naturally in [37, Appendix H2 and Chapter
10.2], [33, Section 6] or [32, Appendix C]. This class of functions is
strictly included in each of the spaces $\mathcal{D}_{s},s>0$ (see Proposition
4.15 and a discussion following it), which we now define.

To define a functional calculus for all $A\in\operatorname{Sect}(\pi/2-)$, we
let $\mathcal{D}_{s},s>-1$, be the linear space of all holomorphic functions
$f$ on $\mathbb{C}_{+}$ such that

(1.6)
$\|f\|_{\mathcal{D}_{s,0}}:=\int_{0}^{\infty}\alpha^{s}\int_{-\infty}^{\infty}\frac{|f^{\prime}(\alpha+i\beta)|}{(\alpha^{2}+\beta^{2})^{(s+1)/2}}\,d\beta\,d\alpha<\infty.$

If $f\in\mathcal{D}_{s}$, then there exists a finite limit
$f(\infty):=\lim_{|z|\to\infty,\,z\in{\Sigma}_{\psi}}\,f(z)$ for all
$\psi\in(0.\pi/2)$.

For every $s>-1$ the space $\mathcal{D}_{s}$ equipped with the norm

$\|f\|_{\mathcal{D}_{s}}:=|f(\infty)|+\|f\|_{\mathcal{D}_{s,0}},\qquad
f\in\mathcal{D}_{s},$

is a Banach space, but not an algebra. However, the spaces $\mathcal{D}_{s}$
increase with $s$, and we prove in Lemma 3.20 that

$\mathcal{D}_{\infty}:=\bigcup_{s>-1}\mathcal{D}_{s}$

is an algebra.

Let $f\in\mathcal{D}_{\infty}$, so $f\in\mathcal{D}_{s}$ for some $s>-1$, and
let $A$ be sectorial with $\theta_{A}<\pi/2$. Define

(1.7)
$f_{\mathcal{D}_{s}}(A):=f(\infty)-\frac{2^{s}}{\pi}\int_{0}^{\infty}\alpha^{s}\int_{-\infty}^{\infty}f^{\prime}(\alpha+i\beta)(A+\alpha-i\beta)^{-(s+1)}\,d\beta\,d\alpha.$

Then $f_{\mathcal{D}_{\sigma}}(A)=f_{\mathcal{D}_{s}}(A)$ whenever $\sigma>s$.
The following result sets out other properties of this functional calculus.
The proof will be given in Section 7.

######  Theorem 1.1.

Let A be a densely defined, closed operator on a Banach space $X$ such that
$\sigma(A)\subset\overline{\mathbb{C}}_{+}$. The following are equivalent:

  1. (i)

$A\in{\rm Sect}(\pi/2-)$,

  2. (ii)

There is an algebra homomorphism $\Psi_{A}:\mathcal{D}_{\infty}\to L(X)$ such
that

$\Psi_{A}((z+\cdot)^{-1})=(z+A)^{-1},\qquad z\in\mathbb{C}_{+},$

and $\Psi_{A}$ is bounded in the sense that there exist constants
$C_{s},\,s>-1$, such that, for every $f\in\mathcal{D}_{s}$,

(1.8) $\|\Psi_{A}(f)\|\leq C_{s}\|f\|_{\mathcal{D}_{s}}.$

When these properties hold, $\Psi_{A}$ is unique, and it is defined by the
formula (1.7):

$\displaystyle\Psi_{A}:\mathcal{D}_{\infty}\mapsto
L(X),\qquad\Psi_{A}(f)=f_{\mathcal{D}_{s}}(A),\quad f\in\mathcal{D}_{s}.$

The homomorphism $\Psi_{A}$ will be called the $\mathcal{D}$-calculus for $A$.
It will be shown in Section 7 that the $\mathcal{D}$-calculus is compatible
with the Hille-Phillips calculus and the holomorphic calculus for sectorial
operators, and a spectral mapping theorem is given in Theorem 9.3. Corollary
7.8 provides a version of this functional calculus based on the Banach algebra
$H^{\infty}(\mathbb{C}_{+})\cap\mathcal{D}_{s}$ for a fixed value of $s$.

The $\mathcal{D}$-calculus defined as above does not take into account the
sectoriality angle of $A\in{\rm Sect}(\pi/2-)$. However, it can be used to
construct a functional calculus which does not have this drawback. To achieve
this aim we introduce the Hardy-Sobolev spaces $\mathcal{H}_{\psi}$, on
sectors $\Sigma_{\psi}$. First, for any $\psi\in(0,\pi)$, we define the Hardy
space $H^{1}(\Sigma_{\psi})$ as the linear space of functions
$f\in\operatorname{Hol}(\Sigma_{\psi})$ such that

(1.9)
$\|f\|_{H^{1}(\Sigma_{\psi})}:=\sup_{|\varphi|<\psi}\,\int_{0}^{\infty}\bigl{(}|f(te^{i\varphi})|+|f(te^{-i\varphi})|\bigr{)}\,dt<\infty.$

Note that $H^{1}(\Sigma_{\pi/2})$ coincides with the classical Hardy space
$H^{1}(\mathbb{C}_{+})$ in the right half-plane $\mathbb{C}_{+}$. It is well-
known that $(H^{1}(\Sigma_{\psi}),\|\cdot\|_{H^{1}(\Sigma_{\psi})})$ is a
Banach space, and every $f\in H^{1}(\Sigma_{\psi})$ has a boundary function on
$\partial\Sigma_{\psi}$. The boundary function exists as the limit of $f$ in
both an $L^{1}$-sense and a pointwise (a.e.) sense. Moreover, the norm of $f$
in $H^{1}(\Sigma_{\psi})$ is attained by the $L^{1}$-norm of its boundary
function. See Section 4.1 for a succinct approach to the Hardy spaces on
sectors.

The space $H^{1}(\Sigma_{\psi})$ induces the corresponding Hardy-Sobolev space
$\mathcal{H}_{\psi}$ on $\Sigma_{\psi}$ as

$\mathcal{H}_{\psi}:=\left\\{f\in\operatorname{Hol}(\Sigma_{\psi}):f^{\prime}\in
H^{1}(\Sigma_{\psi})\right\\}.$

Any function $f\in\mathcal{H}_{\psi}$ has a finite limit
$f(\infty):=\lim_{t\to\infty}f(t)$, and moreover $f\in
H^{\infty}(\Sigma_{\psi})$. Then $\mathcal{H}_{\psi}$ becomes a Banach algebra
in the norm

$\|f\|_{\mathcal{\mathcal{H}}_{\psi}}:=\|f\|_{H^{\infty}(\Sigma_{\psi})}+\|f^{\prime}\|_{H^{1}(\Sigma_{\psi})},\quad
f\in\mathcal{H}_{\psi}.$

The relationship between these spaces and the spaces $\mathcal{D}_{s}$ for all
$s>-1$ is set out in Corollary 4.3 and Lemma 4.13; in particular, for each
$s>-1$, $\mathcal{H}_{\pi/2}$ is contained in $\mathcal{D}_{s}$, and
$\mathcal{D}_{s}$ is embedded in $\mathcal{H}_{\psi}$ for $\psi<\pi/2$.

To make use of the angle of sectoriality of $A$, we can adjust the
$\mathcal{D}$-calculus to sectors as follows. If $f\in\mathcal{H}_{\psi}$
where $\psi\in(\theta_{A},\pi)$, $\gamma=\pi/(2\psi)$, and
$f_{1/\gamma}(z):=f(z^{1/\gamma})$, then $f^{\prime}_{1/\gamma}\in
H^{1}(\mathbb{C}_{+})$ and $f_{1/\gamma}(\infty)=f(\infty)$, and hence
$f_{1/\gamma}\in\mathcal{D}_{0}$. This observation allows us to extend the
$\mathcal{D}$-calculus to the class of all sectorial operators, and makes the
next definition (based on the $\mathcal{D}$-calculus) natural and plausible.

If $A$ is sectorial and $\psi\in(\theta_{A},\pi)$, define

(1.10)
$f_{\mathcal{H}}(A):=f(\infty)-\frac{1}{\pi}\int_{0}^{\infty}\int_{-\infty}^{\infty}f^{\prime}_{1/\gamma}(\alpha+i\beta)(A^{\gamma}+\alpha-i\beta)^{-1}\,d\beta\,d\alpha.$

One can prove (see (8.2)) that

(1.11)
$\|f_{\mathcal{H}}(A)\|\leq|f(\infty)|+\frac{M_{\pi/2}{(A^{\gamma})}}{\pi}\|f_{1/\gamma}\|_{\mathcal{D}_{0}}\leq|f(\infty)|+M_{\pi/2}({A^{\gamma}})\|f\|_{\mathcal{H}_{\psi}}.$

Then (1.10) and (1.11) hold for any $\gamma\in(1,\pi/(2\theta))$, and the
definition of $f_{\mathcal{H}}(A)$ does not depend on the choice of $\psi$.

Now we are able to formalize our extension of the $\mathcal{D}$-calculus as
follows.

######  Theorem 1.2.

Let $A$ be a densely defined operator on a Banach space $X$ such that
$\sigma(A)\subset\overline{\Sigma_{\theta}}$, where $\theta\in[0,\pi)$. The
following are equivalent:

  1. (i)

$A\in\operatorname{Sect}(\theta)$;

  2. (ii)

For each $\psi\in(\theta,\pi)$, there is a bounded Banach algebra homomorphism
$\Upsilon_{A}:\mathcal{\mathcal{H}}_{\psi}\mapsto L(X)$ such that
$\Upsilon_{A}((z+\cdot)^{-1})=(z+A)^{-1},\quad z\in\Sigma_{\pi-\psi}$.

When these properties hold, the homomorphism $\Upsilon_{A}$ is unique for each
value of $\psi$, and it is defined by the formula (1.10):

$\Upsilon_{A}:\mathcal{H}_{\psi}\to
L(X),\qquad\Upsilon_{A}(f)=f_{\mathcal{H}}(A),\quad f\in\mathcal{H}_{\psi}.$

The homomorphism $\Upsilon_{A}$ will be called the $\mathcal{H}$-calculus for
$A$.

The $\mathcal{D}$-calculus can be given a more succinct form, by replacing
(1.10) with the somewhat more transparent formula (1.12) below.

######  Theorem 1.3.

Let $A\in\operatorname{Sect}(\theta)$, $\theta<\psi<\pi$, and
$\gamma=\pi/(2\psi)$. For $f\in\mathcal{H}_{\psi}$, let

$f_{\psi}(s):=\frac{f(e^{i\psi}t)+f(e^{-i\psi}t)}{2},\quad t\geq 0.$

Then

(1.12)
$f_{\mathcal{H}}(A)=f(\infty)-\frac{2}{\pi}\int_{0}^{\infty}f_{\psi}^{\prime}(t){\operatorname{arccot}}(A^{\gamma}/t^{\gamma})\,dt$

where the integral converges in the uniform operator topology, and

(1.13)
$\|f_{\mathcal{H}}(A)\|\leq|f(\infty)|+M_{\psi}(A)\|f_{\psi}^{\prime}\|_{L^{1}(\mathbb{R}_{+})}\leq
M_{\psi}(A)\|f\|_{\mathcal{H}_{\psi}}.$

Thus $\|\Upsilon_{A}\|\leq M_{\psi}(A)$.

See Section 8.2 for details. The $\mathcal{D}$-calculus and the
$\mathcal{H}$-calculus possess natural properties of functional calculi such
as spectral mapping theorems and Convergence Lemmas. These properties are
studied in Section 9.

The strength of the constructed calculi is illustrated by several examples
showing that they lead to sharper estimates than those offered by other
calculi (see Section 12 for one example). Moreover, the theory developed in
this paper is successfully tested by deriving several significant estimates
for functions of sectorial operators from the literature. In particular, in
Section 10, we provide a proof of permanence of the class of sectorial
operators under subordination and we revisit a few basic results from
semigroup theory.

In developing the $\mathcal{D}$\- and $\mathcal{H}$-calculi we prove a number
of results of independent interest in function theory. Apart from the theory
of the spaces $\mathcal{D}_{s}$ and $\mathcal{H}_{\psi}$, their reproducing
formulas, and boundedness of the associated operators elaborated in this
paper, we emphasize the property (4.9) in Corollary 4.3 yielding isometric
coincidence of spaces of Hardy type, Theorem 4.12 on Laplace representability
of Hardy-Sobolev functions, and Theorem 5.10 on the density of rational
functions in Hardy-Sobolev spaces.

##  2\. Preliminaries

### Notation

Throughout the paper, we shall use the following notation:

  1. $\mathbb{R}_{+}:=[0,\infty)$,

  2. $\mathbb{C}_{+}:=\\{z\in\mathbb{C}:\operatorname{Re}z>0\\}$, $\overline{\mathbb{C}}_{+}=\\{z\in\mathbb{C}:\operatorname{Re}z\geq 0\\}$,

  3. $\Sigma_{\theta}:=\\{z\in\mathbb{C}:z\neq 0,|\arg z|<\theta\\}$ for $\theta\in(0,\pi)$.

  4.

For $f:\mathbb{C}_{+}\to\mathbb{C}$, we say that $f$ has a _sectorial limit at
infinity_ if

$\lim_{|z|\to\infty,z\in\Sigma_{\psi}}f(z)$

exists for every $\psi\in(0,\pi/2)$. Similarly, $f$ has a _sectorial limit at
$0$_ if

$\lim_{|z|\to 0,z\in\Sigma_{\psi}}f(z)$

exists for every $\psi\in(0,\pi/2)$. We say that $f$ has a _half-plane limit
at infinity_ if

$\lim_{\operatorname{Re}z\to\infty}f(z)$

exists in $\mathbb{C}$. We say that $f$ has a _full limit at infinity_ or _at
zero_ if

$\lim_{|z|\to\infty,z\in\mathbb{C}_{+}}f(z)\quad\text{or}\quad\lim_{|z|\to
0,z\in\mathbb{C}_{+}}f(z)$

exists in $\mathbb{C}$. The notation $f(\infty)$ and $f(0)$ may denote a
sectorial limit, a half-plane limit, or a full limit, according to context.

For $a\in\overline{\mathbb{C}}_{+}$, we define functions on $\mathbb{C}$ by

$e_{a}(z)=e^{-az};\qquad r_{a}(z)=(z+a)^{-1},\,z\neq-a.$

We use the following notation for spaces of functions or measures, and
transforms, on $\mathbb{R}$ or $\mathbb{R}_{+}$:

  1. $\operatorname{Hol}(\Omega)$ denotes the space of holomorphic functions on an open subset $\Omega$ of $\mathbb{C}$, $H^{\infty}(\Omega)$ is the space of bounded homorphic functions on $\Omega$, and $\|f\|_{H^{\infty}(\Omega)}=\sup_{\Omega}|f(z)|$.

  2. $H^{p}(\mathbb{C}_{+}),\,1\leq p\leq\infty$, are the standard Hardy spaces on the (right) half-plane.

  3. $M(\mathbb{R}_{+})$ denotes the Banach algebra of all bounded Borel measures on $\mathbb{R}_{+}$ under convolution. We identify $L^{1}(\mathbb{R}_{+})$ with a subalgebra of $M(\mathbb{R}_{+})$ in the usual way. We write $\mathcal{L}\mu$ for the Laplace transform of $\mu\in M(\mathbb{R}_{+})$.

  4. $\mathcal{LM}$ is the Hille-Phillips algebra, $\mathcal{LM}:=\\{\mathcal{L}\mu:\mu\in M(\mathbb{R}_{+})\\}$, with the norm $\|\mathcal{L}\mu\|_{\mathrm{HP}}:=|\mu|(\mathbb{R}_{+})$, and $\mathcal{L}L^{1}:=\\{\mathcal{L}f:f\in L^{1}(\mathbb{R}_{+})\\}$.

  5. $dS$ denotes area measure on $\mathbb{C}_{+}$.

For a Banach space $X$, $X^{*}$ denotes the dual space of $X$ and $L(X)$
denotes the space of all bounded linear operators on $X$. The domain, spectrum
and resolvent set of an (unbounded) operator $A$ on $X$ are denoted by $D(A)$,
$\sigma(A)$ and $\rho(A)$, respectively.

If $(\mathcal{X},\|\cdot\|_{\mathcal{X}})$ and
$(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ are normed spaces of holomorphic
functions on domains $\Omega_{\mathcal{X}}$ and $\Omega_{\mathcal{Y}}$, we
will use notation as follows:

  1. $\mathcal{Y}\overset{i}{\hookrightarrow}\mathcal{X}$ if $\Omega_{\mathcal{Y}}=\Omega_{\mathcal{X}}$, $\mathcal{Y}$ is a subset of $\mathcal{X}$ and the inclusion map is continuous;

  2. $\mathcal{Y}\subset\mathcal{X}$ if $\Omega_{\mathcal{Y}}=\Omega_{\mathcal{X}}$, $\mathcal{Y}$ is a subset of $\mathcal{X}$ and $\mathcal{Y}$ inherits the norm from $\mathcal{X}$;

  3. $\mathcal{Y}\overset{r}{\hookrightarrow}\mathcal{X}$ if $\Omega_{\mathcal{Y}}\subset\Omega_{\mathcal{X}}$, and the restriction map $f\mapsto f|_{\Omega_{\mathcal{Y}}}$ is a continuous map from $\mathcal{Y}\to\mathcal{X}$.

### Elementary inequalities

We will need the following lemma which gives elementary inequalities for
$|z+\lambda|$, when $z$, $\lambda\in\mathbb{C}$.

######  Lemma 2.1.

  1. (i)

Let $z=|z|e^{i\psi}$ and $\lambda=|\lambda|e^{i\varphi}\in\mathbb{C}$, where
$|\psi-\varphi|\leq\pi$. Then

(2.1) $|z+\lambda|\geq\cos\left(\frac{\psi-\varphi}{2}\right)(|z|+|\lambda|).$

  2. (ii)

Let $z\in\overline{\Sigma}_{\psi}$ and
$\lambda\in\overline{\Sigma}_{\varphi}$, where $\psi,\varphi>0$ and
$\varphi+\psi<\pi$. Then

(2.2) $|z+\lambda|\geq\cos\left(\frac{\psi+\theta}{2}\right)(|z|+|\lambda|).$

  3. (iii)

Let $z=|z|e^{i\psi}$ and $\lambda=|\lambda|e^{i\varphi}\in\mathbb{C}$, where
$|\psi|<\pi/2$ and $|\varphi|\leq\pi/2$. Then

(2.3) $|z+\lambda|\geq\cos\psi\,|\lambda|,$

and

(2.4) $|z+\lambda|\geq\cos\psi\,|z|.$

###### Proof.

For (2.1), we may assume that $\varphi\geq\psi$ and let
$\theta:=(\pi-\varphi+\psi)/2\in[0,\pi/2]$. By applying a rotation of
$\mathbb{C}$ we may further assume that $\varphi=\pi-\theta$ and
$\psi=\theta$. Then

$|z+\lambda|\geq\operatorname{Im}z+\operatorname{Im}\lambda=\sin\theta(|z|+|\lambda|)=\cos\left(\frac{\varphi-\psi}{2}\right)(|z|+|\lambda|).$

The inequality (2.2) follows from (2.1), since $\psi+\varphi$ is the maximum
value of $|\psi^{\prime}-\varphi^{\prime}|$ for $\psi^{\prime}\in[-\psi,\psi]$
and $\varphi^{\prime}\in[-\varphi,\varphi]$.

The inequality (2.4) is obtained by considering
$\operatorname{Re}(z+\lambda)$. For the inequality (2.3), note that

$\displaystyle|\lambda+z|^{2}-|\lambda|^{2}\cos^{2}\psi$
$\displaystyle=\left(|z|+|\lambda|\cos(\varphi-\psi)\right)^{2}+|\lambda|^{2}\left(\sin^{2}\varphi-\cos^{2}(\varphi-\psi)\right).$

This is clearly non-negative if $\cos(\varphi-\psi)\geq 0$. If
$\cos(\varphi-\psi)<0$, and assuming without loss of generality that
$\sin\varphi\geq 0$, we have

$\sin\varphi-|\cos(\varphi-\psi)|=\sin\varphi\,(1+\sin\psi)+\cos\varphi\cos\psi\geq
0.$

This completes the proof. ∎

### Beta function

The Beta function appears in many places in the paper. It is defined for
$s,t>0$ by

$B(s,t)=B(t,s):=\int_{0}^{1}\tau^{s-1}(1-\tau)^{t-1}\,d\tau=2\int_{0}^{\pi/2}\cos^{2s-1}\psi\sin^{2t-1}\psi\,d\psi.$

In particular, for $s>-1$ we will use the relations

$B\left(\frac{s+1}{2},\frac{1}{2}\right)=\int_{-\pi/2}^{\pi/2}\cos^{s}\psi\,d\psi=\int_{-\infty}^{\infty}\frac{dt}{(1+t^{2})^{(s+1)/2}}=\frac{\sqrt{\pi}\Gamma((s+1)/2)}{\Gamma(s/2+1)},$

see [47, p. 386]. We note also the following limit properties:

$\lim_{s\to-1}\,(s+1)\,B\left(\frac{s+1}{2},\frac{1}{2}\right)=2,\quad\lim_{s\to\infty}\,\sqrt{s}B\left(\frac{s+1}{2},\frac{1}{2}\right)=\sqrt{2\pi}.$

### Proof conventions

We will make extensive use of the dominated convergence theorem, often for
vector-valued functions. With a few exceptions, we will not give details of
the relevant dominating functions, as they are usually easily identified.

We will also use the following elementary lemma on several occasions. See [20,
p.21, Lemma 1] for a proof.

######  Lemma 2.2.

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space, and $(f_{n})_{n\geq
1}\subset L^{p}(\Omega,\mu)$, where $p\in[1,\infty)$. If $f_{n}\to f_{0}$
a.e., and $\|f_{n}\|_{L^{p}(\Omega,\mu)}\to\|f_{0}\|_{L^{p}(\Omega,\mu)}$,
then $\|f_{n}-f_{0}\|_{L^{p}(\Omega,\mu)}\to 0$ as $n\to\infty$.

We will use Vitali’s theorem several times, usually for holomorphic vector-
valued functions. We refer to the version given in [4, Theorem A.5].

Let $\mathcal{X}$ be a Banach space of holomorphic functions on a domain
$\Omega_{\mathcal{X}}$ such that the point evaluations $\delta_{z}:f\mapsto
f(z),\,z\in\Omega$, are continuous on $X$. Let $(\Omega,\mu)$ be either an
interval in $\mathbb{R}$ with length measure or an open set in $\mathbb{C}$
with area measure, and $F:\Omega\to\mathcal{X}$ be a continuous function such
that $\int_{\Omega}\|F(t)\|_{\mathcal{X}}\,d\mu(t)<\infty$. Then the integral

$G:=\int_{I}F(t)\,dt$

exists as a Bochner integral in $\mathcal{X}$ and it can be approximated by
Riemann sums. It follows that $G$ belongs to the closed linear span of
$\\{F(t):t\in I\\}$ in $\mathcal{X}$.

Now assume that $F:\Omega\to\mathcal{X}$ is locally bounded, where $\Omega$ is
an open set in $\mathbb{C}$, and that $\lambda\mapsto F(\lambda)(z)$ is
holomorphic on $\Omega$ for all $z\in\Omega_{\mathcal{X}}$. We shall use the
fact that $F:\Omega\to\mathcal{X}$ is holomorphic in the vector-valued sense,
without further comment. The result at this level of generality can be seen
from [4, Corollary A.7], using the point evaluations as separating
functionals. An alternative is to show that $F$ is continuous, and then apply
Morera’s theorem. If the definition of $F$ is by an integral formula, it may
also be possible to apply a standard corollary of the dominated convergence
theorem which leads to an integral formula for $F^{\prime}$.

##  3\. The Banach spaces $\mathcal{D}_{s}$ and their reproducing formulas

In this section we introduce some spaces of holomorphic functions to which we
will extend the $\mathcal{B}$-calculus of operators in Section 7 onwards.

###  3.1. The spaces $\mathcal{V}_{s}$

Let $s>-1$, $z=\alpha+i\beta$, and let $\mathcal{V}_{s}$ be the Banach space
of (equivalence classes of) measurable functions
$g:\mathbb{C}_{+}\to\mathbb{C}$ such that the norm

(3.1) $\displaystyle\|g\|_{\mathcal{V}_{s}}:$
$\displaystyle=\int_{\mathbb{C}_{+}}\frac{(\operatorname{Re}z)^{s}|g(z)|}{|z|^{s+1}}\,dS(z)$
$\displaystyle=\int_{0}^{\infty}\alpha^{s}\int_{-\infty}^{\infty}\frac{|g(\alpha+i\beta)|}{(\alpha^{2}+\beta^{2})^{(s+1)/2}}\,d\beta\,d\alpha$
$\displaystyle=\int_{-\pi/2}^{\pi/2}\cos^{s}\varphi\int_{0}^{\infty}|g(\rho
e^{i\varphi})|\,d\rho\,d\varphi,$

is finite, where $S$ is the area measure on $\mathbb{C}_{+}$. Note that

(3.2)
$\mathcal{V}_{s}\subset\mathcal{V}_{\sigma}\qquad\text{and}\qquad\|g\|_{\mathcal{V}_{\sigma}}\leq\|g\|_{\mathcal{V}_{s}},\qquad
g\in\mathcal{V}_{s},\;s<\sigma,$

and

(3.3)
$\int_{\Sigma_{\psi}}\frac{|g(z)|}{|z|}\,dS(z)\leq\max\left\\{1,\frac{1}{\cos^{s}\psi}\right\\}\|g\|_{\mathcal{V}_{s}},\qquad
g\in\mathcal{V}_{s},\,\psi\in(0,\pi/2).$

The following property of functions from $\mathcal{V}_{s}$ is an essential
element in the arguments which lead to the representations for functions in
$\mathcal{V}_{s}$ in Proposition 3.7 and for $\mathcal{D}_{s}$ in Corollary
3.9, and eventually to the definition of a functional calculus for operators
in (7.5).

######  Lemma 3.1.

Let $g\in\mathcal{V}_{s}$ be holomorphic, where $s>-1$. For every $k\geq 1$
and every $\psi\in(0,\pi/2)$,

(3.4) $\lim_{|z|\to\infty,\;z\in\Sigma_{\psi}}\,z^{k}g^{(k-1)}(z)=0.$

###### Proof.

Let $g\in\mathcal{V}_{s}$ be holomorphic, $\psi\in(0,\pi/2)$,
$\psi^{\prime}=(\pi/2+\psi)/2$ and
$b_{\psi}={\sin((\pi/2-\psi)/2)}=\cos\psi^{\prime}$. If $z\in\Sigma_{\psi}$,
then

$\\{\lambda\in\mathbb{C}:|\lambda-z|\leq
b_{\psi}|z|\\}\subset\\{\lambda\in\Sigma_{\psi^{\prime}}:|\lambda|\geq(1-b_{\psi})|z|\\}.$

Let $r\in(0,b_{\psi}|z|)$. By Cauchy’s integral formula for derivatives,

$g^{(k-1)}(z)=\frac{(k-1)!}{2\pi
i}\int_{\\{\lambda:\,|\lambda-z|=r\\}}\frac{g(\lambda)}{(\lambda-z)^{k}}\,d\lambda.$

Multiplying by $r^{k}$ and integrating with respect to $r$ over
$(0,b_{\psi}|z|)$ gives

$\frac{(b_{\psi}|z|)^{k+1}}{k+1}|g^{(k-1)}(z)|\leq\frac{(k-1)!}{2\pi}\int_{\\{\lambda:|\lambda-z|\leq
b_{\psi}|z|\\}}|g(\lambda)|\,dS(\lambda),$

and then

$|z|^{k}|g^{(k-1)}(z)|\leq\frac{(k+1)(k-1)!(1+b_{\psi})}{2\pi
b_{\psi}^{k+1}}\int_{\\{\lambda\in\Sigma_{\psi^{\prime}}:|\lambda|\geq(1-b_{\psi})|z|\\}}\frac{|g(\lambda)|}{|\lambda|}\,dS(\lambda).$

It now follows from (3.3) that $|z^{k}g^{(k-1)}(z)|\to 0$ as
$|z|\to\infty,\,z\in\Sigma_{\psi}$. ∎

###  3.2. The spaces $\mathcal{D}_{s}$ and the operators $Q_{s}$

We now define a linear operator $Q_{s}$ on $\mathcal{V}_{s},\,s>-1$. It will
play a similar role to the operator $Q$ on $\mathcal{W}$ considered in [7,
Section 3], where $\mathcal{W}$ is the Banach space of all (equivalence
classes of) measurable functions $g:\mathbb{C}_{+}\to\mathbb{C}$ such that

(3.5)
$\|g\|_{\mathcal{W}}:=\int_{0}^{\infty}\sup_{\beta\in\mathbb{R}}|g(\alpha+i\beta)|\,d\alpha<\infty.$

Indeed the definition of $Q_{1}$ is formally the same as the definition of $Q$
in [7], but the domain $\mathcal{V}_{1}$ of $Q_{1}$ is larger than
$\mathcal{W}$.

For $g\in\mathcal{V}_{s}$, let

(3.6)
$(Q_{s}g)(z):=-\frac{2^{s}}{\pi}\int_{0}^{\infty}\alpha^{s}\int_{-\infty}^{\infty}\frac{g(\alpha+i\beta)}{(z+\alpha-i\beta)^{s+1}}\,d\beta\,d\alpha,\quad
z\in\mathbb{C}_{+}\cup\\{0\\}.$

By (2.3), the integral is absolutely convergent, and

(3.7)
$|(Q_{s}g)(z)|\leq\frac{2^{s}\|g\|_{\mathcal{V}_{s}}}{\pi\cos^{s+1}\psi},\qquad
z\in\Sigma_{\psi},\,\psi\in(0,\pi/2).$

The dominated convergence theorem implies that $Q_{s}g$ is continuous on
$\mathbb{C}_{+}$, with sectorial limits at infinity and $0$:

(3.8)
$\lim_{|z|\to\infty,\,z\in{\Sigma}_{\psi}}\,(Q_{s}g)(z)=0,\qquad\psi\in(0,\pi/2),$

and

(3.9) $\lim_{|z|\to
0,\,z\in{\Sigma}_{\psi}}\,(Q_{s}g)(z)=(Q_{s}g)(0),\qquad\psi\in(0,\pi/2).$

Thus $Q_{s}g$ is bounded and continuous on $\overline{\Sigma}_{\psi}$ for
every $\psi\in(0,\pi/2)$. Moreover, $Q_{s}g$ is holomorphic on
$\mathbb{C}_{+}$ and

(3.10)
$(Q_{s}g)^{\prime}(z)=(s+1)\frac{2^{s}}{\pi}\int_{0}^{\infty}\alpha^{s}\int_{-\infty}^{\infty}\frac{g(\alpha+i\beta)}{(z+\alpha-i\beta)^{s+2}}\,d\beta\,d\alpha,\qquad
z\in\mathbb{C}_{+}.$

Using this, (2.3) and (2.4), we obtain that

(3.11)
$|(Q_{s}g)^{\prime}(z)|\leq\frac{(s+1)2^{s}}{\pi\cos^{s+2}\psi\,|z|}\|g\|_{\mathcal{V}_{s}},\qquad
z\in\Sigma_{\psi},$

and

$|z(Q_{s}g)^{\prime}(z)|\leq\frac{(s+1)2^{s}}{\pi\cos^{s+2}\psi}\int_{\mathbb{C}_{+}}\frac{|z|(\operatorname{Re}\lambda)^{s}\,|g(\lambda)|}{|z+\overline{\lambda}||\lambda|^{s+1}}\,dS(\lambda),\qquad
z\in\Sigma_{\psi}.$

Using the dominated convergence theorem again, we obtain

(3.12) $\lim_{|z|\to 0,z\in\Sigma_{\psi}}z(Q_{s}g)^{\prime}(z)=0.$

We now give another formula for $Q_{s}$. Let $s=n+\delta>-1$ where
$n\in\mathbb{N}\cup\\{-1,0\\}$ and $\delta\in[0,1)$, and let

(3.13)
$C_{s}:=\int_{0}^{\infty}\frac{dt}{(t+1)^{n+2}t^{\delta}}=\int_{0}^{1}\frac{(1-\tau)^{s}}{\tau^{\delta}}\,d\tau=B(1-\delta,s+1).$

Then

$\int_{0}^{\infty}\frac{dt}{(\lambda+t)^{n+2}t^{\delta}}=\frac{C_{s}}{\lambda^{s+1}},\quad\lambda\in\mathbb{C}_{+}.$

Indeed, both sides of this equation are holomorphic functions of
$\lambda\in\mathbb{C}_{+}$, and they coincide for $\lambda\in(0,\infty)$, so
they coincide for all $\lambda\in\mathbb{C}_{+}$, by the identity theorem for
holomorphic functions. Putting $\lambda=z+\alpha-i\beta$, we obtain

(3.14) $(Q_{s}g)(z)=-\frac{2^{s}}{\pi
C_{s}}\int_{0}^{\infty}\int_{0}^{\infty}\int_{-\infty}^{\infty}\frac{\alpha^{s}\,g(\alpha+i\beta)\,d\beta\,d\alpha}{(z+\alpha-i\beta+t)^{n+2}}\,\frac{dt}{t^{\delta}},\quad
z\in\mathbb{C}_{+}.$

For $s>-1$ let $\mathcal{D}_{s}$ be the linear space of all holomorphic
functions $f$ on $\mathbb{C}_{+}$ such that

$f^{\prime}\in\mathcal{V}_{s},$

equipped with the semi-norm

$\|f\|_{\mathcal{D}_{s,0}}:=\|f^{\prime}\|_{\mathcal{V}_{s}},\qquad
f\in\mathcal{D}_{s}.$

If $\sigma>s>-1$, then it is immediate from (3.2) that
$\mathcal{D}_{s}\subset\mathcal{D}_{\sigma}$. We will exhibit some functions
in $\mathcal{D}_{s}$ later in this section and in Section 3.3.

In the rest of this section we will also obtain a reproducing formula (3.27)
for functions from $\mathcal{D}_{s}$ and describe some basic properties which
will be relevant for the sequel. To this aim, we first define and study the
behaviour of operators $Q_{s}$ on the scale of $\mathcal{D}_{s}$-spaces.
Recall that in [7, Proposition 3.1] we showed that $Q$ maps $\mathcal{W}$ into
$\mathcal{B}$. However $Q_{s}$ does not map the whole of $\mathcal{V}_{s}$
into $\mathcal{D}_{s}$. For $s>-1$, a function $g\in\mathcal{V}_{s}$ for which
$Q_{s}g\notin\mathcal{D}_{s}$, can be defined as follows:

$g(\rho
e^{i\varphi}):=\left(\cos^{s+1}\varphi(\rho-\sin\varphi)\log^{2}(\rho-\sin\varphi)\right)^{-1},\\\
1<\rho<2-\sin\varphi,\,\pi/4<\varphi<\pi/2,$

and $g(z)=0$ for all other points in $\mathbb{C}_{+}$. We do not give details
in this paper. Instead we will show in Propositions 3.6 and 3.7 that $Q_{s}$
maps $\mathcal{V}_{s}$ boundedly into $\mathcal{D}_{\sigma}$ for any
$\sigma>s$, and it maps holomorphic functions in $\mathcal{V}_{s}$ into
$\mathcal{D}_{s}$. We need the following auxiliary lemma which will be useful
in a number of instances.

######  Lemma 3.2.

Let $h\in L^{1}[0,1]\cap L^{\infty}[1/2,1]$ be a positive function. Let
$s>-1$, $\beta>1/2$, and

(3.15)
$G_{h,\beta,s}(\varphi):=\int_{-\pi/2}^{\pi/2}\cos^{s}\psi\int_{0}^{1}\frac{h(t)\,dt}{(t^{2}+2t\cos(\varphi+\psi)+1)^{\beta}}\,d\psi,\quad\varphi\in(-\pi/2,\pi/2).$

  1. (a)

If $2\beta-s-2\leq 0$, then

(3.16) $K_{h,\beta,s}:=\sup_{|\varphi|<\pi/2}\,G_{h,\beta,s}(\varphi)<\infty.$

  2. (b)

If $2\beta-s-2\geq 0$, then

(3.17)
$\tilde{K}_{h,\beta,s}:=\sup_{|\varphi|<\pi/2}\,\cos^{2\beta-s-2}\varphi\,G_{h,\beta,s}(\varphi)<\infty.$

###### Proof.

Since $G_{h,\beta,s}(-\varphi)=G_{h,\beta,s}(\varphi)$, we may assume that
$\varphi\in[0,\pi/2)$. Now

$\displaystyle G_{h,\beta,s}(\varphi)$
$\displaystyle=\int_{-\pi/2}^{0}\cos^{s}\psi\int_{0}^{1}\frac{h(t)\,dt}{(t^{2}+2t\cos(\varphi+\psi)+1)^{\beta}}\,d\psi$
$\displaystyle\hbox{}\hskip
30.0pt+\int_{0}^{\pi/2}\cos^{s}\psi\int_{0}^{1}\frac{h(t)\,dt}{(t^{2}+2t\cos(\varphi+\psi)+1)^{\beta}}\,d\psi$
$\displaystyle=:G_{h,\beta,s}^{-}(\varphi)+G_{h,\beta,s}^{+}(\varphi),$

and we estimate these two integrals separately.

Since $\varphi\in[0,\pi/2)$, $\beta>1/2$, and $s>-1$, we have

$\displaystyle G_{h,\beta,s}^{-}(\varphi)$
$\displaystyle\leq\int_{-\pi/2}^{0}\cos^{s}\psi\,d\psi\int_{0}^{1}h(t)\,dt=\frac{\|h\|_{L^{1}[0,1]}}{2}B\left(\frac{s+1}{2},\frac{1}{2}\right).$

For the second integral, note that if $\phi=\varphi+\psi\in[0,\pi)$,

$t^{2}+2t\cos\phi+1\geq\begin{cases}\frac{1}{4},\quad&t\in[0,1/2],\\\
(1-t)^{2}+1+\cos\phi,\quad&t\in[1/2,1].\end{cases}$

Hence

$\displaystyle\hskip-30.0pt\int_{0}^{1}\frac{h(t)}{(t^{2}+2t\cos\phi+1)^{\beta}}\,dt$
$\displaystyle\leq
4^{\beta}\int_{0}^{1/2}h(t)\,dt+\int_{1/2}^{1}\frac{h(t)}{((t-1)^{2}+(1+\cos\phi))^{\beta}}\,dt$
$\displaystyle\leq
4^{\beta}\|h\|_{L^{1}[0,1/2]}+\|h\|_{L^{\infty}[1/2,1]}\int_{0}^{\infty}\frac{d\tau}{(\tau^{2}+2\cos^{2}(\phi/2))^{\beta}}$
$\displaystyle\leq\frac{C_{h,\beta}}{\cos^{2\beta-1}(\phi/2)},$

for some constant $C_{h,\beta}>0$. Replacing $\varphi$ by $\pi/2-\varphi$ and
$\psi$ by $\pi/2-\psi$, and using

$\omega\geq\sin\omega\geq\frac{2}{\pi}\omega,\qquad\omega\in(0,\pi/2),$

we infer that if $\varphi\in[0,\pi/2)$, then

$\displaystyle G^{+}_{h,\beta,s}(\pi/2-\varphi)$ $\displaystyle\leq
C_{h,\beta}\int_{0}^{\pi/2}\frac{\sin^{s}\psi}{\sin^{2\beta-1}((\varphi+\psi)/2)}\,d\psi$
$\displaystyle\leq
C_{h,\beta}\pi^{2\beta-1}\int_{0}^{\pi/2}\frac{\psi^{s}}{(\varphi+\psi)^{2\beta-1}}\,d\psi.$

In case (a), when $2\beta-s-2\leq 0$, we have

$\int_{0}^{\pi/2}\frac{\psi^{s}}{(\varphi+\psi)^{2\beta-1}}\,d\psi\leq\int_{0}^{\pi/2}\psi^{s-2\beta+1}\,d\psi=\frac{\pi^{s-2\beta+2}}{(s-2\beta+2)2^{s-2\beta+2}}<\infty.$

In case (b), when $2\beta-s-2\geq 0$, we obtain

$\int_{0}^{\pi/2}\frac{\psi^{s}}{(\varphi+\psi)^{2\beta-1}}\,d\psi\leq\varphi^{s+2-2\beta}\int_{0}^{\infty}\frac{t^{s}}{(t+1)^{2\beta-1}}\,dt,$

and hence

$\cos^{2\beta-s-2}(\pi/2-\varphi)\,G_{h,\beta,s}(\pi/2-\varphi)\leq\varphi^{2\beta-s-2}G_{h,\beta,s}(\pi/2-\varphi)\\\
\hbox{}\hskip-20.0pt\leq
C_{h,\beta}\pi^{2\beta-1}\int_{0}^{\infty}\frac{t^{s}}{(t+1)^{2\beta-1}}\,dt<\infty,$

for some constant $C_{h,\beta}>0$. ∎

Let $\mathcal{D}_{s,0}$ be the space of all functions $f\in\mathcal{D}_{s}$
such that $f$ has a sectorial limit $0$ at infinity. Then
$(\mathcal{D}_{s,0},\|\cdot\|_{\mathcal{D}_{s,0}})$ is a normed space, and we
shall see in Corollary 3.10 that it is a Banach space.

The following basic examples will play roles in several estimates later in the
paper. We start with the the resolvent functions and their powers.

######  Example 3.3.

Let $\lambda=|\lambda|e^{i\varphi}\in\mathbb{C}_{+}$, and
$r_{\lambda}(z)=(z+\lambda)^{-1}$. Let $\gamma>0$, and consider

$r_{\lambda}^{\gamma}(z):=(z+\lambda)^{-\gamma},\quad z\in\mathbb{C}_{+}.$

Let $s>-1$. Then

$\displaystyle\|r_{\lambda}^{\gamma}\|_{\mathcal{D}_{s,0}}$
$\displaystyle=\gamma\int_{-\pi/2}^{\pi/2}\cos^{s}\psi\int_{0}^{\infty}\frac{d\rho}{|\rho
e^{i\psi}+re^{i\varphi}|^{\gamma+1}}\,d\psi$
$\displaystyle=\frac{\gamma}{|\lambda|^{\gamma}}\int_{-\pi/2}^{\pi/2}\cos^{s}\psi\int_{0}^{\infty}\frac{d\rho}{|\rho+e^{i(\varphi-\psi)}|^{\gamma+1}}\,d\psi$
$\displaystyle=\frac{\gamma}{|\lambda|^{\gamma}}\int_{-\pi/2}^{\pi/2}\cos^{s}\psi\int_{0}^{1}\frac{1+t^{\gamma-1}}{(t^{2}+2t\cos(\varphi+\psi)+1)^{(\gamma+1)/2}}\,dt\,d\psi,$

where we have put $t=\rho^{-1}$ for $\rho>1$. Now we apply Lemma 3.2 with
$h(t)=1+t^{\gamma-1}$, $\beta=(\gamma+1)/2$, so $2\beta-s-2=\gamma-s-1$. Thus
we obtain

(3.18)
$\|r_{\lambda}^{\gamma}\|_{\mathcal{D}_{s,0}}\leq\begin{cases}\dfrac{\gamma
K_{h,(\gamma+1)/2,s}}{|\lambda|^{\gamma}},&s>\gamma-1>-1,\\\ \vskip
8.0pt\cr\dfrac{\gamma\tilde{K}_{h,(\gamma+1)/2,s}}{|\lambda|^{\gamma}\cos^{\gamma-s-1}\varphi},&\gamma-1>s>-1.\end{cases}$

In particular, taking $\gamma=1$ and a fixed $s>0$,

(3.19)
$\|r_{\lambda}\|_{\mathcal{D}_{s,0}}=\int_{-\pi/2}^{\pi/2}\int_{0}^{\infty}\frac{\cos^{s}\psi}{|\lambda+\rho
e^{i\psi}|^{2}}\,d\rho\,d\psi\leq\frac{C_{s}}{|\lambda|},\qquad\lambda\in\mathbb{C}_{+}.$

This estimate will play a crucial role in the proof of Theorem 1.1.

Next we consider some functions which appear frequently in the studies of
holomorphic $C_{0}$-semigroups.

######  Example 3.4.

Let

$f_{\nu}(z):=z^{\nu}e^{-z},\quad z\in\mathbb{C}_{+},\quad\nu\geq 0.$

We shall show here that $f_{\nu}\in\mathcal{D}_{s}$ if and only if $s>\nu$.
Moreover, if $s>\nu$, then

(3.20) $\|f_{\nu}\|_{\mathcal{D}_{s,0}}\leq
2B\left(\frac{s-\nu}{2},\frac{1}{2}\right)\Gamma(\nu+1).$

This estimate will be crucial for operator estimates in Section 10.

We have

$f_{\nu}^{\prime}(z)=e^{-z}\left(\nu{z^{\nu-1}}-z^{\nu}\right)$

and

$\displaystyle\|f_{\nu}\|_{\mathcal{D}_{s,0}}$
$\displaystyle=\int_{-\pi/2}^{\pi/2}\cos^{s}\varphi\int_{0}^{\infty}{e^{-\rho\cos\varphi}}{\rho^{\nu-1}}\left|{\rho
e^{i\varphi}}-\nu\right|\,d\rho\,d\varphi$
$\displaystyle=\int_{-\pi/2}^{\pi/2}\cos^{s-\nu-1}\varphi\int_{0}^{\infty}e^{-r}r^{\nu-1}|re^{-i\varphi}-\nu\cos\varphi|\,dr\,d\varphi.$

We use the estimates

$r|\sin\varphi|\leq|re^{-i\varphi}-\nu\cos\varphi|\leq r+\nu.$

If $s>\nu$, we obtain

$\displaystyle\|f_{\nu}\|_{\mathcal{D}_{s,0}}$ $\displaystyle\leq
2\int_{0}^{\pi/2}\cos^{s-\nu-1}\varphi\,d\varphi\int_{0}^{\infty}e^{-r}r^{\nu-1}(r+\nu)\,dr$
$\displaystyle=2B\left(\frac{s-\nu}{2},\frac{1}{2}\right)\Gamma(\nu+1).$

If $s\leq\nu$, then

$\|f_{\nu}\|_{\mathcal{D}_{s,0}}\geq
2\int_{0}^{\pi/2}\cos^{s-\nu-1}\varphi\,\sin\varphi\,d\varphi\int_{0}^{\infty}e^{-\tau}\tau^{\nu}\,d\tau=\infty.$

This establishes the claims above.

Finally we consider a function which will play an important role in our
constructions of functional calculi in Section 8.

######  Example 3.5.

The function $\operatorname{arccot}$ is defined by

(3.21)
$\operatorname{arccot}(z)=\frac{1}{2i}\log\left(\frac{z+i}{z-i}\right),\quad
z\in\mathbb{C}_{+}.$

Then $|\operatorname{Re}\operatorname{arccot}(z)|\leq\pi/2$,
$\operatorname{arccot}$ has sectorial limit $0$ at infinity, and its
derivative is $-(z^{2}+1)^{-1}$. It is easy to see that
$\operatorname{arccot}\in\mathcal{D}_{s}$ for all $s>-1$. For $s=0$, we have

(3.22) $\displaystyle\|\operatorname{arccot}\|_{\mathcal{D}_{0,0}}$
$\displaystyle=\int_{-\pi/2}^{\pi/2}\int_{0}^{\infty}\frac{d\rho}{|1+\rho^{2}e^{2i\varphi}|}\,d\varphi=\int_{0}^{\infty}\int_{0}^{\pi}\frac{d\psi}{(\rho^{4}+2\rho^{2}\cos\psi+1)^{1/2}}\,d\rho$
$\displaystyle\leq\sqrt{\pi}\int_{0}^{\infty}\left(\int_{0}^{\pi}\frac{d\psi}{\rho^{4}+2\rho^{2}\cos\psi+1}\right)^{1/2}\,d\rho=\sqrt{\pi}\int_{0}^{\infty}\left(\frac{\pi}{|\rho^{4}-1|}\right)^{1/2}\,d\rho$
$\displaystyle=\frac{\pi}{2}B(1/4,1/2)<3\pi.$

See [47, 2.5.16, (38)] for the evaluation of the integral with respect to
$\psi$.

######  Proposition 3.6.

Let $\sigma>s>-1$. The following hold.

  1. (i)

$\mathcal{D}_{s}$ is continuously embedded in $\mathcal{D}_{\sigma}$.

  2. (ii)

The restriction of $Q_{\sigma}$ to $\mathcal{V}_{s}$ is in
$L(\mathcal{V}_{s},\mathcal{D}_{s})$.

  3. (iii)

$Q_{s}\in L(\mathcal{V}_{s},\mathcal{D}_{\sigma})$.

###### Proof.

The first statement is immediate from the definitions of the spaces and (3.2).

For the second statement, let $g\in\mathcal{V}_{s}$. From (3.1) and (3.10), we
have

$\displaystyle\frac{\pi}{2^{\sigma}(\sigma+1)}\|Q_{\sigma}g\|_{\mathcal{D}_{s,0}}\hskip
10.0pt$
$\displaystyle\leq\frac{1}{\sigma+1}\int_{-\pi/2}^{\pi/2}\cos^{s}\psi\int_{0}^{\infty}\int_{-\pi/2}^{\pi/2}\cos^{\sigma}\varphi\int_{0}^{\infty}\frac{r^{\sigma+1}|g(te^{i\varphi})|}{|\rho
e^{i\psi}+te^{-i\varphi}|^{\sigma+2}}\,d\rho\,d\varphi\,dt\,d\psi$
$\displaystyle=\int_{-\pi/2}^{\pi/2}\cos^{\sigma}\varphi\int_{0}^{\infty}t^{\sigma+1}\|r_{te^{-i\varphi}}^{\sigma+1}\|_{\mathcal{D}_{s,0}}|g(te^{i\varphi})|\,dt\,d\varphi$
$\displaystyle\leq K_{h,(s+2)/2,\sigma}\|g\|_{\mathcal{V}_{s}},$

where $h(t)=1+t^{s}$. Here we have used the first case of (3.18) with $s$ and
$\sigma$ interchanged. This establishes the second statement.

For the third statement, the same estimation, but with $s$ and $\sigma$
interchanged, setting $h(t)=1+t^{\sigma}$, and using the second case of
(3.18), shows that

$\frac{\pi}{2^{s}(s+1)}\|Q_{s}g\|_{\mathcal{D}_{\sigma,0}}\leq\tilde{K}_{h,(s+2)/2,\sigma}\|g\|_{\mathcal{V}_{s}}.$

This establishes the third statement. ∎

For insight on why $\|r_{\lambda}^{\sigma+1}\|_{\mathcal{D}_{s,0}}$ appears in
the proof above, we refer the reader to the proof of Theorem 5.1.

######  Proposition 3.7.

Let $g\in\mathcal{V}_{s}$ be holomorphic, where $s>-1$. Then
$Q_{s}g\in\mathcal{D}_{s}$ and $(Q_{s}g)^{\prime}=g$.

###### Proof.

First we consider the case when $g\in\mathcal{V}_{n}$, where
$n\in\mathbb{N}\cup\\{0\\}$. Then $Q_{n}g$ is holomorphic and (3.6) holds for
$s=n$. It suffices to show that $(Q_{n}g)^{\prime}=g$. Let

$\displaystyle I(\alpha)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{|g(\alpha+i\beta)|}{(\alpha^{2}+\beta^{2})^{(n+1)/2}}\,d\beta,$
$\displaystyle J(\beta)$
$\displaystyle=\int_{0}^{\infty}\frac{\alpha^{n}|g(\alpha+i\beta)|}{(\alpha^{2}+\beta^{2})^{(n+1)/2}}\,d\alpha.$

From (3.1) and Fubini’s theorem, we see that
$\int_{0}^{\infty}\alpha^{n}I(\alpha)\,d\alpha<\infty$. Hence
$I(\alpha)<\infty$ for almost all $\alpha>0$ and it follows that there exists
a sequence $(\alpha_{j})_{j\geq 1}$ such that

$\lim_{j\to\infty}\alpha_{j}=\infty,\qquad\lim_{j\to\infty}I(\alpha_{j})=0.$

Similarly $\int_{-\infty}^{\infty}J(\beta)\,d\beta<\infty$, and so there exist
sequences $(\beta_{k}^{\pm})_{k\geq 1}$ such that

$\lim_{k\to\infty}\beta_{k}^{\pm}=\pm\infty,\qquad\lim_{k\to\infty}J(\beta_{k}^{\pm})=0.$

Let $z\in\mathbb{C}_{+}$ be fixed. Take $\alpha>0$ with $I(\alpha)<\infty$,
and let $j$ be sufficiently large that
$\alpha_{j}>2\alpha+\operatorname{Re}z$, and $k$ be sufficiently large that
$\beta_{k}^{-}<\operatorname{Im}z<\beta_{k}^{+}$. We may apply the Cauchy
integral formula around the rectangle with vertices $\alpha+i\beta_{k}^{\pm}$
and $\alpha_{j}+i\beta_{k}^{\pm}$, and we obtain

$\displaystyle\frac{2\pi}{n!}g^{(n)}(2\alpha+z)$
$\displaystyle=\int_{\beta_{k}^{-}}^{\beta_{k}^{+}}\frac{g(\alpha_{j}+i\beta)}{(\alpha_{j}+i\beta-z-2\alpha)^{n+1}}d\beta-\int_{\beta_{k}^{-}}^{\beta_{k}^{+}}\frac{g(\alpha+i\beta)}{(\alpha+i\beta-z-2\alpha)^{n+1}}d\beta$
$\displaystyle\hbox{}\hskip
10.0pt-i\int_{\alpha}^{\alpha_{j}}\frac{g(s+i\beta_{k}^{-})}{(s+i\beta_{k}^{-}-z-2\alpha)^{n+1}}\,ds+i\int_{\alpha}^{\alpha_{j}}\frac{g(s+i\beta_{k}^{+})}{(s+i\beta_{k}^{+}-z-2\alpha)^{n+1}}\,ds.$

Letting $k\to\infty$, we obtain

$(-1)^{n}\frac{2\pi}{n!}g^{(n)}(2\alpha+z)=\int_{-\infty}^{\infty}\frac{g(\alpha+i\beta)}{(z+\alpha-i\beta)^{n+1}}\,d\beta-\int_{-\infty}^{\infty}\frac{g(\alpha_{j}+i\beta)}{(z+\alpha_{j}-i\beta)^{n+1}}\,d\beta.$

Letting $j\to\infty$, we obtain

(3.23)
$(-1)^{n}\frac{2\pi}{n!}g^{(n)}(2\alpha+z)=\int_{-\infty}^{\infty}\frac{g(\alpha+i\beta)}{(z+\alpha-i\beta)^{n+1}}\,d\beta.$

This holds for almost all $\alpha>0$. Substituting this into (3.6), and then
integrating by parts and using Lemma 3.1, we infer that

(3.24) $\displaystyle(Q_{n}g)(z)$
$\displaystyle=(-1)^{n+1}\frac{2^{n+1}}{n!}\int_{0}^{\infty}\alpha^{n}g^{(n)}(2\alpha+z)\,d\alpha$
$\displaystyle=\frac{(-1)^{n+1}}{n!}\int_{0}^{\infty}\alpha^{n}g^{(n)}(\alpha+z)\,d\alpha=-\int_{0}^{\infty}g(\alpha+z)\,d\alpha.$

Here the first two integrals are absolutely convergent, by Fubini’s theorem.
The final integral may be improper, but it follows that $(Q_{n}g)^{\prime}=g$.

Now consider the case when $s=n+\delta>-1$, where
$n\in\mathbb{N}\cup\\{-1,0\\}$, $\delta\in(0,1)$, and $g\in\mathcal{V}_{n}$ is
holomorphic. Then $g\in\mathcal{V}_{n+1}$, $n+1\in\mathbb{N}\cup\\{0\\}$, and
(3.23) gives

$\int_{-\infty}^{\infty}\frac{g(\alpha+i\beta)}{(z+\alpha-i\beta+t)^{n+2}}\,d\beta=(-1)^{n+1}\frac{2\pi}{(n+1)!}g^{(n+1)}(2\alpha+z+t),$

for $z\in\mathbb{C}_{+}$, $t>0$ and almost all $\alpha>0$. We obtain from
(3.14) and (3.13) that

$\displaystyle C_{s}(Q_{s}g)(z)$
$\displaystyle=-\frac{2^{s}}{\pi}\int_{0}^{\infty}\int_{0}^{\infty}\int_{-\infty}^{\infty}\frac{\alpha^{s}\,g(\alpha+i\beta)\,d\beta\,d\alpha}{(z+\alpha-i\beta+t)^{n+2}}\,\frac{dt}{t^{\delta}}$
$\displaystyle=(-1)^{n}\frac{2^{s+1}}{(n+1)!}\int_{0}^{\infty}\int_{0}^{\infty}\alpha^{s}g^{(n+1)}(2\alpha+z+t)\,d\alpha\,\frac{dt}{t^{\delta}}$
$\displaystyle=\frac{(-1)^{n}}{(n+1)!}\int_{0}^{\infty}\int_{t}^{\infty}(\tau-t)^{s}g^{(n+1)}(\tau+z)\,d\tau\,\frac{dt}{t^{\delta}}$
$\displaystyle=C_{s}\frac{(-1)^{n}}{(n+1)!}\int_{0}^{\infty}\tau^{n+1}\,g^{(n+1)}(\tau+z)\,d\tau.$

As in (3.24), it follows that $(Q_{s}g)^{\prime}=g$. ∎

######  Corollary 3.8.

If $g\in\mathcal{V}_{s}$ is holomorphic, then $Q_{\sigma}g=Q_{s}g$ for all
$\sigma\geq s$.

###### Proof.

This is immediate from Proposition 3.7 and (3.8). ∎

The following representation of functions in $\mathcal{D}_{s}$ has appeared in
[2, Corollary 4.2] (see also [3, Lemma 3.13.2] for the case $s=1$).

######  Corollary 3.9.

Let $f\in\mathcal{D}_{s}$, $s>-1$. Then the sectorial limits

(3.25) $\displaystyle f(\infty):$
$\displaystyle=\lim_{z\to\infty,\;z\in\Sigma_{\psi}}\,f(z),$ (3.26)
$\displaystyle f(0):$ $\displaystyle=\lim_{z\to 0,\;z\in\Sigma_{\psi}}\,f(z),$

exist in $\mathbb{C}$ for every $\psi\in(0,\pi/2)$. Moreover, for all
$z\in\mathbb{C}_{+}\cup\\{0\\}$,

(3.27) $\displaystyle f(z)$ $\displaystyle=f(\infty)+(Q_{s}f^{\prime})(z)$
$\displaystyle=f(\infty)-\frac{2^{s}}{\pi}\int_{0}^{\infty}\alpha^{s}\int_{-\infty}^{\infty}\frac{f^{\prime}(\alpha+i\beta)\,d\beta}{(z+\alpha-i\beta)^{s+1}}\,d\alpha.$

With $f(0)$ defined as above, $f\in C(\overline{\Sigma}_{\psi})$ for every
$\psi\in(0,\pi/2)$.

###### Proof.

It follows from Proposition 3.7 that $(Q_{s}f^{\prime})^{\prime}=f^{\prime}$.
The statements follow from (3.8) and (3.9). ∎

######  Corollary 3.10.

For every $s>-1$ the space $\mathcal{D}_{s}$ equipped with the norm

$\|f\|_{\mathcal{D}_{s}}:=|f(\infty)|+\|f\|_{\mathcal{D}_{s,0}},\qquad
f\in\mathcal{D}_{s},$

is a Banach space.

###### Proof.

Let $s>-1$ be fixed and let $(f_{k})_{k=1}^{\infty}$ be a Cauchy sequence in
$\mathcal{D}_{s}$. Then (3.7) and Vitali’s theorem imply that
$(Q_{s}f_{k}^{\prime})_{k=1}^{\infty}$ converges uniformly on each
$\Sigma_{\psi}$ to a limit $g$ which is holomorphic on $\mathbb{C}_{+}$.
Moreover $(f_{k}(\infty))_{k=1}^{\infty}$ converges to a limit
$\zeta\in\mathbb{C}$. It follows from Proposition 3.7 that
$(f_{k})_{k=1}^{\infty}$ converges uniformly on $\Sigma_{\psi}$ to
$f:=\zeta+g$. Then $(f^{\prime}_{k})_{k=1}^{\infty}$ converges pointwise on
$\mathbb{C}_{+}$ to $g^{\prime}=f^{\prime}$. Applying Fatou’s lemma to the
sequences
$(\|f^{\prime}_{k}-f^{\prime}_{n}\|_{\mathcal{V}_{s}})_{n=k}^{\infty}$ for
fixed $k$, one sees that $\|f_{k}^{\prime}-f^{\prime}\|_{\mathcal{V}_{s}}\to
0$. So $f^{\prime}\in\mathcal{V}_{s}$ and $f\in\mathcal{D}_{s}$. By (3.8),
$f(\infty)=\zeta$ and so $\|f_{k}-f\|_{\mathcal{D}_{s}}\to 0$, $k\to\infty$. ∎

The argument used in the proof of Corollary 3.10 also provides the following
lemma of Fatou type (see also Lemma 4.11).

######  Corollary 3.11.

Let $s>-1$ and $(f_{k})_{k=1}^{\infty}\subset\mathcal{D}_{s}$ be such that
$\sup_{k\geq 1}\|f_{k}\|_{\mathcal{D}_{s}}<\infty$ and
$f(z)=\lim_{k\to\infty}f_{k}(z)$ exists for all $z\in\mathbb{C}_{+}$. Then
$f\in\mathcal{D}_{s}$.

Now employing (3.27), the estimates (3.7), (3.11) and (3.12), and Lemma 3.1,
we obtain the following estimates.

######  Corollary 3.12.

Let $\psi\in(0,\pi/2)$. For all $f\in\mathcal{D}_{s}$, $s>-1$,

$|f(z)|\leq\max\left(1,\frac{2^{s}}{\pi\cos^{s+1}\psi}\right)\|f\|_{\mathcal{D}_{s}},\qquad
z\in\Sigma_{\psi},$

and

$|f^{\prime}(z)|\leq\frac{(s+1)2^{s}}{\pi|z|\cos^{s+2}\psi}\|f^{\prime}\|_{\mathcal{V}_{s}},\qquad
z\in\Sigma_{\psi}.$

Moreover, $f$ is continuous on $\Sigma_{\psi}\cup\\{0\\}$ and

$\lim_{|z|\to 0,z\in\Sigma_{\psi}}zf^{\prime}(z)=0.$

######  Remark 3.13.

Corollary 3.12 implies that the point evaluation functionals
$\delta_{z},\,z\in\mathbb{C}_{+}$, are continuous on $\mathcal{D}_{s},\,s>-1$.
Using (3.18) and the principle set out in Section 2, we see that the function
$\lambda\mapsto r_{\lambda}^{\gamma}$ is holomorphic from $\mathbb{C}_{+}$ to
$\mathcal{D}_{s}$, for any $s>-1$, $\gamma>0$.

###  3.3. More functions in $\mathcal{D}_{s}$ and their properties

In this section we give more examples of functions from $\mathcal{D}_{s}$ and
note some additional elementary properties which will be relevant for the
sequel.

######  Proposition 3.14.

For $s>0$, $\mathcal{B}\overset{i}{\hookrightarrow}\mathcal{D}_{s}$.

###### Proof.

For $s>0$ and $f\in\mathcal{B}$ we have

$\displaystyle\|f\|_{\mathcal{D}_{s,0}}$
$\displaystyle=|f(\infty)|+\int_{0}^{\infty}\alpha^{s}\int_{-\infty}^{\infty}\frac{|f^{\prime}(\alpha+i\beta)|}{|\alpha+i\beta|^{s+1}}\,d\beta\,d\alpha$
$\displaystyle\leq\|f\|_{\infty}+2\int_{0}^{\infty}\sup_{\beta\in\mathbb{R}}\,|f^{\prime}(\alpha+i\beta)|\,\int_{0}^{\infty}\frac{dt}{(t^{2}+1)^{(s+1)/2}}\,d\alpha$
$\displaystyle\leq\max\\{1,B(1/2,s/2)\\}\|f\|_{\mathcal{B}}.$

Thus $\mathcal{B}$ is continuously included in $\mathcal{D}_{s}$. ∎

######  Remark 3.15.

The representation (3.27) in Corollary 3.9 extends the reproducing formula for
$\mathcal{B}$, i.e., (3.27) for $s=1$, to a larger class of functions.

Recall that $\mathcal{LM}\overset{i}{\hookrightarrow}\mathcal{B}$, see [6,
Section 2.4], Thus, in view of Proposition 3.14, we have

$\mathcal{LM}\overset{i}{\hookrightarrow}\mathcal{B}\overset{i}{\hookrightarrow}\mathcal{D}_{s},\qquad
s>0.$

We will show in Corollary 5.2 that $\mathcal{B}$ is dense in $\mathcal{D}_{s}$
for every $s>0$, and hence $\mathcal{D}_{s}$ is dense in
$\mathcal{D}_{\sigma}$ for all $\sigma>s>0$. On the other hand, we will show
in Corollary 5.2 that $\mathcal{B}$ is not dense in $\mathcal{D}^{\infty}_{s}$
for $s>0$.

For $f\in\operatorname{Hol}(\mathbb{C}_{+})$, let

$\tilde{f}(z):=f(1/z),\qquad f_{t}(z):=f(tz),\qquad t>0,\quad
z\in\mathbb{C}_{+}.$

######  Lemma 3.16.

Let $s>-1$ and $t>0$. Then

  1. (i)

$f\in\mathcal{D}_{s}$ if and only if $\tilde{f}\in\mathcal{D}_{s}$ and

$\|f-f(\infty)\|_{\mathcal{D}_{s}}=\|\tilde{f}-f(0)\|_{\mathcal{D}_{s}}.$

  2. (ii)

If $f$ is bounded away from $0$ and $f\in\mathcal{D}_{s}$, then
$1/f\in\mathcal{D}_{s}$.

  3. (iii)

$f\in\mathcal{D}_{s}$ if and only if $f_{t}\in\mathcal{D}_{s}$, and
$\|f\|_{\mathcal{D}_{s}}=\|f_{t}\|_{\mathcal{D}_{s}}$.

###### Proof.

Note that

$\displaystyle\|\tilde{f}^{\prime}\|_{\mathcal{V}_{s}}$
$\displaystyle=\int_{-\pi/2}^{\pi/2}\cos^{s}\varphi\int_{0}^{\infty}\frac{|f^{\prime}(\rho^{-1}e^{-i\varphi})|}{\rho^{2}}\,d\rho\,d\phi$
$\displaystyle=\int_{-\pi/2}^{\pi/2}\cos^{s}\varphi\int_{0}^{\infty}|f^{\prime}(re^{i\varphi})|\,dr\,d\varphi$
$\displaystyle=\|f^{\prime}\|_{\mathcal{V}_{s}}.$

Moreover, by Corollary 3.9, $\tilde{f}(\infty)=f(0)$. This proves (i). The
other statements are very easy. ∎

######  Remarks 3.17.

1\. Neither of the spaces $\mathcal{B}$ and $\mathcal{D}_{0}$ is contained in
the other. Indeed, the function $e^{-z}\in\mathcal{LM}\subset\mathcal{B}$ but
$e^{-z}\not\in\mathcal{D}_{0}$ (see Example 3.4). On the other hand, there are
bounded functions in $\mathcal{D}_{0}$ which are not in $\mathcal{B}$, for
example the function $\exp(\operatorname{arccot}z)\in\mathcal{D}_{0}$ and is
bounded but it is not in $\mathcal{B}$ (see Example 3.18).

More generally, for $\nu\geq 0$, let
$f_{\nu}(z)=z^{\nu}e^{-z},\,z\in\mathbb{C}_{+}$. Then
$f_{\nu}\in\mathcal{D}_{s}$ if and only if $s>\nu$ (see Example 3.4). Note
that, if $\nu>0$, $f_{\nu}$ is not bounded on any right half-plane. One can
show that if $f\in\mathcal{D}_{s,0}$, then

$|f(z)|\leq\frac{2^{s}}{\pi}\|f\|_{\mathcal{D}_{s,0}}\left(1+\frac{4\beta^{2}}{\alpha^{2}}\right)^{(s+1)/2},\quad
z=\alpha+i\beta\in\mathbb{C}_{+}.$

The function $\log(1+z)e^{-z}$ is in $\mathcal{D}_{s},s>0$, but is also
unbounded on every right half-plane.

2\. Since $e^{-z}\in\mathcal{D}_{s}$ for $s>0$, it follows from Lemma 3.16
that the functions $e^{-t/z}$ are in $\mathcal{D}_{s}$ for all $t>0,\,s>0$.
This shows that functions $f\in\mathcal{D}_{s}$ may not have full limits at
infinity or at zero. However, the properties (3.25) and (3.26) in Corollary
3.9 establish values for $f$ at infinity and at zero as sectorial limits.

3\. The spaces $\mathcal{D}_{s},\,s>-1$, are invariant under shifts given by

$(T(\tau)f)(z)=f(z+\tau),\qquad
f\in\mathcal{D}_{s},\,\tau\in\mathbb{C}_{+},\,z\in\mathbb{C}_{+}.$

Indeed these operators form a bounded $C_{0}$-semigroup on $\mathcal{D}_{s}$.
See Section 11 for a proof. On the other hand, $\mathcal{D}_{s}$ are not
invariant under the vertical shifts when $\tau\in i\mathbb{R}$, as we see in
the following example.

######  Example 3.18.

As stated in Example 3.5, the function $\operatorname{arccot}$ is in
$\mathcal{D}_{s}$ for all $s>-1$. Let

$g(z)=\exp(\operatorname{arccot}(z)),\qquad\quad z\in\mathbb{C}_{+},$

Since $|\operatorname{Re}\operatorname{arccot}(z)|\leq\pi/2$,
$\|g\|_{\infty}=\exp(\pi/2)$. For $s>-1$, we have

$\displaystyle\|g\|_{\mathcal{D}_{s}^{\infty}}$
$\displaystyle\leq\|g\|_{\infty}+\|g\cdot(\operatorname{arccot})^{\prime}\|_{\mathcal{V}_{s}}\leq\exp(\pi/2)(1+\|\operatorname{arccot}\|_{\mathcal{D}_{s}}).$

Thus $g\in\mathcal{D}_{s}^{\infty}$ for all $s>-1$.

However the boundary function of $g$ is not continuous at $z=\pm i$. Indeed,
for a fixed $\epsilon>0$,

$\operatorname{arccot}(i+i\epsilon)=\frac{1}{2i}\log\left(1+\frac{2}{\epsilon}\right),$

hence

$g(i+i\epsilon)=\exp{(-i\log(1+2/\epsilon)^{1/2})}$

does not have a limit as $\epsilon\to 0+$.

Note that $g(\epsilon+i)$ does not converge as $\epsilon\to 0+$. This means
that if $f(z):=g(z-i)$, then $f$ does not have a sectorial limit at 0 and
therefore does not belong to $\mathcal{D}_{s}$ for any $s>-1$. Thus
$\mathcal{D}_{s}$ and $\mathcal{D}_{s}^{\infty}$ are not invariant under
vertical shifts.

###  3.4. Bernstein functions and $\mathcal{D}_{s}$

Recall that a holomorphic function $g:\mathbb{C}_{+}\to\mathbb{C}_{+}$ is a
_Bernstein function_ if it is of the form

(3.28) $g(z)=a+bz+\int_{(0,\infty)}(1-e^{-zs})\,d\mu(s),$

where $a\geq 0$, $b\geq 0$ and $\mu$ is a positive Borel measure on
$(0,\infty)$ such that $\int_{(0,\infty)}\frac{s}{1+s}\,d\mu(s)<\infty$. The
following properties of Bernstein functions $g$ will be used (these properties
differ slightly from those used in [5]):

  1. (B1)

$g$ maps $\Sigma_{\psi}$ into $\Sigma_{\psi}$ for each $\psi\in[0,\pi/2]$, see
[50, Proposition 3.6] or [5, Proposition 2.1(1)].

  2. (B2)

$g$ is increasing on $(0,\infty)$.

  3. (B3)

For all $z\in\mathbb{C}_{+}$,

$g(\operatorname{Re}z)\leq\operatorname{Re}g(z)\leq|g(z)|,\qquad|g^{\prime}(z)|\leq
g^{\prime}(\operatorname{Re}z).$

Here the first inequality follows from taking the real parts in (3.28), and
the second inequality is shown in [6, Section 3.5, (B3)].

Further information on Bernstein functions can be found in [50].

######  Lemma 3.19.

Let $g$ be a Bernstein function, $\lambda\in\mathbb{C}_{+}$, and

$f(z;\lambda):=(\lambda+g(z))^{-1},\qquad z\in\mathbb{C}_{+}.$

Then $f(\cdot;\lambda)\in\mathcal{D}_{s}$ for $s>2$ and

$\|f(\cdot;\lambda)\|_{\mathcal{D}_{s,0}}\leq\frac{2^{s}}{(s-2)|\lambda|}.$

###### Proof.

We have

$f^{\prime}(z;\lambda)=-\frac{g^{\prime}(z)}{(\lambda+g(z))^{2}},$

and then, for $\psi\in(-\pi/2,\pi/2)$, using Lemma 2.1, (B2) and (B3),

$\displaystyle\int_{0}^{\infty}|f^{\prime}(\rho e^{i\psi};\lambda)|\,d\rho$
$\displaystyle=\int_{0}^{\infty}\frac{|g^{\prime}(\rho
e^{i\psi})|}{|\lambda+g(\rho e^{i\psi})|^{2}}\,d\rho$
$\displaystyle\leq\frac{1}{\cos^{2}((|\psi|+\pi/2)/2)}\int_{0}^{\infty}\frac{g^{\prime}(\rho\cos\psi)}{(|\lambda|+|g(\rho
e^{i\psi})|)^{2}}\,d\rho$
$\displaystyle\leq\frac{1}{\cos^{2}((|\psi|+\pi/2)/2)}\int_{0}^{\infty}\frac{g^{\prime}(\rho\cos\psi)}{(|\lambda|+g(\rho\cos\psi))^{2}}\,d\rho$
$\displaystyle\leq\frac{1}{\cos^{2}((|\psi|+\pi/2)/2)\cos\psi}\int_{0}^{\infty}\frac{dt}{(|\lambda|+t)^{2}}$
$\displaystyle=\frac{1}{\cos^{2}((|\psi|+\pi/2)/2)\cos\psi}\cdot\frac{1}{|\lambda|}.$

If $s>2$, then

$\displaystyle|\lambda|\|f(\cdot;\lambda)\|_{\mathcal{D}_{s,0}}$
$\displaystyle=|\lambda|\int_{-\pi/2}^{\pi/2}\cos^{s}\psi\int_{0}^{\infty}|f^{\prime}(\rho
e^{i\psi};\lambda)|\,d\rho\,d\psi$ $\displaystyle\leq
2\int_{0}^{\pi/2}\frac{\cos^{s-1}\psi}{\cos^{2}((\psi+\pi/2)/2)}\,d\psi=4\int_{0}^{\pi/2}\frac{\cos^{s-1}\psi}{1-\sin\psi}\,d\psi$
$\displaystyle=4\int_{0}^{1}(1+t)^{s-2}(1-t)^{s-3}\,dt<\frac{2^{s}}{s-2}.\qed$

###  3.5. Algebras associated with $\mathcal{D}_{s}$

The spaces $\mathcal{D}_{s},\,s>-1$, are not algebras, but there are some
related algebras. Consider the Banach spaces
$\mathcal{D}_{s}^{\infty}:=\mathcal{D}_{s}\cap H^{\infty}(\mathbb{C}_{+})$
equipped with the norm

$\|f\|_{\mathcal{D}_{s}^{\infty}}:=\|f\|_{\infty}+\|f^{\prime}\|_{\mathcal{V}_{s}}.$

Thus $\mathcal{D}_{s}^{\infty}$ is the space of bounded holomorphic functions
on $\mathbb{C}_{+}$ such that

$f(\infty):=\lim_{|z|\to\infty,z\in\Sigma_{\psi}}\,f(z)$

exists for every $\psi\in(0,\pi/2)$, and

$\|f\|_{\mathcal{D}_{s}^{\infty}}:=\|f\|_{\infty}+\int_{0}^{\infty}\alpha^{s}\int_{-\infty}^{\infty}\frac{|f^{\prime}(\alpha+i\beta)|}{(\alpha^{2}+\beta^{2})^{(s+1)/2}}\,d\alpha\,d\beta<\infty.$

Then $(\mathcal{D}_{s}^{\infty},\|\cdot\|_{\mathcal{D}_{s}})$ is a Banach
algebra, and in particular

(3.29)
$\|fg\|_{\mathcal{D}_{s}^{\infty}}\leq\|f\|_{\mathcal{D}_{s}^{\infty}}\|g\|_{\mathcal{D}_{s}^{\infty}},\quad
f,g\in\mathcal{D}_{s}^{\infty}.$

By Proposition 3.14,
$\mathcal{B}\overset{i}{\hookrightarrow}\mathcal{D}_{s}^{\infty}$ for $s>0$,
and the embeddings are continuous.

It follows from Lemma 3.16 that

$f\in\mathcal{D}_{s}^{\infty}\quad\text{if and only
if}\quad\tilde{f}(z):=f(1/z)\in\mathcal{D}_{s}^{\infty},$

and

$\|f\|_{\mathcal{D}_{s}^{\infty}}=\|\tilde{f}\|_{\mathcal{D}_{s}^{\infty}}.$

Moreover, the spectrum of $f$ in $\mathcal{D}_{s}^{\infty}$ is the closure of
the range of $f$. In particular, the spectral radius of $f$ is
$\|f\|_{\infty}$.

Now we consider the linear space

$\mathcal{D}_{\infty}:=\bigcup_{s>-1}\mathcal{D}_{s}.$

We will show that $\mathcal{D}_{\infty}$ is an algebra, which opens the way to
an operator functional calculus on $\mathcal{D}_{\infty}$.

######  Lemma 3.20.

For $s,\sigma>-1$, let $f\in\mathcal{D}_{s}$ and $g\in\mathcal{D}_{\sigma}$.
Then

$h:=fg\in\mathcal{D}_{s+\sigma+1},$

and

(3.30)
$\|h\|_{\mathcal{D}_{s+\sigma+1}}\leq\left(2+\frac{2^{s}+2^{\sigma}}{\pi}\right)\|f\|_{\mathcal{D}_{s}}\,\|g\|_{\mathcal{D}_{\sigma}}.$

Hence, $\mathcal{D}_{\infty}$ is an algebra.

###### Proof.

By Corollary 3.12, for $\rho>0$ and $|\varphi|<\pi/2$, we have

$\displaystyle|f(\rho e^{i\varphi})|$
$\displaystyle\leq\left(1+\frac{2^{s}}{\pi\cos^{s+1}\varphi}\right)\|f\|_{\mathcal{D}_{s}},$
$\displaystyle|g(\rho e^{i\varphi})|$
$\displaystyle\leq\left(1+\frac{2^{\sigma}}{\pi\cos^{\sigma+1}\varphi}\right)\|g\|_{\mathcal{D}_{\sigma}}.$

Hence

$\displaystyle\|h\|_{\mathcal{D}_{s+\sigma+1,0}}$
$\displaystyle=\int_{-\pi/2}^{\pi/2}\cos^{s+\sigma+1}\varphi\int_{0}^{\infty}\left|f^{\prime}(\rho
e^{i\varphi})g(\rho e^{i\varphi})+f(\rho e^{i\varphi})g^{\prime}(\rho
e^{i\varphi})\right|\,d\rho\,d\varphi$
$\displaystyle\leq\left(1+\frac{2^{\sigma}}{\pi}\right)\|f\|_{\mathcal{D}_{s,0}}\,\|g\|_{\mathcal{D}_{\sigma}}+\left(1+\frac{2^{s}}{\pi}\right)\|f\|_{\mathcal{D}_{s}}\,\|g\|_{\mathcal{D}_{\sigma,0}}.$

This shows that $h\in\mathcal{D}_{q}$, and (3.30) follows easily. ∎

###  3.6. Derivatives of functions in $\mathcal{D}_{s}$

This section further clarifies the behaviour of the derivatives of functions
from $\mathcal{D}_{s}$, and Lemma 3.21 is of independent interest. Corollary
3.22 will be used in Section 10.3. Here $zf^{\prime}$ denotes the function
$z\mapsto zf^{\prime}(z)$.

######  Lemma 3.21.

Let $f\in\mathcal{D}_{s}$, $s>-1$. Then $zf^{\prime}\in\mathcal{D}_{s+1}$, and
there exists $C_{s}$ (independent of $f$) such that

(3.31)
$\|zf^{\prime}(z)\|_{\mathcal{D}_{s+1}}\leq{C}_{s}\|f\|_{\mathcal{D}_{s}}.$

and

(3.32) $\|f(tz)-f(\tau
z)\|_{\mathcal{D}_{s+1}}\leq\frac{C_{s}\|f\|_{\mathcal{D}_{s}}}{\min\\{t,\tau\\}}|t-\tau|,\qquad
t,\tau>0.$

###### Proof.

Note that

$\|zf^{\prime}\|_{\mathcal{D}_{s+1}}\leq\|f^{\prime}\|_{\mathcal{V}_{s+1}}+\|zf^{\prime\prime}\|_{\mathcal{V}_{s+1}}\leq\|f\|_{\mathcal{D}_{s}}+\|zf^{\prime\prime}\|_{\mathcal{V}_{s+1}}.$

So, for (3.31), it suffices to consider
$\|zf^{{}^{\prime\prime}}\|_{\mathcal{V}_{s+1}}$. The argument is similar to
Example 3.3 and the proof of Proposition 3.6. By Corollary 3.9, for fixed
$\sigma>s$,

$f^{\prime\prime}(z)=-c_{\sigma}\int_{-\pi/2}^{\pi/2}\cos^{\sigma}\varphi\int_{0}^{\infty}\frac{\rho^{\sigma+1}f^{\prime}(\rho
e^{i\varphi})}{(z+\rho e^{-i\varphi})^{\sigma+3}}\,d\rho\,d\varphi,\quad
z\in\mathbb{C}_{+},$

where $c_{\sigma}=(\sigma+1)(\sigma+2)\frac{2^{\sigma}}{\pi}$. Then we have

$\displaystyle c_{\sigma}^{-1}\|zf^{\prime\prime}(z)\|_{\mathcal{V}_{s+1}}$
$\displaystyle=\int_{-\pi/2}^{\pi/2}\cos^{s+1}\psi\int_{0}^{\infty}r\left|\int_{-\pi/2}^{\pi/2}\cos^{\sigma}\varphi\int_{0}^{\infty}\frac{\rho^{\sigma+1}f^{\prime}(\rho
e^{i\varphi})}{(re^{i\psi}+\rho
e^{-i\varphi})^{\sigma+3}}\,d\rho\,d\varphi\right|\,dr\,d\psi$
$\displaystyle\leq\int_{-\pi/2}^{\pi/2}\cos^{\sigma}\varphi\int_{-\pi/2}^{\pi/2}\cos^{s+1}\psi\int_{0}^{\infty}\int_{0}^{\infty}\frac{t|f^{\prime}(\rho
e^{i\varphi})|\,dt\,d\rho}{(t^{2}+2t\cos(\varphi+\psi)+1)^{(\sigma+3)/2}}\,d\psi\,d\varphi$
$\displaystyle=\int_{-\pi/2}^{\pi/2}\cos^{\sigma-s}\varphi\,G_{h,\beta,s+1}(\varphi)\int_{0}^{\infty}|f^{\prime}(\rho
e^{i\varphi})|\,d\rho\,d\varphi,$

where $h(t)=t+t^{\sigma},\,t\in(0,1)$, $\beta=(\sigma+3)/2$ and
$G_{h,\beta,s+1}(\varphi)$ is defined in (3.15), noting that
$2\beta-(s+1)-2=\sigma-s$. Now the estimate (3.31) follows from Lemma 3.2(b).

For (3.32), let $g(z)=f(\tau z)-f(tz)$. Without loss, assume that $0<t<\tau$.
Then

$g^{\prime}(z)=\tau f^{\prime}(\tau
z)-tf^{\prime}(tz)=\int_{t}^{\tau}\left(f^{\prime}(rz)+rzf^{\prime\prime}(rz)\right)\,dr=\int_{t}^{\tau}\frac{d}{dz}(zf^{\prime}(rz))\,dr.$

Hence, by Fubini’s theorem,

$\displaystyle\|g\|_{\mathcal{D}_{s+1}}$
$\displaystyle\leq\int_{\mathbb{C}_{+}}\int_{t}^{\tau}\frac{(\operatorname{Re}z)^{s+1}}{|z|^{s+2}}\left|\frac{d}{dz}(zf^{\prime}(rz))\right|\,dr\,dS(z)$
$\displaystyle=\int_{t}^{\tau}\frac{\|zf^{\prime}(z)\|_{\mathcal{D}_{s+1}}}{r}\,dr\leq\frac{C_{s}\|f\|_{\mathcal{D}_{s}}}{t}(\tau-t),$

since the $\mathcal{D}_{s+1}$-norm is invariant under the change of variable
$z\mapsto rz$ (Lemma 3.16(iii)). ∎

The following corollary is easily proved by induction.

######  Corollary 3.22.

If $f\in\mathcal{D}_{\infty}$ and $n\in\mathbb{N}$, then
$z^{n}f^{(n)}\in\mathcal{D}_{\infty}$.

######  Remark 3.23.

Lemma 3.21 is sharp in the sense that for any $s>0$ and $\sigma\in(-1,s+1)$,
there exist functions $f\in\mathcal{D}_{s}$ for which
$zf^{\prime}\notin\mathcal{D}_{\sigma}$. For example, the function
$f_{\nu}(z):=z^{\nu}e^{-z}$ has these properties if
$\max\\{0,\sigma-1\\}<\nu<s$. This follows directly from Example 3.4.

##  4\. Hardy-Sobolev algebras on sectors

###  4.1. $H^{p}$-spaces on the right half-plane and their norms

In this section and in Section 4.2 we will study the Hardy spaces
$H^{1}(\Sigma_{\psi})$ defined on sectors $\Sigma_{\psi},\,\psi\in(0,\pi)$.
The properties of such spaces are similar to the properties of the classical
Hardy space $H^{1}(\mathbb{C}_{+})$, though their theory seems to be more
involved. The Hardy spaces $H^{p}(\Sigma_{\psi})$ have been studied, mostly
for $p>1$, but the results are scattered around various places in the
literature, which is often obscure, and some proofs contain rather
complicated, incomplete or vague arguments. We propose a streamlined (and
probably new) approach avoiding the use of Carleson measures or log-convexity,
and we obtain a new result (Corollary 4.3) on the way. The case $p=1$ does not
require any significant adjustments as we illustrate below. Standard
references for the theory of Hardy spaces on the right half-plane are [20] and
[28].

We set out the situation when $\psi=\pi/2$ in this section, and the case of
general $\psi$ in Section 4.2. Although we are mainly interested in
$H^{1}$-spaces, we present statements which are valid for $H^{p}$-spaces with
$p\in[1,\infty)$, since the arguments are the same for all such $p$.

Let $1\leq p<\infty$. The classical Hardy space $H^{p}(\mathbb{C}_{+})$ in the
right half-plane $\mathbb{C}_{+}$ is defined as

$H^{p}(\mathbb{C}_{+})=\left\\{g\in\mbox{Hol}(\mathbb{C}_{+}):\|g\|_{p}:=\sup_{\alpha>0}\,\left(\int_{-\infty}^{\infty}|g(\alpha+i\beta)|^{p}\,d\beta\right)^{1/p}<\infty\right\\}.$

It is well-known that $\|\cdot\|_{p}$ is a norm on $H^{p}(\mathbb{C}_{+})$ and
$(H^{p}(\mathbb{C}_{+}),\|\cdot\|_{p})$ is a Banach space. Moreover, for
almost every $t\in\mathbb{R}$ there exists a sectorial limit
$g(it):=\lim_{z\to it}g(z)$ in $\mathbb{C}$. For every $g\in
H^{p}(\mathbb{C}_{+})$ one has $\lim_{\alpha\to 0}g(\alpha+i\cdot)=g(i\cdot)$
in $L^{p}(\mathbb{R})$, and
$\|g\|_{H^{p}(\mathbb{C}_{+})}:=\|g\|_{p}=\|g(i\cdot)\|_{L^{p}(\mathbb{R})}$.

One may also consider the normed space
$(H^{p}(\Sigma_{\pi/2}),\|\cdot\|_{H^{p}(\Sigma_{\pi/2})})$ as the space of
all $g\in\operatorname{Hol}(\mathbb{C}_{+})$ such that

$\|g\|_{H^{p}(\Sigma_{\pi/2})}:=\sup_{|\varphi|<\pi/2}\,\left(\int_{0}^{\infty}\left(|g(te^{i\varphi})|^{p}+|g(te^{-i\varphi})|^{p}\right)\,dt\right)^{1/p}<\infty.$

and $(H^{p}_{*}(\Sigma_{\pi/2}),\|\cdot\|_{H^{p}_{*}(\Sigma_{\pi/2})})$ as

$\left\\{g\in\operatorname{Hol}(\mathbb{C}_{+}):\|g\|_{H^{p}_{*}(\Sigma_{\pi/2})}:=\sup_{|\varphi|<\pi/2}\left(\int_{0}^{\infty}|g(te^{i\varphi})|^{p}\,dt\right)^{1/p}<\infty\right\\}.$

It is clear that $H^{p}(\Sigma_{\pi/2})$ and $H^{p}_{*}(\Sigma_{\pi/2})$
coincide as vector spaces, and

$\|g\|_{H^{p}_{*}(\Sigma_{\pi/2})}\leq\|g\|_{H^{p}(\Sigma_{\pi/2})}\leq
2^{1/p}\|g\|_{H^{p}_{*}(\Sigma_{\pi/2})}.$

######  Lemma 4.1.

Let $p\in[1,\infty)$. Then $H^{p}(\Sigma_{\pi/2})\subset
H^{p}(\mathbb{C}_{+})$ and

(4.1) $\|g\|_{p}\leq\|g\|_{H^{p}(\Sigma_{\pi/2})},\qquad g\in
H^{p}(\Sigma_{\pi/2}).$

###### Proof.

Fix $p\in[1,\infty)$. For fixed $\gamma\in(1/2,1)$ define

$g_{\gamma}(z):=(\gamma
z^{\gamma-1})^{1/p}g(z^{\gamma})\in\operatorname{Hol}(\Sigma_{\pi/(2\gamma)}).$

Note that

(4.2)
$\int_{0}^{\infty}|g_{\gamma}(te^{i\varphi})|^{p}\,dt=\int_{0}^{\infty}|g(te^{i\gamma\varphi})|^{p}\,dt,$

and

$\int_{-\pi/2}^{\pi/2}\int_{0}^{\infty}|g_{\gamma}(te^{i\varphi})|^{p}\,dt\,d\varphi\leq\frac{\pi}{2}\|g\|^{p}_{{H^{p}(\Sigma_{\pi/2})}}.$

By Fubini’s theorem and Hölder’s inequality there exist sequences
$(t_{1,n})_{n\geq 1}$ and $(t_{2,n})_{n\geq 1}$ such that $0<t_{1,n}<t_{2,n}$,
$t_{1,n}\to 0$, $t_{2,n}\to\infty$ as $n\to\infty$, and

(4.3)
$\displaystyle\lim_{n\to\infty}\,t_{1,n}\int_{-\pi/2}^{\pi/2}|g_{\gamma}(t_{1,n}e^{i\varphi})|\,d\varphi$
$\displaystyle=0,$ (4.4)
$\displaystyle\lim_{n\to\infty}\,\int_{-\pi/2}^{\pi/2}|g_{\gamma}(t_{2,n}e^{i\varphi})|\,d\varphi$
$\displaystyle=0.$

Let $\Omega_{n}:=\\{z\in\mathbb{C}_{+}:\,t_{1,n}<|z|<t_{2,n}\\}$. By Cauchy’s
formula,

(4.5) $g_{\gamma}(z)=\frac{\alpha}{\pi
i}\int_{\partial\Omega_{n}}\frac{g_{\gamma}(\lambda)}{(\lambda-z)(\lambda+\overline{z})}\,d\lambda,\quad
z=\alpha+i\beta\in\Omega_{n},$

for large $n$. Passing to the limit in (4.5) as $n\to\infty$, and using (4.2),
(4.3) and (4.4), we obtain the Poisson formula

(4.6)
$g_{\gamma}(z)=\frac{\alpha}{\pi}\int_{-\infty}^{\infty}\frac{g_{\gamma}(it)}{(t-\beta)^{2}+\alpha^{2}}\,dt.$

Hence, by Young’s inequality and (4.2), for every $\alpha>0$,

$\|g_{\gamma}(\alpha+i\cdot)\|_{L^{p}(\mathbb{R})}\leq\|g_{\gamma}(i\cdot)\|_{L^{p}(\mathbb{R})}\leq\|g\|_{H^{p}(\Sigma_{\pi/2})}.$

Letting $\gamma\to 1$, Fatou’s Lemma implies (4.1). ∎

######  Lemma 4.2.

Let $p\in[1,\infty)$. Then $H^{p}(\mathbb{C}_{+})\subset
H^{p}(\Sigma_{\pi/2})$ and

(4.7) $\|g\|_{H^{p}(\Sigma_{\pi/2})}\leq\|g\|_{p},\qquad g\in
H^{p}(\mathbb{C}_{+}).$

###### Proof.

First let $p=2$ so that $g\in H^{2}(\mathbb{C}_{+})$. Then, by [21, Ch.VIII,
p.508], there exists $f\in L^{1}(\mathbb{R},e^{\pi|t|}\,dt)$ such that $f\geq
0$ on $\mathbb{R}$, and for all $|\varphi|\leq\pi/2$,

$\int_{0}^{\infty}|g(te^{i\varphi})|^{2}\,dt=\int_{-\infty}^{\infty}e^{2\varphi
t}f(t)\,dt.$

Hence

$\displaystyle\int_{0}^{\infty}\bigl{(}|g(te^{i\varphi})|^{2}+|g(te^{-i\varphi})|^{2}\bigr{)}\,dt$
$\displaystyle=2\int_{-\infty}^{\infty}\cosh(2\varphi t)f(t)\,dt$
$\displaystyle\leq 2\int_{-\infty}^{\infty}\cosh(\pi t)f(t)\,dt$
$\displaystyle=\int_{0}^{\infty}\bigl{(}|g(it)|^{2}+|g(-it)|^{2}\bigr{)}\,dt,$

and (4.1) holds for $p=2$.

Let $p\in[1,\infty),\,p\not=2$, be fixed, and $g\in H^{p}(\mathbb{C}_{+})$.
Then $g(z)=B(z)\tilde{g}(z)$, $z\in\mathbb{C}_{+}$, where $B$ is the Blaschke
product associated with $g$ and $\tilde{g}$ has no zeros in $\mathbb{C}_{+}$.
Then there is a well-defined holomorphic function
$g_{p}(z)=[\tilde{g}(z)]^{p/2}$ on $\mathbb{C}_{+}$ and $g_{p}\in
H^{2}(\mathbb{C}_{+})$. Using (4.1) for $p=2$, for all $|\varphi|<\pi/2$, we
have

(4.8)
$\displaystyle\int_{0}^{\infty}\bigl{(}|g(te^{i\varphi})|^{p}+|g(te^{-i\varphi})|^{p}\bigr{)}\,dt=\int_{0}^{\infty}\bigl{(}|g_{p}(te^{i\varphi})|^{2}+|g_{p}(te^{-i\varphi})|^{2}\bigr{)}\,dt$
$\displaystyle\leq\int_{-\infty}^{\infty}|g_{p}(it)|^{2}\,dt=\int_{-\infty}^{\infty}|g(it)|^{p}\,dt=\|g\|_{p}^{p},$

and (4.7) follows. ∎

Lemmas 4.1 and 4.2 imply the next statement.

######  Corollary 4.3.

Let $p\in[1,\infty)$. Then $H^{p}(\Sigma_{\pi/2})=H^{p}(\mathbb{C}_{+})$, and
for every $g\in H^{p}(\mathbb{C}_{+})$,

(4.9) $\|g\|_{H^{p}(\Sigma_{\pi/2})}=\|g\|_{H^{p}(\mathbb{C}_{+})},$

and

(4.10) $\|g\|_{H^{p}_{*}(\Sigma_{\pi/2})}\leq\|g\|_{H^{p}(\mathbb{C}_{+})}\leq
2^{1/p}\|g\|_{H^{p}_{*}(\Sigma_{\pi/2})}.$

Note that the two-sided estimate (4.10) was proved in [49] and [41] in a more
complicated way (see also [1], [18], [44], [53], [54]). The coincidence of
norms in (4.9) seems not to have been noted before. It appears to be quite
useful, as we will see in the proof of Corollary 4.5 below.

######  Remark 4.4.

The two-sided estimate (4.10) is sharp (and cannot be improved). Indeed, let
$p\in[1,\infty)$ and let

$f_{k}(z):=\frac{1}{\pi^{1/p}(z+1+ik)^{2/p}},\qquad k\in\mathbb{N}.$

Then for all $k$, we have $\|f_{k}\|_{H^{p}(\mathbb{C}_{+})}=1$ and, by direct
estimates,

$\displaystyle\left(\|f_{k}\|_{H^{p}_{*}(\Sigma_{\pi/2})}\right)^{p}=\frac{1}{\pi}\int_{0}^{\infty}|f_{k}(te^{-i\pi/2})|^{p}\,dt$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\infty}\frac{dt}{(t-k)^{2}+1}\,dt=\frac{1}{2}+\frac{\arctan
k}{\pi}.$

[In fact, for all $f\in\mathcal{H}^{p}_{*}(\Sigma_{\pi/2})$,
$\|f\|_{H^{p}_{*}(\Sigma_{\pi/2})}$ is attained at the boundary of
$\Sigma_{\psi/2}$.] Thus,

$\|f_{0}\|_{H^{p}(\mathbb{C}_{+})}=2^{1/p}\|f_{0}\|_{H^{p}_{*}(\Sigma_{\pi/2})}\qquad\text{and}\qquad\lim_{k\to\infty}\,\frac{\|f_{k}\|_{H^{p}_{*}(\Sigma_{\pi/2})}}{\|f_{k}\|_{H^{p}(\mathbb{C}_{+})}}=1.$

######  Corollary 4.5.

Let $g\in H^{p}(\Sigma_{\pi/2})$, $p\in[1,\infty)$. Then there exist $g(\pm
it):=\lim_{\varphi\to\pm\pi/2}g(te^{\pm i\varphi})$ for a.e.
$t\in\mathbb{R}_{+}$, $g(\pm i\cdot)\in L^{p}(\mathbb{R}_{+})$, and

(4.11) $\lim_{\varphi\to\pm\pi/2}\,\int_{0}^{\infty}|g(te^{i\varphi})-g(\pm
it)|^{p}\,dt=0.$

As a consequence, for every $g\in H^{p}(\Sigma_{\pi/2})$,

(4.12) $\|g\|_{H^{p}(\Sigma_{\pi/2})}=\|g(i\cdot)\|_{L^{p}(\mathbb{R})}.$

###### Proof.

Let $p\in[1,\infty)$ be fixed. By Corollary 4.3, it suffices to prove (4.11)
for $g\in H^{p}(\mathbb{C}_{+})$. If $g\in H^{p}(\mathbb{C}_{+})$, then as
recalled above, for almost all $t\in\mathbb{R}$ there exists a sectorial limit
$g(it):=\lim_{z\to it}g(z)$, and $g(i\cdot)\in L^{p}(\mathbb{R})$. Therefore,
we also have $\lim_{\varphi\to\pm\pi/2}g(te^{\pm i\varphi})=g(\pm it)$ for
almost all $t$. Using this and Fatou’s Lemma, we infer from (4.8) that

$\displaystyle\limsup_{\varphi\to\pi/2}\,$
$\displaystyle\int_{0}^{\infty}\bigl{(}|g(te^{i\varphi})|^{p}+|g(te^{-i\varphi})|^{p}\bigr{)}\,dt\leq\int_{-\infty}^{\infty}|g(it)|^{p}\,dt$
$\displaystyle\leq\liminf_{\varphi\to\pi/2}\,\int_{0}^{\infty}\bigl{(}|g(te^{i\varphi})|^{p}+|g(te^{-i\varphi})|^{p}\bigr{)}\,dt,$

and hence

(4.13)
$\lim_{\varphi\to\pi/2}\,\int_{0}^{\infty}\bigl{(}|g(te^{i\varphi})|^{p}+|g(te^{-i\varphi})|^{p}\bigr{)}\,dt=\int_{-\infty}^{\infty}|g(it)|^{p}\,dt.$

Then, by (4.13) and Lemma 2.2, using once again the pointwise a.e. convergence
of $g(te^{\pm i\varphi})$ to $g(\pm it)$ as $\varphi\to\pm\pi/2$, we obtain
(4.11). Since
$\|g\|_{H^{p}(\mathbb{C}_{+})}=\|g(i\cdot)\|_{L^{p}(\mathbb{R})},$ we get
(4.12) as well. ∎

For (formally) more general versions of (4.9) and (4.10) see (4.15) and (4.17)
below.

###  4.2. The spaces $H^{1}(\Sigma_{\psi})$

Now using the results of Section 4.1 for $\psi=\pi/2$, we develop basic
properties of $H^{1}(\Sigma_{\psi})$ for any $\psi\in(0,\pi)$. Define the
Hardy space $H^{1}(\Sigma_{\psi})$ on the sector $\Sigma_{\psi}$ to be the
space of all functions $f\in\operatorname{Hol}(\Sigma_{\psi})$ such that

(4.14)
$\|f\|_{H^{1}(\Sigma_{\psi})}:=\sup_{|\varphi|<\psi}\int_{0}^{\infty}\bigl{(}|f(te^{i\varphi})|+|f(te^{-i\varphi})|\bigr{)}\,dt<\infty.$

We shall also consider a non-symmetric version of $H^{1}(\Sigma_{\psi})$,
defined as

$H^{1}_{*}(\Sigma_{\psi}):=\left\\{f\in\operatorname{Hol}\,(\Sigma_{\psi}):\|f\|_{H^{1}_{*}(\Sigma_{\psi})}:=\sup_{|\varphi|<\psi}\,\int_{0}^{\infty}|f(te^{i\varphi})|\,dt<\infty\right\\}.$

######  Theorem 4.6.

Let $\psi,\psi_{1},\psi_{2}\in(0,\pi)$.

  1. (i)

$f\in H^{1}(\Sigma_{\psi})$ if and only if $f\in H^{1}_{*}(\Sigma_{\psi})$,
and then

(4.15)
$2^{-1}\|f\|_{H^{1}(\Sigma_{\psi})}\leq\|f\|_{H^{1}_{*}(\Sigma_{\psi})}\leq\|f\|_{H^{1}(\Sigma_{\psi})}.$

  2. (ii)

For any $\psi_{1},\psi_{2}\in(0,\pi)$, the map

$\displaystyle H^{1}(\Sigma_{\psi_{1}})$ $\displaystyle\to$ $\displaystyle
H^{1}(\Sigma_{\psi_{2}})$ $\displaystyle f(z)$ $\displaystyle\mapsto$
$\displaystyle\frac{\psi_{1}}{\psi_{2}}z^{(\psi_{1}/\psi_{2})-1}f(z^{\psi_{1}/\psi_{2}})$

is an isometric isomorphism of $H^{1}(\Sigma_{\psi_{1}})$ onto
$H^{1}(\Sigma_{\psi_{2}})$, and of $H^{1}_{*}(\Sigma_{\psi_{1}})$ onto
$H^{1}_{*}(\Sigma_{\psi_{2}})$.

  3. (iii)

If $f\in H^{1}(\Sigma_{\psi})$, then the limits $f(re^{\pm
i\psi}):=\lim_{\varphi\to\pm\psi}f(re^{i\varphi})$ exist a.e., and in the
$L^{1}$-sense with respect to $r$. Moreover

(4.16) $f(z)=\frac{1}{2\pi
i}\int_{\partial\Sigma_{\psi}}\,\frac{f(\lambda)}{\lambda-z}\,d\lambda,\quad
z\in\Sigma_{\psi},$

where the integral is taken downwards.

  4. (iv)

If $f\in H^{1}(\Sigma_{\psi})$, then

(4.17) $\displaystyle\|f\|_{H^{1}(\Sigma_{\psi})}$
$\displaystyle=\int_{0}^{\infty}\bigl{(}|f(te^{i\psi})|+|f(te^{-i\psi})|\bigr{)}\,dt.$

  5. (v)

$H^{1}(\Sigma_{\psi})$ and $H^{1}_{*}(\Sigma_{\psi})$ are Banach spaces.

###### Proof.

The proof of (i) is clear, and (ii) is a direct verification. For
$\psi=\pi/2$, the statements (iii) and (iv), excluding (4.16), were proved in
Corollaries 4.3 and 4.5, and (v) is well-known. Then the general cases are
reduced to the case when $\psi=\pi/2$, by means of (ii).

The Cauchy formula (4.16) is well-known for $\psi=\pi/2$ (see, for example,
[20, Theorem 11.8]). For general $\psi$, we may argue similarly to the proof
of Lemma 4.1, as follows.

Since $f\in H^{1}(\Sigma_{\psi})$,

$\int_{0}^{\infty}\int_{-\psi}^{\psi}|f(te^{i\varphi})|\,d\varphi\,dt<\infty.$

Hence there exist sequences $(t_{1,n})_{n\geq 1}$ and $(t_{2,n})_{n\geq 1}$
such that $0<t_{1,n}<t_{2,n}$, $t_{1,n}\to 0,\,t_{2,n}\to\infty$ as
$n\to\infty$, and

$\lim_{n\to\infty}\int_{-\psi}^{\psi}t_{1,n}|f(t_{1,n}e^{i\varphi})|\,d\varphi=\lim_{n\to\infty}\int_{-\psi}^{\psi}|f(t_{2,n}e^{i\varphi})|\,d\varphi=0.$

By applying Cauchy’s theorem around the boundary of

$\left\\{z\in\partial\Sigma_{\psi-n^{-1}}:t_{1,n}<|z|<t_{2,n}\right\\}$

for large $n$, and taking the limit, we obtain (4.16). ∎

######  Remark 4.7.

In addition to (4.17), it is also possible to prove that

$\|f\|_{H^{1}_{*}(\Sigma_{\psi})}=\max\left(\int_{0}^{\infty}|f(te^{-i\psi})|\,dt,\int_{0}^{\infty}|f(te^{i\psi})|\,dt\right).$

This requires additional techniques, and it is not used in this paper.

###  4.3. Functions with derivatives in $H^{1}(\Sigma_{\psi})$

For $\psi\in(0,\pi)$, let us introduce the space

$\mathcal{H}_{\psi}:=\left\\{f\in\operatorname{Hol}(\Sigma_{\psi}):f^{\prime}\in
H^{1}(\Sigma_{\psi})\right\\}.$

In view of Corollary 4.3,

(4.18)
$\mathcal{H}_{\pi/2}=H^{1,1}(\mathbb{C}_{+}):=\\{f\in\operatorname{Hol}(\mathbb{C}_{+}):f^{\prime}\in
H^{1}(\mathbb{C}_{+})\\},$

and we may sometimes use the notation $H^{1,1}(\mathbb{C}_{+})$ instead of
$\mathcal{H}_{\pi/2}$.

Such function spaces are often called Hardy-Sobolev spaces, and we shall also
use this terminology sporadically. Spaces more general than
$\mathcal{H}_{\psi}$ appear in [19]. Namely, for
$f\in\operatorname{Hol}(\Sigma_{\psi})$ it was required in [19] that the
boundary values of $f$ exist and belong to a Besov space
$B^{s}_{\infty,1},\,s>0$. One can develop a similar approach to those spaces,
but we do not see much advantage in such generality within the present
context.

######  Theorem 4.8.

Let $f\in\mathcal{H}_{\psi},\,\psi\in(0,\pi)$.

  1. (i)

The function $f$ extends to a continuous bounded function on
$\overline{\Sigma}_{\psi}$.

  2. (ii)

The limit

$f(\infty):=\lim_{|z|\to\infty,z\in\Sigma_{\psi}}\,f(z)$

exists.

  3. (iii)

One has

(4.19)
$\|f\|_{H^{\infty}(\Sigma_{\psi})}\leq|f(\infty)|+\|f^{\prime}\|_{H^{1}(\Sigma_{\psi})}.$

In particular, the evaluation functionals $\delta_{z},\,z\in\mathbb{C}_{+}$,
are continuous on $\mathcal{H}_{\psi}$.

###### Proof.

Let $\psi=\pi/2$. Since
$\mathcal{H}_{\pi/2}=H^{1,1}(\mathbb{C}_{+})\subset\mathcal{B}$ by (4.18) and
[6, Proposition 2.4], the statement (i) follows from [6, Proposition 2.2(iv)],
and (ii) follows from [6, Proposition 2.4]. In the general case, the map
$f(z)\mapsto f(z^{2\psi/\pi})$ is an isomorphism of $\mathcal{H}_{\psi}$ onto
$H^{1,1}(\mathbb{C}_{+})$, by Theorem 4.6(ii), so (i) and (ii) hold for
$\mathcal{H}_{\psi}$. The statement (iii) is easily seen. ∎

It follows from Theorem 4.8 that $\mathcal{H}_{\psi}$ is an algebra for every
$\psi\in(0,\pi)$. We define a norm on $\mathcal{H}_{\psi}$ by

(4.20)
$\|f\|_{\mathcal{H}_{\psi}}:=\|f\|_{H^{\infty}(\Sigma_{\psi})}+\|f^{\prime}\|_{H^{1}(\Sigma_{\psi})},\qquad
f\in\mathcal{H}_{\psi}.$

This is easily seen to be an algebra norm. Theorem 4.8(iii) shows that

(4.21)
$\|f\|^{\prime}_{\mathcal{H}_{\psi}}:=|f(\infty)|+\|f^{\prime}\|_{H^{1}(\Sigma_{\psi})}$

is an equivalent norm on $\mathcal{H}^{1}(\Sigma_{\psi})$.

The following lemma is simple, but crucial for our theory. The completeness of
the norm is a standard fact, the scale-invariance is trivial, and the final
isomorphism follows from Theorem 4.6(ii).

######  Lemma 4.9.

For every $\psi\in(0,\pi)$, the space
$(\mathcal{H}_{\psi},\|\cdot\|_{\mathcal{H}_{\psi}})$ is a Banach algebra. For
$t>0$, the map $f(z)\mapsto f(tz)$ is an isometric algebra isomorphism on
$\mathcal{H}_{\psi}$. Moreover, for any $\psi_{1},\psi_{2}\in(0,\pi)$, the map

$\displaystyle\mathcal{H}_{\psi_{1}}$ $\displaystyle\to$
$\displaystyle\mathcal{H}_{\psi_{2}}$ $\displaystyle f(z)$
$\displaystyle\mapsto$ $\displaystyle f(z^{\psi_{1}/\psi_{2}})$

is an isometric algebra isomorphism.

We now give some examples of functions in $\mathcal{H}_{\psi}$ which will play
important roles in subsequent sections of this paper. The first example is of
similar type to Example 3.3.

######  Examples 4.10.

Let $\psi\in(0,\pi/2)$, and
$\lambda=|\lambda|e^{i\varphi}\in\Sigma_{\pi-\psi}$.

  1. 1.

Let $\gamma>0$, and consider the function
$r_{\lambda}^{\gamma}(z)=(\lambda+z)^{-\gamma},\,z\in\Sigma_{\psi}$. Then
$r_{\lambda}^{\gamma}\in\mathcal{H}_{\psi}$ and

(4.22)
$\|r_{\lambda}^{\gamma}\|_{\mathcal{H}_{\psi}}=\int_{\partial\Sigma_{\psi}}\frac{\gamma|dz|}{|z+\lambda|^{\gamma+1}}\leq\frac{2}{\sin^{\gamma+1}((\varphi-\psi)/2)\,|\lambda|^{\gamma}},$

where we have used Lemma 2.1. Thus
$r_{\lambda}^{\gamma}\in\mathcal{H}_{\psi}$, and there exists
$C_{\varphi,\psi,\nu}$ such that

(4.23)
$\|r_{\lambda}^{\gamma}\|^{\prime}_{\mathcal{H}_{\psi}}\leq\frac{C_{\varphi,\psi,\gamma}}{|\lambda|^{\nu}},\qquad\lambda\in\Sigma_{\varphi},\,\varphi<\pi-\psi.$

In particular, if $\gamma=1$, then

(4.24)
$\|r_{\lambda}\|_{\mathcal{H}_{\psi}}\leq\frac{2}{\sin^{2}((\varphi-\psi)/2)\,|\lambda|},\qquad\lambda\in\Sigma_{\pi-\varphi}.$

This property will be important for the proof of Theorem 1.2 (see Theorem
8.2). A more general estimate will be given in Corollary 4.16.

  2. 2.

Let $\gamma\in(0,\pi/(2\psi)$ and $e_{\gamma,\lambda}(z):=e^{-\lambda
z^{\gamma}},\,z\in\Sigma_{\psi}$. Then
$e_{\gamma,\lambda}\in\mathcal{H}_{\psi}$ and

$\|e_{\gamma,\lambda}\|^{\prime}_{\mathcal{H}_{\psi}}=\|e_{1,\lambda}\|^{\prime}_{\mathcal{H}_{\gamma\psi}}\leq\int_{\partial\Sigma_{\psi}}|\lambda|e^{-\operatorname{Re}\lambda
z}\,|dz|\leq\frac{1}{\cos(\varphi+\psi)}+\frac{1}{\cos(\varphi-\psi)}.$

More examples can be found in Sections 4.5 and 4.6. In particular, Lemma 4.13
shows that the restriction of any function in $\mathcal{D}_{\infty}$ to
$\Sigma_{\psi},\,\psi\in(0,\pi/2)$, belongs to $\mathcal{H}_{\psi}$.

The following lemma is a result of Fatou type closely related to Corollary
3.11.

######  Lemma 4.11.

Let $\psi\in(0,\pi)$ and $(f_{k})_{k=1}^{\infty}\subset\mathcal{H}_{\psi}$ be
such that $\sup_{k\geq 1}\|f_{k}\|_{\mathcal{H}_{\psi}}<\infty$ and
$f(z)=\lim_{k\to\infty}f_{k}(z)$ exists for all $z\in\mathbb{C}_{+}$. Then
$f\in\mathcal{H}_{\psi}$.

###### Proof.

By Theorem 4.8, the functions $\\{f_{k}:k\geq 1\\}$ are uniformly bounded on
$\Sigma_{\psi}$. By Vitali’s theorem, $f$ is holomorphic, and
$f_{k}^{\prime}(z)\to f^{\prime}(z)$ as $k\to\infty$ for each
$z\in\Sigma_{\psi}$. By Fatou’s Lemma, for $|\varphi|<\psi$,

$\int_{0}^{\infty}|f^{\prime}(te^{i\varphi})|\,dt\leq\liminf_{k\to\infty}\int_{0}^{\infty}|f_{k}^{\prime}(te^{i\varphi})|\,dt\leq\sup_{k\geq
1}\|f_{k}\|_{\mathcal{H}_{\psi}}.$

Thus $f\in\mathcal{H}_{\psi}$. ∎

###  4.4. The spaces $H^{1,1}(\mathbb{C}_{+})$ and $\mathcal{L}L^{1}$

In [6, Proposition 2.4], we showed that
$\mathcal{H}_{\pi/2}\overset{i}{\hookrightarrow}\mathcal{B}$. We will now show
a stronger result that
$H^{1,1}(\mathbb{C}_{+})\overset{i}{\hookrightarrow}\mathcal{L}L^{1}+\mathbb{C}\subset\mathcal{LM}$,
where $(\mathcal{LM},\|\cdot\|)_{\mathrm{HP}}$ is the Hille-Phillips algebra
as in Section 2. In particular, it shows that the Laplace transforms of
singular measures on $(0,\infty)$ are not in $H^{1,1}(\mathbb{C}_{+})$, which
may be of interest.

######  Theorem 4.12.

If $f\in H^{1,1}(\mathbb{C}_{+})$ then there exists $g\in
L^{1}(\mathbb{R}_{+})$ such that $f=f(\infty)+\mathcal{L}g$. Moreover, there
is an absolute constant $C$ such that

(4.25) $\|f\|_{\mathrm{HP}}\leq C\|f\|_{H^{1,1}(\mathbb{C}_{+})}.$

###### Proof.

Let $f\in H^{1,1}(\mathbb{C}_{+})$, and, for $n\in\mathbb{N}$, let

$\displaystyle f_{n}(z)$ $\displaystyle:=f(z)-f(z+n),\qquad
z\in\mathbb{C}_{+},$ $\displaystyle g_{n}(t)$
$\displaystyle:=-\frac{2}{\pi}t(1-e^{-nt})\int_{0}^{\infty}\alpha e^{-\alpha
t}\int_{-\infty}^{\infty}f^{\prime}(\alpha+i\beta)e^{-i\beta
t}\,d\beta\,d\alpha,\quad t>0.$

Then $f_{n}\in H^{1,1}(\mathbb{C}_{+})$, and
$\|f_{n}\|_{H^{1,1}(\mathbb{C}_{+})}\leq 2\|f\|_{H^{1,1}(\mathbb{C}_{+})}$.
Moreover,

$\displaystyle|g_{n}(t)|$
$\displaystyle\leq\frac{2}{\pi}t(1-e^{-nt})\int_{0}^{\infty}\alpha e^{-\alpha
t}\int_{-\infty}^{\infty}|f^{\prime}(\alpha+i\beta)|\,d\beta\,d\alpha$
$\displaystyle\leq\frac{2}{\pi}\|f^{\prime}\|_{H^{1}(\mathbb{C}_{+})}t(1-e^{-nt})\int_{0}^{\infty}\alpha
e^{-\alpha
t}\,d\alpha=\frac{2}{\pi}\|f^{\prime}\|_{H^{1}(\mathbb{C}_{+})}\frac{(1-e^{-nt})}{t}$
$\displaystyle\leq\frac{2n}{\pi}\|f^{\prime}\|_{H^{1}(\mathbb{C}_{+})}.$

By the reproducing formula for $\mathcal{B}$ (see Remark 3.15 or [6,
Proposition 2.20]) and Fubini’s theorem,

$\displaystyle f_{n}(z)=$
$\displaystyle-\frac{2}{\pi}\int_{0}^{\infty}\alpha\int_{-\infty}^{\infty}f^{\prime}(\alpha+i\beta)\left(\frac{1}{(z+\alpha-i\beta)^{2}}-\frac{1}{(z+n+\alpha-i\beta)^{2}}\right)d\beta\,d\alpha$
$\displaystyle=$
$\displaystyle-\frac{2}{\pi}\int_{0}^{\infty}\alpha\int_{-\infty}^{\infty}f^{\prime}(\alpha+i\beta)\left(\int_{0}^{\infty}e^{-(z+\alpha-i\beta)t}t(1-e^{-nt})\,dt\right)\,d\beta\,d\alpha$
$\displaystyle=$
$\displaystyle-\frac{2}{\pi}\int_{0}^{\infty}e^{-zt}g_{n}(t)\,dt,\quad
z\in\mathbb{C}_{+}.$

It follows that $(\pi/2)f_{n}^{\prime}$ is the Laplace transform of
$tg_{n}(t)$, and then $tg_{n}(t)$ is the inverse Fourier transform of
$(\pi/2)f_{n}^{\prime}(i\cdot)\in L^{1}(\mathbb{R})$. By Hardy’s inequality in
the form of [20, p.198],

(4.26)
$\int_{0}^{\infty}|g_{n}(t)|\,dt=\int_{0}^{\infty}\frac{|tg_{n}(t)|}{t}\,dt\leq\frac{\pi}{4}\|f_{n}^{\prime}\|_{H^{1}(\mathbb{C}_{+})}\leq\frac{\pi}{2}\|f^{\prime}\|_{H^{1}(\mathbb{C}_{+})}.$

Moreover,

$f(z)=f(\infty)+\lim_{n\to\infty}f_{n}(z),\qquad
z\in\overline{\mathbb{C}}_{+},$

and then by [48, Theorem 1.9.2] we obtain that $f\in\mathcal{LM}$ and (4.25)
holds.

Now let

$u(z):=f^{\prime}(z+1)-\int_{0}^{\infty}e^{-zt}e^{-t}t\,d\mu(t),\qquad
z\in\mathbb{C}_{+}.$

Since $f\in H^{\infty}(\mathbb{C}_{+})$ and $f^{\prime}\in
H^{1}(\mathbb{C}_{+})$, we have that $u\in H^{1}(\mathbb{C}_{+})\cap
H^{\infty}(\mathbb{C}_{+})\subset H^{2}(\mathbb{C}_{+})$. Hence
$u=\mathcal{L}h$ for some $h\in L^{2}(\mathbb{R}_{+})$, so

$u(z)=\int_{0}^{\infty}e^{-zt}h(t)\,dt,\qquad z\in\mathbb{C}_{+}.$

From the uniqueness properties of Laplace transforms it follows that

$-e^{-t}t\,\mu(dt)=h(t)\,dt.$

Thus $\mu$ is absolutely continuous on $(0,\infty)$, with Radon-Nikodym
derivative $g(t)$ on $(0,\infty)$. Since $\mu$ is a bounded measure, $g\in
L^{1}(\mathbb{R}_{+})$, and

$\mu(dt)=f(\infty)\delta_{0}+g(t)\,dt.$

Hence $f=f(\infty)+\mathcal{L}g$. ∎

###  4.5. The spaces $\mathcal{H}_{\psi}$ and $\mathcal{D}_{s}$

Since $H^{1,1}(\mathbb{C}_{+})=\mathcal{H}_{\pi/2}$ (by Corollary 4.3), we
have shown in Theorem 4.12 that

$\mathcal{H}_{\pi/2}\overset{i}{\hookrightarrow}\mathcal{L}L^{1}+\mathbb{C}\subset\mathcal{LM}\overset{i}{\hookrightarrow}\mathcal{B}\overset{i}{\hookrightarrow}\mathcal{D}_{s}$

if $s>0$. In the next lemma, we show that, for all $s>-1$,
$\mathcal{H}_{\pi/2}\overset{i}{\hookrightarrow}\mathcal{D}_{s}$ and
$\mathcal{D}_{s}\overset{r}{\hookrightarrow}\mathcal{H}_{\psi}$ for every
$\psi\in(0,\pi/2)$. Moreover it follows that
$\mathcal{D}_{s}^{\infty}\overset{r}{\hookrightarrow}\mathcal{H}_{\psi}$.

######  Lemma 4.13.

  1. (i)

If $f\in\mathcal{H}_{\pi/2}$, then $f\in\mathcal{D}_{s}^{\infty}$ for every
$s>-1$, and

(4.27) $\displaystyle\|f\|_{\mathcal{D}_{s,0}}$ $\displaystyle\leq
B\left(\frac{s+1}{2},\frac{1}{2}\right)\|f^{\prime}\|_{H^{1}(\mathbb{C}_{+})},$
$\displaystyle\|f\|_{\mathcal{D}_{s}^{\infty}}$
$\displaystyle\leq\max\left\\{1,B\left(\frac{s+1}{2},\frac{1}{2}\right)\right\\}\|f\|_{\mathcal{H}_{\pi/2}}.$

  2. (ii)

If $f\in\mathcal{D}_{s}$, $s>-1$, then $f\in\mathcal{H}_{\psi}$ for every
$\psi\in(0,\pi/2)$, and

(4.28)
$\|f^{\prime}\|_{H^{1}(\Sigma_{\psi})}\leq\frac{2^{s}}{\pi\cos^{s+2}(\psi/2+\pi/4)}\|f\|_{\mathcal{D}_{s,0}}.$

Thus, for all $\psi\in(0,\pi/2)$ and $s>-1$, there are natural continuous
embeddings

$\mathcal{H}_{\pi/2}\overset{i}{\hookrightarrow}\mathcal{D}_{s}\overset{r}{\hookrightarrow}\mathcal{H}_{\psi}.$

Note that the estimates (4.23) and (4.27) for functions $r_{\lambda}^{\gamma}$
reproduce the estimate (3.18), with different constants.

###### Proof.

Let $s>-1$ be fixed, and let $f\in\mathcal{H}_{\pi/2}$. Using (4.15), we have

$\displaystyle\|f\|_{\mathcal{D}_{s,0}}$
$\displaystyle=\int_{-\pi/2}^{\pi/2}\cos^{s}\varphi\int_{0}^{\infty}|f^{\prime}(te^{i\varphi})|\,dt\,d\varphi$
$\displaystyle\leq\left(\int_{-\pi/2}^{\pi/2}\cos^{s}\varphi\,d\varphi\right)\sup_{|\varphi|\leq\pi/2}\int_{0}^{\infty}|f^{\prime}(te^{i\varphi})|\,dt$
$\displaystyle=B\left(\frac{s+1}{2},\frac{1}{2}\right)\|f^{\prime}\|_{H^{1}(\mathbb{C}_{+})},$

and (i) follows.

To prove (ii), note that if $f\in\mathcal{D}_{s}$, then by Corollary 3.9,

$f^{\prime}(z)=\frac{(s+1)2^{s}}{\pi}\int_{0}^{\infty}\alpha^{s}\int_{-\infty}^{\infty}\frac{f^{\prime}(\alpha+i\beta)}{(z+\alpha-i\beta)^{s+2}}\,d\beta\,d\alpha,\quad
z\in\mathbb{C}_{+}.$

Hence using (2.1), for every $\psi\in(0,\pi/2)$, we obtain

$\displaystyle\hskip-10.0pt2^{-s}\int_{0}^{\infty}|f^{\prime}(te^{\pm
i\psi})|\,dt$
$\displaystyle\leq\frac{(s+1)}{\pi}\int_{0}^{\infty}\int_{0}^{\infty}\alpha^{s}\int_{-\infty}^{\infty}\frac{|f^{\prime}(\alpha+i\beta)|}{|te^{\pm
i\psi}+\alpha-i\beta|^{s+2}}\,d\beta\,d\alpha\,dt$
$\displaystyle\leq\frac{s+1}{\pi\cos^{s+2}(\psi/2+\pi/4)}\int_{0}^{\infty}\alpha^{s}\int_{-\infty}^{\infty}\left(\int_{0}^{\infty}\frac{|f^{\prime}(\alpha+i\beta)|}{(t+|\alpha+i\beta|)^{s+2}}\right)\,dt\,d\beta\,d\alpha$
$\displaystyle=\frac{1}{\pi\cos^{s+2}(\psi/2+\pi/4)}\int_{0}^{\infty}\alpha^{s}\int_{-\infty}^{\infty}\frac{|f^{\prime}(\alpha+i\beta)|}{|\alpha+i\beta|^{s+1}}\,d\beta\,d\alpha$
$\displaystyle=\frac{1}{\pi\cos^{s+2}(\psi/2+\pi/4)}\|f^{\prime}\|_{\mathcal{V}_{s}},$

and (4.28) follows. ∎

For a function $f\in\operatorname{Hol}(\Sigma_{\psi})$, $\gamma>0$ and
$0<\varphi\leq\min\\{\pi,\psi/\gamma\\}$ , define
$f_{\gamma}\in\operatorname{Hol}(\Sigma_{\varphi})$ by

$f_{\gamma}(z)=f(z^{\gamma}),\quad z\in\Sigma_{\varphi}.$

######  Corollary 4.14.

Let $f\in\mathcal{D}_{s},\,s>-1$, and let $\gamma\in(0,1)$. Then
$f_{\gamma}\in\mathcal{D}_{\sigma}^{\infty}\cap\mathcal{H}_{\pi/2}$ for all
$\sigma>-1$. Moreover, for each $s>-1$ and $\sigma>-1$, there exist constants
$C_{s,\sigma,\gamma}$ and $\tilde{C}_{s,\gamma}$ such that

$\|f_{\gamma}\|_{\mathcal{D}_{\sigma}^{\infty}}\leq
C_{s,\sigma,\gamma}\|f\|_{\mathcal{D}_{s}}\quad\text{and}\quad\|f_{\gamma}\|_{\mathcal{H}_{\pi/2}}\leq\tilde{C}_{s,\gamma}\|f\|_{\mathcal{D}_{s}},\qquad
f\in\mathcal{D}_{s}.$

###### Proof.

Using Lemma 4.13, (i) and (ii), and Lemma 4.9, we see firstly that
$f\in\mathcal{H}_{\pi\gamma/2}$, and then
$f_{\gamma}\in\mathcal{H}_{\pi/2}\subset\mathcal{D}_{\sigma}^{\infty}$.
Moreover each of the embeddings from Lemma 4.13 is continuous, and the map
$f\mapsto f_{\gamma}$ is isometric. ∎

Now we relate the spaces $\mathcal{H}_{\psi}$ and $\mathcal{D}_{s}$ to another
class of spaces used in the literature on functional calculi. For
$\psi\in(0,\pi)$, let

(4.29)
$\mathcal{E}_{\psi}:=\left\\{f\in{\operatorname{Hol}}(\Sigma_{\psi}):\|f\|_{\psi}:=\sup_{\varphi\in(0,\psi)}\int_{\partial\Sigma_{\varphi}}\frac{|f(z)|}{|z|}|dz|<\infty\right\\}.$

It is easy to see that $(\mathcal{E}_{\psi},\|\cdot\|_{\psi})$ is a Banach
space, and that

$\mathcal{E}_{\psi}=\\{f\in{\operatorname{Hol}}(\Sigma_{\psi}):f(z)/z\in
H^{1}(\Sigma_{\psi})\\}.$

######  Proposition 4.15.

Let $f\in\mathcal{E}_{\psi}$ and let $g(z):=f(z)/z$. Then, for every
$\varphi\in(0,\psi)$,

(4.30)
$\|f^{\prime}\|_{H^{1}(\Sigma_{\varphi})}\leq\frac{1}{2\pi}\left(\frac{\pi-\psi-\varphi}{\sin(\psi+\varphi)}+\frac{\pi-\psi+\varphi}{\sin(\psi-\varphi)}\right)\|g\|_{H^{1}(\Sigma_{\psi})}.$

Thus,
$\mathcal{E}_{\psi}\overset{i}{\hookrightarrow}\mathcal{H}_{\varphi},\,\varphi\in(0,\psi)$,
and $\mathcal{E}_{\pi/2}\overset{i}{\hookrightarrow}\mathcal{D}_{s},\,s>0$.

###### Proof.

By Cauchy’s theorem, for every $z\in\Sigma_{\psi}$,

$f^{\prime}(z)=\frac{1}{2\pi i}\int_{\partial\Sigma_{\psi}}\,\frac{\lambda
g(\lambda)\,d\lambda}{(z-\lambda)^{2}},$

Hence, for every $\varphi\in(0,\psi)$, by Fubini’s theorem,

$\displaystyle\int_{0}^{\infty}\left(|f^{\prime}(\rho
e^{i\varphi})|+|f^{\prime}(\rho
e^{-i\varphi})|\right)\,d\rho\leq\frac{1}{2\pi}\int_{\partial\Sigma_{\psi}}|g(\lambda)|\,J(\lambda,\varphi)\,|d\lambda|,$

where

$\displaystyle J(\lambda,\varphi)$
$\displaystyle:=\int_{0}^{\infty}\left(\frac{|\lambda|}{|\rho
e^{i\varphi}-\lambda|^{2}}+\frac{|\lambda|}{|\rho
e^{-i\varphi}-\lambda|^{2}}\right)\ d\rho$
$\displaystyle=\int_{0}^{\infty}\frac{d\rho}{\rho^{2}-2\rho\cos(\psi+\varphi)+1}+\int_{0}^{\infty}\frac{d\rho}{\rho^{2}-2\rho\cos(\psi-\varphi)+1}$
$\displaystyle=\frac{\pi-\psi-\varphi}{\sin(\psi+\varphi)}+\frac{\pi-\psi+\varphi}{\sin(\psi-\varphi)},$

in view of [47, p.311]. Hence $f\in\mathcal{H}_{\varphi}$ and (4.30) follows.
Thus, $\mathcal{E}_{\psi}\overset{r}{\hookrightarrow}\mathcal{H}_{\varphi}$
for all $\varphi\in(0,\psi)$. Recalling Lemma 4.13 and that
$\mathcal{H}_{\pi/2}=H^{1,1}(\mathbb{C}_{+})$, we have proved the inclusion
$\mathcal{E}_{\pi/2}\overset{i}{\hookrightarrow}\mathcal{D}_{s},\,s>0$, as
well. ∎

Note that the inclusions in Lemma 4.15 are strict. Indeed, if
$f(z)=z(z+1)^{-1}$ then one has $f\in\mathcal{H}_{\psi}$ for every
$\psi\in(0,\pi)$, but $f\notin\mathcal{E}_{\psi}$ for any $\psi\in(0,\pi)$.
Moreover, if

$f(z)=\frac{z\,e^{-z}}{(z+1)\log^{2}(z+2)},$

then $z^{-1}f\in H^{1}(\mathbb{C}_{+})$, and $f\in\mathcal{D}_{0}$, but
$f\not\in\mathcal{D}_{s}$ for any $s\in(-1,0)$.

The spaces $\mathcal{E}_{\psi}$ are studied in [37, Chapter 10]; see also [37,
Appendix H2], [33, Section 6] and [32, Appendix C]. To ensure the algebra
property and to relate the spaces to the $H^{\infty}$-calculus, the authors
considered the algebras $H^{\infty}(\Sigma_{\psi})\cap\mathcal{E}_{\psi}$.
Lemma 4.15 shows that the spaces $\mathcal{E}_{\psi}$ are fully covered within
the framework of the algebras $\mathcal{D}_{\infty}$ and $\mathcal{H}_{\psi}$.
These algebras will be associated to the more powerful functional calculi
constructed in Corollary 7.8 and Theorem 8.2.

###  4.6. Bernstein functions and $\mathcal{H}_{\psi}$

To illustrate the relevance of the Hardy-Sobolev spaces, we show that the
“resolvent” of a Bernstein function belongs to an appropriate Hardy-Sobolev
space. This observation will be used in Section 10 to provide a new proof of
the permanence of subordination for holomorphic semigroups, one of the main
results of [30]; see also [5] and [6].

Let $g$ be a Bernstein function, $\psi\in(0,\pi/2)$, and
$\lambda\in\Sigma_{\pi-\psi}$. Let

$f(z,\lambda):=(\lambda+g(z))^{-1},\qquad z\in\Sigma_{\psi}.$

If $\lambda\in\mathbb{C}_{+}$, it follows from Lemmas 3.19 and 4.13 that
$f(\cdot,\lambda)\in\mathcal{\mathcal{H}}_{\psi}$, and
$\|f(\cdot,\lambda)\|_{\mathcal{H}_{\psi}}\leq C_{\psi}/|\lambda|$, where
$C_{\psi}$ is independent of $g$ and $\lambda\in\mathbb{C}_{+}$. In order to
obtain the correct angle, we will need to extend this to
$\lambda\in\Sigma_{\pi-\varphi}$, where $\varphi\in(\psi,\pi/2)$.

######  Corollary 4.16.

Let $g$ be a Bernstein function, $\psi\in(0,\pi/2)$, $\varphi\in(\psi,\pi)$
and $\lambda\in\Sigma_{\pi-\varphi}$. Then

(4.31) $\|f(\cdot,\lambda)\|_{\mathcal{\mathcal{H}}_{\psi}}\leq
2\left(\frac{1}{\sin(\min(\varphi,\pi/2))}+\frac{2}{\cos\psi\sin^{2}((\varphi-\psi)/2)}\right)\frac{1}{|\lambda|}.$

###### Proof.

For fixed $\psi\in(0,\pi/2)$, $\varphi\in(\psi,\pi)$, and
$\lambda\in\Sigma_{\pi-\varphi}$, observe that

$\|f^{\prime}(\cdot,\lambda)\|_{H^{1}(\Sigma_{\psi})}\leq\int_{0}^{\infty}\left(\frac{|g^{\prime}(te^{i\psi})|}{|\lambda+g(te^{i\psi})|^{2}}+\frac{|g^{\prime}(te^{-i\psi})|}{|\lambda+g(te^{-i\psi})|^{2}}\right)\,dt.$

Using the property (B1) for Bernstein functions and (2.1), we have

$|\lambda+g(te^{\pm i\psi})|\geq\sin((\varphi-\psi)/2)(|\lambda|+|g(te^{\pm
i\psi})|),\qquad t\geq 0.$

Moreover, in view of (B3), for all $t\geq 0$,

$\displaystyle|g(te^{\pm i\psi})|\geq\operatorname{Re}g(te^{\pm i\psi})\geq
g(t\cos\psi)\quad\text{and}\quad|g^{\prime}(e^{\pm i\psi}t)|$
$\displaystyle\leq g^{\prime}(t\cos\psi).$

Using (B2), we have

$\displaystyle\|f^{\prime}(\cdot,\lambda)\|_{H^{1}(\Sigma_{\psi})}$
$\displaystyle\leq\frac{2}{\sin^{2}((\varphi-\psi)/2)}\int_{0}^{\infty}\frac{g^{\prime}(t\cos\psi)}{(|\lambda|+g(t\cos\psi))^{2}}\,dt$
$\displaystyle\leq\frac{2}{\cos\psi\sin^{2}((\varphi-\psi)/2)}\int_{0}^{\infty}\frac{ds}{(|\lambda|+s)^{2}}$
$\displaystyle=\frac{2}{\cos\psi\sin^{2}((\varphi-\psi)/2)}\frac{1}{|\lambda|}.$

Thus $f(\cdot,\lambda)\in\mathcal{H}_{\psi}$ and
$f(\infty,\lambda)=(\lambda+g(\infty))^{-1}$. Since
$|\arg\lambda|<\pi-\varphi$ and $g(\infty)\in[0,\infty]$,

(4.32) $|f(\infty,\lambda)|\leq\frac{1}{\sin(\min(\varphi,\pi/2))|\lambda|}.$

Now (4.31) follows from Theorem 4.8(ii). ∎

###  4.7. Representations for functions in $\mathcal{H}_{\psi}$

In this section we derive a reproducing formula for functions from
$\mathcal{H}_{\psi}$ and obtain certain alternative representations for its
kernel.

######  Proposition 4.17.

Let $f\in\mathcal{H}_{\psi},\,\psi\in(0,\pi)$, and let

$\gamma:=\frac{2\psi}{\pi}\in(0,2),\quad f_{\gamma}(z):=f(z^{\gamma}),\quad
z\in\mathbb{C}_{+}.$

Then

(4.33)
$f(z)=f(\infty)-\frac{1}{\pi}\int_{0}^{\infty}\int_{-\infty}^{\infty}\frac{f_{\gamma}^{\prime}(\alpha+i\beta)}{z^{1/\gamma}+\alpha-i\beta}\,d\beta\,d\alpha,\qquad
z\in\Sigma_{\psi}\cup\\{0\\}.$

###### Proof.

Since $f\in\mathcal{H}_{\psi}$, Lemma 4.9 implies that $f^{\prime}_{\gamma}\in
H^{1}(\mathbb{C}_{+})$. Hence, by Lemma 4.13 and Theorem 4.8, we have
$f_{\gamma}\in\mathcal{D}_{0}\cap C(\overline{\mathbb{C}}_{+})$. Then, in view
of Corollary 3.9,

$f_{\gamma}(z)=f_{\gamma}(\infty)-\frac{1}{\pi}\int_{0}^{\infty}\int_{-\infty}^{\infty}\frac{f_{\gamma}^{\prime}(\alpha+i\beta)}{z+\alpha-i\beta}\,d\beta\,d\alpha,\qquad
z\in\mathbb{C}_{+}\cup\\{0\\},$

and (4.33) follows. ∎

######  Corollary 4.18.

Let $f\in\mathcal{D}_{\infty}$. For $\gamma\in(0,1)$, set
$f_{\gamma}(z):=f(z^{\gamma}),\,z\in\mathbb{C}_{+}$. Then

(4.34)
$f(z)=f(\infty)-\frac{1}{\pi}\int_{0}^{\infty}\int_{-\infty}^{\infty}\frac{f_{\gamma}^{\prime}(\alpha+i\beta)}{z^{1/\gamma}+\alpha-i\beta}\,d\beta\,d\alpha,\quad
z\in\Sigma_{\pi\gamma/2}\cup\\{0\\}.$

###### Proof.

By Lemma 4.13, $f_{\gamma}\in\mathcal{H}_{\pi\gamma/2}$, so (4.34) follows
from (4.33). ∎

The next reproducing formula for functions in $\mathcal{H}_{\psi}$ resembles
[11, Lemma 7.4], and it was, in fact, inspired by [11, Lemma 7.4]. In
particular, this formula replaces the double (area) integral in (4.33) with a
line integral, it involves boundary values of $f^{\prime}$ rather than
scalings of $f^{\prime}$ (such as in (4.33)), and it offers a different kernel
which might sometimes be easier to deal with. The $\operatorname{arccot}$
function has been defined in (3.21).

######  Proposition 4.19.

Let $f\in\mathcal{\mathcal{H}}_{\psi}$, $\psi\in(0,\pi)$. Let
$\nu:=\pi/(2\psi)$, and

(4.35) $f_{\psi}(t):=\frac{f(e^{i\psi}t)+f(e^{-i\psi}t)}{2},\quad t>0.$

Then

(4.36)
$f(z)=f(\infty)-\frac{2}{\pi}\int_{0}^{\infty}f_{\psi}^{\prime}(t)\,{\operatorname{arccot}}(z^{\nu}/t^{\nu})\,dt,\qquad
z\in\Sigma_{\psi}\cup\\{0\\}.$

###### Proof.

Since $\operatorname{arccot}\lambda\in\mathcal{D}_{0}$ (see Example 3.5),
(3.27) shows that

(4.37)
$\operatorname{arccot}(\lambda)=\frac{1}{\pi}\int_{0}^{\infty}\int_{-\infty}^{\infty}\frac{(\lambda+u-iv)^{-1}}{(u+iv)^{2}+1}\,dv\,du,\quad\lambda\in\mathbb{C}_{+}.$

Let $\gamma=1/\nu$, and $f_{\gamma}(\lambda):=f(\lambda^{\gamma})$,
$\lambda\in\mathbb{C}_{+}$. By Proposition 4.17, for
$z\in\Sigma_{\psi}\cup\\{0\\}$,

(4.38)
$f(z)-f(\infty)=\frac{1}{\pi}\int_{0}^{\infty}\int_{-\infty}^{\infty}\frac{f_{\gamma}^{\prime}(\alpha+i\beta)}{z^{\nu}+\alpha-i\beta}\,d\beta\,d\alpha.$

It follows from Lemma 4.9 that $f_{\gamma}^{\prime}\in
H^{1}(\Sigma_{\pi/2})=H^{1}(\mathbb{C}_{+})$, so by Cauchy’s formula for
functions in $H^{1}(\mathbb{C}_{+})$ [20, Theorem 11.8], we have, for
$\lambda\in\mathbb{C}_{+}$,

$\displaystyle f_{\gamma}^{\prime}(\lambda)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{f_{\gamma}^{\prime}(it)}{\lambda-
it}\,dt+\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{f_{\gamma}^{\prime}(it)}{\lambda+it}\,dt$
$\displaystyle=-\frac{i}{\pi}\int_{-\infty}^{\infty}\frac{tf_{\gamma}^{\prime}(it)}{\lambda^{2}+t^{2}}\,dt$
$\displaystyle=\frac{i\gamma}{\pi}\int_{0}^{\infty}\frac{e^{i\pi(\gamma-1)/2}f^{\prime}(e^{i\pi\gamma/2}t^{\gamma})t^{\gamma}-e^{-i\pi(\gamma-1)/2}f^{\prime}(e^{-i\pi\gamma/2}t^{\gamma})t^{\gamma}}{\lambda^{2}+t^{2}}\,dt$
$\displaystyle=\frac{\gamma}{\pi}\int_{0}^{\infty}\left(e^{i\psi}f^{\prime}(e^{i\psi}t^{\gamma})+e^{-i\psi}f^{\prime}(e^{-i\psi}t^{\gamma})\right)\frac{t^{\gamma}}{\lambda^{2}+t^{2}}\,dt$
$\displaystyle=\frac{2}{\pi}\int_{0}^{\infty}\frac{f_{\psi}^{\prime}(s)s^{\nu}}{\lambda^{2}+s^{2\nu}}\,ds.$

Therefore, by (4.38) and (4.37), we obtain

$\displaystyle f(z)-f(\infty)$
$\displaystyle=-\frac{2}{\pi^{2}}\int_{0}^{\infty}f^{\prime}_{\psi}(s)\left(\int_{0}^{\infty}\int_{-\infty}^{\infty}\frac{s^{\nu}\,d\beta\,d\alpha}{(z^{\nu}+\alpha-i\beta)((\alpha+i\beta)^{2}+s^{2\nu})}\right)\,ds$
$\displaystyle=-\frac{2}{\pi^{2}}\int_{0}^{\infty}f^{\prime}_{\psi}(s)\left(\int_{0}^{\infty}\int_{-\infty}^{\infty}\frac{d\beta\,d\alpha}{((z/s)^{\nu}+\alpha-i\beta)((\alpha+i\beta)^{2}+1)}\right)\,ds$
$\displaystyle=-\frac{2}{\pi}\int_{0}^{\infty}f^{\prime}_{\psi}(s)\operatorname{arccot}(z^{\nu}/s^{\nu})\,ds,$

so (4.36) holds for $z\in\Sigma_{\psi}\cup\\{0\\}$. ∎

Proposition 4.19 motivates a more careful study of the kernel
$\operatorname{arccot}(z^{\nu})$. The integral representation of this kernel
will be crucial in deriving fine estimates for the $\mathcal{H}$-calculus for
operators in Section 8.2.

######  Lemma 4.20.

Let $\psi\in(0,\pi)$ and $\nu={\pi}/{(2\psi)}$. Then

(4.39) $\displaystyle\operatorname{arccot}(z^{\nu})$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{\infty}V_{\psi}(z,t)\log\left|\frac{1+t^{\nu}}{1-t^{\nu}}\right|\,\frac{dt}{t}+\frac{1}{4i}\int_{\Gamma_{\psi}}\,\frac{d\lambda}{\lambda-z},\quad
z\in\Sigma_{\psi},$

where

(4.40) $\displaystyle V_{\psi}(z,t)$
$\displaystyle:=-\frac{t}{2}\left(\frac{e^{-i\psi}}{z-te^{-i\psi}}+\frac{e^{i\psi}}{z-te^{i\psi}}\right),$
$\displaystyle\Gamma_{\psi}$
$\displaystyle:=\\{\lambda:|\lambda|=1,\,\arg\lambda\in(\psi,2\pi-\psi)\\}.$

###### Proof.

We have

$\displaystyle\operatorname{arccot}(z^{\nu})=\frac{1}{2i}\log\left|\frac{z^{\nu}+i}{z^{\nu}-i}\right|+\frac{1}{2}\arg\left(\frac{z^{\nu}+i}{z^{\nu}-i}\right),\qquad
z\in\Sigma_{\psi}.$

Since $\nu\psi=\pi/2$, for every $t>0,\,t\neq 1$,

$\lim_{z\in\Sigma_{\psi},\,z\to
te^{i\psi}}\operatorname{arccot}(z^{\nu})=\frac{1}{2i}\log\left|\frac{1+t^{\nu}}{1-t^{\nu}}\right|+\frac{1}{2}\arg\left(\frac{t^{\nu}+1}{t^{\nu}-1}\right),$

and

$\lim_{z\in\Sigma_{\psi},\,z\to
te^{-i\psi}}\operatorname{arccot}(z^{\nu})=-\frac{1}{2i}\log\left|\frac{1+t^{\nu}}{1-t^{\nu}}\right|+\frac{1}{2}\arg\left(\frac{t^{\nu}-1}{t^{\nu}+1}\right).$

Here,

$\arg\left(\frac{t^{\nu}+1}{t^{\nu}-1}\right)=\arg\left(\frac{t^{\nu}-1}{t^{\nu}+1}\right)=\pi\chi_{(0,1)}(t),\quad
t>0,\,t\neq 1,$

where $\chi_{[0,1]}$ is the characteristic function of $(0,1)$. So,

(4.41) $\lim_{z\in\Sigma_{\psi},\,z\to te^{\pm
i\psi}}\operatorname{arccot}(z^{\nu})=\mp\frac{i}{2}\log\left|\frac{1+t^{\nu}}{1-t^{\nu}}\right|+\frac{\pi}{2}\chi_{[0,1]}(t),$

Now fix $z\in\Sigma_{\psi}$. Using
$\limsup_{|\lambda|\to\infty,\,\lambda\in\mathbb{C}_{+}}\,|\lambda\operatorname{arccot}\lambda|<\infty$,
it follows from Cauchy’s theorem that

$\operatorname{arccot}(z^{\nu})=-\frac{1}{2\pi
i}\int_{\partial\Sigma_{\psi}}\,\frac{\operatorname{arccot}(\lambda^{\nu})}{\lambda-z}\,d\lambda,\quad\int_{\lambda\in\partial\Sigma_{\psi},|\lambda|<1}\frac{d\lambda}{\lambda-z}=-\int_{\Gamma_{\psi}}\frac{d\lambda}{\lambda-z}.$

Thus by (4.41),

$\displaystyle\operatorname{arccot}(z^{\nu})$
$\displaystyle=-\frac{1}{2\pi}\int_{0}^{\infty}\frac{\operatorname{Im}\operatorname{arccot}(it^{\nu})}{t-e^{-i\psi}z}\,dt+\frac{1}{2\pi}\int_{0}^{\infty}\frac{\operatorname{Im}\operatorname{arccot}(-it^{\nu})}{t-e^{i\psi}z}\,dt$
$\displaystyle\hbox{}\hskip
30.0pt-\frac{1}{4i}\int_{\lambda\in\partial\Sigma_{\psi},\,|\lambda|\leq
1}\,\frac{d\lambda}{\lambda-z}$
$\displaystyle=\frac{1}{4\pi}\int_{0}^{\infty}\log\left|\frac{1+t^{\nu}}{1-t^{\nu}}\right|\left(\frac{1}{t-e^{i\psi}z}+\frac{1}{t-e^{-i\psi}z}\right)\,dt+\frac{1}{4i}\int_{\Gamma_{\psi}}\,\frac{d\lambda}{\lambda-z}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{\infty}V_{\psi}(z,t)\log\left|\frac{1+t^{\nu}}{1-t^{\nu}}\right|\,\frac{dt}{t}+\frac{1}{4i}\int_{\Gamma_{\psi}}\,\frac{d\lambda}{\lambda-z}.\qed$

######  Remark 4.21.

Letting $z\to 0$ in (4.39) we obtain

$\displaystyle\frac{\pi}{2}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{\infty}\log\left|\frac{1+t^{\nu}}{1-t^{\nu}}\right|\frac{dt}{t}+\frac{\pi-\psi}{2},$

hence

(4.42)
$\frac{1}{2\pi}\int_{0}^{\infty}\log\left|\frac{1+t^{\nu}}{1-t^{\nu}}\right|\frac{dt}{t}=\frac{\psi}{2}.$

##  5\. Dense sets in $\mathcal{D}_{s}$ and $\mathcal{H}_{\psi}$

In this section we establish some results concerning density and
approximations in our spaces.

###  5.1. Dense subsets of $\mathcal{D}_{s}$ and some applications

Let $\mathcal{R}(\mathbb{C}_{+})$ be the linear span of
$\\{r_{\lambda}:\lambda\in\mathbb{C}_{+}\\}$, and
$\widetilde{\mathcal{R}}(\mathbb{C}_{+})$ be the sum of
$\mathcal{R}(\mathbb{C}_{+})$ and the constant functions. Using Example
4.10(1) and Lemma 4.13, we have

(5.1)
$\mathcal{\widetilde{R}}(\mathbb{C}_{+})\subset\mathcal{H}_{\pi/2}\overset{i}{\hookrightarrow}\mathcal{D}_{s},\quad
s>-1.$

######  Theorem 5.1.

The space

$\widetilde{\mathcal{R}}(\mathbb{C}_{+}):=\left\\{a_{0}+\sum_{k=1}^{n}a_{k}(\lambda_{k}+z)^{-1}:n\in\mathbb{N},\;a_{k}\in\mathbb{C},\;\lambda_{k}\in\mathbb{C}_{+}\right\\}$

is dense in $\mathcal{D}_{s}$ for each $s>-1$.

###### Proof.

Let $\mathcal{R}_{\mathcal{D}_{s}}(\mathbb{C}_{+})$ be the closure of
$\mathcal{R}(\mathbb{C}_{+})$ in $\mathcal{D}_{s}$, F and let
$f\in\mathcal{D}_{s}$. It follows from Example 3.3 and Remark 3.13 (or a
direct estimate) that the function

$(R_{f})(\lambda):=-\frac{2^{s}}{\pi}f^{\prime}(\lambda)r_{\overline{\lambda}}$

is continuous from $\mathbb{C}_{+}$ to $\mathcal{D}_{s}$, and it is Bochner
integrable with respect to area measure $S$ on $\mathbb{C}_{+}$. Since point
evaluations are continuous on $\mathcal{D}_{s}$ (Corollary 3.12), it follows
from Corollary 3.9 that

$Q_{s}f^{\prime}=\int_{\mathbb{C}_{+}}R_{f}(\lambda)\,dS(\lambda)$

as the Bochner integral of a continuous function. Hence $Q_{s}f^{\prime}$
belongs to the closure in $\mathcal{D}_{s}$ of the linear span of the range of
the integrand, which is contained in
$\mathcal{R}_{\mathcal{D}_{s}}(\mathbb{C}_{+})$. Now
$f=f(\infty)+Q_{s}f^{\prime}$ which is in the closure of
$\widetilde{\mathcal{R}}(\mathbb{C}_{+})$ in $\mathcal{D}_{s}$. ∎

From Proposition 3.6, we have the continuous inclusion

$\mathcal{D}_{s}\overset{i}{\hookrightarrow}\mathcal{D}_{\sigma}\quad\text{if
$\sigma>s>-1$},$

and from (5.1), Theorem 4.12 and Proposition 3.14, we have

$\widetilde{\mathcal{R}}(\mathbb{C}_{+})\subset
H^{1,1}(\mathbb{C}_{+})\overset{i}{\hookrightarrow}\mathcal{L}L^{1}+\mathbb{C}\subset\mathcal{LM}\overset{i}{\hookrightarrow}\mathcal{B}\overset{i}{\hookrightarrow}\mathcal{D}_{s}^{\infty}\overset{i}{\hookrightarrow}\mathcal{D}_{s}\quad\text{if
$s>0$}.$

Here $\mathcal{L}L^{1}+\mathbb{C}$ is the sum of $\mathcal{L}L^{1}$ and the
constant functions, and it is a closed subspace of $\mathcal{LM}$. The
following density results hold.

######  Corollary 5.2.

  1. 1.

If $\sigma>s>-1$, then $\mathcal{D}_{s}$ is dense in $\mathcal{D}_{\sigma}$.

  2. 2.

For $s>-1$, the spaces $H^{1,1}(\mathbb{C}_{+})$ and
$\mathcal{D}_{s}^{\infty}$ are dense in $\mathcal{D}_{s}$.

  3. 3.

For $s>0$, the spaces $\mathcal{L}L^{1}+\mathbb{C}$, $\mathcal{LM}$ and
$\mathcal{B}$ are dense in $\mathcal{D}_{s}$.

  4. 4.

For $s>0$, the spaces $\widetilde{\mathcal{R}}(\mathbb{C}_{+})$,
$H^{1,1}(\mathbb{C}_{+})$, $\mathcal{L}L^{1}+\mathbb{C}$, $\mathcal{LM}$ and
$\mathcal{B}$ are not dense in $\mathcal{D}_{s}^{\infty}$.

###### Proof.

The first three statements are immediate from Theorem 5.1.

Since any function in $\mathcal{B}$ extends continuously to $i\mathbb{R}$, the
same holds for the closure of $\mathcal{B}$ in $\mathcal{D}_{s}^{\infty}$ when
$s>0$. The function $f(z)=e^{-1/z}\in\mathcal{D}_{s}^{\infty}$ for $s>0$, (see
Remarks 3.17 and Example 3.4 with $\nu=0$), but $f$ is not continuous at
$z=0$. This establishes the final statement. ∎

The function $g(z)=\exp(\operatorname{arccot}z)$ considered in Example 3.18
provides another example of a function from $\mathcal{D}_{0}^{\infty}$ which
is discontinuous on $i\mathbb{R}$, and so does not belong to the closure of
$\mathcal{B}$ in $\mathcal{D}^{\infty}_{s}$ for $s>0$.

In order to obtain operator norm-estimates for functions $f^{(n)}$ applied to
semigroup generators (see Theorem 10.3), we will need a stronger version of
Corollary 3.22 on differentiability of $t\to f(t\cdot)$ in the $D_{s}$-norm.
We first prove a lemma, and we present the stronger statement in Corollary
5.5.

######  Lemma 5.3.

Let $\lambda\in\mathbb{C}_{+}$, $\tau>0$, $t\in(\tau/2,2\tau)$, and define

$g_{t,\tau,\lambda}(z):=\frac{r_{\lambda}(tz)-r_{\lambda}(\tau
z)}{t-\tau}-zr_{\lambda}^{\prime}(\tau z),\qquad z\in\mathbb{C}_{+}.$

Then, for every $s>-1$,

(5.2) $\lim_{t\to\tau}\,\|g_{t,\tau,\lambda}\|_{\mathcal{D}_{s}}=0.$

###### Proof.

We have

$\displaystyle g_{t,\tau,\lambda}^{\prime}(z)$
$\displaystyle=\left(-\frac{t}{(tz+\lambda)^{2}}+\frac{\tau}{(\tau
z+\lambda)^{2}}\right)\frac{1}{t-\tau}+\frac{1}{(\tau
z+\lambda)^{2}}-\frac{2\tau z}{(\tau z+\lambda)^{3}}$
$\displaystyle=\frac{\tau tz^{2}-\lambda^{2}}{(tz+\lambda)^{2}(\tau
z+\lambda)^{2}}+\frac{\lambda-\tau z}{(\tau z+\lambda)^{3}}\to 0,\qquad
t\to\tau.$

Hence

$|g_{t,\tau,\lambda}^{\prime}(z)|\leq\frac{C_{\tau,\lambda}}{1+|z|^{2}},\qquad
t\in(\tau/2,2\tau),\,z\in\mathbb{C}_{+},$

for some $C_{\tau,\lambda}$. Since

$\int_{-\pi/2}^{\pi/2}\cos^{s}\psi\int_{0}^{\infty}\frac{d\rho}{1+\rho^{2}}\,d\psi<\infty$

for any $s>-1$, the dominated convergence theorem implies (5.2). ∎

######  Corollary 5.4.

Let $f\in\mathcal{D}_{s},$ $s>-1$. For every $\tau>0$,

$\lim_{t\to\tau}\left\|\frac{f(tz)-f(\tau z)}{t-\tau}-zf^{\prime}(\tau
z)\right\|_{\mathcal{D}_{s+1}}=0.$

###### Proof.

Let $\tau>0$ be fixed, and

$(R_{t,\tau}f)(z):=\frac{f(tz)-f(\tau z)}{t-\tau}-zf^{\prime}(\tau z),\qquad
f\in\mathcal{D}_{s},\,t>\tau/2.$

By Lemma 3.21, $\\{R_{t,\tau}:t>\tau/2\\}$ is a bounded subset of
$L(\mathcal{D}_{s},\mathcal{D}_{s+1})$. By Lemma 5.3,

$\lim_{t\to\tau}\|R_{t,\tau}r_{\lambda}\|_{\mathcal{D}_{s+1}}=0,\qquad\lambda\in\mathbb{C}_{+}.$

Since the linear span of the functions $r_{\lambda}$ and the constants is
dense in $\mathcal{D}_{s}$ (see Theorem 5.1), the assertion follows. ∎

######  Corollary 5.5.

Let $f\in\mathcal{D}_{s}$, $s>-1$, and let

$G(t)(z):=f(tz),\qquad F_{n}(t)(z):=z^{n}f^{(n)}(tz),\quad
n\in\mathbb{N},\,t>0,\,z\in\mathbb{C}_{+}.$

Then $G$ and $F_{n}$ map $(0,\infty)$ into $\mathcal{D}_{s+n}$, $G$ is
$n$-times differentiable as a function from $(0,\infty)$ to
$\mathcal{D}_{s+n}$, and

$F_{n}=G^{(n)}.$

###### Proof.

Firstly, $f(tz)\in\mathcal{D}_{s}\subset\mathcal{D}_{s+n}$, so $G$ maps
$(0,\infty)$ into $\mathcal{D}_{s+n}$.

The proof is by induction on $n$. The case $n=1$ is given by Corollary 5.4.
Assume that $G^{(n)}=F_{n}$ with values in $\mathcal{D}_{s+n}$, and let
$f_{n}(z)=z^{n}f^{(n)}(z)$. Then

$G^{(n)}(t)(z)=F_{n}(t)(z)=t^{-n}f_{n}(tz).$

By Corollary 5.4 applied to $f_{n}\in\mathcal{D}_{s+n}$, $G^{(n)}$ is
differentiable with respect to $t$, when considered as a function with values
in $\mathcal{D}_{s+n+1}$. Finally,

$G^{(n+1)}(t)(z)=\frac{d}{dt}(z^{n}f^{(n)}(tz))=z^{n+1}f^{(n+1)}(tz)=F_{n+1}(t)(z).\qed$

###  5.2. Approximations via change of variables

Here we consider approximations of $f$ from $\mathcal{D}_{s}$ and
$\mathcal{H}_{\psi}$ by the functions $f_{\gamma}(z):=f(z^{\gamma})$ as
$\gamma\to 1-$.

######  Proposition 5.6.

Let $\gamma\in(0,1)$. The following hold.

  1. 1.

Let $s>-1$, and $f\in\mathcal{D}_{s}$. Then

(5.3) $\|f_{\gamma}\|_{\mathcal{D}_{s}}\leq\|f\|_{\mathcal{D}_{s}},$

and

(5.4) $\lim_{\gamma\to 1-}\|f_{\gamma}-f\|_{\mathcal{D}_{s}}=0.$

  2. 2.

Let $\psi\in(0,\pi)$and $g\in\mathcal{H}_{\psi}$. Then

(5.5) $\lim_{\gamma\to 1-}\|g_{\gamma}-g\|_{\mathcal{H}_{\psi}}=0.$

###### Proof.

1\. First let $g\in H^{1,1}(\mathbb{C}_{+})$. If $0<\psi<\varphi\leq\pi/2$,
then it follows from Theorem 4.6(iv) that

(5.6)
$\int_{0}^{\infty}\left(|g^{\prime}(te^{i\varphi})|+|g^{\prime}(te^{-i\varphi})|\right)\,dt\geq\int_{0}^{\infty}\left(|g^{\prime}(te^{i\psi})|+|g^{\prime}(te^{-i\psi})|\right)\,dt.$

Hence

$\displaystyle\|g\|_{\mathcal{D}_{s}}$
$\displaystyle=|g(\infty)|+\int_{0}^{\pi/2}\cos^{s}\varphi\int_{0}^{\infty}\left(|g^{\prime}(te^{i\varphi})|+|g(te^{-i\varphi})|\right)\,dt\,d\varphi$
$\displaystyle\geq|g(\infty)|+\int_{0}^{\pi/2}\cos^{s}\varphi\int_{0}^{\infty}\left(|g^{\prime}(te^{i\gamma\varphi})|+|g^{\prime}(te^{-i\gamma\varphi})|\right)\,dt\,d\varphi$
$\displaystyle=\|g_{\gamma}\|_{\mathcal{D}_{s}}.$

Since $\mathcal{H}_{\pi/2}$ is dense in $\mathcal{D}_{s}$ (see Corollary 5.2),
it follows that the map $g\mapsto g_{\gamma}$ extends to a contraction on
$\mathcal{D}_{s}$, and this contraction maps $f$ to $f_{\gamma}$. Then (5.3)
holds.

Now (5.4) follows from (5.5), (5.3), and the fact that $\mathcal{H}_{\pi/2}$
is continuously and densely embedded in $\mathcal{D}_{s}$ (see Proposition
4.13 and Corollary 5.2).

2\. Since the norms $\|\cdot\|_{\mathcal{H}_{\psi}}$ and
$\|\cdot\|^{\prime}_{\mathcal{H}_{\psi}}$ are equivalent and
$g_{\gamma}(\infty)=g(\infty)$, it suffices to show that

$\|g^{\prime}_{\gamma}-g^{\prime}\|_{H^{1}(\Sigma_{\psi})}=\int_{\partial\Sigma_{\psi}}|g_{\gamma}^{\prime}(z)-g(z)|\,|dz|\to
0.$

By Lemma 4.9, we have

$\int_{\partial\Sigma_{\psi}}|g^{\prime}_{\gamma}(z)|\,|dz|=\int_{\partial\Sigma_{\gamma\psi}}|g^{\prime}(z)|\,|dz|.$

Applying Theorem 4.6(iii) to $g^{\prime}$,

$\lim_{\gamma\to
1-}\int_{\partial\Sigma_{\gamma\psi}}|g^{\prime}(z)|\,|dz|=\int_{\partial\Sigma_{\psi}}|g^{\prime}(z)|\,|dz|$

and

$\lim_{\gamma\to 1-}g^{\prime}_{\gamma}(z)=g^{\prime}(z),\quad\text{for almost
all $z\in\partial\Sigma_{\psi}$}.$

Now the statement (5.5) follows from Lemma 2.2. ∎

###  5.3. Density of rational functions in $\mathcal{H}_{\psi}$

In Theorem 5.1 and Corollary 5.2, we established that
$\widetilde{\mathcal{R}}(\mathbb{C}_{+})$ and several larger spaces are dense
in $\mathcal{D}_{s}$, for $s>-1$ or $s>0$. In particular, we noted that
$H^{1,1}(\mathbb{C}_{+})$ is dense in $\mathcal{D}_{s}$.

Let $\psi\in(0,\pi)$ and

$\mathcal{H}_{\psi,0}=\\{f\in\mathcal{H}_{\psi}:\,f(\infty)=0\\},$

with the norm

$\|f\|_{\mathcal{H}_{\psi,0}}=\|f^{\prime}\|_{H^{1}(\Sigma_{\psi})}.$

By (4.19), this norm is equivalent to $\|\cdot\|_{\mathcal{H}_{\psi}}$ on
$\mathcal{H}_{\psi,0}$.

Let $\psi\in(0,\pi)$ and $\mathcal{R}(\Sigma_{\psi})$ be the linear span of
$\\{r_{\lambda}:\lambda\in\Sigma_{\pi-\psi}\\}$. Let
$\mathcal{R}_{\mathcal{H}}(\Sigma_{\psi})$ be the closure of
$\mathcal{R}(\Sigma_{\psi})$ in $\mathcal{H}_{\psi,0}$. We shall prove that
$\mathcal{H}_{\psi,0}=\mathcal{R}_{\mathcal{H}}(\Sigma_{\psi})$. Thus the
rational functions which vanish at infinity and have simple poles outside
$\overline{\Sigma}_{\psi}$ are dense in $\mathcal{H}_{\psi}$ modulo constants.
This fact may be known, but we did not find it in the literature. Our proof
involves several lemmas given below and it may be of interest as a piece of
function theory. The following lemma, relating to the function spaces
$\mathcal{E}_{\varphi}$ from Definition (4.29), is the key step in our proof.

######  Lemma 5.7.

Let $\psi\in(0,\pi)$, $\varphi\in(\psi,\pi)$ and let $f\in
H^{1}(\Sigma_{\varphi})$. If

$\int_{\partial\Sigma_{\varphi}}\frac{|f(\lambda)|}{|\lambda|}\,|d\lambda|<\infty,$

then

(5.7) $f\in\mathcal{R}_{\mathcal{H}}(\Sigma_{\psi}).$

###### Proof.

From (4.22) for $\gamma=1$, the function $F(\lambda):=r_{\lambda}$ maps
$\Sigma_{\pi-\psi}$ into $\mathcal{H}_{\psi,0}$, and is locally bounded.
Moreover $F$ is holomorphic (see Theorem 4.8(iii) and Section 2) and its
derivative is $-r_{\lambda}^{2}$.

The Cauchy integral formula (4.16) may be written as

$f(z)=\frac{1}{2\pi
i}\int_{\partial\Sigma_{\varphi}}f(\lambda)F(-\lambda)(z)\,d\lambda,\quad
z\in\Sigma_{\psi}.$

From (4.22), we obtain

$\int_{\partial\Sigma_{\varphi}}\|f(\lambda)F(-\lambda)\|_{\mathcal{H}_{\psi,0}}\,|d\lambda|\leq\frac{1}{2\pi\cos^{2}((\varphi-\psi)/2)}\int_{\partial\Sigma_{\varphi}}\frac{|f(\lambda)|}{|\lambda|}\,d\lambda<\infty.$

Thus

$f=\frac{1}{2\pi
i}\int_{\partial\Sigma_{\varphi}}f(\lambda)F(-\lambda)\,d\lambda$

as a Bochner integral in $\mathcal{H}_{\psi,0}$, with continuous integrand, so
it may be approximated in the $\mathcal{H}_{\psi,0}$-norm by Riemann sums of
the integrand, which lie in $\mathcal{R}(\Sigma_{\psi})$. Hence
$f\in\mathcal{R}_{\mathcal{H}}(\Sigma_{\psi})$. ∎

The next step in the proof is to construct a family of functions which serve
as an approximate identity for $\mathcal{H}_{\varphi,0}$ when restricted to
any sector smaller than $\Sigma_{\varphi}$.

######  Lemma 5.8.

Let $\varphi\in(0,\pi)$ and $\epsilon\in(0,1)$, and let

(5.8) $g_{\epsilon}(z):=\frac{2z^{\epsilon}}{1+z^{\epsilon}}(1+\epsilon
z)^{-2},\qquad z\in\mathbb{C}\setminus(-\infty,0].$

Then $g_{\epsilon}\in\mathcal{H}_{\varphi,0}\cap H^{1}(\Sigma_{\varphi})$ and

(5.9)
$\sup_{\epsilon\in(0,1)}\|g_{\epsilon}^{\prime}\|_{H^{1}(\Sigma_{\varphi})}<\infty.$

Moreover,

(5.10) $\lim_{\epsilon\to 0}\,g_{\epsilon}(z)=1,\quad
z\in\mathbb{C}\setminus(-\infty,0],$

and, for $0<a<b<\infty$, there exists $C_{\varphi,a,b}$ such that

(5.11) $|g^{\prime}_{\epsilon}(z)|\leq C_{\varphi,a,b}\cdot\epsilon,\quad
z\in\partial\Sigma_{\varphi},\,|z|\in(a,b).$

###### Proof.

It is clear that $g_{\epsilon}\in H^{1}(\Sigma_{\varphi})$ and
$g_{\epsilon}(\infty)=0$. Moreover

$g^{\prime}_{\epsilon}(z)=\frac{2\epsilon
z^{\epsilon}}{z(1+z^{\epsilon})^{2}(1+\epsilon z)^{2}}-\frac{4\epsilon
z^{\epsilon}}{(1+z^{\epsilon})(1+\epsilon z)^{3}}.$

Applying Lemma 2.1, there is a constant $C_{\varphi}$ such that, for
$z\in\partial\Sigma_{\varphi}$ and $t=|z|$,

(5.12) $|g^{\prime}_{\epsilon}(z)|\leq C_{\varphi}\epsilon
t^{\epsilon}\left(\frac{2}{t(1+t^{\epsilon})^{2}(1+\epsilon
t)^{2}}+\frac{4}{(1+t^{\epsilon})(1+\epsilon t)^{3}}\right).$

Hence

$\displaystyle\|g^{\prime}_{\epsilon}\|_{H^{1}(\Sigma_{\varphi})}$
$\displaystyle\leq 2\epsilon
C_{\varphi}\int_{0}^{\infty}\frac{t^{\epsilon}\,dt}{t(1+t^{\epsilon})^{2}(1+\epsilon
t)^{2}}+4\epsilon
C_{\varphi}\int_{0}^{\infty}\frac{t^{\epsilon}\,dt}{(1+t^{\epsilon})(1+\epsilon
t)^{3}}$ $\displaystyle\leq 2\epsilon
C_{\varphi}\int_{1}^{\infty}\frac{dt}{(1+\epsilon t)^{2}}+2\epsilon
C_{\varphi}\int_{0}^{1}t^{\epsilon-1}\,dt+4\epsilon
C_{\varphi}\int_{0}^{\infty}\frac{dt}{(1+\epsilon t)^{3}}$
$\displaystyle=6C_{\varphi}.$

This yields (5.9).

The property (5.10) is straightforward, and (5.11) follows from (5.12). ∎

Lemmas 5.7 and 5.8 enable us to show that any function
$f\in\mathcal{H}_{\varphi,0}$, when restricted to
$\Sigma_{\psi},\,\psi\in(0,\varphi)$, can be approximated by rational
functions in $\mathcal{H}_{\psi,0}$.

######  Lemma 5.9.

Let $\psi\in(0,\pi)$, $\varphi\in(\psi,\pi)$, and let
$f\in\mathcal{H}_{\varphi,0}.$ Then

(5.13) $f\in\mathcal{R}_{\mathcal{H}}(\Sigma_{\psi}).$

###### Proof.

Assume first that $f\in H_{\varphi,0}$ and $f(0)=0$, and let $g_{\epsilon}$ be
defined by (5.8). Then $fg_{\epsilon}\in\mathcal{H}_{\varphi,0}\cap
H^{1}(\Sigma_{\varphi})$, and

$\int_{\partial\Sigma_{\varphi}}\frac{|f(z)g_{\epsilon}(z)|}{|z|}\,|dz|<\infty.$

By Lemma 5.7, $fg_{\epsilon}\in\mathcal{R}_{\mathcal{H}}(\Sigma_{\psi})$.

Note that

$\|f(1-g_{\epsilon})\|_{\mathcal{H}_{\varphi,0}}\leq\|f^{\prime}(1-g_{\epsilon})\|_{H^{1}(\Sigma_{\varphi})}+\|fg^{\prime}_{\epsilon}\|_{H^{1}(\Sigma_{\varphi})}.$

We will prove that

(5.14) $\lim_{\epsilon\to
0}\,\|f^{\prime}(1-g_{\epsilon})\|_{H^{1}(\Sigma_{\varphi})}=0,$

and

(5.15) $\lim_{\epsilon\to
0}\,\|fg^{\prime}_{\epsilon}\|_{H^{1}(\Sigma_{\varphi})}=0.$

By (4.19) and (5.9),

$\sup_{\epsilon\in(0,1)}\|g_{\epsilon}\|_{H^{\infty}(\Sigma_{\varphi})}\leq\sup_{\epsilon\in(0,1)}\|g^{\prime}_{\epsilon}\|_{H^{1}(\Sigma_{\varphi})}<\infty,$

so, using (5.10) and the dominated convergence theorem, we have

$\|f^{\prime}(1-g_{\epsilon})\|_{H^{1}(\Sigma_{\varphi})}=\int_{\partial\Sigma_{\varphi}}|f^{\prime}(\lambda)(1-g_{\epsilon}(\lambda))|\,|d\lambda|\to
0,\quad\epsilon\to 0.$

For $0<a<b<\infty$,

$\displaystyle\|fg^{\prime}_{\epsilon}\|_{H^{1}(\Sigma_{\varphi})}$
$\displaystyle=\int_{\partial\Sigma_{\varphi}}|f(\lambda)||g_{\epsilon}^{\prime}(\lambda)|\,|d\lambda|$
$\displaystyle\leq\sup_{|z|>b,\,z\in\partial\Sigma_{\varphi}}\,|f(z)|\,\int_{|\lambda|>b,\,\lambda\in\partial\Sigma_{\varphi}}|g^{\prime}_{\epsilon}(\lambda)|\,|d\lambda|$
$\displaystyle\hbox{}\hskip
20.0pt+\sup_{|z|<a,\,z\in\partial\Sigma_{\varphi}}\,|f(z)|\,\int_{|\lambda|<a,\,\lambda\in\partial\Sigma_{\varphi}}|g^{\prime}_{\epsilon}(\lambda)|\,|d\lambda|$
$\displaystyle\hbox{}\hskip
20.0pt+\|f\|_{H^{\infty}(\Sigma_{\varphi})}\int_{|\lambda|\in(a,b),\,\lambda\in\partial\Sigma_{\varphi}}|g^{\prime}_{\epsilon}(\lambda)|\,|d\lambda|$
$\displaystyle\leq\left(\sup_{|z|>b,\,z\in\partial\Sigma_{\varphi}}\,|f(z)|+\sup_{|z|<a,\,z\in\partial\Sigma_{\varphi}}\,|f(z)|\right)\|g^{\prime}_{\epsilon}\|_{H^{1}(\Sigma_{\varphi})}$
$\displaystyle\hbox{}\hskip
20.0pt+2(b-a)\|f\|_{H^{\infty}(\Sigma_{\varphi})}\sup_{|z|\in(a,b),\,z\in\partial\Sigma_{\varphi}}\,|g^{\prime}_{\epsilon}(z)|.$

Letting first $\epsilon\to 0$, using (5.9)-(5.11) along with Vitali’s theorem,
and then letting $a\to 0$ and $b\to\infty$, using $f(0)=f(\infty)=0$, we
obtain

$\lim_{\epsilon\to 0}\|fg^{\prime}_{\epsilon}\|_{H^{1}(\Sigma_{\varphi})}=0.$

We have now proved the assertions (5.14) and (5.15). Thus we obtain (5.13)
under the additional assumption that $f(0)=0$.

Now let $f\in\mathcal{H}_{\varphi,0}$ be arbitrary. Then consider

$f_{0}(z):=f(z)-2f(0)\left(\frac{1}{z+1}-\frac{1}{z+2}\right),$

and note that

$f_{0}\in\mathcal{H}_{\varphi,0}\cap H^{1}(\Sigma_{\varphi}),\qquad
f_{0}(0)=0.$

Then, by the above,

$f_{0}\in\mathcal{R}_{\mathcal{H}}(\Sigma_{\varphi}),$

and hence (5.13) holds. ∎

We now approximate functions $f\in\mathcal{H}_{\psi,0}$ by functions from
$\mathcal{H}_{\psi^{\prime},0},\,\psi^{\prime}>\psi$, using the change of
variables from Proposition 5.6.

For $f\in\mathcal{H}_{\psi,0}$ and $f_{\gamma}(z):=f(z^{\gamma})$ for
$\gamma\in(0,1)$, we now have