Notation

Throughout the article, we will use the following notation:

• $\mathbb {R}_+ :=[0,\infty )$ ,

• $\mathbb {C}_+ := \{z \in \mathbb {C}: \operatorname {Re}z>0\}$ , $\overline {\mathbb {C}}_+ = \{z \in \mathbb {C}: \operatorname {Re}z\ge 0\}$ ,

• $\Sigma _{\theta } := \{z\in \mathbb {C}: z \ne 0, |\arg z|<\theta \}$ for $\theta \in (0,\pi )$ .

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

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

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

$$ \begin{align*} \lim_{|z|\to0, z \in \Sigma_{\psi}} f(z) \end{align*} $$

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

$$ \begin{align*} \lim_{\operatorname{Re}z \to \infty} f(z) \end{align*} $$

exists in $\mathbb {C}$ . We say that f has a full limit at infinity or at zero if

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

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

$$ \begin{align*} e_a(z) = e^{-az}; \qquad r_a(z) = (z+a)^{-1}, \, z\ne-a. \end{align*} $$

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

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

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

• $M(\mathbb {R}_+)$ 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 $\mu \in M(\mathbb {R}_+)$ .

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

• $dS$ 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:

• $\mathcal {Y} \overset {i}{\hookrightarrow } \mathcal {X}$ if $\Omega _{\mathcal {Y}} = \Omega _{\mathcal {X}}$ , $\mathcal {Y}$ is a subset of $\mathcal {X}$ and the inclusion map is continuous;

• $\mathcal {Y} \subset \mathcal {X}$ if $\Omega _{\mathcal {Y}} = \Omega _{\mathcal {X}}$ , $\mathcal {Y}$ is a subset of $\mathcal {X}$ and $\mathcal {Y}$ inherits the norm from $\mathcal {X}$ ;

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

Boundaries of all of the sectors appearing in this article will be oriented from top to bottom.

Elementary inequalities

We will need the following elementary lemma, which gives lower bounds for $|z+\lambda |$ in terms of $|z|$ and $|\lambda |$ , for z, $\lambda \in \mathbb {C}$ .

Proof. For (2.1), we may assume that $\varphi \ge \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

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

The inequality (2.2) follows from (2.1), because $\psi +\varphi $ is the maximum value of $|\psi '-\varphi '|$ for $\psi '\in [-\psi ,\psi ]$ and $\varphi ' \in [-\varphi ,\varphi ]$ .

The inequality (2.4) is obtained by considering $\operatorname {Re}(z+\lambda )$ . For the inequality (2.3), we assume without loss of generality that $\sin \varphi \ge 0$ . Note that

(2.5) $$ \begin{align} |\lambda+z|^2 -|\lambda|^2\cos^2\psi &= \left(|z|+|\lambda|\cos(\varphi-\psi)\right)^2 + |\lambda|^2 \left(\sin^2\varphi - \cos^2(\varphi-\psi)\right) \\ &= |z|^2 + 2|z|\,|\lambda| \cos(\varphi-\psi) + |\lambda|^2 \sin^2 \varphi. \nonumber \end{align} $$

If $\cos (\varphi -\psi )<0$ , we have

$$ \begin{align*} \sin\varphi - |\cos(\varphi-\psi)| = \sin\varphi \, (1+\sin\psi) + \cos \varphi\cos\psi \ge 0. \end{align*} $$

Then the expression on the right-hand side of the first line of (2.5) is clearly nonnegative. If $\cos (\varphi -\psi )\ge 0$ , then the expression in the second line is clearly nonnegative. This completes the proof.

Beta function

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

$$ \begin{align*} B(s,t) = B(t,s) := \int_0^1 \tau^{s-1} (1-\tau)^{t-1} \, d\tau = 2 \int_0^{\pi/2} \cos^{2s-1} \psi \sin^{2t-1} \psi \, d\psi. \end{align*} $$

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

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

see [Reference Prudnikov, Brychkov and Marichev48, items 2.5.3,(1) and 2.2.3,(5)]. We note also the following limit properties:

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

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, because they are usually easily identified.

We will also use the following elementary lemma on several occasions. See [Reference Duren21, p.21, Lemma 1] for a proof.

We will use Vitali’s theorem several times, usually for holomorphic vector-valued functions. We refer to the version given in [Reference Arendt, Batty, Hieber and Neubrander4, Theorem A.5].

Let $\mathcal {X}$ be a Banach space of holomorphic functions on a domain $\Omega _{\mathcal {X}}$ such that the point evaluations $\delta _z : f \mapsto f(z), \, z \in \Omega _{\mathcal {X}}$ , are continuous on $\mathcal {X}$ . 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

$$ \begin{align*} G := \int_{\Omega} F(t) \, d\mu(t) \end{align*} $$

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 $\{F(t) : t\in \Omega \}$ 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 will 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 [Reference Arendt, Batty, Hieber and Neubrander4, 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 that leads to an integral formula for the derivative $F'$ .

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) $$ \begin{align} \|g\|_{\mathcal{V}_s}:&= \int_{\mathbb{C}_+} \frac{(\operatorname{Re}z)^s |g(z)|}{|z|^{s+1}} \, dS(z) \\ &= \int_0^{\infty} \alpha^s\int_{-\infty}^{\infty}\frac{|g(\alpha +i\beta)|}{(\alpha^2+\beta^2)^{(s+1)/2}}\,d\beta\,d\alpha \notag\\ &=\int_{-\pi/2}^{\pi/2}\cos^s\varphi\int_0^{\infty} |g(\rho e^{i\varphi})|\,d\rho\,d\varphi\notag \end{align} $$

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

(3.2) $$ \begin{align} \mathcal{V}_s\subset \mathcal{V}_{\sigma} \qquad \text{and} \qquad \|g\|_{\mathcal{V}_{\sigma}}\le \|g \|_{\mathcal{V}_s}, \qquad g\in \mathcal{V}_s,\; s < \sigma, \end{align} $$

and

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

The following property of functions from $\mathcal V_s$ is an essential element in the arguments that lead to the representations for functions in $\mathcal {V}_s$ in Proposition 3.7 and for $\mathcal {D}_s$ in Corollary 3.10, 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 $k\ge 1$ and every $\psi \in (0,\pi /2)$ ,

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

Proof. Let $g \in \mathcal V_s$ be holomorphic, $\psi \in (0,\pi /2)$ , $\psi ' = (\pi /2 + \psi )/2$ and $b_{\psi }={\sin ((\pi /2-\psi )/2)} = \cos \psi '$ . If $z \in \Sigma _{\psi }$ , then

$$ \begin{align*} \{\lambda\in\mathbb{C}: |\lambda-z|\le b_{\psi}|z|\}\subset \{\lambda\in\Sigma_{\psi'}: |\lambda|\ge(1-b_{\psi})|z|\}. \end{align*} $$

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

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

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

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

and then

$$ \begin{align*} |z|^k |g^{(k-1)}(z)| \le \frac{(k+1)(k-1)!(1+b_{\psi})}{2\pi b_{\psi}^{k+1}} \int_{\{\lambda\in\Sigma_{\psi'}: |\lambda|\ge (1-b_{\psi})|z|\}} \frac{|g(\lambda)|}{|\lambda|} \, dS(\lambda). \end{align*} $$

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 role similar to the operator Q on $\mathcal {W}$ considered in [Reference Batty, Gomilko and Tomilov8, 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) $$ \begin{align} \|g\|_{\mathcal{W}} := \int_0^{\infty} \sup_{\beta\in \mathbb{R}} |g(\alpha+i\beta)| \, d\alpha < \infty. \end{align} $$

Indeed, the definition of $Q_1$ is formally the same as the definition of Q in [Reference Batty, Gomilko and Tomilov8], but the domain $\mathcal {V}_1$ of $Q_1$ is larger than $\mathcal {W}$ .

For $g \in \mathcal {V}_s$ , let

(3.6) $$ \begin{align} (Q_s g)(z):= - \frac{2^s}{\pi}\int_0^{\infty} \alpha^s\int_{-\infty}^{\infty}\frac{g(\alpha+i\beta)}{(z+\alpha-i\beta)^{s+1}} \, d\beta\,d\alpha, \quad z\in \mathbb{C}_{+} \cup \{0\}. \end{align} $$

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

(3.7) $$ \begin{align} |(Q_s g)(z)|\le \frac{2^s\|g\|_{\mathcal{V}_s}}{\pi \cos^{s+1}\psi}, \qquad z \in \Sigma_{\psi}, \, \psi \in (0,\pi/2). \end{align} $$

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

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

and

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

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) $$ \begin{align} (Q_s g)'(z) = (s+1)\frac{2^s}{\pi}\int_0^{\infty} \alpha^s\int_{-\infty}^{\infty}\frac{g(\alpha + i\beta)}{(z+\alpha-i\beta)^{s+2}}\,d\beta \, d\alpha, \qquad z\in \mathbb{C}_{+}. \end{align} $$

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

(3.11) $$ \begin{align} |(Q_s g)'(z)|\le \frac{(s+1)2^s}{\pi\cos^{s+2}\psi \,|z|}\|g\|_{\mathcal{V}_s},\qquad z\in \Sigma_{\psi}, \end{align} $$

and

$$ \begin{align*} |z(Q_s g)'(z)|\le \frac{(s+1)2^s}{\pi\cos^{s+2}\psi}\int_{\mathbb{C}_{+}} \frac{|z|(\operatorname{Re}\lambda)^s\,|g(\lambda)|}{|z+\overline{\lambda}||\lambda|^{s+1}}\,dS(\lambda), \qquad z\in \Sigma_{\psi}. \end{align*} $$

Using the dominated convergence theorem again, we obtain

(3.12) $$ \begin{align} \lim_{|z|\to0,z\in\Sigma_{\psi}} z(Q_sg)'(z) = 0. \end{align} $$

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) $$ \begin{align} C_s:=\int_0^{\infty}\frac{dt}{(t+1)^{n+2}t^{\delta}}= \int_0^1\frac{(1-\tau)^s}{\tau^{\delta}}\,d\tau=B(1-\delta,s+1). \end{align} $$

Then

$$ \begin{align*} \int_0^{\infty} \frac{dt}{(\lambda+t)^{n+2}t^{\delta}} = \frac{C_s}{\lambda^{s+1}},\quad \lambda\in \mathbb{C}_{+}. \end{align*} $$

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) $$ \begin{align} (Q_s g)(z) =- \frac{2^s}{\pi C_s}\int_0^{\infty} \int_0^{\infty}\int_{-\infty}^{\infty} \frac{\alpha^s\,g(\alpha+i\beta)\,d\beta\,d\alpha}{(z+\alpha-i\beta+t)^{n+2}}\,\frac{dt}{t^{\delta}},\quad z\in \mathbb{C}_{+}. \end{align} $$

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

$$ \begin{align*} f'\in \mathcal{V}_s, \end{align*} $$

equipped with the semi-norm

$$ \begin{align*} \|f\|_{\mathcal D_{s,0}}:=\|f'\|_{\mathcal{V}_s}, \qquad f \in \mathcal D_s. \end{align*} $$

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 Subsection 3.3.

In the rest of this section we will obtain a reproducing formula (3.29) for functions from $\mathcal {D}_s$ and we will describe some basic properties that 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 [Reference Batty, Gomilko and Tomilov8, 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 $Q_sg \notin \mathcal {D}_s$ can be defined as follows:

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

and $g(z)= 0$ for all other points in $\mathbb {C}_+$ . We do not give details in this article. 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) $$ \begin{align} G_{h,\beta,s}(\varphi):= \int_{-\pi/2}^{\pi/2}\cos^s\psi \int_0^1\frac{h(t)\,dt}{(t^2+2t\cos(\varphi+\psi)+1)^{\beta}}\,d\psi,\quad \varphi\in (-\pi/2,\pi/2). \end{align} $$

1. (a) If $2\beta -s-2 < 0$ , then

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

2. (b) If $2\beta -s-2> 0$ , then

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

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

$$ \begin{align*} G_{h,\beta,s}(\varphi) &= \int_{-\pi/2}^{0}\cos^s\psi \int_0^1\frac{h(t)\,dt}{(t^2+2t\cos(\varphi+\psi)+1)^{\beta}}\,d\psi\\ &\hskip30pt + \int_0^{\pi/2}\cos^s\psi\int_0^1\frac{h(t)\,dt}{(t^2+2t\cos(\varphi+\psi)+1)^{\beta}}\,d\psi\\ &=: G_{h,\beta, s}^{-}(\varphi)+G_{h,\beta, s}^+(\varphi), \end{align*} $$

and we estimate these two integrals separately.

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

$$ \begin{align*} G_{h,\beta, s}^{-}(\varphi) &\le \int_{-\pi/2}^{0}\cos^s\psi \,d\psi\int_0^1 h(t)\,dt = \frac{\|h\|_{L^1[0,1]}}{2}B\left(\frac{s+1}{2},\frac{1}{2}\right). \end{align*} $$

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

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

Hence,

$$ \begin{align*} &{\int_0^1\frac{h(t)}{(t^2+2t\cos\phi+1)^{\beta}}\,dt } \\ &\qquad \le 4^{\beta}\int_0^{1/2} h(t)\,dt+ \int_{1/2}^1\frac{h(t)}{((t-1)^2+(1+\cos\phi))^{\beta}}\,dt\\ &\qquad \le 4^{\beta} \|h\|_{L^1[0,1/2]}+ \|h\|_{L^{\infty}[1/2,1]}\int_0^{\infty}\frac{d\tau}{(\tau^2+ 2\cos^2(\phi/2))^{\beta}} \\ &\qquad \le \frac{C_{h,\beta}}{\cos^{2\beta-1}(\phi/2)}, \end{align*} $$

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

$$ \begin{align*} \omega\ge \sin \omega\ge \frac{2}{\pi}\omega,\qquad \omega\in (0,\pi/2), \end{align*} $$

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

$$ \begin{align*} G^+_{h,\beta,s}(\pi/2-\varphi) &\le C_{h,\beta} \int_0^{\pi/2}\frac{\sin^s\psi}{\sin^{2\beta-1}((\varphi+\psi)/2)} \,d\psi\\ &\le C_{h,\beta}\pi^{2\beta-1} \int_0^{\pi/2}\frac{\psi^s}{(\varphi+\psi)^{2\beta-1}} \,d\psi. \end{align*} $$

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

$$ \begin{align*} \int_0^{\pi/2} \frac{\psi^s}{(\varphi+\psi)^{2\beta-1}} \,d\psi \le \int_0^{\pi/2} \psi^{s-2\beta+1} \,d\psi = \frac{\pi^{s-2\beta+2}}{(s-2\beta+2)2^{s-2\beta+2}}<\infty. \end{align*} $$

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

$$ \begin{align*} \int_0^{\pi/2} \frac{\psi^s}{(\varphi+\psi)^{2\beta-1}} \,d\psi \le \varphi^{s+2-2\beta} \int_0^{\infty} \frac{t^s}{(t+1)^{2\beta-1}} \,dt, \end{align*} $$

and hence

$$ \begin{align*} \cos^{2\beta-s-2}(\pi/2-\varphi) \, G_{h,\beta,s}(\pi/2-\varphi)& \le \varphi^{2\beta-s-2} G_{h,\beta,s}(\pi/2-\varphi)\\ & \le C_{h,\beta} \pi^{2\beta-1} \int_0^{\infty} \frac{t^s}{(t+1)^{2\beta-1}} \,dt < \infty, \end{align*} $$

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 will see in Corollary 3.11 that it is a Banach space.

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

Example 3.3. Let $\lambda =|\lambda |e^{i\varphi }\in \mathbb {C}_{+}$ , and $r_{\lambda }(z)= (z+\lambda )^{-1},\; z \in \mathbb {C}_+$ . Let $\gamma>0$ , and consider

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

Let $s>-1$ . Then

(3.18) $$ \begin{align} \|r_{\lambda}^{\gamma}\|_{\mathcal{D}_{s,0}} &=\gamma \int_{-\pi/2}^{\pi/2} \cos^s\psi\int_0^{\infty} \frac{d\rho}{\left|\rho e^{i\psi}+ |\lambda| e^{i\varphi}\right|{}^{\gamma+1}}\,d\psi\\ &=\frac{\gamma}{|\lambda|^{\gamma}}\int_{-\pi/2}^{\pi/2} \cos^s\psi\int_0^{\infty} \frac{d\rho}{|\rho+e^{i(\varphi-\psi)}|^{\gamma+1}}\,d\psi \nonumber \\ &=\frac{\gamma}{|\lambda|^{\gamma}} \int_{-\pi/2}^{\pi/2} \cos^s\psi\int_0^1 \frac{1+t^{\gamma-1}} {(t^2+2t\cos(\varphi+\psi)+1)^{(\gamma+1)/2}}\,dt\,d\psi, \nonumber \end{align} $$

where we have put $t=\rho $ for $\rho \le 1$ and $t = \rho ^{-1}$ 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

(3.19) $$ \begin{align} \|r_{\lambda}^{\gamma}\|_{\mathcal{D}_{s,0}} \le \begin{cases} \dfrac{\gamma K_{h,(\gamma+1)/2,s}}{|\lambda|^{\gamma}}, &s>\gamma-1>-1, \\ &\\ \dfrac{\gamma \tilde{K}_{h,(\gamma+1)/2,s}}{|\lambda|^{\gamma}\cos^{\gamma-s-1}\varphi}, &\gamma-1> s>-1. \end{cases} \end{align} $$

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

(3.20) $$ \begin{align} \|r_{\lambda}\|_{\mathcal{D}_{s,0}} = \int_{-\pi/2}^{\pi/2} \int_0^{\infty} \frac{\cos^s \psi}{|\lambda+\rho e^{i\psi}|^2} \,d\rho \,d\psi \le \frac{C_s}{|\lambda|}, \qquad\lambda \in \mathbb{C}_+. \end{align} $$

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

Next we consider some functions that appear frequently in the studies of holomorphic $C_0$ -semigroups.

Example 3.4. Let

$$ \begin{align*} f_{\nu}(z):=z^{\nu} e^{-z},\quad z\in \mathbb{C}_{+},\quad \nu\ge0. \end{align*} $$

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

(3.21) $$ \begin{align} \|f_{\nu}\|_{\mathcal{D}_{s,0}}\le 2 B\left(\frac{s-\nu}{2},\frac{1}{2}\right)\Gamma(\nu+1). \end{align} $$

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

We have

$$ \begin{align*} f_{\nu}'(z)=e^{-z}\left(\nu{z^{\nu-1}}-z^{\nu}\right) \end{align*} $$

and

$$ \begin{align*} \|f_{\nu}\|_{\mathcal{D}_{s,0}}&=\int_{-\pi/2}^{\pi/2} \cos^s\varphi\int_0^{\infty} {e^{-\rho\cos\varphi}}{\rho^{\nu-1}}\left|{\rho e^{i\varphi}}-\nu\right|\,d\rho\,d\varphi\\ &=\int_{-\pi/2}^{\pi/2} \cos^{s-\nu-1}\varphi\int_0^{\infty} e^{-r}r^{\nu-1}| r e^{i\varphi}-\nu\cos\varphi|\,dr\,d\varphi. \end{align*} $$

We use the estimates

$$ \begin{align*} r |\sin\varphi| \le |r e^{i\varphi}-\nu\cos\varphi| \le r+\nu. \end{align*} $$

If $s> \nu $ , we obtain

$$ \begin{align*} \|f_{\nu}\|_{\mathcal{D}_{s,0}} &\le 2\int_0^{\pi/2} \cos^{s-\nu-1}\varphi\,d\varphi \int_0^{\infty} e^{-r} r^{\nu-1}(r+\nu) \,dr \\ &= 2 B\left(\frac{s-\nu}{2},\frac{1}{2}\right)\Gamma(\nu+1). \notag \end{align*} $$

If $s \le \nu $ , then

$$ \begin{align*} \|f_{\nu}\|_{\mathcal{D}_{s,0}} \ge 2\int_0^{\pi/2} \cos^{s-\nu-1}\varphi \, \sin\varphi\,d\varphi \int_0^{\infty} e^{-\tau} \tau^{\nu} \,d\tau = \infty. \end{align*} $$

This establishes the claims above.

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

Example 3.5. The function $\operatorname {\mathrm {arccot}}$ is defined by

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

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

(3.23) $$ \begin{align} &{\|\operatorname{\mathrm{arccot}}\|_{\mathcal{D}_{0,0}}} \\ & \ =\int_{-\pi/2}^{\pi/2}\int_0^{\infty} \frac{d\rho}{|1+\rho^2e^{2i\varphi}|}\,d\varphi =\int_0^{\infty} \int_0^{\pi}\frac{d\psi}{(\rho^4+2\rho^2\cos\psi+1)^{1/2}}\,d\rho \notag\\ & \ \le \sqrt{\pi}\int_0^{\infty} \left(\int_0^{\pi} \frac{d\psi}{\rho^4+2\rho^2\cos\psi+1}\right)^{1/2}\,d\rho = \sqrt{\pi}\int_0^{\infty} \left(\frac{\pi}{|\rho^4-1|}\right)^{1/2}\,d\rho\notag\\ & \ =\frac{\pi}{2} B(1/4,1/2) < 3 \pi.\notag \end{align} $$

See [Reference Prudnikov, Brychkov and Marichev48, item 2.5.16, (38)] for the evaluation of the integral with respect to $\psi $ .

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), (3.10) and the second case of (3.19) with $\gamma =\sigma +1$ and $h(t)=1+t^{\sigma }$ , we have

$$ \begin{align*} &{\frac{\pi}{2^{\sigma}(\sigma+1)} \|Q_{\sigma}g\|_{\mathcal{D}_{s,0}}\hskip10pt}\\ & \le \int_{-\pi/2}^{\pi/2}\cos^{s}\psi \int_0^{\infty} \int_{-\pi/2}^{\pi/2} \cos^{\sigma} \varphi \int_0^{\infty} \frac{t^{\sigma+1}|g(t e^{i\varphi})|}{|\rho e^{i\psi}+t e^{-i\varphi}|^{\sigma+2}}\,d\rho\,d\varphi \,dt\,d\psi\\ & = \frac{1}{\sigma+1}\int_{-\pi/2}^{\pi/2} \cos^{\sigma} \varphi \int_0^{\infty} t^{\sigma+1}\|r_{t e^{-i\varphi}}^{\sigma+1}\|_{\mathcal{D}_{s,0}} |g(t e^{i\varphi})|\,dt\,d\varphi \\ & \le \int_{-\pi/2}^{\pi/2} \cos^{\sigma} \varphi \int_0^{\infty} \frac{\tilde{K}_{h,(\sigma+2)/2,s}}{\cos^{\sigma-s}\varphi} |g(t e^{i\varphi})|\,dt\,d\varphi \\ & = \tilde{K}_{h,(\sigma+2)/2,s} \|g\|_{\mathcal{V}_s}. \end{align*} $$

This establishes the second statement.

For the third statement, the same estimation but with s and $\sigma $ interchanged and using the first case of (3.19) with $\gamma =s+1$ and $h(t)=1+t^s$ shows that

$$ \begin{align*} \frac{\pi}{2^s(s+1)} \|Q_{s}g\|_{\mathcal{D}_{\sigma,0}}\le {K}_{h,(s+2)/2,\sigma} \|g\|_{\mathcal{V}_s}. \end{align*} $$

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.

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 $(Q_ng)' = g$ . Let

$$ \begin{align*} I(\alpha) &:= \int_{-\infty}^{\infty} \frac{|g(\alpha+i\beta)|}{(\alpha^2+\beta^2)^{(n+1)/2}} \,d\beta, \qquad \alpha>0,\\ J(\beta) &:= \int_0^{\infty} \frac{\alpha^n|g(\alpha+i\beta)|}{(\alpha^2+\beta^2)^{(n+1)/2}} \,d\alpha, \qquad \beta \in \mathbb{R}. \end{align*} $$

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 $(\alpha _j)_{j\ge 1}$ such that

$$ \begin{align*} \lim_{j \to \infty}\alpha_j =\infty, \qquad \lim_{j\to\infty} I(\alpha_j) = 0. \end{align*} $$

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

$$ \begin{align*} \lim_{k \to \infty}\beta_k^{\pm} = \pm\infty, \qquad \lim_{k\to\infty} J(\beta_k^{\pm}) = 0. \end{align*} $$

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

$$ \begin{align*} &{\frac{2\pi}{n!} g^{(n)}(2\alpha+z)} \\ &= \int_{\beta_k^-}^{\beta_k^+} \frac{g(\alpha_j+i\beta)}{(\alpha_j+i\beta - z -2\alpha)^{n+1}} d\beta - \int_{\beta_k^-}^{\beta_k^+} \frac{g(\alpha+i\beta)}{(\alpha+i\beta - z -2\alpha)^{n+1}} d\beta \\ &\quad -i \int_{\alpha}^{\alpha_j}\frac{g(s+i\beta_k^{-})}{(s+i\beta_k^{-}- z-2\alpha)^{n+1}}\,ds + i \int_{\alpha}^{\alpha_j}\frac{g(s+i\beta_k^{+})}{(s+i\beta_k^{+} - z -2\alpha)^{n+1}}\,ds. \end{align*} $$

Letting $k \to \infty $ , we obtain

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

Letting $j \to \infty $ , we obtain

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

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

(3.25) $$ \begin{align} (Q_n g)(z) &= (-1)^{n+1} \frac{2^{n+1}}{n!} \int_0^{\infty} \alpha^{n} g^{(n)}(2\alpha+z)\,d\alpha \\ &=\frac{(-1)^{n+1}}{n!}\int_0^{\infty} \alpha^n g^{(n)}(\alpha+z)\,d\alpha = - \int_0^{\infty} g(\alpha+z)\,d\alpha. \nonumber \end{align} $$

By Lemma 3.1, the integral $\int _{0}^{\infty } g'(\alpha +z)\,d\alpha $ converges absolutely and uniformly for z in compact subsets of $\mathbb {C}_+$ . So, differentiating under the integral sign we get $(Q_n g)' = g$ .

Now consider the case when $s=n+\delta>-1$ , where $n \in \mathbb {N}\cup \{-1,0\}$ , $\delta \in (0,1)$ and $g \in \mathcal {V}_s$ is holomorphic. Then $g\in \mathcal {V}_{n+1}$ , $n+1\in \mathbb {N}\cup \{0\}$ , and (3.24) gives

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

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

$$ \begin{align*} C_s (Q_s g)(z) &=- \frac{2^s}{\pi}\int_0^{\infty} \int_0^{\infty}\int_{-\infty}^{\infty} \frac{\alpha^s\,g(\alpha+i\beta)\,d\beta\,d\alpha}{(z+\alpha-i\beta+t)^{n+2}}\,\frac{dt}{t^{\delta}}\\ &=(-1)^n\frac{2^{s+1}}{(n+1)!}\int_0^{\infty} \int_0^{\infty} \alpha^s g^{(n+1)}(2\alpha+z+t)\, d\alpha\,\frac{dt}{t^{\delta}}\\ &=\frac{(-1)^n}{(n+1)!}\int_0^{\infty} \int_t^{\infty} (\tau-t)^s g^{(n+1)}(\tau+z) \, d\tau\,\frac{dt}{t^{\delta}}\\ &=C_s\frac{(-1)^n}{(n+1)!}\int_0^{\infty} \tau^{n+1} \,g^{(n+1)}(\tau+z)\,d\tau. \end{align*} $$

As in (3.25), it follows that $(Q_sg)' = g$ .

Remark 3.9. The proof of the property $(Q_ng)'=g', n \ge 0,$ in Proposition 3.7 uses just improper convergence of the integrals $\int _{0}^{\infty } \alpha ^{k} g^{(k)}(\alpha +z)\,d\alpha $ for $0 \le k \le n-1.$ It is instructive to note that if $g \in \mathcal V_n$ is holomorphic and $z \in \mathbb C_+$ , then

(3.26) $$ \begin{align} \int_{0}^{\infty} \alpha^{k} |g^{(k)}(\alpha+z)|\,d\alpha<\infty \end{align} $$

for all $ k \ge 0.$ Indeed, if $g \in \mathcal V_n,$ then (3.26) holds for $k=n$ by (3.24) and the definition of norm in $\mathcal V_n.$ If $n \ge 1,$ then using Lemma 3.1, we infer that $g^{(n-1)}(\alpha +z)=-\int _{\alpha }^{\infty } g^{(n)}{(s+z)\,ds},$ and hence by Fubini’s theorem,

$$ \begin{align*} \int_0^{\infty} \alpha^{n-1} |g^{(n-1)}(\alpha+z)|\,d\alpha \le \frac{1}{n} \int_0^{\infty} \alpha^n | g^{(n)}(\alpha +z)| d\alpha <\infty. \end{align*} $$

Repeating this argument, we conclude that (3.26) holds also for k such that $0 \le k < n.$ If $k>n,$ then (3.26) follows directly from (3.24) and the inclusion $\mathcal V_n \subset \mathcal V_k.$

The following representation of functions in $\mathcal {D}_s$ has appeared in [Reference Aleksandrov2, Corollary 4.2] (see also [Reference Aleksandrov and Peller3, Lemma 3.13.2] for the case $s=1$ ).

Corollary 3.10. Let $f\in \mathcal {D}_s$ , $s>-1 $ . Then the sectorial limits

(3.27) $$ \begin{align} f(\infty): =\lim_{z\to\infty ,\;z\in \Sigma_{\psi}}\,f(z), \end{align} $$

(3.28) $$ \begin{align} f(0):= \lim_{z\to 0,\;z\in \Sigma_{\psi}}\,f(z) \end{align} $$

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

(3.29) $$ \begin{align} f(z)&=f(\infty)+(Q_s f')(z)\\ &=f(\infty)- \frac{2^s}{\pi}\int_0^{\infty} \alpha^s\int_{-\infty}^{\infty} \frac{f'(\alpha+i\beta)\,d\beta}{(z+\alpha-i\beta)^{s+1}} \, d\alpha.\notag \end{align} $$

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

Corollary 3.11. For every $s>-1$ the space $\mathcal D_s$ equipped with the norm

$$ \begin{align*} \|f\|_{\mathcal D_s}:= |f(\infty)|+\|f\|_{\mathcal D_{s,0}}, \qquad f \in\mathcal D_s, \end{align*} $$

is a Banach space.

The argument used in the proof of Corollary 3.11 also provides the following corollary of Fatou type (see also Lemma 4.11).

Now employing (3.29), the estimates (3.7), (3.11) and (3.12) and Lemma 3.1, we obtain the following estimates.

Corollary 3.13. Let $\psi \in (0,\pi /2)$ . For all $f\in \mathcal {D}_s$ , $s>-1$ ,

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

and

$$ \begin{align*} |f'(z)|\le \frac{(s+1)2^s}{\pi |z| \cos^{s+2}\psi}\|f'\|_{\mathcal{V}_s},\qquad z\in\Sigma_{\psi}. \end{align*} $$

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

$$ \begin{align*} \lim_{|z|\to0, z \in \Sigma_{\psi}} z f'(z) = 0. \end{align*} $$

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 that will be relevant for the sequel.

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

$$ \begin{align*} \|f\|_{\mathcal{D}_{s,0}}&=|f(\infty)| +\int_0^{\infty} \alpha^s\int_{-\infty}^{\infty}\frac{|f'(\alpha+i\beta)|} {|\alpha+i\beta|^{s+1}} \,d\beta\,d\alpha \\ &\le \|f\|_{\infty} +2\int_0^{\infty} \sup_{\beta\in \mathbb{R}}\,|f'(\alpha+i\beta)|\, \int_{0}^{\infty}\frac{dt} {(t^2+1)^{(s+1)/2}}\,d\alpha\notag\\ &\le \max\{1, B(1/2,s/2)\}\|f\|_{\mathcal{B}}.\notag \end{align*} $$

Thus, $\mathcal {B}$ is continuously included in $\mathcal {D}_s$ .

Recall that $\mathcal {LM} \overset {i}{\hookrightarrow } \mathcal {B}$ ; see [Reference Batty, Gomilko and Tomilov7, Section 2.4], Thus, in view of Proposition 3.15, we have

$$ \begin{align*} \mathcal{LM} \overset{i}{\hookrightarrow} \mathcal{B} \overset{i}{\hookrightarrow} \mathcal{D}_s, \qquad s>0. \end{align*} $$

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

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

Proof. Note that

$$ \begin{align*} \|\tilde{f}'\|_{\mathcal{V}_s}&=\int_{-\pi/2}^{\pi/2}\cos^s\varphi\int_0^{\infty} \frac{|f'(\rho^{-1}e^{-i\varphi})|}{\rho^2}\,d\rho\,d\phi\\ &=\int_{-\pi/2}^{\pi/2}\cos^s\varphi\int_0^{\infty} |f'(r e^{i\varphi})|\,dr\,d\varphi\\ &=\|f'\|_{\mathcal{V}_s}. \end{align*} $$

Moreover, by Corollary 3.10, $\tilde {f}(\infty )=f(0)$ . This proves (i). The other statements are very easy.

Remarks 3.18. 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 $\mathcal {D}_0$ that are not in $\mathcal {B}$ ; for example, the function $\exp (\operatorname {\mathrm {arccot}} z) \in \mathcal D_0$ and is bounded but it is not in $\mathcal {B}$ (see Example 3.19).

More generally, for $\nu \ge 0$ , 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

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

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

2. Because $e^{-z} \in \mathcal {D}_s$ for $s>0$ , it follows from Lemma 3.17 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 (3.27) and (3.28) in Corollary 3.10 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

$$ \begin{align*} (T(\tau)f)(z) = f(z+\tau), \qquad f \in \mathcal{D}_s, \, \tau \in \mathbb{C}_+, \, z \in \mathbb{C}_+. \end{align*} $$

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

$$ \begin{align*} g(z)=\exp(\operatorname{\mathrm{arccot}}(z)), \qquad \quad z\in \mathbb{C}_{+}. \end{align*} $$

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

$$ \begin{align*} \|g\|_{\mathcal{D}_s}&= |g(\infty)|+\|g \cdot (\operatorname{\mathrm{arccot}})'\|_{\mathcal{V}_s} \le \exp(\pi/2)(1+\|\operatorname{\mathrm{arccot}}\|_{\mathcal{D}_s}). \end{align*} $$

Thus, $g \in \mathcal {D}_s$ for all $s>-1$ .

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

$$ \begin{align*} \operatorname{\mathrm{arccot}} (i+i\epsilon)=\frac{1}{2i}\log\left(1+\frac{2}{\epsilon}\right); \end{align*} $$

hence,

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

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, $\mathcal {D}_s$ is 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

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

where $a\ge 0$ , $b\ge 0$ 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 [Reference Batty, Gomilko and Tomilov6]):

1. (B1) g maps $\Sigma _{\psi }$ into $\Sigma _{\psi }$ for each $\psi \in [0,\pi /2]$ ; see [Reference Schilling, Song and Vondraček50, Proposition 3.6] or [Reference Batty, Gomilko and Tomilov6, Proposition 2.1(1)].

2. (B2) g is increasing on $(0,\infty )$ .

3. (B3) For all $z \in \mathbb {C}_+$ ,

   $$ \begin{align*} g(\operatorname{Re}z) \le \operatorname{Re}g(z) \le |g(z)|, \qquad |g'(z)| \le g'(\operatorname{Re}z). \end{align*} $$
   Here the first inequality follows from taking the real parts in (3.30), and the second inequality is shown in [Reference Batty, Gomilko and Tomilov7, Section 3.5, (B3)].

Further information on Bernstein functions can be found in [Reference Schilling, Song and Vondraček50].

Lemma 3.20. Let g be a Bernstein function, $\lambda \in \mathbb {C}_+$ , and

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

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

$$ \begin{align*} \|f(\cdot;\lambda)\|_{\mathcal{D}_{s,0}} \le \frac{2^s}{(s-2)|\lambda|}. \end{align*} $$

Proof. We have

$$ \begin{align*} f'(z;\lambda)=-\frac{g'(z)}{(\lambda+g(z))^2}, \end{align*} $$

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

$$ \begin{align*} \int_0^{\infty} |f'(\rho e^{i\psi};\lambda)|\,d\rho &=\int_0^{\infty} \frac{|g'(\rho e^{i\psi})|} {|\lambda+g(\rho e^{i\psi})|^2} \,d\rho\\ &\le \frac{1}{\cos^2((|\psi|+\pi/2)/2)} \int_0^{\infty} \frac{g'(\rho\cos\psi)} {(|\lambda|+|g(\rho e^{i\psi})|)^2} \,d\rho\\ &\le\frac{1}{\cos^2((|\psi|+\pi/2)/2)} \int_0^{\infty} \frac{g'(\rho\cos\psi)} {(|\lambda|+g(\rho\cos\psi))^2} \,d\rho\\ &\le \frac{1}{\cos^2((|\psi|+\pi/2)/2)\cos\psi} \int_0^{\infty} \frac{dt} {(|\lambda|+t)^2}\\ &= \frac{1}{\cos^2((|\psi|+\pi/2)/2)\cos\psi}\cdot \frac{1}{|\lambda|}. \end{align*} $$

If $s>2$ , then

$$ \begin{align*} |\lambda|\|f(\cdot;\lambda)\|_{\mathcal{D}_{s,0}} &=|\lambda|\int_{-\pi/2}^{\pi/2}\cos^s\psi \int_0^{\infty} |f'(\rho e^{i\psi};\lambda)|\,d\rho\,d\psi\\ &\le 2\int_0^{\pi/2} \frac{\cos^{s-1}\psi}{\cos^2((\psi+\pi/2)/2)}\,d\psi = 4\int_0^{\pi/2} \frac{\cos^{s-1}\psi}{1-\sin\psi} \,d\psi\\&\le 16\int_0^1(1-t^2)^{s-3}\,dt=\frac{8\sqrt{\pi} \Gamma(s-2)}{\Gamma(s-3/2)}\le \frac{16 s^{1/2}}{s-2}, \end{align*} $$

where the latter estimate follows from Wendel’s inequality for the Gamma function [Reference Wendel56].

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

$$ \begin{align*} \|f\|_{\mathcal{D}_s^{\infty}}:=\|f\|_{\infty}+\|f'\|_{\mathcal{V}_s}. \end{align*} $$

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

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

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

$$ \begin{align*} \|f\|_{\mathcal{D}_s^{\infty}}:=\|f\|_{\infty}+\int_0^{\infty} \alpha^s \int_{-\infty}^{\infty} \frac{|f'(\alpha+i\beta)|}{(\alpha^2+\beta^2)^{(s+1)/2}}\,d\alpha\,d\beta < \infty. \end{align*} $$

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

(3.31) $$ \begin{align} \|fg\|_{\mathcal{D}_s^{\infty}}\le \|f\|_{\mathcal{D}_s^{\infty}} \|g\|_{\mathcal{D}_s^{\infty}},\quad f,g\in \mathcal{D}_s^{\infty}. \end{align} $$

By Proposition 3.15, $\mathcal {B}\overset {i}{\hookrightarrow } \mathcal {D}_s^{\infty }$ for $s>0$ , and the embeddings are continuous.

Example 3.19 shows that the function $g(z) := \exp (\operatorname {\mathrm {arccot}} z)$ is in $\mathcal {D}_s^{\infty }$ for all $s>-1$ , with $\|g\|_{\mathcal {D}_s^{\infty }} = \|g\|_{\mathcal {D}_s}$ , and consequently $\mathcal {D}_s^{\infty }$ is not invariant under vertical shifts.

It follows from Lemma 3.17 that

$$ \begin{align*} f\in \mathcal{D}_s^{\infty} \quad \text{if and only if}\quad \tilde{f}(z):=f(1/z)\in \mathcal{D}_s^{\infty}, \end{align*} $$

and

$$ \begin{align*} \|f\|_{\mathcal{D}_s^{\infty}}=\|\tilde{f}\|_{\mathcal{D}_s^{\infty}}. \end{align*} $$

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

$$ \begin{align*} \mathcal D_{\infty}:=\bigcup_{s>-1} \mathcal{D}_s. \end{align*} $$

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

Lemma 3.21. For $s,\sigma>-1$ , let $f\in \mathcal {D}_s$ and $g\in \mathcal {D}_{\sigma }$ . Then

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

and

(3.32) $$ \begin{align} \|h\|_{\mathcal{D}_{s+\sigma+1}} \le \left(2+\frac{2^s+2^{\sigma}}{\pi}\right)\|f\|_{\mathcal{D}_s}\,\|g\|_{\mathcal{D}_{\sigma}}. \end{align} $$

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

Proof. By Corollary 3.13, for $\rho>0$ and $|\varphi |< \pi /2$ , we have

$$ \begin{align*} |f(\rho e^{i\varphi})| &\le \left(1+\frac{2^s}{\pi\cos^{s+1}\varphi}\right)\|f\|_{\mathcal{D}_s},\\ |g(\rho e^{i\varphi})| &\le \left(1+\frac{2^{\sigma}}{\pi\cos^{\sigma+1}\varphi}\right)\|g\|_{\mathcal{D}_{\sigma}}. \end{align*} $$

Hence,

$$ \begin{align*} \|h\|_{\mathcal{D}_{s+\sigma+1,0}} &= \int_{-\pi/2}^{\pi/2} \cos^{s+\sigma+1}\varphi \int_0^{\infty} \left| f'(\rho e^{i\varphi}) g(\rho e^{i\varphi}) + f(\rho e^{i\varphi}) g'(\rho e^{i\varphi})\right| \,d\rho\,d\varphi \\ &\le \left(1+\frac{2^{\sigma}}{\pi}\right)\|f\|_{\mathcal{D}_{s,0}}\,\|g\|_{\mathcal{D}_{\sigma}} +\left(1+\frac{2^s}{\pi}\right)\|f\|_{\mathcal{D}_s}\,\|g\|_{\mathcal{D}_{\sigma,0}}. \end{align*} $$

This shows that $h \in \mathcal {D}_q$ , and (3.32) 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 3.22 is of independent interest. Corollary 3.23 will be used in Subsection 10.3. For $m,n \in \mathbb {N}$ , the notation $z^mf^{(n)}$ denotes the function mapping z to $z^mf^{(n)}(z)$ . Moreover, $f_t$ is the function mapping z to $f(tz)$ .

Lemma 3.22. Let $f\in \mathcal {D}_s$ , $s>-1$ . Then $zf'\in \mathcal {D}_{s+1}$ , and there exists $C^{\prime }_{s}$ (independent of f) such that

(3.33) $$ \begin{align} \|zf'\|_{\mathcal{D}_{s+1}}\le {C}^{\prime}_{s}\|f\|_{\mathcal{D}_s} \end{align} $$

and

(3.34) $$ \begin{align} \|f_t-f_{\tau}\|_{\mathcal{D}_{s+1}} \le \frac{C^{\prime}_s \|f\|_{\mathcal{D}_s}}{\min\{t,\tau\}} |t-\tau|, \qquad t,\tau>0. \end{align} $$

Proof. Note that

$$ \begin{align*} \|zf'\|_{\mathcal{D}_{s+1}}\le \|f'\|_{\mathcal{V}_{s+1}}+\|zf''\|_{\mathcal{V}_{s+1}} \le \|f\|_{\mathcal{D}_{s}}+\|zf''\|_{\mathcal{V}_{s+1}}. \end{align*} $$

So, for (3.33), it suffices to consider $\|zf^{''}\|_{\mathcal V_{s+1}}$ . The argument is similar to Example 3.3 and the proof of Proposition 3.6. By Corollary 3.10, for fixed $\sigma>s$ ,

$$ \begin{align*} f''(z)=-c_{\sigma} \int_{-\pi/2}^{\pi/2} \cos^{\sigma} \varphi \int_0^{\infty} \frac{\rho^{\sigma+1} f'(\rho e^{i\varphi})}{(z+\rho e^{-i\varphi})^{\sigma+3}}\,d\rho\,d\varphi,\quad z\in \mathbb{C}_{+}, \end{align*} $$

where $c_{\sigma }=(\sigma +1)(\sigma +2)\frac {2^{\sigma }}{\pi }$ . Then estimates similar to those in Example 3.3 and Proposition 3.6 give

$$ \begin{align*}&{c_{\sigma}^{-1}\|zf''\|_{\mathcal{V}_{s+1}}}\\ &\ =\int_{-\pi/2}^{\pi/2}\cos^{s+1}\psi \int_0^{\infty} r \left| \int_{-\pi/2}^{\pi/2} \cos^{\sigma}\varphi \int_0^{\infty} \frac{\rho^{\sigma+1}f'(\rho e^{i\varphi})}{(r e^{i\psi}+\rho e^{-i\varphi})^{\sigma+3}}\,d\rho\,d\varphi\right| \,dr\,d\psi\\&\ \le \int_{-\pi/2}^{\pi/2} \cos^{\sigma} \varphi \int_{-\pi/2}^{\pi/2} \cos^{s+1}\psi \int_{0}^{\infty}\int_0^{\infty} \frac{t |f'(\rho e^{i\varphi})|\,dt\,d\rho}{(t^2+2t\cos(\varphi+\psi)+1)^{(\sigma+3)/2}} \,d\psi\,d\varphi \\&\ = \int_{-\pi/2}^{\pi/2} \cos^{\sigma}\varphi \, G_{h,\beta,s+1}(\varphi) \int_0^{\infty} |f'(\rho e^{i\varphi})| \,d\rho\,d\varphi, \end{align*} $$

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 (3.33) follows from Lemma 3.2(b).

For (3.34), let $g = f_{\tau } - f_t$ . Without loss, assume that $0 < t < \tau $ . Then

$$ \begin{align*} g'(z) = \tau f'(\tau z) - t f'(tz) = \int_t^{\tau} \left(f'(rz) + rz f''(rz)\right) \, dr = \int_t^{\tau} \frac{d}{dz} (zf'(rz)) \,dr. \end{align*} $$

Hence, by Fubini’s theorem,

$$ \begin{align*} \|g\|_{\mathcal{D}_{s+1}} &\le \int_{\mathbb{C}_+} \int_t^{\tau} \frac{(\operatorname{Re}z)^{s+1}}{|z|^{s+2}} \left|\frac{d}{dz} (zf'(rz))\right|\,dr \,dS(z)\\ &= \int_t^{\tau} \frac{\|zf'\|_{\mathcal{D}_{s+1}}}{r} \,dr \le \frac{C^{\prime}_s\|f\|_{\mathcal{D}_s}}{t} (\tau-t), \end{align*} $$

because the $\mathcal {D}_{s+1}$ -norm is invariant under the change of variable $z\mapsto rz$ (Lemma 3.17(iii)).

The following corollary is easily proved by induction.

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

In this section and in Subsection 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 here. Standard references for the theory of Hardy spaces on the right half-plane are [Reference Duren21] and [Reference Garnett29].

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

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

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

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

$$ \begin{align*}\|g\|_{H^p(\Sigma_{\pi/2})}:=\sup_{|\varphi|<\pi/2}\, \left(\int_0^{\infty} \left(|g(te^{i\varphi})|^p+ |g(te^{-i\varphi})|^p\right)\,dt\right)^{1/p}<\infty \end{align*} $$

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

$$ \begin{align*} \left \{g\in \operatorname{Hol}(\mathbb{C}_{+}): \|g\|_{H^p_*(\Sigma_{\pi/2})}:=\sup_{|\varphi|<\pi/2}\left(\int_0^{\infty} |g(te^{i\varphi})|^p\,dt\right)^{1/p} < \infty \right\}. \end{align*} $$

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

$$ \begin{align*} \|g\|_{H^p_*(\Sigma_{\pi/2})} \le \|g\|_{H^p(\Sigma_{\pi/2})}\le 2^{1/p}\|g\|_{H^p_*(\Sigma_{\pi/2})}. \end{align*} $$

Lemma 4.1. Let $p\in [1,\infty )$ . Then $H^{p}(\Sigma _{\pi /2})\subset H^{p}(\mathbb {C}_{+})$ and

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

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

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

Note that

(4.2) $$ \begin{align} \int_0^{\infty} |g_{\gamma}(te^{i\varphi})|^p\,dt= \int_0^{\infty} |g(te^{i\gamma\varphi})|^p\,dt, \end{align} $$

and

$$ \begin{align*} \int_{-\pi/2}^{\pi/2} \int_0^{\infty} |g_{\gamma}(te^{i\varphi})|^p\,dt \, d\varphi\le \frac{\pi}{2} \|g\|^p_{{H^{p}(\Sigma_{\pi/2})}}. \end{align*} $$

By Fubini’s theorem and Hölder’s inequality there exist sequences $(t_{1,n})_{n\ge 1}$ and $(t_{2,n})_{n\ge 1}$ 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) $$ \begin{align} \lim_{n\to\infty}\, t_{1,n} \int_{-\pi/2}^{\pi/2} |g_{\gamma} (t_{1,n}e^{i\varphi})|\,d\varphi =0, \end{align} $$

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

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

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

for large n. Passing to the limit in (4.5) as $n \to \infty $ and using (4.2), (4.3) and (4.4), we infer that $g_{\gamma }$ satisfies the Poisson formula

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

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

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

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) $$ \begin{align} \|g\|_{H^{p}(\Sigma_{\pi/2})}\le \|g\|_{p}, \qquad g\in H^{p}(\mathbb{C}_{+}). \end{align} $$

Proof. First let $p=2$ so that $g\in H^2(\mathbb {C}_{+})$ . Then, by [Reference Džrbašjan22, Ch.VIII, p.508], there exists $f\in L^1(\mathbb {R}, e^{\pi |t|}\,dt)$ such that $f \ge 0$ on $\mathbb R$ , and for all $|\varphi |\le \pi /2$ ,

$$ \begin{align*} \int_0^{\infty} |g(te^{i\varphi})|^2\,dt= \int_{-\infty}^{\infty} e^{2\varphi t} f(t)\,dt. \end{align*} $$

Hence,

$$ \begin{align*} \int_0^{\infty} \bigl(|g(te^{i\varphi})|^2+|g(t e^{-i\varphi})|^2\bigr)\,dt&= 2\int_{-\infty}^{\infty} \cosh(2\varphi t)f(t)\,dt \\ \le 2\int_{-\infty}^{\infty} \cosh(\pi t)f(t)\,dt&=\int_0^{\infty} \bigl(|g(it)|^2+|g(-it)|^2 \bigr)\,dt,\notag \end{align*} $$

and (4.7) 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 (4.7) for $p=2$ , for all $|\varphi |<\pi /2$ we have

(4.8) $$ \begin{align} \int_0^{\infty} \bigl(|g(te^{i\varphi})|^p+|g(te^{-i\varphi})|^p\bigr)\,dt = \int_0^{\infty} \bigl(|g_p(te^{i\varphi})|^2+| g_p(te^{-i\varphi})|^2\bigr)\,dt \end{align} $$

$$\begin{align*} \le \int_{-\infty}^{\infty} |g_p(it)|^2\,dt=\int_{-\infty}^{\infty} |g(it)|^p\,dt=\|g\|_p^p, \notag \end{align*}$$

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

and

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

Note that the two-sided estimate (4.10) was proved in [Reference Sedleckii52] and [Reference Luxemburg42] in a more complicated way (see also [Reference Akopjan1], [Reference Dilanian19], [Reference Martirosian45], [Reference von Keviczky54], [Reference von Keviczky55]). 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.

Remark 4.4. The two-sided estimate (4.10) is sharp (and cannot be improved). Indeed, let $p\in [1,\infty )$ and let

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

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

$$ \begin{align*} \left( \|f_k\|_{H^p_{*}(\Sigma_{\pi/2})} \right)^p&=\int_0^{\infty} |f_k(te^{-i\pi/2})|^p\,dt\\ &=\frac{1}{\pi}\int_0^{\infty} \frac{dt}{(t-k)^2+1}\,dt =\frac{1}{2}+\frac{\arctan k}{\pi}. \end{align*} $$

Thus,

$$ \begin{align*} \|f_0\|_{H^p(\mathbb{C}_{+})}=2^{1/p}\|f_0\|_{H^p_{*}(\Sigma_{\pi/2})} \qquad \text{and}\qquad \lim_{k\to\infty}\,\frac{\|f_k\|_{H^p_{*}(\Sigma_{\pi/2})}}{\|f_k\|_{H^p(\mathbb{C}_{+})}}=1. \end{align*} $$

In fact, for all $f \in \mathcal {H}^p_{*}(\Sigma _{\pi /2})$ , the norm $\|f\|_{H^p_{*}(\Sigma _{\pi /2})}$ is attained at the boundary of $\Sigma _{\pi /2}$ .

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) $$ \begin{align} \lim_{\varphi\to\pm \pi/2}\, \int_0^{\infty} |g(te^{i\varphi})-g(\pm it)|^p\,dt=0. \end{align} $$

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

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

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

$$ \begin{align*} \limsup_{\varphi\to \pi/2}\,&\int_0^{\infty} \bigl(|g(te^{i\varphi})|^p+|g(te^{-i\varphi})|^p \bigr)\,dt\le \int_{-\infty}^{\infty} |g(it)|^p\,dt\\ &\le \liminf_{\varphi\to\pi/2}\,\int_0^{\infty} \bigl(|g(te^{i\varphi})|^p+|g(te^{-i\varphi})|^p\bigr)\,dt, \end{align*} $$

and hence

(4.13) $$ \begin{align} \lim_{\varphi\to\pi/2}\,\int_0^{\infty} \bigl(|g(te^{i\varphi})|^p+|g(te^{-i\varphi})|^p \bigr)\,dt= \int_{-\infty}^{\infty} |g(it)|^p\,dt. \end{align} $$

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). Because $\|g\|_{H^p(\mathbb {C}_+)}=\|g(i\cdot )\|_{L^p(\mathbb R)},$ we get (4.12) as well.

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

4.2. The spaces $H^1(\Sigma _{\psi })$

Now using the results of Subsection 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) $$ \begin{align} \|f\|_{H^1(\Sigma_{\psi})} := \sup_{|\varphi|<\psi} \int_0^{\infty} \bigl(|f(te^{i\varphi})|+ |f(te^{-i\varphi})| \bigr) \,dt <\infty. \end{align} $$

We will also consider a nonsymmetric version of $H^1(\Sigma _{\psi })$ , defined as

$$ \begin{align*} H^1_{*}(\Sigma_{\psi}):=\left\{f \in \operatorname{Hol}\, (\Sigma_{\psi}): \|f\|_{H^1_{*}(\Sigma_{\psi})}:=\sup_{|\varphi|< \psi}\,\int_0^{\infty} |f(t e^{i\varphi})|\,dt<\infty\right\}. \end{align*} $$

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, [Reference Duren21, Theorem 11.8]). For general $\psi $ , we may argue similarly to the proof of Lemma 4.1, as follows.

Because $f \in H^1(\Sigma _{\psi })$ ,

$$ \begin{align*} \int_0^{\infty} \int_{-\psi}^{\psi} |f(te^{i\varphi})| \,d\varphi\,dt < \infty. \end{align*} $$

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

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

By applying Cauchy’s theorem around the boundary of

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

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

Remark 4.7. In addition to (4.17), it is possible to prove that

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

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

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

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

$$ \begin{align*}\mathcal{H}_{\psi}:=\left\{ f \in \operatorname{Hol}(\Sigma_{\psi}): f' \in H^1 (\Sigma_{\psi})\right\}.\end{align*} $$

In view of Corollary 4.3,

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

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 will also use this terminology sporadically. Spaces more general than $\mathcal {H}_{\psi }$ appear in [Reference Domelevo, Kriegler and Petermichl20]. Namely, for $f \in \operatorname {Hol}(\Sigma _{\psi })$ it was required in [Reference Domelevo, Kriegler and Petermichl20] that the boundary values of f exist and belong (after an appropriate ‘rescaling’) 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.

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) $$ \begin{align} \|f \|_{\mathcal{H}_{\psi}}:=\|f\|_{H^{\infty}(\Sigma_{\psi})}+\|f'\|_{H^1(\Sigma_{\psi})}, \qquad f \in \mathcal{H}_{\psi}. \end{align} $$

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

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

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

$$ \begin{align*} \mathcal{H}_{\psi_1}&\to \mathcal{H}_{\psi_2}\\ f(z)&\mapsto f(z^{\psi_1/\psi_2}) \end{align*} $$

is an isometric algebra isomorphism.

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

More examples can be found in Subsections 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 3.12.

Proof. By Theorem 4.8, the functions $\{f_k: k\ge 1\}$ are uniformly bounded on $\Sigma _{\psi }$ . By Vitali’s theorem, f is holomorphic, and $f_k'(z) \to f'(z)$ as $k \to \infty $ for each $z \in \Sigma _{\psi }$ . By Fatou’s lemma, for $|\varphi | < \psi $ ,

$$ \begin{align*} \int_0^{\infty} |f'(te^{i\varphi})| \,dt \le \liminf_{k\to\infty} \int_0^{\infty} |f_k'(te^{i\varphi})| \,dt \le \sup_{k\ge1} \|f_k\|_{\mathcal{H}_{\psi}}. \end{align*} $$

Thus, $f \in \mathcal {H}_{\psi }$ .

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

In [Reference Batty, Gomilko and Tomilov7, 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 $(\mathcal {LM},\|\cdot\|_{{\rm HP}})$ is the HP 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) $$ \begin{align} \|f\|_{\text{HP}} \le C\|f\|_{H^{1,1}(\mathbb{C}_+)}. \end{align} $$

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

$$ \begin{align*} f_n(z) &:=f(z)-f(z+n),\qquad z \in \mathbb{C}_+, \\ g_n(t) &:= -\frac{2}{\pi} t(1-e^{-nt})\int_0^{\infty} \alpha e^{-\alpha t} \int_{-\infty}^{\infty} f'(\alpha+i\beta) e^{i\beta t}\,d\beta\,d\alpha, \quad t>0. \end{align*} $$

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

$$ \begin{align*} |g_n(t)| &\le \frac{2}{\pi}t(1-e^{-nt})\int_0^{\infty} \alpha e^{-\alpha t} \int_{-\infty}^{\infty} |f'(\alpha+i\beta)|\,d\beta\,d\alpha\\ &\le \frac{2}{\pi} \|f'\|_{H^1(\mathbb{C}_{+})}t(1-e^{-nt})\int_0^{\infty} \alpha e^{-\alpha t} \,d\alpha= \frac{2}{\pi}\|f'\|_{H^1(\mathbb{C}_{+})}\frac{(1-e^{-nt})}{t}\\ &\le \frac{2n}{\pi} \|f'\|_{H^1(\mathbb{C}_{+})}. \end{align*} $$

By the reproducing formula for $\mathcal {B}$ (see Remark 3.16 or [Reference Batty, Gomilko and Tomilov7, Proposition 2.20]) and Fubini’s theorem,

$$ \begin{align*} f_n(z)=&-\frac{2}{\pi}\int_0^{\infty} \alpha\int_{-\infty}^{\infty} f'(\alpha+i\beta) \left(\frac{1}{(z+\alpha-i\beta)^{2}}- \frac{1}{(z+n+\alpha-i\beta)^{2}}\right) d\beta\,d\alpha\\ =&-\frac{2}{\pi} \int_0^{\infty} \alpha \int_{-\infty}^{\infty} f'(\alpha+i\beta) \left(\int_0^{\infty} e^{-(z+\alpha-i\beta)t} t(1-e^{-nt})\,dt\right)\,d\beta\,d\alpha\\ =&-\frac{2}{\pi}\int_0^{\infty} e^{-zt} g_n(t)\,dt,\quad z\in \mathbb{C}_{+}. \end{align*} $$

It follows that $(\pi /2) f_n'$ is the Laplace transform of $t g_n(t)$ , and then $tg_n(t)$ is the inverse Fourier transform of $(\pi /2)f_n'(i\cdot )\in L^1(\mathbb {R})$ . By Hardy’s inequality in the form of [Reference Duren21, p.198],

(4.26) $$ \begin{align} \int_0^{\infty} |g_n(t)|\,dt =\int_0^{\infty} \frac{|tg_n(t)|}{t}\,dt \le \frac{\pi}{4}\|f_n'\|_{H^1(\mathbb{C}_{+})} \le \frac{\pi}{2}\|f'\|_{H^1(\mathbb{C}_{+})}. \end{align} $$

Moreover,

$$ \begin{align*} f(z)=f(\infty)+\lim_{n\to\infty} f_n(z),\qquad z\in \overline{\mathbb{C}}_{+}, \end{align*} $$

and then by [Reference Rudin49, Theorem 1.9.2] we infer that $f=\mathcal L (\mu)$ for some $\mu \in M(\mathbb R_+),$ and (4.25) holds.

Now let

$$ \begin{align*} u(z) := f'(z+1)=-\int_0^{\infty} e^{-zt}e^{-t}t \, d\mu(t),\qquad z\in \mathbb{C}_{+}. \end{align*} $$

Because $f\in H^{\infty }(\mathbb {C}_{+})$ and $f'\in H^1(\mathbb {C}_{+})$ , 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

$$ \begin{align*} u(z) = \int_0^{\infty} e^{-zt} h(t) \, dt, \qquad z \in \mathbb{C}_+. \end{align*} $$

From the uniqueness properties of Laplace transforms it follows that

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

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

$$ \begin{align*} \mu(dt)=f(\infty)\delta_0 +g(t)\,dt. \end{align*} $$

Hence, $f = f(\infty ) + \mathcal {L} g$ .

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

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

$$ \begin{align*} \mathcal{H}_{\pi/2} \overset{i}{\hookrightarrow} \mathcal{L} L^1 + \mathbb{C} \subset \mathcal{LM} \overset{i}{\hookrightarrow} \mathcal{B} \overset{i}{\hookrightarrow} \mathcal{D}_s \end{align*} $$

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) $$ \begin{align} \|f\|_{\mathcal D_{s,0}} &\le B\left(\frac{s+1}{2},\frac{1}{2}\right)\|f'\|_{H^1(\mathbb{C}_+)}, \\ \|f\|_{\mathcal D_{s}^{\infty} } &\le \max \left\{1, B\left(\frac{s+1}{2},\frac{1}{2}\right)\right\}\|f\|_{\mathcal{H}_{\pi/2}}. \notag \end{align} $$

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) $$ \begin{align} \|f'\|_{H^1(\Sigma_{\psi})}\le \frac{2^{s+1}}{\pi\cos^{s+2}(\psi/2+\pi/4)}\|f\|_{\mathcal{D}_{s,0}}. \end{align} $$

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

$$ \begin{align*} \mathcal{H}_{\pi/2} \overset{i}{\hookrightarrow} \mathcal D_s \overset{r}{\hookrightarrow} \mathcal{H}_{\psi}. \end{align*} $$

Note that the estimates (4.23) and (4.27) for functions $r_{\lambda }^{\gamma }$ reproduce the estimate (3.19), with different constants.

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

$$ \begin{align*} \|f\|_{\mathcal D_{s,0}}&= \int_{-\pi/2}^{\pi/2}\cos^s\varphi \int_0^{\infty} |f'(te^{i\varphi})|\,dt\,d\varphi \\ &\le \left(\int_{-\pi/2}^{\pi/2}\cos^s\varphi\,d\varphi\right) \sup_{|\varphi|\le \pi/2}\int_0^{\infty} |f'(te^{i\varphi})|\,dt \\ &= B\left(\frac{s+1}{2},\frac{1}{2}\right)\|f'\|_{H^{1}(\mathbb{C}_+)}, \end{align*} $$

and (i) follows.

To prove (ii), note that if $f \in \mathcal D_s$ , then by Corollary 3.10,

$$ \begin{align*} f'(z)=\frac{(s+1)2^s}{\pi}\int_0^{\infty} \alpha^s\int_{-\infty}^{\infty} \frac{f'(\alpha+i\beta)}{(z+\alpha-i\beta)^{s+2}}\,d\beta\,d\alpha, \quad z\in \mathbb{C}_{+}. \end{align*} $$

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

$$ \begin{align*} &{2^{-s}\int_0^{\infty} |f'(te^{\pm i\psi})|\,dt}\\ &\quad \le \frac{(s+1)}{\pi}\int_0^{\infty} \int_0^{\infty} \alpha^s\int_{-\infty}^{\infty} \frac{|f'(\alpha+i\beta)|}{|te^{\pm i\psi}+\alpha-i\beta|^{s+2}}\,d\beta\,d\alpha\,dt\\ &\quad \le \frac{s+1}{\pi\cos^{s+2}(\psi/2+\pi/4)} \int_0^{\infty} \alpha^s\int_{-\infty}^{\infty} \left( \int_0^{\infty} \frac{|f'(\alpha+i\beta)|}{(t+|\alpha+i\beta|)^{s+2}}\right) \,dt\,d\beta\,d\alpha\\ &\quad = \frac{1}{\pi\cos^{s+2}(\psi/2+\pi/4)} \int_0^{\infty} \alpha^s\int_{-\infty}^{\infty} \frac{|f'(\alpha+i\beta)|}{|\alpha+i\beta|^{s+1}}\,d\beta\,d\alpha\\ &\quad =\frac{1}{\pi\cos^{s+2}(\psi/2+\pi/4)}\|f'\|_{\mathcal{V}_s}, \end{align*} $$

and (4.28) follows.

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

$$ \begin{align*} f_{\gamma}(z):= f(z^{\gamma}), \quad z \in \Sigma_{\varphi}. \end{align*} $$

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

$$ \begin{align*} \|f_{\gamma}\|_{\mathcal{D}_{\sigma}^{\infty}} \le C_{s,\sigma,\gamma} \|f\|_{\mathcal{D}_s} \quad \text{and} \quad \|f_{\gamma}\|_{\mathcal{H}_{\pi/2}} \le \tilde C_{s,\gamma} \|f\|_{\mathcal{D}_s}, \qquad f \in \mathcal{D}_s. \end{align*} $$

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

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

$$ \begin{align*} \mathcal E_{\psi}=\{f \in {\operatorname{Hol}} (\Sigma_{\psi}): f(z)/z \in H^1(\Sigma_{\psi})\}. \end{align*} $$

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) $$ \begin{align} \|f'\|_{H^{1}(\Sigma_{\varphi})} \le \frac{1}{2\pi} \left(\frac{\pi-\psi-\varphi}{\sin(\psi+\varphi)}+ \frac{\pi-\psi+\varphi}{\sin(\psi-\varphi)}\right)\|g\|_{H^1(\Sigma_{\psi})}, \end{align} $$

where we set $\frac {0}{\sin 0}:= 1.$ Thus, $\mathcal E_{\psi } \overset {r}{\hookrightarrow } \mathcal H_{\varphi }, \, \varphi \in (0,\psi )$ , and $\mathcal E_{\pi /2} \overset {i}{\hookrightarrow } \mathcal D_s,\, s>0$ .

Proof. By Cauchy’s theorem, for every $z\in \Sigma _{\psi }$ ,

$$ \begin{align*} f'(z) =\frac{1}{2\pi i}\int_{\partial\Sigma_{\psi}}\, \frac{\lambda g(\lambda)\,d\lambda}{(z-\lambda)^2}. \end{align*} $$

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

$$ \begin{align*} \int_0^{\infty} \left(|f'(\rho e^{i\varphi})| + |f'(\rho e^{-i\varphi})|\right)\,d\rho \le\frac{1}{2\pi} \int_{\partial\Sigma_{\psi}}|g(\lambda)| \, J(\lambda, \varphi) \, |d\lambda|, \end{align*} $$

where

$$ \begin{align*} J(\lambda, \varphi)&:= \int_0^{\infty}\left(\frac{|\lambda|}{|\rho e^{i\varphi}-\lambda|^2}+ \frac{|\lambda|}{|\rho e^{-i\varphi}-\lambda|^2}\right)\ d\rho\\ &=\int_0^{\infty}\frac{d\rho}{\rho^2-2\rho\cos(\psi+\varphi)+1} +\int_0^{\infty}\frac{d\rho}{\rho^2-2\rho\cos(\psi-\varphi)+1} \\ &=\frac{\pi-\psi-\varphi}{\sin(\psi+\varphi)}+ \frac{\pi-\psi+\varphi}{\sin(\psi-\varphi)}, \end{align*} $$

in view of [Reference Prudnikov, Brychkov and Marichev48, item 2.2.9, (25)]. 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 Proposition 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

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

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 [Reference Hytönen, van Neerven, Veraar and Weis38, Chapter 10]; see also [Reference Hytönen, van Neerven, Veraar and Weis38, Appendix H2], [Reference Haase34, Section 6] and [Reference Haak and Haase33, 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 [Reference Gomilko and Tomilov31]; see also [Reference Batty, Gomilko and Tomilov6] and [Reference Batty, Gomilko and Tomilov7].

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

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

If $\lambda \in \mathbb {C}_+$ , it follows from Lemmas 3.20 and 4.13 that $f(\cdot , \lambda ) \in \mathcal {H}_{\psi }$ and $\|f(\cdot ,\lambda )\|_{\mathcal {H}_{\psi }} \le C_{\psi }/|\lambda |$ , 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 $\varphi \in (\psi, \pi)$ .

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

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

$$ \begin{align*} \|f'(\cdot,\lambda)\|_{H^1(\Sigma_{\psi})} = \int_0^{\infty} \left( \frac{|g'(te^{i\psi})|}{|\lambda+g(te^{i\psi})|^2} + \frac{|g'(te^{-i\psi})|} {|\lambda+g(te^{-i\psi})|^2} \right) \,dt. \end{align*} $$

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

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

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

$$ \begin{align*} |g(te^{\pm i\psi})|\ge \operatorname{Re}g(te^{\pm i\psi})\ge g(t\cos\psi)\quad \text{and} \quad | g'(e^{\pm i\psi}t)|&\le g'(t\cos\psi). \end{align*} $$

Using (B2), we have

$$ \begin{align*} \|f'(\cdot, \lambda)\|_{H^1(\Sigma_{\psi})} &\le \frac{2}{\sin^2((\varphi-\psi)/2)} \int_0^{\infty}\frac{g'(t\cos\psi)} {(|\lambda|+g(t\cos\psi ))^2} \,dt\\ &\le \frac{2}{\cos\psi \sin^2((\varphi-\psi)/2)} \int_0^{\infty}\frac{ds} {(|\lambda|+s)^2}\\ &= \frac{2}{\cos\psi \sin^2((\varphi-\psi)/2)}\frac{1}{|\lambda|}. \end{align*} $$

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

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

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

(4.33) $$ \begin{align} f_{\gamma}(z):=f(z^{\gamma}),\qquad z\in \mathbb{C}_{+}. \end{align} $$

Then

(4.34) $$ \begin{align} f(z)=f(\infty)-\frac{1}{\pi}\int_0^{\infty} \int_{-\infty}^{\infty} \frac{f_{\gamma}'(\alpha+i\beta)}{z^{1/\gamma}+\alpha-i\beta}\,d\beta\,d\alpha,\qquad z\in \Sigma_{\psi} \cup \{0\}. \end{align} $$

Proof. Because $f \in \mathcal {H}_{\psi }$ , 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 3.10,

$$ \begin{align*} f_{\gamma}(z)=f_{\gamma}(\infty)-\frac{1}{\pi}\int_0^{\infty} \int_{-\infty}^{\infty} \frac{f_{\gamma}'(\alpha+i\beta)}{z+\alpha-i\beta}\,d\beta\,d\alpha, \qquad z\in \mathbb{C}_{+}\cup \{0\}, \end{align*} $$

and (4.34) follows.

Corollary 4.18. Let $f\in \mathcal {D}_{\infty }$ , and $\gamma \in (0,1)$ . If $f_{\gamma }$ is given by (4.33), then

(4.35) $$ \begin{align} f(z)=f(\infty)-\frac{1}{\pi}\int_0^{\infty} \int_{-\infty}^{\infty} \frac{f_{\gamma}'(\alpha+i\beta)}{z^{1/\gamma}+\alpha-i\beta}\,d\beta\,d\alpha,\quad z\in \Sigma_{\pi\gamma/2}\cup\{0\}. \end{align} $$

The next reproducing formula for functions in $\mathcal {H}_{\psi }$ resembles [Reference Boyadzhiev12, Lemma 7.4], and it was, in fact, inspired by [Reference Boyadzhiev12, Lemma 7.4]. In particular, this formula replaces the double (area) integral in (4.34) with a line integral, it involves boundary values of $f'$ rather than scalings of $f'$ (such as in (4.34)), and it offers a different kernel that might sometimes be easier to deal with. The $\operatorname {\mathrm {arccot}}$ function has been defined in (3.22).

Proposition 4.19. Let $f \in \mathcal {H}_{\psi }$ , $\psi \in (0,\pi )$ . Let $\nu =\pi /(2\psi )$ , and

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

Then

(4.37) $$ \begin{align} f(z)=f(\infty)-\frac{2}{\pi}\int_0^{\infty} f_{\psi}'(t)\, {\operatorname{\mathrm{arccot}}}(z^{\nu}/t^{\nu})\,dt,\qquad z\in \Sigma_{\psi}\cup \{0\}. \end{align} $$

Proof. Because $\operatorname {\mathrm {arccot}} \lambda \in \mathcal D_0$ (see Example 3.5), (3.29) shows that

(4.38) $$ \begin{align} \operatorname{\mathrm{arccot}}(\lambda)=\frac{1}{\pi}\int_0^{\infty}\int_{-\infty}^{\infty} \frac{(\lambda+u-iv)^{-1}}{(u+iv)^2+1}\,dv\, du, \quad \lambda\in \mathbb{C}_{+}. \end{align} $$

Let $\gamma =1/\nu $ and $f_{\gamma }$ be given by (4.33). By Proposition 4.17, for $z \in \Sigma _{\psi } \cup \{0\}$ ,

(4.39) $$ \begin{align} f(z)-f(\infty)= -\frac{1}{\pi}\int_0^{\infty} \int_{-\infty}^{\infty} \frac{f_{\gamma}'(\alpha+i\beta)}{z^{\nu}+\alpha-i\beta}\,d\beta\,d\alpha. \end{align} $$

It follows from Lemma 4.9 that $f_{\gamma }' \in H^1(\Sigma _{\pi /2}) = H^1(\mathbb {C}_+)$ , so by Cauchy’s formula for functions in $H^1(\mathbb {C}_+)$ [Reference Duren21, Theorem 11.8], we have, for $\lambda \in \mathbb {C}_{+}$ ,

$$ \begin{align*} f_{\gamma}'(\lambda) &= \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{f_{\gamma}'(it)}{\lambda-it} \,dt - \frac{1}{2\pi}\int_{-\infty}^{\infty} \frac{f_{\gamma}'(it)}{\lambda+it} \,dt \\ &= \frac{i}{\pi}\int_{-\infty}^{\infty} \frac{tf_{\gamma}'(it)}{\lambda^2+t^2} \,dt\\ &=\frac{i\gamma}{\pi }\int_0^{\infty} \frac{e^{i\pi(\gamma-1)/2}f'(e^{i\pi\gamma/2}t^{\gamma})t^{\gamma} - e^{-i\pi(\gamma-1)/2}f'(e^{-i\pi\gamma/2}t^{\gamma})t^{\gamma}}{\lambda^2+t^2} \,dt\\ &=\frac{\gamma}{\pi}\int_0^{\infty} \left(e^{i\psi}f'(e^{i\psi}t^{\gamma}) +e^{-i\psi}f'(e^{-i\psi}t^{\gamma})\right)\frac{t^{\gamma}}{\lambda^2+t^2} \,dt\\ &=\frac{2}{\pi}\int_0^{\infty} \frac{f_{\psi}'(s) s^{\nu}}{\lambda^2+s^{2\nu}} \,ds. \end{align*} $$

Therefore, by (4.39) and (4.38), we obtain

$$ \begin{align*} &f(z)-f(\infty)\\ &= -\frac{2}{\pi^2}\int_0^{\infty} f^{\prime}_{\psi}(s) \left(\int_0^{\infty} \int_{-\infty}^{\infty} \frac{s^{\nu}\,d\beta\, d\alpha}{(z^{\nu}+\alpha-i\beta)((\alpha+i\beta)^2+s^{2\nu})}\right)\,ds\\ &=-\frac{2}{\pi^2}\int_0^{\infty} f^{\prime}_{\psi}(s) \left(\int_0^{\infty} \int_{-\infty}^{\infty} \frac{d\beta\, d\alpha}{((z/s)^{\nu}+\alpha-i\beta)((\alpha+i\beta)^2+1)}\right)\,ds\\ &=-\frac{2}{\pi}\int_0^{\infty} f^{\prime}_{\psi}(s)\operatorname{\mathrm{arccot}}(z^{\nu}/s^{\nu})\,ds, \end{align*} $$

so (4.37) holds for $z \in \Sigma _{\psi } \cup \{0\}$ .

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

Lemma 4.20. Let $\psi \in (0,\pi )$ and $\nu ={\pi }/{(2\psi )}$ . Then

(4.40) $$ \begin{align} \operatorname{\mathrm{arccot}}(z^{\nu})&=\frac{1}{2\pi}\int_0^{\infty} V_{\psi}(z,t)\log\left|\frac{1+t^{\nu}}{1-t^{\nu}}\right| \,\frac{dt}{t}+\frac{1}{4 i}\int_{\Gamma_{\psi}}\, \frac{d\lambda}{\lambda-z},\quad z\in \Sigma_{\psi}, \end{align} $$

where

(4.41) $$ \begin{align} V_{\psi}(z,t)&:=-\frac{t}{2} \left(\frac{ e^{-i\psi}} {z-t e^{-i\psi}} +\frac{ e^{i\psi}}{z-t e^{i\psi}}\right), \\ \Gamma_{\psi}&:= \{\lambda: |\lambda|=1,\,\arg\lambda \in (\psi,2\pi-\psi)\}. \notag \end{align} $$

Proof. We have

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

Because $\nu \psi =\pi /2$ , for every $t>0, \, t\ne 1$ ,

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

and

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

Here,

$$ \begin{align*} \arg\left(\frac{t^{\nu}+1} {t^{\nu}-1}\right)=\arg\left(\frac{t^{\nu}-1} {t^{\nu}+1}\right)=\pi\chi_{(0,1)}(t),\quad t>0, \, t\ne1, \end{align*} $$

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

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

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

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

Thus, by (4.42),

$$ \begin{align*} \operatorname{\mathrm{arccot}}(z^{\nu}) &=\frac{1}{4\pi}\int_0^{\infty} \log\left|\frac{1+t^{\nu}}{1-t^{\nu}}\right| \left(\frac{1}{t-e^{i\psi}z} +\frac{1}{t-e^{-i\psi}z}\right)\,dt +\frac{1}{4 i}\int_{\Gamma_{\psi}}\, \frac{d\lambda}{\lambda-z}\\ &=\frac{1}{2\pi}\int_0^{\infty} V_{\psi}(z,t)\log\left|\frac{1+t^{\nu}}{1-t^{\nu}}\right| \,\frac{dt}{t}+\frac{1}{4 i}\int_{\Gamma_{\psi}}\, \frac{d\lambda}{\lambda-z}. \\[-40pt] \end{align*} $$