Generalizations of Koga’s version of the Wiener-Ikehara theoremThanks: B. Chen gratefully acknowledges support by the China Scholarship Council (CSC) and the Research Foundation–Flanders, through an FWO postdoctoral fellowshipThanks: The work of J. Vindas was supported by the Research Foundation–Flanders, through the FWO-grant number G067621N, and by Ghent University, through the grant number bof/baf/4y/2024/01/155

Abstract.

We establish new versions of the Wiener-Ikehara theorem where only boundary assumptions on the real part of the Laplace transform are imposed. Our results generalize and improve a recent theorem of T. Koga [J. Fourier Anal. Appl. 27 (2021), Article No. 18]. As an application, we give a quick Tauberian proof of Blackwell’s renewal theorem.

1. Introduction

The Wiener-Ikehara theorem [32] is a foundational result in complex Tauberian theory. Originally devised to significantly simplify an early result of Landau [22] and so deliver one of the quickest deductions of the prime number theorem, it has found countless applications in diverse areas of mathematics such as operator theory, partial differential equations, and number theory. The interested reader is referred to the books [8, Chapter 10], [19, Chapter III], and [30, Chapter II.7] for excellent accounts on the subject and the recent articles [3, 6, 7, 15, 17, 24, 26, 33] for some developments during the last decade. See also [1, 4, 21, 29] for closely related complex Tauberian theorems for Laplace transforms, such as the Ingham-Karamata theorem.

In one of its many forms, the Wiener-Ikehara theorem states that if a non-decreasing function $S$ has convergent Laplace transform $\mathcal{L}\{S;s\}=\int^{\infty}_{0}e^{-sx}S(x)\mathrm{d}x$ for $\Re e\>s>1$ and if there is a constant $a\in\mathbb{R}$ such that the analytic function

(1.1) $$G(s)=\mathcal{L}\{S;s\}-\frac{a}{s-1}$$

admits $L^{1}_{loc}$-boundary behavior on the whole boundary line $1+i\mathbb{R}$, then

(1.2) $$S(x)\sim ae^{x}\qquad\mbox{as }x\to\infty.$$

Naturally, the hypothesis of $L^{1}_{loc}$-boundary behavior covers the case of continuous extension, and in particular that of analytic continuation. On the other hand, we point out that the boundary requirements on the Laplace transform can further be taken to a minimum if one employs the so-called local pseudofunction boundary behavior (cf. [6, 7, 20]). The pseudofunction approach plays a major role in modern complex Tauberian theory (cf. [19, Chapter III]).

Very recently [17], Koga has obtained an interesting generalization of this version of the Wiener-Ikehara theorem with $L^{1}_{loc}$-boundary behavior, where only the boundary properties of the real part of the Laplace transform are needed. His result also weakens the non-decreasing hypothesis on $S$ to log-linear slow decrease, a Tauberian condition that was introduced and studied in [6, 33] and that is intimately connected with exact Wiener-Ikehara theorems, that is, complete Laplace transform characterizations of the asymptotic behavior (1.2). We call a function $S$ log-linearly slowly decreasing (at $\infty$) if for each $\varepsilon$ there are $h,x_{0}>0$ such that

$$\frac{S(y)-S(x)}{e^{x}}\geq-\varepsilon\qquad\mbox{for }x\leq y\leq x+h\mbox{ and }x\geq x_{0}.$$

Koga’s main motivation to establish a novel version of the Wiener-Ikehara theorem was to provide a Dirichlet series generalization of the Kolmogorov-Erdős-Feller-Pollard renewal theorem [9, 18] (cf. [13, Sections XIII.3 and XIII.11]). Moreover, he also obtained a Tauberian theorem for power series and applied it to give a new proof of the classical quoted renewal theorem. Upon a minor reformulation (cf. Remark 7.2), Koga’s Tauberian theorem for Laplace transforms reads:

Theorem 1.1 ([17, Theorem 2]).

Let $S\in L^{1}_{loc}[0,\infty)$ be log-linearly slowly decreasing and satisfy

(1.3) $$\int_{1}^{\infty}\frac{|S(x)|}{x^{2}e^{x}}\>\mathrm{d}x<\infty.$$

Let $U(s)=\Re e\>\mathcal{L}\{S;s\}$. Assume there are $\lambda>0$ and $g\in L^{1}(-\lambda,\lambda)$ such that

(1.4) $$U(\sigma+it)\geq g(t),\qquad\mbox{for a.e. }t\in(-\lambda,\lambda)\mbox{ and }\sigma\in(1,2].$$

If in addition $U$ has $L^{1}_{loc}$-boundary behavior on the boundary open subset $1+i(\mathbb{R}\setminus\{0\})$, namely, if there is $f\in L^{1}_{loc}(\mathbb{R}\setminus\{0\})$ such that on any finite interval $I$ not containing the origin we have

(1.5) $$\lim_{\sigma\to 1^{+}}\int_{I}|U(\sigma+it)-f(t)|\mathrm{d}t=0,$$

then (1.2) must hold for some constant $a\in\mathbb{R}$.

The aim of this paper is to considerably improve Koga’s theorem by showing that it still holds true if one removes condition (1.3) from its set of hypotheses. In addition to hold under weaker assumptions, we shall also consider a new useful alternative hypothesiss for the boundary behavior of $\Re e\>\mathcal{L}\{S;s\}$ near $s=1$.

Theorem 1.2 (Laplace transforms).

Let $S\in L^{1}_{loc}[0,\infty)$ be log-linearly slowly decreasing and have convergent Laplace transform on $\Re e\>s>1.$ Suppose that the harmonic function $U(s)=\Re e\>\mathcal{L}\{S;s\}$ has $L^{1}_{loc}$-boundary behavior on $1+i(\mathbb{R}\setminus\{0\})$ and that there is some $\lambda>0$ such that one of the following two conditions holds:

• there is $g\in L^{1}(-\lambda,\lambda)$ such that (1.4) holds;

• $\displaystyle\sup_{1<\sigma<2}\int_{-\lambda}^{\lambda}|U(\sigma+it)|\mathrm{d}t<\infty$.

Then

(1.6) $$S(x)\sim ae^{x}\qquad\mbox{as }x\to\infty,$$

where $a\in\mathbb{R}$ is in fact given by

(1.7) $$a=\lim_{\sigma\to{1}^{+}}(\sigma-1)U(\sigma).$$

When $S$ is non-decreasing, it is clearly automatically log-linearly slowly decreasing. In this case however, it is more natural to work with its Laplace-Stieltjes transform $\mathcal{L}\{\mathrm{d}S;s\}=\int_{0^{-}}^{\infty}e^{-sx}\mathrm{d}S(x)$ instead of the Laplace transform of the function. We shall show the following version of our Tauberian theorem for Laplace-Stieltjes transforms.

Theorem 1.3 (Laplace-Stieltjes transforms).

Let $S$ be log-linearly decreasing and of local bounded variation on $[0,\infty)$ with convergent Laplace-Stieltjes transform on $\Re e\>s>1$. Suppose that the hypotheses of Theorem 1.2 are satisfied with $U(s)=\Re e\>\mathcal{L}\{\mathrm{d}S;s\}$ instead of $\Re e\>\mathcal{L}\{S;s\}$. Then (1.6) and (1.7) still hold true.

Working with this new formulation has great practical value as in certain situations it is easier to apply than Theorem 1.2. In fact, we will deduce the following corollary of Ingham-Karamata type from Theorem 1.3. We shall exemplify its usefulness in Section 2 by giving a quick Tauberian proof of Blackwell’s renewal theorem [2]. We also give there a simpler treatment of Koga’s renewal theorem for Dirichlet series [17, Theorem 5] based on Theorem 1.3.

Corollary 1.4 (Ingham-Karamata type theorem).

Let $T$ be a function on $[0,\infty)$ such that $T(x)+Mx$ is non-decreasing, for some constant $M>0$, and such that its Laplace-Stieltjes transform converges on $\Re e\>s>0$. Suppose that $U(s)=\Re e\>\int_{0^{-}}^{\infty}e^{-sx}\mathrm{d}T(x)$ has $L^{1}_{loc}$-boundary behavior on $i\mathbb{R}\setminus\{0\}$ and that there is some $\lambda>0$ such that one of the following two conditions holds:

• there is $g\in L^{1}(-\lambda,\lambda)$ and $k\in\mathbb{N}$ such that $U(\sigma+it)\geq g(t)$ for a.e. $t\in(-\lambda,\lambda)$ and $\sigma\in(0,1]$;

• $\displaystyle\sup_{0<\sigma<1}\int_{-\lambda}^{\lambda}|U(\sigma+it)|\mathrm{d}t<\infty$;

Then, there is a function $\tau(x)=o(1)$ as $x\to\infty$ such that

(1.8) $$T(x)=ax+\int_{0}^{x}\tau(u)\mathrm{d}u+o(1)\qquad\mbox{as }x\to\infty,$$

where $a$ is given by

(1.9) $$a=\lim_{\sigma\to{0}^{+}}\sigma U(\sigma).$$

We shall also prove the next Tauberian theorem for power series, which improves upon [17, Theorem 3].

Theorem 1.5 (Power series).

Let $F(z)=\sum_{n=0}^{\infty}c_{n}z^{n}$ be analytic on the unit disc $\mathbb{D}$ with real coefficients $\{c_{n}\}_{n=0}^{\infty}$. Suppose that the harmonic function $U(z)=\Re e\>F(z)$ has $L^{1}_{loc}$-boundary behavior on $\partial\mathbb{D}\setminus\{1\}$ and there is some $\theta_{0}\in(0,\pi)$ such that one of the following two conditions holds:

• there is $g\in L^{1}(-\theta_{0},\theta_{0})$ such that $U(re^{i\theta})\geq g(\theta)$ for a.e. $\theta\in(-\theta_{0},\theta_{0})$ and $r\in[0,1)$;

• $\displaystyle\sup_{0<r<1}\int_{-\theta_{0}}^{\theta_{0}}|U(re^{\theta})|\mathrm{d}\theta<\infty$;

Then $\{c_{n}\}_{n=0}^{\infty}$ is convergent. In particular, its limit is given by

(1.10) $$\lim_{n\to\infty}c_{n}=\lim_{r\to 1^{-}}(1-r)U(r).$$

The plan of the article is as follows. We discuss in Section 2 how Theorem 1.3, Corollary 1.4, and Theorem 1.5 can be applied to renewal theory [14, Chapter XI]; our applications emphasize the role of the assumptions (B.1), (B₀.1), and (b.1) in the corresponding cases, which make the theorems relatively simple to apply. In Section 3, we obtain a slight extension of the exact Wiener-Ikehara Tauberian theorem [6, Theorem 3.6], where we shall show that (1.2) holds if and only if $S$ is log-linearly slowly decreasing, its Laplace transform converges for $\Re e\>s>1$, and the real part of the function $G$ given by (1.1) has so-called local pseudofunction boundary behavior on $1+i\mathbb{R}$. Section 4 is devoted to the proofs of Theorem 1.2 and Theorem 1.3; our approach there will be to reduce them to the exact Wiener-Ikehara theorem from Section 3. Theorem 1.5 will be shown in Section 5, while a proof of Corollary 1.4 will be given in Section 6. Finally, we close the article with some remarks and further extensions of our Tauberian theorems, which will be discussed in Section 7.

2. Application: Renewal theorems

Before showing our new versions of the Wiener-Ikehara theorem, we illustrate their usefulness with some applications. Our first application is to probability theory. We will give in this section a quick simple Tauberian proof of a fundamental result in renewal theory, namely, the renewal theorem [14].

Let $\mathrm{d}P$ be a probability measure on $[0,\infty)$ that is continuous at the origin, namely, $P(0)=0$. Its renewal function $Q$ is determined by the convolution equation

(2.1) $$\mathrm{d}Q=\delta+\mathrm{d}Q\ast\mathrm{d}P,$$

where hereafter $\delta$ stands for the Dirac delta measure concentrated at $0$. In fact, the solution to (2.1) is given by the convergent series $\mathrm{d}Q=\sum_{n=0}^{\infty}\mathrm{d}P^{\ast n}$.

We shall distinguish two cases for $\mathrm{d}P$. We say that it is lattice if there is $\alpha>0$ such that $\mathrm{d}P$ is concentrated on $\alpha\mathbb{N}=\{\alpha,2\alpha,3\alpha\dots\}$ (when $\alpha$ is maximal we call it its span); otherwise, we shall call $\mathrm{d}P$ non-lattice.

Theorem 2.1 (The renewal theorem [2, 9, 18]).

If $\mathrm{d}P$ is non-lattice, then, for each $h>0$,

(2.2) $$Q(h+x)-Q(x)\to\frac{h}{\int_{0}^{\infty}x\>\mathrm{d}P(x)},\qquad x\to\infty.$$

For lattice $\mathrm{d}P$ with span $\alpha>0$, the relation (2.2) holds for all $h=n\alpha$, $n\in\mathbb{N}$.

Proof.

We divide the proof into the corresponding two cases.

Non-lattice $\mathrm{d}P$ (Blackwell’s renewal theorem). Let $F(s):=\mathcal{L}\{\mathrm{d}Q;s\}$. Laplace transforming (2.1), we obtain

(2.3) $$F(s)=\frac{1}{1-G(s)},\qquad\Re e\>s>0.$$

with $G(s)=\mathcal{L}\{\mathrm{d}P;s\}$. The function $G(s)$ clearly extends continuously to the imaginary axis $i\mathbb{R}$ (because $\mathrm{d}P$ is a finite measure), and, with the exception of $s=0$, we have $G(s)\neq 1$ for all other points of $\{s:\Re e\>s\geq 0\}$ (since otherwise $\mathrm{d}P$ would necessarily be lattice). We conclude that $F$ has a continuous extension to $i\mathbb{R}\setminus\{0\}$ and in particular has $L^{1}_{loc}$-behavior on this boundary subset. Furthermore,

$$\Re e\>F(s)=\frac{1-\int_{0^{-}}^{\infty}e^{-\sigma x}\cos(tx)\mathrm{d}P(x)}{|1-G(s)|^{2}}>0,\qquad\sigma=\Re e\>s>0.$$

Corollary 1.4 applied to the non-decreasing function $Q$ then yields

$$Q(x)=ax+\int_{0}^{x}\tau(u)\mathrm{d}u+o(1)$$

with (see (1.9))

$$a=\lim_{\sigma\to 0^{+}}\sigma F(\sigma)=\lim_{\sigma\to 0^{+}}\frac{\sigma}{1-\int_{0^{-}}^{\infty}e^{-\sigma x}\mathrm{d}P(x)}=\frac{1}{\int_{0}^{\infty}x\mathrm{d}P(x)}$$

and some function $\tau(x)=o(1)$, whence (2.2) follows at once.

Lattice $\mathrm{d}P$ (the Kolmogorov-Erdős-Feller-Pollard renewal theorem). In this case $\mathrm{d}P(x)=\sum_{n=0}^{\infty}p_{n}\delta(x-n\alpha)$ and $\mathrm{d}Q(x)=\sum_{n=0}^{\infty}q_{n}\delta(x-n\alpha)$ with $q_{0}=1$, $p_{0}=0$, and $\sum_{n=1}^{\infty}p_{n}=1$. Furthermore, these non-negative sequences are linked by the convolution relation

(2.4) $$q_{n}=\sum_{k=1}^{n}p_{k}q_{n-k},\qquad n\geq 1.$$

Since we assumed $\alpha$ to be maximal, we have $1=\mathrm{gcd}\{n:p_{n}\neq 0\}$, which implies that $G(re^{i\theta})\neq 1$ for all $\theta\in[-\pi,\pi]\setminus\{0\}$. Here $G$ stands for the power series $G(z)=\sum_{n=1}^{\infty}p_{n}z^{n}$, which is continuous on the closed unit disc. Due to (2.4), we obtain $F(z)=\sum_{n=0}^{\infty}q_{n}z^{n}=(1-G(z))^{-1}$. As in the previous the case, we also have $\Re e\>F(z)>0$ for all $z\in\mathbb{D}$ (since $\Re e\>G(z)<1$ on $\mathbb{D}$). Hence, Theorem 1.5 allows us to conclude that

$$\lim_{n\to\infty}q_{n}=\lim_{r\to 1^{-}}\frac{1-r}{1-\sum_{n=1}^{\infty}r^{n}p_{n}}=\frac{1}{\sum_{n=1}^{\infty}np_{n}},$$

which completes the proof of the renewal theorem. ∎

We can also give a simpler proof than Koga’s original one for his version of the renewal theorem for Dirichlet series. The symbol $\star$ below stands for the Dirichlet convolution [30] of two arithmetic functions, while $e$ denotes the identity of this convolution, namely, the arithmetic function $e:\mathbb{N}\to\mathbb{R}$ given by $e(1)=1$ and $e(n)=0$ for $n\geq 2$.

Theorem 2.2 ([17, Theorem 5]).

Let $g:\mathbb{N}\to[0,\infty)$ be such that $\sum_{n=2}^{\infty}{g(n)}/n=1$, $g(1)=0$, and no set $\{d^{k}:k\in\mathbb{N}\}$ with $d\geq 2$ entirely contains $\{n:\>g(n)\neq 0\}$. If $f$ is defined through $f=e+f\star g$, then

$$\lim_{x\to\infty}\frac{1}{x}\sum_{n\leq x}f(n)=\displaystyle\frac{1}{\sum_{n=2}^{\infty}\frac{g(n)\log n}{n}}\>.$$

Proof.

We set $S(x)=\sum_{n\leq e^{x}}f(n)$, so that $F(s)=\mathcal{L}\{\mathrm{d}S;s\}=\sum_{n=1}^{\infty}f(n)/n^{s}$. The familiar properties of Dirichlet series and $f=e+f\star g$ yield (2.3) with now $G$ given by the Dirichlet series of $g$, i.e., $G(s)=\sum_{n=2}^{\infty}g(n)/n^{s}$, which continuously extends to $\Re e\>s=1$. In view of [17, Lemma 11], the assumption on $\{n:\>g(n)\neq 0\}$ implies that $G(1+it)\neq 1$ for $t\neq 0$. Obviously, $\Re e\>F(s)>0$ on the entire open half-plane $\Re e\>s>1$. An application of Theorem 1.3 thus shows that

$$\lim_{x\to\infty}\frac{S(x)}{e^{x}}=\lim_{\sigma\to 1^{+}}(\sigma-1)F(\sigma)=\lim_{\sigma\to 1^{+}}\frac{(\sigma-1)}{1-G(\sigma)}=\displaystyle\frac{1}{\sum_{n=2}^{\infty}\frac{g(n)\log n}{n}}\>.$$

∎

3. Exact Wiener-Ikehara theorem revisited

One of the exact Wiener-Ikehara theorems from [6] states that for $S\in L^{1}_{loc}[0,\infty)$ to satisfy $S(x)\sim ae^{x}$ is necessary and sufficient that $S$ is log-linearly slowly decreasing, its Laplace transform converges for $\Re e\>s>1$, and

$$G(s)=\mathcal{L}\{S;s\}-\frac{a}{s-1}$$

has local pseudofunction boundary behavior on $1+i\mathbb{R}$. We wish to replace $G$ by its real part in this characterization. This will be done in fact in Corollary 3.2 below, but before we move on, let us briefly recall what is meant by local pseudofunction boundary behavior.

In this and the next sections, we shall make use of Schwartz distribution theory. Our notation for calculus with distributions is as in the standard textbooks [31] or [12]; in particular, we make use of dummy variables of evaluation to facilitate our manipulations. As usual, $\mathcal{D}(I)$ stands for the space of smooth test functions with compact supports on an open set $I\subset\mathbb{R}$, while $\mathcal{D}^{\prime}(I)$ is the space of distributions on $I$. We say that $f\in\mathcal{D}^{\prime}(I)$ is a local pseudofunction if for every $\varphi\in\mathcal{D}(I)$ the (distributional) Fourier transform of $\varphi f$ (which is entire by the Paley-Wiener theorem) is a continuous function that vanishes at $\pm\infty$. We then write $f\in\operatorname*{PF}_{loc}(I)$. Note that $L^{1}_{loc}(I)\subset\operatorname*{PF}_{loc}(I)$, thanks to the Riemann-Lebesgue lemma. In what follows we exploit that $\mathcal{D}^{\prime}$ and $PF_{loc}$ are both (fine) sheaves, which allows us to work with localizations.

Let $I\subset\mathbb{R}$ be open. A harmonic function $U$ on $\Re e\>s>1$ is said to have distributional boundary values on $1+iI$ if there is $u\in\mathcal{D}^{\prime}(I)$ such that

$$\lim_{\sigma\to 1^{+}}U(\sigma+it)=u(t)\qquad\mbox{in }\mathcal{D}^{\prime}(I),$$

that is, if for each test function $\varphi\in\mathcal{D}(I)$,

$$\lim_{\sigma\to 1^{+}}\int_{-\infty}^{\infty}U(\sigma+it)\varphi(t)\mathrm{d}t=\langle u(t),\varphi(t)\rangle.$$

We say that $U$ has local pseudofunction boundary behavior on $1+iI$ if it has distributional boundary values there and its boundary distribution $u\in\operatorname*{PF}_{loc}(I)$. We refer to [11, 23] for the theory of boundary values of harmonic functions in distribution spaces (the article [23] actually deals with the general case of distributional boundary values for zero solutions of partially hypoelliptic constant coefficient partial differential operators, such as the Laplacian in our case). We point out that the harmonic function $U$ has distributional boundary values on $1+iI$ if and only if for each compact $K\subset I$ one can find $k=k(K)$ such that

(3.1) $$U(\sigma+it)=O\left(\frac{1}{(\sigma-1)^{k}}\right)$$

for $t\in K$ and, say, $1<\sigma<2$.

The next lemma is our most important technical tool in this section.

Lemma 3.1.

Let $U$ be a real-valued harmonic function on the half-plane $\Re e\>s>1$ and let $I\subseteq\mathbb{R}$ be open. If $U$ has local pseudofunction boundary behavior on the boundary set $1+iI$, so does any harmonic conjugate to $U$.

Proof.

$Let$ $V$ be a harmonic conjugate to $U$. Note that since $U$ has distributional boundary values, then $V$ should also admit a boundary distribution. It suffices to see that the analytic function $F=U+iV$ has local pseudofunction boundary behavior on $1+iI$. We first show this under the additional assumption $U(\bar{s})=U(s)$. Using the Cauchy-Riemann equations, we see that $(V(s)-V(\bar{s}))/2$ must also be harmonic conjugate to $U$. Therefore, dropping a constant summand, we may assume that $V$ satisfies $V(\bar{s})=-V(s)$. We might also assume that $I$ is symmetric about the origin. Set

$$u(t)=\lim_{\sigma\to 1^{+}}U(\sigma+it)\in{\operatorname*{PF}}_{loc}(I)\qquad\mbox{ and }\qquad f(t)=\lim_{\sigma\to 1^{+}}F(\sigma+it)\in\mathcal{D}^{\prime}(I).$$

By [6, Proposition 2.1], $\langle f(t),\varphi(t)e^{iht}\rangle=o(1)$ as $h\to-\infty$, for each $\varphi\in\mathcal{D}(I)$ and our assumption is $\langle u(t),\varphi(t)e^{iht}\rangle=o(1)$ as $|h|\to\infty$. We also notice that $u$ is an even distribution, while $f(t)-u(t)$ is odd. For $h>0$ and $\varphi\in\mathcal{D}(I)$ real-valued and even,

$$\langle f(t),\varphi(t)e^{iht}\rangle=\langle f(t),\varphi(t)(e^{iht}+e^{-iht})\rangle+o(1)=\langle u(t),\varphi(t)(e^{iht}+e^{-iht})\rangle+o(1)=o(1)$$

as $h\to\infty$. Likewise, for $\varphi\in\mathcal{D}(I)$ real-valued and odd,

$$\langle f(t),\varphi(t)e^{iht}\rangle=\langle u(t),\varphi(t)(e^{iht}-e^{-iht})\rangle+o(1)=o(1)\qquad\mbox{as }h\to\infty.$$

Decomposing an arbitrary test function into real and imaginary parts, and then each of them into the sum of their even and odd parts, we obtain that $\langle f(t),\varphi(t)e^{iht}\rangle=o(1)$ as $|h|\to\infty$ for each $\varphi\in\mathcal{D}(I)$, namely, $f\in{\operatorname*{PF}}_{loc}(I).$

A small variant of the above argument also applies when $U$ satisfies $U(s)=-U(\bar{s})$. Finally, the general case follows from these two particular ones by writing $U(s)=(U(s)+U(\bar{s}))/2+(U(s)-U(\bar{s}))/2$. ∎

Lemma 3.1 and [6, Theorem 3.6] together thus yield:

Corollary 3.2.

Let $S\in L^{1}_{loc}[0,\infty)$. Then, $S(x)\sim ae^{x}$ holds if and only if $S$ is log-linearly slowly decreasing, its Laplace transform is convergent on $\Re e\>s>1$, and the harmonic function

(3.2) $$\Re e\>\left(\mathcal{L}\{S;s\}-\frac{a}{s-1}\right)$$

admits local pseudofunction boundary behavior on the whole line $\Re e\>s=1$.

Remark 3.3.

Let $S$ be of local bounded variation, so that $\mathcal{L}\{\mathrm{d}S;s\}=s\mathcal{L}\{S;s\}$. Since smooth functions are multipliers for local pseudofunctions, $\mathcal{L}\{S;s\}-a/(s-1)$ has local pseudofunction boundary behavior on a given boundary subset if and only if $\mathcal{L}\{\mathrm{d}S;s\}-a/(s-1)$ does it. Employing Lemma 3.1 once more, we might replace (3.2) by the hypothesis that the real part of $\mathcal{L}\{\mathrm{d}S;s\}-a/(s-1)$ has local pseudofunction boundary behavior on $\Re e\>s=1$.

Remark 3.4.

Lemma 3.1 highlights a key advantage of the local pseudofunction approach over $L^{1}_{loc}$-boundary behavior. In fact, it is well-known that if a harmonic function has $L^{1}_{loc}$-boundary behavior, the distributional boundary values of its harmonic conjugate functions do not necessarily belong to $L^{1}_{loc}$ (see e.g. [16, p. 73])

4. Proof of Theorem 1.2 (and Theorem 1.3)

We are now ready to show Theorem 1.2. We will do so with the aid of Corollary 3.2. Observe that it suffices to prove (1.6), since once this is established (1.7) automatically holds by the familiar real Abelian result for Laplace transforms.

The first part of our proof consists in showing that any of our assumptions imply that $U$ admits a boundary distribution on $1+i\mathbb{R}$. This is actually our hypothesis away from the boundary point $s=1$, where we even have the stronger $L_{loc}^{1}$-boundary behavior. So, we must then still establish the existence of a boundary distribution in a boundary neighborhood of 1. We need some preparation for it.

Let $\Phi:\mathbb{D}\to\Omega$ be a conformal equivalence between the unit disc and a region $\Omega\subset\{s:\>\Re e\>s>1\}$ whose boundary is a smooth (namely, $C^{\infty}$) Jordan curve that meets the line $\Re e\>s=1$ in a closed segment containing the interval $1+i(-\lambda,\lambda)$. Classical results from the theory of conformal maps (see [25, Theorem 2.6, p. 24; Theorem 3.5, p. 48; Theorem 3.6, p. 49] guarantee that $\Phi$ extends to a smooth diffeomorphism $\Phi:\overline{\mathbb{D}}\to\overline{\Omega}$, and we may assume that $\Phi(1)=1.$ Moreover, if $\Phi(J)=1+i[-\lambda,\lambda]$, then $\Phi$ has analytic extension through an open circular arc containing $J$, as one infers from the Schwarz reflection principle for analytic arcs. We consider the harmonic function $V=U\circ\Phi$.

Claim 4.1.

The function $V$ belongs to the harmonic Hardy space $h^{1}(\mathbb{D})$.

Proof.

If (B.2) is satisfied, we directly get $V\in h^{1}(\mathbb{D})$, as inferred from [10, Theorem 10.1, p. 168] and [25, Theorem 3.5, p. 48]. Assume now that (B.1) holds. Applying again [10, Theorem 10.1, p. 168], we conclude that $V$ has $L^{1}_{loc}$-boundary behavior on any boundary arc that does not contain the boundary point $1$. Combining the latter fact with a.e. existence of the radial boundary limits of $V$, we obtain that $V$ belongs to the harmonic Hardy space $h^{1}(S)$ for any disc sector of the form

$$S=\{z\in\mathbb{D}:\>\arg z\notin J^{\prime}\}\qquad\mbox{with a closed subarc }1\in J^{\prime}\subsetneq J.$$

Let us fix such a sector $S$. We also write $J=\{e^{i\theta}:\theta\in[\alpha,\beta]\}$. We have

$$\sup_{0<r<1}\int_{[-\pi,\pi]\setminus{(\alpha,\beta)}}|V(re^{i\theta})|\mathrm{d}\theta<\infty,$$

because $V\in h^{1}(S)$. Therefore, we just need to bound the integral of $|V(re^{i\theta})|$ over $[\alpha,\beta]$ as $r\to 1^{-}$. Using the mean value property of harmonic functions and condition (B.1), we obtain

$$\int_{\alpha}^{\beta}|V(re^{i\theta})|\mathrm{d\theta} \leq\int_{\alpha}^{\beta}|g(\Im m\>(\Phi(re^{i\theta})))|\mathrm{d}\theta+\int_{\alpha}^{\beta}[V(re^{i\theta})-g(\Im m\>(\Phi(re^{i\theta})))]\mathrm{d}\theta$$

$$\leq 2\int_{\alpha}^{\beta}|g(\Im m\>(\Phi(re^{i\theta})))|\mathrm{d}\theta+2\pi V(0)+\int_{[-\pi,\pi]\setminus{(\alpha,\beta)}}|V(re^{i\theta})|\mathrm{d}\theta$$

$$=2\int_{\alpha}^{\beta}|g(\Im m\>(\Phi(e^{i\theta})))|\mathrm{d}\theta+o(1)+2\pi V(0)+\int_{[-\pi,\pi]\setminus{(\alpha,\beta)}}|V(re^{i\theta})|\mathrm{d}\theta$$

$$=O(1),\qquad r\to 1^{-},$$

where in the third line we have used that $\Phi(re^{i\theta})\to\Phi(ie^{\theta})$ and $(\Phi(re^{i\theta}))^{\prime}\to(\Phi(ie^{\theta}))^{\prime}$ uniformly on $[\alpha,\beta]$. This shows that $V\in h^{1}(\mathbb{D})$, also under (B.1). ∎

We can now establish our original claim:

Claim 4.2.

$U$ admits a boundary distribution on a boundary neighborhood of 1.

Proof.

In view of [10, Theorem 1.1, p. 2] and Claim 4.1, the harmonic function $V$ is a Poisson-Stieltjes integral, whence we readily obtain the bound

$$V(z)=O\left(\frac{1}{1-|z|}\right),\qquad|z|<1.$$

Since $\Phi$ extends to a diffeomorphism between complex neighborhoods of $J$ and $1+i[-\lambda,\lambda]$, we also have

$$U(\sigma+it)=O\left(\frac{1}{\sigma-1}\right),\qquad\sigma+it\in(1,2]\times[-\lambda,\lambda].$$

The latter bound yields the claim (see Section 3). ∎

We can now move to the second part of proof. Let

$$u(t)=\lim_{\sigma\to 1^{+}}U(\sigma+it)\qquad\mbox{in }\mathcal{D}^{\prime}(\mathbb{R}).$$

By assumption $u=f$ on $\mathbb{R}\setminus\{0\}$ with $f\in L^{1}_{loc}(\mathbb{R}\setminus\{0\})$. If (B.1) holds, then $u-g$ should be a non-negative measure on $(-\lambda,\lambda)$. When (B.2) is satisfied, $u$ is a measure on $(-\lambda,\lambda)$, as follows from the Banach-Alaoglu theorem (or Helly’s selection principle as better known in the Lebesgue-Stieltjes measure context), because (B.2) tells us that $\{U(\sigma+i\>\cdot\>)\>:1<\sigma<2\}$ is bounded in the dual of $C[-\lambda,\lambda]$. Summarizing, in every case we have shown that the distribution $u$ is a Radon measure on $\mathbb{R}$. Using the Lebesgue decomposition of $u$ (Lebesgue-Radon-Nikodym theorem [28, p. 121]), we conclude that $f\in L^{1}_{loc}(\mathbb{R})$ and that $f$ is its absolutely continuos part, while its singular part must have point support at 0. Hence, $u=f+\pi a\delta$ for some constant $a\in\mathbb{R}$, where as usual $\delta$ stands for the Dirac delta distribution. Finally, using the well-known formula (cf. [12, Eq. (2.17), p. 58])

$$\lim_{\sigma\to 1^{+}}\frac{1}{\sigma-1+it}=\frac{-i}{t-i0}=\pi\delta(t)-i\>\mathrm{p.v.}\left(\frac{1}{t}\right),$$

we deduce from Corollary 3.2 that $S(x)\sim ae^{x}$ as $x\to\infty$, because (3.2) has boundary value distribution $f\in L^{1}_{loc}(\mathbb{R})\subset PF_{loc}(\mathbb{R})$ . This completes the proof of Theorem 1.2.

The proof of Theorem 1.3 is exactly the same as the one we just gave, but now making use of Remark 3.3.

5. The power series case: proof of Theorem 1.5

The proof of Theorem 1.5 is similar to that of Theorem 1.2, but simpler since we can avoid using Corollary 3.2 via a more direct argument. It suffices to show that $\{c_{n}\}_{n=0}^{\infty}$ converges to some finite limit, because then necessarily (1.10) should hold due to Abel’s classical limit theorem for power series. As in Section 4, the assumptions imply that $U(re^{i\theta})$ converges distributionally to a boundary measure, which is absolutely continuous with respect to the Lebesgue measure off $2\pi\mathbb{Z}$. Therefore, there are $f\in L^{1}[-\pi,\pi]$ and $a\in\mathbb{R}$ such that

$$\lim_{r\to{1}^{-}}U(re^{i\theta})=a\pi\delta(\theta)+f(\theta)$$

in, say, the dual of $C^{\infty}[-\pi,\pi].$ Now,

$$c_{n}r^{n} =\frac{1}{2\pi}\int_{-\pi}^{\pi}F(r{e}^{i\theta})e^{-in\theta}\mathrm{d}\theta=\frac{1}{2\pi}\int_{-\pi}^{\pi}F(r{e}^{i\theta})(e^{-in\theta}+e^{in\theta})\mathrm{d}\theta$$

$$=\frac{1}{\pi}\int_{-\pi}^{\pi}U(r{e}^{i\theta})\cos n\theta\>\mathrm{d}\theta,$$

where we have used that $\{c_{k}\}_{k=0}^{\infty}$ is real. Taking $r\to 1^{-}$,

$$c_{n}=a+\frac{1}{\pi}\int_{-\pi}^{\pi}f(\theta)\cos n\theta\>\mathrm{d}\theta=a+o(1)\qquad\mbox{as }n\to\infty,$$

in view of the Riemann-Lebesgue lemma.

6. Proof of Corollary 1.4

We set $S(x)=\int_{0^{-}}^{x}e^{u}\mathrm{d}Q(u)$. Using that $Q(x)+Mx$ is non-decreasing, we see that $S$ is log-linearly slowly decreasing; in fact, for $y\in[x,x+h]$,

$$\frac{S(y)-S(x)}{e^{x}}=e^{-x}\int_{x^{+}}^{y}e^{u}(\mathrm{d}Q(u)+M\mathrm{d}u)-Me^{-x}\int_{x}^{y}e^{u}\mathrm{d}u\geq M(1-e^{h})=o(1),$$

$h\to 0^{+}$. The hypotheses (B₀.1) and (B₀.2) translate into the conditions (B.1) and (B.2), respectively, for the real part of the Laplace-Stieljes transform of $S$. Theorem 1.3 then yields $S(x)\sim ae^{x}$ with $a$ given by (1.9). Writing $\tau(x)=e^{-x}S(x)-a=o(1)$, noticing that $\mathrm{d}Q(x)=e^{-x}\mathrm{d}S(x)$, and integrating by parts, we obtain,

$$Q(x)=a(x+1)+\tau(x)+\int_{0}^{x}\tau(u)\mathrm{d}u=ax+a+\int_{0}^{x}\tau(u)\mathrm{d}u+o(1).$$

The asymptotic formula (1.8) now follows upon redefining $\tau$ on a finite interval so that the constant $a$ gets absorbed into its integral on such an interval.

7. Concluding remarks

We end this article with some remarks.

Remark 7.1.

Theorem 1.1 is also directly covered by (B.2). In fact, suppose that (1.4) and (1.3) are satisfied. Let $\varphi$ be a real-valued even non-negative smooth function with support on $(-\lambda,\lambda)$ such that $\varphi(t)=1$ for $t\in[-\lambda/2,\lambda/2]$. Let $\widehat{\varphi}(x)=\int_{-\infty}^{\infty}\varphi(t)e^{-itx}\mathrm{d}x$, so that $\widehat{\varphi}$ is a Schwartz function. Then, since $U(\sigma+it)-g(t)\geq 0$ in the considered range,

$$\int_{-\lambda/2}^{\lambda/2}|U(\sigma+it)|\mathrm{d}t \leq\int_{-\lambda/2}^{\lambda/2}|g(t)|\mathrm{d}t+\int_{-\infty}^{\infty}(U(\sigma+it)-g(t))\varphi(t)\mathrm{d}t$$

$$\leq 2\int_{\infty}^{\infty}|g(t)|\varphi(t)\mathrm{d}t+\int_{-\infty}^{\infty}\mathcal{L}\{S;(\sigma+it)\}\varphi(t)\mathrm{d}t$$

$$=2\int_{\infty}^{\infty}|g(t)|\varphi(t)\mathrm{d}t+\int_{0}^{\infty}e^{-x}S(x)e^{-(\sigma-1)x}\widehat{\varphi}(x)\mathrm{d}x$$

$$\leq 2\int_{\infty}^{\infty}|g(t)|\varphi(t)\mathrm{d}t+\int_{0}^{\infty}e^{-x}|S(x)||\widehat{\varphi}(x)|\mathrm{d}x<\infty.$$

Remark 7.2.

Koga originally stated his Tauberian theorem (cf. [17, Theorem 2]) by only imposing the boundary requirements for the Laplace transform on a sequence tending to $1^{+}$. More precisely, in addition to (1.3), he assumes the existence of $\sigma_{n}\to 1^{+}$ such that

(7.1) $$\lim_{n\to\infty}\int_{I}|U(\sigma_{n}+it)-f(t)|\mathrm{d}t=0,$$

for some $f\in L^{1}_{loc}(\mathbb{R}\setminus\{0\})$ and any finite interval $I$ not containing the origin, and

(7.2) $$U(\sigma_{n}+it)\geq g(t),\qquad\mbox{for a.e. }t\in(-\lambda,\lambda)\mbox{ and }n\in\mathbb{N},$$

for some $\lambda>0$ and $g\in L^{1}(-\lambda,\lambda)$.

We have however that (7.1) is equivalent to (1.5) in our case. In fact, since (1.3), which allows us to view $e^{-x}S(x)$ as a tempered distribution, ensures [31, Section 6.6.9, p. 100] that $\mathcal{L}\{S;s\}$ has distributional boundary values on $\Re e\>s>1$, the relation (1.5) might be inferred from (7.1) by using a standard localization argument together with the (distributional) Schwarz reflection principle [27] (see also [5, 23] for generalized reflection principles). Likewise, the condition (7.2) follows from (1.4) and (1.3) (where one might need to subtract a constant from $g$ resulting from the application of the reflection principle). Alternatively, one can also see that Koga’s original set of hypotheses is covered by Theorem 1.2 via the same argument employed in Remark 7.1, which clearly yields, say,

$$\sup_{n\in\mathbb{N}}\int_{-2\lambda/3}^{2\lambda/3}|U(\sigma_{n}+it)|\mathrm{d}t<\infty.$$

The latter condition in turn implies (via localization and the reflection principle once more) that (B.2) holds with $\lambda$ replaced by, say, $\lambda/2$.

Remark 7.3.

As Koga, we could also have only assumed in Theorem 1.2 and Theorem 1.3 that (1.4) just holds on a sequence $\sigma=\sigma_{n}\to 1^{+}$. Exactly the same argument given in Section 4 would then still yield that the boundary distribution of $U$ is a Radon measure. This comment also applies to Corollary 1.4 and Theorem 1.5.

Remark 7.4.

Sometimes one is just interested in deducing an upper bound

(7.3) $$S(x)=O(e^{x})$$

from relatively mild regularity boundary properties of the Laplace transform. For instance, such criteria play an important role in abstract analytic number theory (see e.g. [8, Chapter 11] for applications to Beurling primes). The following extension of [6, Proposition 3.1] (cf. [8, Theorem 10.1]) could then be useful in that respect. We call $S$ log-linearly boundedly decreasing if there is $h>0$ such that

$$\liminf_{x\to\infty}\inf_{y\in[x,x+h]}\frac{S(y)-S(x)}{e^{x}}>-\infty.$$

We also use the notation $A_{c}(\mathbb{R})$ for the subspace of compactly supported elements of the Wiener algebra, namely, those compactly supported continuous functions such that their Fourier transforms belong to $L^{1}(\mathbb{R})$.

Proposition 7.5.

Let $S\in L^{1}_{loc}[0,\infty)$ and let $\varphi\in A_{c}(\mathbb{R})\setminus\{0\}$ be even real-valued and have non-negative Fourier transform. Then, (7.3) holds if and only if $S$ is log-linearly boundedly decreasing, has convergent Laplace transform on $\Re e\>s>1$, and there is a sequence $\sigma_{n}\to 1^{+}$ such that

$$\mathfrak{I}_{\varphi}(h)=\lim_{n\to\infty}\int_{-\infty}^{\infty}\left(\Re e\mathcal{L}\{S;\sigma_{n}+it\}\right)\varphi(t)\cos ht\>\mathrm{d}t$$

exists for each $h>0$ and $\mathfrak{I}_{\varphi}(h)=O(1)$ as $h\to\infty$.

Proof.

That the conditions are necessary is easy to verify. Let us show their sufficiency for (7.3). Set $\widehat{\varphi}(x)=\int_{-\infty}^{\infty}\varphi(t)e^{-itx}\mathrm{d}x$ and $\Delta(x)=e^{-x}S(x)$. As shown inside the proof of [6, Proposition 3.1], $\Delta(x)=O(1)$ as $x\to\infty$ would follow from the log-linear bounded decrease if we show that $(\Delta\ast\widehat{\varphi})(h)=O(1)$ as $h\to\infty$. Also, it is shown there that we may assume without any loss of generality that $\Delta\geq 0$. Thus (the use of Parseval’s relation is justified by [6, Lemma 3.4])

$$0\leq\int_{0}^{\infty}e^{(1-\sigma_{n})x}\Delta(x)\widehat{\varphi}(x-h)\mathrm{d}x \leq\int_{0}^{\infty}e^{(1-\sigma_{n})x}\Delta(x)\left((\widehat{\varphi}(x-h)+\widehat{\varphi}(x+h)\right)\mathrm{d}x$$

$$=\int_{-\infty}^{\infty}\mathcal{L}\{S;\sigma_{n}+it\}\varphi(t)\left(e^{iht}+e^{-iht}\right)\mathrm{d}t$$

$$=2\int_{-\infty}^{\infty}\Re e\mathcal{L}\{S;\sigma_{n}+it\}\>\varphi(t)\cos ht\>\mathrm{d}t.$$

Applying the Beppo Levi theorem, $0\leq(\Delta\ast\widehat{\varphi})(h)\leq 2\mathfrak{I}_{\varphi}(h)=O(1)$, $h\to\infty$. ∎

For example, Proposition 7.5 could have been used in Section 4 to directly show that $U(s)$ has distributional boundary values under (B.2) without having to pass through the conformal map argument, because once $T(x)=e^{-x}S(x)=O(1)$, the Laplace transform $\mathcal{L}\{S;s\}$ tends to the (distributional) Fourier transform of $T$ on $\Re e\>s=1$.