Criteria for analyticity of subordinate semigroups

TL;DR

This paper provides new criteria and sufficient conditions for Bernstein functions to ensure that the associated operator generates a quasibounded holomorphic semigroup, extending previous work by Carasso and Kato.

Contribution

It offers an alternative to Carasso-Kato's criteria and introduces several new sufficient conditions for Bernstein functions to generate holomorphic semigroups.

Findings

Derived alternative criteria for Bernstein functions to generate holomorphic semigroups.

Established several new sufficient conditions for these functions.

Extended the theoretical understanding of semigroup generation in Banach spaces.

Abstract

Let $\psi$ be a Bernstein function. A.~Carasso and T.~Kato obtained necessary and sufficient conditions for $\psi$ to have a property that $\psi(A)$ generates a quasibounded holomorphic semigroup for every generator $A$ of a bounded $C_0$-semigroup in a Banach space, in terms of some convolution semigroup of measures associated with $\psi$. We give an alternative to Carasso-Kato's criterium, and derive several sufficient conditions for $\psi$ to have the above-mentioned property.

Equations

$$\psi (z)=c_0+\underset{{\mathbb{R}}_{+}}{\int }(e^{zu}-1)u^{-1}d\rho (u),Rez\le 0$$

$$g_t(z):=e^{t\psi (z)}=\underset{{\mathbb{R}}_{+}}{\int }{e}^{zu}d{\nu }_t(u),Rez\le 0$$

$$\psi (A)x=c_{0}x+\underset{{\mathbb{R}}_{+}}{\int }(T(u)-I)x{u}^{-1}d\rho (u).$$

$$T(t)X\subset D(A),t>0,\mathrm{and}\lim \underset{t\downarrow 0}{\sup }(t\Vert AT(t)\Vert )<\infty ,$$

$$g(s)=La(s):=\underset{{\mathbb{R}}_{+}}{\int }{e}^{su}da(u)(s\le 0)$$

$$g(A)x=\underset{{\mathbb{R}}_{+}}{\int }T(u)xda(u),x\in X.$$

$$p(t)=\sum _{j=1}^n{c}_j{e}^{s_{j}t},c_j\in \mathbb{C},s_j<0,$$

$$|f_n(x)-p_n(x)|<n^{-1},\mathrm{and}|f_n^{\prime }(x)-p_n^{\prime }(x)|<n^{-1},x\in [0,1].$$

$$|q(-\log x)-p_n(x)|<n^{-1},x\in (0,1],n>-\log x.$$

$$|q^{\prime }(-\log x)(-x^{-1})-p_n^{\prime }(x)|<n^{-1},x\in (0,1],n>-\log x.$$

$$K(a,\rho )=\underset{\varphi \in S}{\sup }\left|\underset{{\mathbb{R}}_{+}}{\int }\underset{{\mathbb{R}}_{+}}{\int }\varphi (r)d_r(a(r-u)-a(r))u^{-1}d\rho (u)\right|$$

$$g(s)=\underset{{\mathbb{R}}_{+}}{\int }{e}^{sr}da(r)=(-s)\underset{{\mathbb{R}}_{+}}{\int }{e}^{sr}a(r)dr.$$

$$(e^{su}-1)g(s)=(e^{su}-1)(-s)\underset{{\mathbb{R}}_{+}}{\int }{e}^{sr}a(r)dr$$

$$=(-s)\left(\underset{{\mathbb{R}}_{+}}{\int }{e}^{s(r+u)}a(r)dr-\underset{{\mathbb{R}}_{+}}{\int }{e}^{sr}a(r)dr\right)=(-s)\underset{{\mathbb{R}}_{+}}{\int }{e}^{sr}(a(r-u)-a(r))dr=L{b}^u(s),$$

$$h(s)=\underset{{\mathbb{R}}_{+}}{\int }(e^{su}-1)g(s)u^{-1}d\rho (u)=\underset{{\mathbb{R}}_{+}}{\int }\underset{{\mathbb{R}}_{+}}{\int }{e}^{sr}d{b}^u(r)u^{-1}d\rho (u).$$

$$b(\varphi ):=\underset{{\mathbb{R}}_{+}}{\int }{b}^u(\varphi )u^{-1}d\rho (u)=\underset{{\mathbb{R}}_{+}}{\int }\underset{{\mathbb{R}}_{+}}{\int }\varphi (r)d_r(a(r-u)-a(r))u^{-1}d\rho (u)$$

$$b=\underset{{\mathbb{R}}_{+}}{\int }{b}^u{u}^{-1}d\rho (u)$$

$$b(q)=\underset{{\mathbb{R}}_{+}}{\int }{b}^u(q)u^{-1}d\rho (u).$$

$$p_n(u)=\underset{{\mathbb{R}}_{+}}{\int }{q}_n(r)d_r(a(r-u)-a(r))=\underset{{\mathbb{R}}_{+}}{\int }(q_n(r+u)-q_n(r))da(r).$$

$$\underset{{\mathbb{R}}_{+}}{\int }{p}_n(u)u^{-1}d\rho (u)\to \underset{{\mathbb{R}}_{+}}{\int }p(u)u^{-1}d\rho (u)(n\to \infty ).$$

$$\underset{{\mathbb{R}}_{+}}{\int }{p}_n(u)u^{-1}d\rho (u)=\underset{{\mathbb{R}}_{+}}{\int }{b}^u(q_n)u^{-1}d\rho (u)=b(q_n)\to b(q)(n\to \infty ).$$

$$f\left(\underset{{\mathbb{R}}_{+}}{\int }Txdb\right)=\underset{{\mathbb{R}}_{+}}{\int }f(T(r)x)db(r)=\underset{{\mathbb{R}}_{+}}{\int }f\left(\underset{{\mathbb{R}}_{+}}{\int }T(r)xd{b}^u(r)\right)u^{-1}d\rho (u).$$

$$\underset{{\mathbb{R}}_{+}}{\int }\left(\underset{{\mathbb{R}}_{+}}{\int }T(r)xd{b}^u(r)\right)u^{-1}d\rho (u)=\underset{{\mathbb{R}}_{+}}{\int }Txdb.$$

$$\underset{{\mathbb{R}}_{+}}{\int }T(r)xd{b}^u(r)=\underset{[u,\infty )}{\int }T(r)x{d}_{r}a(r-u)-\underset{{\mathbb{R}}_{+}}{\int }T(r)xda(r)$$

$$=\underset{{\mathbb{R}}_{+}}{\int }T(r+u)xda(r)-\underset{{\mathbb{R}}_{+}}{\int }T(r)xda(r)=(T(u)-I)g(A)x.$$

$$h(A)x=\underset{{\mathbb{R}}_{+}}{\int }T(r)xdb(r)=\underset{{\mathbb{R}}_{+}}{\int }(T(u)-I)g(A)x{u}^{-1}d\rho (u)=\psi (A)g(A)x.$$

$$\Vert h(A)\Vert \le \underset{{\mathbb{R}}_{+}}{\int }\Vert T(r)\Vert d|b|(r)\le M\Vert b\Vert =MK(a,\rho ).$$

$$K({\nu }_t,\rho )=O(t^{-1}),t\downarrow 0$$

$$\Vert h_t(A)\Vert =\Vert \psi (A)g_t(A)\Vert \le MK({\nu }_t,\rho ).$$

$$y(v):=g_t(A)x(v)=\underset{{\mathbb{R}}_{+}}{\int }x(v+r)d{\nu }_t(r)=-\underset{{\mathbb{R}}_{+}}{\int }{x}^{\prime }(v+r){\nu }_t(r)dr.$$

Full text

**Criteria for analyticity of subordinate semigroups **

A. R. Mirotin 111The author was supported in part by the State Program of Fundamental Research of Republic of Belarus under the contract number 20061473.

Abstract

Let $\psi$ be a Bernstein function. A. Carasso and T. Kato obtained necessary and sufficient conditions for $\psi$ to have a property that $\psi(A)$ generates a quasibounded holomorphic semigroup for every generator $A$ of a bounded $C_{0}$-semigroup in a Banach space, in terms of some convolution semigroup of measures associated with $\psi$. We give an alternative to Carasso-Kato’s criterium, and derive several sufficient conditions for $\psi$ to have the above-mentioned property.

2000 Mathematics Subject Classification: 47A60, 47D03

Key words and phrases: strongly continuous semigroup, holomorphic semigroup, subordination, functional calculus, function of an operator.

1. Introduction

The well known theorem due to Yosida [17] states that for every generator $A$ of a bounded $C_{0}$-semigroup on a Banach space $X$ its fractional power $-(-A)^{\alpha},0<\alpha<1$ is a generator of a holomorphic semigroup on $X$. The present paper is devoted to some generalizations and analogs of Yosida’s Theorem in terms of so-called Bochner-Phillips calculus [1, 14] (see also [5, Chap. XIII]; [8, 15, 11, 2]). Though the majority of works on Bochner-Phillips calculus use the class ${\cal B}$ of (positive) Bernstein functions, we prefer the class ${\cal T}$ of negative one. The corresponding reformulation of Bochner-Phillips calculus is trivial in view of the fact that $\phi(x)\in{\cal B}$ if and only if $-\phi(-s)\in{\cal T}$.

We say that the function $\psi:(-\infty,0]\to(-\infty,0]$ belongs to the class ${\cal T}$ of negative Bernstein functions if $\psi\in C^{\infty}((-\infty,0))\cap C((-\infty,0])$ and its derivative is absolutely monotonic, i.e. $\psi^{(n)}\geq 0$ for all $n\in\mathbb{N}$. It is known that in this case $\psi$ extends analytically to the left half-plane $\Pi_{-}=\{{\rm Re}z<0\}$, the extension is continuous on $\{{\rm Rez}\leq 0\}$, and has the following integral representation

[TABLE]

where $c_{0}=\psi(0)$, the positive measure $\rho$ on $\mathbb{R}_{+}$ is uniquely determined by $\psi$ and $\int_{[0,1]}d\rho<\infty,\int_{[1,\infty)}u^{-1}d\rho(u)<\infty$; the integrand in (1) is defined for $u=0$ to be equal to $z$ .

Moreover, there is a convolution semigroup $(\nu_{t})_{t\geq 0}$ of sub-probability measures on $\mathbb{R}_{+}$ with the Laplace transform

[TABLE]

(see [16], [5, Chap. XIII]).

The class ${\cal T}$ is a cone which is closed with respect to compositions and pointwise convergence on $(-\infty,0]$, and contains a number of important functions, including (up to affine changes of variable) fractional powers, the logarithm, the inverse hyperbolic cosine, and polylogarithms $Li_{p}$ of all orders $p\in\mathbb{N}$ [12].

For a negative Bernstein function $\psi$ with integral representation (1) and a generator $A$ of a bounded $C_{0}$-semigroup $T$ on a complex Banach space $X$ the value of $\psi$ at $A$ for $x\in D(A)$, the domain of $A$, is defined by the Bochner integral

[TABLE]

The closure of this operator, which is also denoted by $\psi(A)$, is a generator of a bounded $C_{0}$-semigroup $g_{t}(A)$ on $X$ (the ”subordinate semigroup”), too. (For the multidimensional version of this calculus see, e.g., [9], [10], [11].)

In the following, without loss of generality we shall assume that $c_{0}=0$. The corresponding subclass of ${\cal T}$ will be denoted by ${\cal T}_{0}$. We shall denote also by ${\cal M}^{b}(\mathbb{R}_{+},\mathbb{C})({\cal M}(\mathbb{R}_{+},\mathbb{R}_{+}))$ the space of all bounded complex valued (respectively positive) measures on $\mathbb{R}_{+}$, and by $C_{0}(\mathbb{R}_{+})$ the space of all continuous complex valued functions on $\mathbb{R}_{+}$ which vanish at infinity; $X$ stands for a complex Banach space.

Another result by Yosida [18] asserts that if the bounded $C_{0}$-semigroup $T$ with generator $A$ on $X$ satisfies

[TABLE]

then for any $\beta>0$, $e^{-\beta t}T(t)$ can be extended to a bounded holomorphic semigroup on $X$.

We shall denote by ${\cal T}_{Y}$ the set of all $\psi\in{\cal T}$ such that $\psi(A)$ generates a bounded $C_{0}$-semigroup with property (Y) for every generator $A$ of a bounded $C_{0}$-semigroup in a Banach space. The class ${\cal T}_{Y}$ is a cone [3, Theorem 6]. Moreover, it is clear that the composition $\psi_{1}\circ\psi_{2}\in{\cal T}_{Y}$ if $\psi_{1}\in{\cal T}_{Y}$, $\psi_{2}\in{\cal T}$. But the class ${\cal T}_{Y}$ is not closed with respect to pointwise convergence.

A. Carasso and T. Kato [3, Theorem 4] obtained necessary and sufficient conditions for a function $\psi$ to be in ${\cal T}_{Y}$ in terms of the semigroup $(\nu_{t})_{t\geq 0}$. They also gave two necessary conditions in terms of $\psi$ itself. Y. Fujita [6] obtained sufficient conditions for $\psi$ to be in ${\cal T}_{Y}$ in terms of analytical continuation of $\psi$ and regular variation.

We proceed as follows. First we prove the multiplication rule which connects the Bochner-Phillips and Hille-Phillips calculi and then derive the alternative to [3] necessary and sufficient conditions for the inclusion $\psi\in{\cal T}_{Y}$ (see Theorem 2 below; the variant of this theorem with $C_{00}(\mathbb{R}_{+})$ instead of $E(\mathbb{R}_{+})$ (for the definition of the last class see below) first appeared in [13]). Then we deduce two theorems from this criterium that give sufficient conditions for $\psi$ to be in ${\cal T}_{Y}$ in terms of $\psi$. It should ne noted that the assumptions of Theorem 4 below contain necessary conditions, obtained by Carasso and Kato (the idea to employ the Hausdorff-Young inequality in this context belongs to Carasso and Kato, too). Finally, we give one more condition, that is sufficient for the inclusion $\psi\in{\cal T}_{Y}$. Several examples have been considered.

2. The multiplication rule for the Bochner-Phillips and Hille-Phillips calculi, and the criterium for $\psi$ to be in ${\cal T}_{Y}$

In [7, Chap.XV] the functional calculus (the Hille-Phillips calculus) of generators of $C_{0}$-semigroups have been constructed. In particular let $a\in{\cal M}^{b}(\mathbb{R}_{+},\mathbb{C})$ and

[TABLE]

be the Laplace transform of $a$. Then for a generator $A$ of a bounded $C_{0}$-semigroup $T$ on a complex Banach space $X$ the value of $g$ at $A$ is the bounded operator on $X$ defined by the Bochner integral

[TABLE]

Our Theorem 1 connects the Bochner-Phillips and Hille-Phillips calculi. It is a generalization of Lemma 1 in [13]. But first we need the following approximation lemma. We shall denote by $E(\mathbb{R}_{+})$ the complex space of exponential polynomials of the form

[TABLE]

endowed with $\sup$-norm on $\mathbb{R}_{+}$.

Lemma 1. For every bounded function $q\in C^{1}(\mathbb{R}_{+})$ with bounded derivative there exists a sequence $q_{n}\in E(\mathbb{R}_{+})$ such that

1)* $q_{n}\to q$, and $q_{n}^{\prime}\to q^{\prime}$ pointwise on $\mathbb{R}_{+}$;*

2)* $(q_{n})$ and $(q_{n}^{\prime})$ are uniformly bounded on $\mathbb{R}_{+}$.*

Proof. Let us pick a sequence $\tilde{q}_{n}\in C^{1}(\mathbb{R}_{+})$ such that $\tilde{q}_{n}(t)=q(t)$ for $t\in[0,n]$, $\tilde{q}_{n}(t)=0$ for $t\in[n+1,\infty)$, and $(\tilde{q}_{n})$ and $(\tilde{q}_{n}^{\prime})$ are uniformly bounded, $|\tilde{q}_{n}|<C_{1}$, $|\tilde{q}_{n}^{\prime}|<C_{1}$. Define $f_{n}(x)=\tilde{q}_{n}(-\log x)$ for $x\in[0,1]\quad(f_{n}(0)=0)$. Then $f_{n}\in C^{1}([0,1]),\quad|f_{n}(x)|<C_{1}$ for $x\in[0,1]$, and $|f_{n}^{\prime}(x)|<C_{1}x^{-1}$ for $x\in(0,1]$. It is well known (see, e. g., [4, Theorem 8.4.1]) that for every natural $n$ the algebraic polynomial $p_{n}$ exists such that

[TABLE]

Then $|p_{n}(0)|<n^{-1},|p_{n}(x)|<C_{1}+1,\quad{\rm and}\quad|p_{n}^{\prime}(x)|<C_{1}x^{-1}+1\quad{\rm for}\quad x\in(0,1]$. Since $f_{n}(x)=q(-\log x)$ for $x\in(0,1]$, and $n>-\log x$, we have

[TABLE]

Let $q_{n}(t):=p_{n}(e^{-t})-p_{n}(0)$. Then $q_{n}\in E(\mathbb{R}_{+}),q_{n}\to q\quad{\rm on}\quad\mathbb{R}_{+}$, and $(q_{n})$ and $(q_{n}^{\prime})$ are uniformly bounded on $\mathbb{R}_{+}$. Finally

[TABLE]

Putting hear $x=e^{-t}$ we have for all natural $n>t\quad(t\in\mathbb{R}_{+})$ that $|q^{\prime}(t)-q_{n}^{\prime}(t)|<n^{-1}$. This completes the proof.

For measures $a\in{\cal M}^{b}(\mathbb{R}_{+},\mathbb{C}),\quad{\rm and}\quad\rho\in{\cal M}(\mathbb{R}_{+},\mathbb{R}_{+})$ let

[TABLE]

(if the right hand side exists), where $S$ is the unit sphere of the space $E(\mathbb{R}_{+})$ with respect to $\sup$-norm on $\mathbb{R}_{+}$ . Here we assume that $a=0$ on $(-\infty;0)$. See the proof of Theorem 5 for an estimate for $K(a,\rho)$ with bounded positive measure $a$, but $K(\delta,\delta)=\infty$.

Theorem 1. Let $g=La,\quad a\in{\cal M}^{b}(\mathbb{R}_{+},\mathbb{C}),\quad and\quad\psi\in{\cal T}_{0}$ has integral representation (1). If $K(a,\rho)<\infty$, then

1)* the function $h:=\psi g$ has the form $h=Lb$, where $b\in{\cal M}^{b}(\mathbb{R}_{+},\mathbb{C})$, $\|b\|=K(a,\rho)$;*

2)* $g(A)X\subset D(\psi(A))$, $h(A)=\psi(A)g(A)$, and $\|h(A)\|\leq MK(a,\rho)$ for every operator $A$ in a Banach space $X$, which generates a bounded $C_{0}$-semigroup $T$ with $\|T(t)\|\leq M$.*

Proof. Let $a(r)$ denotes the distribution function for $a$, $a(r)=0$ for $r\in(-\infty,0]$. Then for $s<0$

[TABLE]

Thus for $u\geq 0$ and $s<0$ we have

[TABLE]

[TABLE]

where $b^{u}(r)=a(r-u)-a(r)$ has bounded variation and is concentrated on $\mathbb{R}_{+}$. Therefore for $\psi\in{\cal T}_{0}$ with integral representation (1) we get

[TABLE]

For $\phi\in E(\mathbb{R}_{+})$ let

[TABLE]

be the linear functional on $E(\mathbb{R}_{+})$ (we use the notation $b^{u}(\phi)$ for $\int\phi db^{u}$). By the hypothesis of the theorem $\|b\|=K(a,\rho)<\infty$, and since $E(\mathbb{R}_{+})$ is dense in $C_{0}(\mathbb{R}_{+})$ by Stone-Weierstrass Theorem, $b$ extends to a measure $b\in{\cal M}^{b}(\mathbb{R}_{+},\mathbb{C})$. Furthermore,

[TABLE]

(the weak integral; ${\cal M}^{b}(\mathbb{R}_{+},\mathbb{C})$ is endowed with vague topology).

We claim that for every bounded function $q\in C^{1}(\mathbb{R}_{+})$ with bounded derivative the following equality holds (we write $b(q)$ instead of $\int_{\mathbb{R}_{+}}qdb$ in the rest of the proof)

[TABLE]

In fact, let $(q_{n})$ be as in Lemma 1, and $|q_{n}|<C$, $|q_{n}^{\prime}|<C$ for some constant $C>0$. Putting $p_{n}(u):=b^{u}(q_{n})$ we have

[TABLE]

Now let $p(u):=b^{u}(q)$. Then $p_{n}(u)\to p(u)\quad(n\to\infty)$ pointwise by Lebesgue Theorem. We have $|q_{n}(r+u)-q_{n}(r)|\leq Cu$, and $\leq 2C$. If we take $w(u)=\min\{u,1\}$, then $w\in L^{1}(u^{-1}d\rho(u))$ and (5) implies that $|p_{n}(u)|\leq 2\|a\|w(u)$. Thus by the Lebesgue Theorem

[TABLE]

On the other hand,

[TABLE]

Then $b(q)=\int_{\mathbb{R}_{+}}p(u)u^{-1}d\rho(u)$, i. e. (4) holds. In particular, for $q(r)=e^{sr}\quad(s\leq 0)$ (4) and (3) imply the equality $h=Lb$ which proves the first statement of the theorem.

To prove the second one, fix a bounded linear functional $f\in X^{\prime}$, vector $x\in D(A)$, and let $q(r)=f(T(r)x)$. Then $q\in C^{1}(\mathbb{R}_{+})$ and $q$ is bounded together with the derivative $q^{\prime}(r)=f(T(r)Ax)\quad(r\geq 0)$. For such $q$ equation (4) implies that

[TABLE]

So by the definition of the weak integral

[TABLE]

In addition, the interior integral in the left hand side here exists in the sense of Bochner, and

[TABLE]

[TABLE]

Therefore for $x\in D(A)$ we have

[TABLE]

Since the operator $h(A)$ is bounded, and, on the other hand, the operator $\psi(A)g(A)$ is closed (as the product of a closed and a bounded operators), the last equality holds for all $x\in X$. In particular, $g(A)X\subset D(\psi(A))$. Finally

[TABLE]

The theorem is proved.

Theorem 2. Let $\psi\in{\cal T}_{0}$. Then $\psi\in{\cal T}_{Y}$ if and only if

[TABLE]

holds (see formulas (1) and (2) for the definitions of $\rho$ and $\nu_{t}$).

Proof. Let (6) holds. Putting $a=\nu_{t}$ in Theorem 1 we get that for sufficiently small $t>0$ the function $h_{t}=\psi g_{t}$ has the form $h_{t}=Lb_{t}$, where $b_{t}$ is a bounded measure on $\mathbb{R}_{+}$, $\|b_{t}\|=K(\nu_{t},\rho)$. In addition, $g_{t}(A)X\subset D(\psi(A))$ for all $t>0$ ($\psi(A)=$ generator of the semigroup $g_{t}(A)$) and

[TABLE]

Now (6) implies (Y) with $g_{t}(A)$ instead of $T(t)$.

To prove the converse, consider $X=C_{0}(\mathbb{R}_{+})$ with $\sup$-norm, let $\psi\in{\cal T}_{Y}$, and let $T$ be the $C_{0}$-semigroup of shifts on $X$, $(T(r)x)(v)=x(v+r)$ (in this concrete situation $A$ is a derivation with appropriate domain). Then, for each $x\in C^{1}(\mathbb{R}_{+})\cap C_{0}(\mathbb{R}_{+}),\quad t>0$ integration by parts gives

[TABLE]

Therefore

[TABLE]

[TABLE]

Since $\nu_{t}$ is concentrated on $\mathbb{R}_{+}$, we get

[TABLE]

and thus

[TABLE]

But integration by parts gives since $\nu_{t}(0)=\nu_{t}(-u)=0$,

[TABLE]

Finally, for each $x\in C^{1}(\mathbb{R}_{+})\cap C_{0}(\mathbb{R}_{+}),\quad v\geq 0$

[TABLE]

Taking into account that $t\|\psi(A)g_{t}(A)\|\leq C$ for some $C>0$ and all $t\in(0,1]$ we have for our $x$ with $\|x\|=1$ that $|\psi(A)g_{t}(A)x(v)|\leq Ct^{-1}$. So for each $v\geq 0$, $t\in(0,1]$

[TABLE]

Since $C^{1}(\mathbb{R}_{+})\cap C_{0}(\mathbb{R}_{+})$ is dense in $C_{0}(\mathbb{R}_{+})$, it follows for $v=0$ that $K(\nu_{t},\rho)=O(t^{-1}),\quad t\downarrow 0$, as desired.

3. Sufficient conditions for $\psi$ to be in ${\cal T}_{Y}$ in terms of $\psi$

In the following we shall denote by ${\cal F}$ the Fourier transform on $\mathbb{R}$,

[TABLE]

and by ${\cal F}^{-1}$ the inverse of ${\cal F}$. Let

[TABLE]

The restriction $F_{t}|\mathbb{R}$ will be also denoted by $F_{t}$.

Theorem 3. Let $\psi\in{\cal T}_{0}$. Assume that

(i)* the derivative $\partial/\partial yF_{t}(y)$ exists for a.e. $y\in\mathbb{R}$ and each sufficiently small $t>0$;*

(ii)* for some $p\in(1,2]$ functions $F_{t}$ and $\partial/\partial yF_{t}$ both belong to $L^{p}(\mathbb{R})$ for each sufficiently small $t>0$;*

(iii)* ${\cal F}F_{t}$ is concentrated on $\mathbb{R}_{+}$ for each sufficiently small $t>0$;*

(iv)* $\|F_{t}\|_{p}^{1/q}\|\partial/\partial yF_{t}\|_{p}^{1/p}=O(t^{-1})\quad as\quad t\downarrow 0\quad(p^{-1}+q^{-1}=1)$.*

Then $\psi\in{\cal T}_{Y}$.

Proof. First we prove that $f_{t}:={\cal F}F_{t}\in L^{1}(\mathbb{R}_{+})$, and $F_{t}={\cal F}^{-1}f_{t}$. Indeed, $f_{t}\in L^{q}(\mathbb{R})$, and ${\cal F}(\partial/\partial yF_{t})(y)=iyf_{t}(y)\in L^{q}(\mathbb{R})$. By Hölder’s inequality $f_{t}(y)=(iyf_{t}(y))(iy)^{-1}\in L^{1}(\{|y|>1\})$, and so $f_{t}\in L^{1}(\mathbb{R})$. Now by the Inverse Theorem for the Fourier transform, $F_{t}(y)={\cal F}^{-1}f_{t}(y)$ a.e. $y\in\mathbb{R}$, and by the continuity the last equality holds for all $y\in\mathbb{R}$. Therefore we have for the Laplace transform

[TABLE]

because both sides here are analytic on the left half-plane $\Pi_{-}$, continuous on its closure, and coincide on its boundary $i\mathbb{R}$. In particular, $Lf_{t}(s)=\sqrt{2\pi}e^{t\psi(s)}\psi(s)$ for all $s\leq 0$. It follows that for an arbitrary exponential polynomial $\phi\in E(\mathbb{R}_{+}),\quad\phi(r)=\sum_{j}c_{j}e^{s_{j}r}\quad(c_{j}\in\mathbb{C},s_{j}<0)$ we have

[TABLE]

On the other hand,

[TABLE]

[TABLE]

and thus

[TABLE]

Now we conclude that ($E(\mathbb{R}_{+})$ is dense in $C_{0}(\mathbb{R}_{+})$)

[TABLE]

Let $k_{t}(u):=iuf_{t}(u)$. Then $k_{t}={\cal F}(\partial/\partial yF_{t})$, and using the Hausdorff-Young inequality we obtain

[TABLE]

Next, for any $v>0$ Hölder’s inequality gives

[TABLE]

[TABLE]

Then for any $v>0$

[TABLE]

Therefore, on choosing $v=(p-1)^{1/q}\|\partial/\partial yF_{t}\|_{p}/\|F_{t}\|_{p}$, it follows that

[TABLE]

Application of Theorem 2 completes the proof.

Before formulating the next theorem we note that by [3, Theorem 4] every $\psi\in{\cal T}_{Y}\cap{\cal T}_{0}$ maps $\Pi_{-}$ into a truncated sector

[TABLE]

for some $\beta\geq 0,\theta\in(0,\pi/2)$, and there exist constants $k,k>0$, and $\gamma,\gamma\in(0,1)$, such that $|\psi(z)|\leq k|z|^{\gamma},\quad|z|\geq 1,\quad{\rm Re}z\leq 0$. The problem is what one can add to this conditions to obtain (necessary and) sufficient conditions for $\psi$ to be in ${\cal T}_{Y}$. Now we shall deduce the partial answer to this question from Theorem 3.

Theorem 4. Let $\psi\in{\cal T}_{0}$, and assume that the following conditions hold:

(i)* $\psi:\Pi_{-}\to S(\theta,\beta)$ for some $\beta\geq 0,\theta\in(0,\pi/2)$;*

there exist such positive constants $k,b,\alpha,\gamma$ and $R$, that $\alpha\leq\gamma<1$, $R\geq 1$, and

(ii)* $b|z|^{\alpha}\leq|\psi(z)|\leq k|z|^{\gamma}$ for $z\in\Pi_{-},\quad|z|\geq R$;*

(iii)* the function $y\mapsto\psi(iy)$ is differentiable for a. e. $y\in\mathbb{R}$ and*

[TABLE]

for some $\delta\in(\alpha-\gamma-1,2\alpha-\gamma-1)$ if $\alpha<\gamma$, and $\delta=\gamma-1$ if $\alpha=\gamma$;

(iv)* $\psi^{\prime}(iy)\in L^{p}([0,R])$ for some $p\in(1,2]$ such that $p=\min\{2,(\alpha-\gamma-\delta)^{-1},(\alpha-\delta-1)/(\gamma-\alpha)\}$ if $\alpha<\gamma$, and $p<\min\{2,(1-\gamma)^{-1}\}$ if $\alpha=\gamma$.*

Then $\psi\in{\cal T}_{Y}$.

Proof. We shall verify all the conditions of Theorem 3 for $\psi$. Let $a_{1}=\max\{|\psi(z)||z\in\Pi_{-},|z|\leq R\}$, $m_{1}=\min\{|\psi(z)|-b|z|^{\alpha}|z\in\Pi_{-},|z|\leq R\}$. Then $b|z|^{\alpha}+a_{2}\leq|\psi(z)|\leq k|z|^{\gamma}+a_{1}$ for $z\in\Pi_{-}$, where $a_{2}=\min\{0,m_{1}\}$. Since $\psi(iy)-\beta\in S(\theta,0)$, we have $-{\rm Re}\psi(iy)+\beta\geq\cos\theta(|\psi(iy)|-\beta)$, and ${\rm Re}\psi(iy)\leq-c_{1}|y|^{\alpha}+c_{2}$, where $c_{1}=b\cos\theta>0,c_{2}\in\mathbb{R}$. It follows that

[TABLE]

and ($p\geq 1$)

[TABLE]

Putting $x=ty^{\alpha}$ we get for some constant $c_{3}>0$

[TABLE]

The integral converges for all $t\geq 0,p\geq 1$, and by B. Levi’s Theorem

[TABLE]

Let $\alpha<\gamma$, $p=\min\{2,(\alpha-\gamma-\delta)^{-1},(\alpha-\delta-1)/(\gamma-\alpha)\}$, $\delta\in(\alpha-\gamma-1,2\alpha-\gamma-1)$. Since

[TABLE]

we have

[TABLE]

[TABLE]

Putting $x=ty^{\alpha}$ in the second integral we get

[TABLE]

[TABLE]

The second integral in (8) converges for all $t\geq 0$ because $(\gamma+\delta)p/\alpha+1/\alpha-1>-1$ for our $p$ and $\delta$. Note that $(\gamma+\delta)/\alpha+1/\alpha p-1\geq 0$. Therefore (8) implies

[TABLE]

It follows from (7) and (9) that for our $\delta$ we have

[TABLE]

because $\gamma/\alpha+1/\alpha p+(\delta/\alpha-1)/p\leq 1$.

The case $\gamma=\alpha,\delta=\gamma-1,1<p<\min\{2,(1-\gamma)^{-1}\}$ can be examined in the same manner.

Finally since $\psi(i\lambda)-\beta\in S(\theta,0)$ for $\lambda\in\mathbb{C}$ with ${\rm Im}\lambda\geq 0$, we have for such $\lambda$ (as above)

[TABLE]

Then for $t>0$ ($\lambda=s+iy,y>0$)

[TABLE]

[TABLE]

But

[TABLE]

Furthermore

[TABLE]

Thus $F_{t}$ belongs to the Hardy class $H^{1}(\{{\rm Im}\lambda>0\})$ for all $t>0$ and therefore ${\cal F}F_{t}$ is concentrated on $\mathbb{R}_{+}$. This completes the proof.

Corollary 1. Let $\psi\in{\cal T}_{0}$, and assume that the following conditions hold:

(i)* $\psi:\Pi_{-}\to S(\theta,\beta)$ for some $\beta\geq 0,\theta\in(0,\pi/2)$;*

(ii)* $\psi(z)\asymp z^{\gamma}$ for some $\gamma\in(0,1)$ ($z\to\infty,\quad z\in\Pi_{-}$);*

(iii)* the function $y\mapsto\psi(iy)$ is differentiable for a. e. $y\in\mathbb{R}$ and*

[TABLE]

*Then $\psi\in{\cal T}_{Y}$. *

Example 1 [17]. Let $\psi(z)=c^{\alpha}-(c-z)^{\alpha},\quad\alpha\in(0,1),\quad c\geq 0$. In this case, all the conditions of Corollary 1 (and hence of Theorems 3 and 4) are clear.

Now we shall give an example of a function $\psi\in{\cal T}_{0}$ that satisfies all the conditions of Theorem 4, but conditions of the Theorem in [6] do not hold for $-\psi(-x)$.

Example 2. Let $0<\alpha<\beta<1$, and

[TABLE]

Since the summands map $\Pi_{-}$ into a sector and into a truncated sector respectively, the condition (i) of Theorem 4 holds. It is easy to verify that $\psi(z)\sim z^{\alpha}\quad{\rm as}\quad z\to\infty,z\in\Pi_{-}$, $\psi^{\prime}(iy)\sim\alpha|y|^{\alpha-1}$ as $y\to\infty$. Finally (iv) holds for $p\in(1,\min\{2,(1-\alpha)^{-1}\})$. At the same time, $-\psi(-x)$ is not regularly varying.

4. Further sufficient conditions for $\psi$ to be in ${\cal T}_{Y}$

In this section, we shall deduce further conditions from Theorem 2, that are sufficient for $\psi\in{\cal T}_{Y}$.

Theorem 5. Let $\psi\in{\cal T}_{0}$ and the function $r\mapsto\nu_{t}([r-u,r))$ is monotone decreasing on $[u,+\infty)$ $(u\geq 0)$ for each sufficiently small $t>0$. If

[TABLE]

then $\psi\in{\cal T}_{Y}$.

Proof. Let $a\in{\cal M}^{b}(\mathbb{R}_{+},\mathbb{R}_{+})$, and the function $r\mapsto a([r-u,r))$ is monotone decreasing on $[u,+\infty)$ $(u\geq 0)$. Since $\lim_{r\to+\infty}a([r-u,r))=0$ for every $u>0$, for all and $\phi\in E(\mathbb{R}_{+})$ with $\sup|\phi|\leq 1$ we find ($a(r)=a([0,r))$ for $r>0$, and $a(r)=0$ for $r\in(-\infty,0]$)

[TABLE]

[TABLE]

Thus

[TABLE]

It remains to put here $a=\nu_{t}$ and to apply Theorem 2.

Example 3 (cf. [3, Example 1]). Let for $b>0$

[TABLE]

It is well known that $d\rho(u)=e^{-bu}du$ and

[TABLE]

So $d\nu_{t}(r)=b^{t}\Gamma(t)^{-1}r^{t-1}e^{-br}dr$, and $\nu_{t}$ has monotone decreasing density for $t\in(0,1)$. Therefore

[TABLE]

[TABLE]

Thus $\psi\in{\cal T}_{Y}$ by Theorem 5.

Example 4 (cf. [13]). Let

[TABLE]

Since $\psi\in{\cal T}_{Y}$ implies $-\psi(-c)+\psi(s-c)\in{\cal T}_{Y}$ for all $c\geq 0$, one can to restrict ourselves to the case $b=1$. In this case, $\psi\in{\cal T}_{0}$ with $d\rho(u)=e^{-u}I_{0}(u)du$ (the corresponding integral representation (1) can be verified by differentiation under the integral sign), and $e^{t\psi(s)}=Lf_{t}(s)$ with $f_{t}(r)=tr^{-1}e^{-r}I_{t}(r),\quad r>0$ ($I_{t}$ denotes the Bessel function of the first kind). Hence, $d\nu_{t}(r)=f_{t}(r)dr$, and $\nu_{t}$ has monotone decreasing density for $t\in(0,1)$ (see [13]). The calculations from Example 3 in [13] show, that the conditions of Theorem 5 hold. So, $\psi\in{\cal T}_{Y}$.

Acknowledgement. The author is sincerely grateful to the referee for helpful comments and suggestions.