Contractive inequalities for mixed norm spaces and the Beta function

Abstract

For a wide range of pairs of mixed norm spaces such that one space is contained in another, we characterize all cases when contractive norm inequalities hold. In particular, this yields such results for many pairs of weighted Bergman spaces. Some inequalities of this type are motivated by their applications in Number Theory and in Mathematical Physics.

Introduction

Let $0<p$, $q\le \infty$, $0<a<\infty$. The mixed norm space $H(p,q,a)$ is defined as the set of all functions f that are analytic in the unit disk $\mathbb{D}$ and satisfy the condition

$${\Vert f\Vert }_{p,q,a}={(aq\underset{0}{\overset{1}{\int }}2\rho {(1-{\rho }^2)}^{aq-1}{M}_p^q(\rho ;f)d\rho )}^{1/q}<\infty ,0<q<\infty ,$$

$${\Vert f\Vert }_{p,\infty ,a}=\underset{0\le \rho <1}{\sup }{(1-{\rho }^2)}^a{M}_p(\rho ;f)<\infty ,$$

where

$$M_p(\rho ;f)={(\underset{0}{\overset{2\pi }{\int }}|f(\rho e^{i\theta }){|}^p\frac{d\theta }{2\pi })}^{1/p}$$

denotes the usual integral means of f of order p on the circle $\{z:|z|=\rho \}$. An important special case is the standard weighted Bergman space $A_{\alpha }^p=H(p,p,\frac{\alpha +1}{p})$, $-1<\alpha <\infty$, with

$${\Vert f\Vert }_{A_{\alpha }^p}={(\underset{\mathbb{D}}{\int }(\alpha +1){(1-|z{|}^2)}^{\alpha }|f(z){|}^{p}dA(z))}^{1/p}<\infty ,$$

where $dA(z)=\frac{1}{\pi }dxdy$ is the normalized Lebesgue area measure on $\mathbb{D}$. The classical Hardy space $H^p$ consisting of all functions analytic in $\mathbb{D}$ for which the finite limit

$${\Vert f\Vert }_{H^p}=\underset{r\to 1^{-}}{\lim }{M}_p(r;f)=\underset{0<r<1}{\sup }{M}_p(r;f)$$

exists can be understood as the limit case $H(p,\infty ,0)$.

Integrals as in (1) had been considered already by Hardy and Littlewood but the space $H(p,q,a)$ was only formally defined and systematically studied later, first by Hardy's student Thomas Flett [14], [15] and later in [1], [6], among many other papers. Inclusions between different mixed normed spaces in many cases have been known, the information being scattered in the literature. The classification below was completed to include the missing cases in [2], [3]. In both theorems below, we assume that $0<a$, $b<\infty$ and $0<p$, q, u, $v\le \infty$.

Theorem A

If $p\ge u$, then $H(p,q,a)\subset H(u,v,b)$ if and only if either $a<b$ or $a=b$ and $q\le v$.

Theorem B

If $p<u$, then $H(p,q,a)\subset H(u,v,b)$ if and only if either $a+\frac{1}{p}<b+\frac{1}{u}$ or $a+\frac{1}{p}=b+\frac{1}{u}$ and $q\le v$.

As a corollary for the weighted Bergman spaces, we have the well-known results (see, for example, [9, Theorem 1.3]): when $p\ge q$, the inclusion $A_{\alpha }^p\subset A_{\beta }^q$ holds only in the trivial case $p=q$ and $\alpha =\beta$ or when $\frac{\alpha +1}{p}<\frac{\beta +1}{q}$. When $p<q$, we have the inclusion $A_{\alpha }^p\subset A_{\beta }^q$ if and only if $\frac{\alpha +2}{p}<\frac{\beta +2}{q}$.

An important and natural question is: when does the contractive inequality:

$${\Vert f\Vert }_{u,v,b}\le {\Vert f\Vert }_{p,q,a}$$

hold for all $f\in H(p,q,a)$? Note that if it holds, it must be sharp since the constant function one has unit norm in all mixed norm spaces. A more specific question is when is the inclusion between two weighted Bergman spaces contractive.

As the main result of this note, we prove that, if $p\ge u$, the inclusion $H(p,q,a)\subset H(u,v,b)$ is contractive if and only if either (1) $q\le v$ and $a\le b$, or (2) $q>v$ and $aq\le bv$.

As a corollary, whenever $p\ge q$, we obtain that the contractive inequality for weighted Bergman spaces: ${\Vert f\Vert }_{A_{\beta }^q}\le {\Vert f\Vert }_{A_{\alpha }^p}$ holds if and only if $\alpha \le \beta$.

Some motivation for this study is in order. Several papers in the literature have been devoted to the study of contractive inequalities under the conditions of Theorem B. For example, a very natural question is whether the following special class of inequalities as above for weighted Bergman norms

$${\Vert f\Vert }_{A_{cp-2}^p}\le {\Vert f\Vert }_{A_{cq-2}^q},\frac{1}{c}<q<p<\infty ,$$

holds for every fixed f analytic in $\mathbb{D}$. This problem turns out to be extremely difficult but its solution would be of interest for several applications. For example, a conjecture of Lieb and Solovej [16] related to certain inequalities for entropies can be reformulated for the disk as the particular case $c=\alpha +\frac{1}{2}$ and $q=2$ of (4), that is:

$${\Vert f\Vert }_{A_{\frac{2\alpha +1}{2}p-2}^p}\le {\Vert f\Vert }_{A_{2\alpha -1}^2},2\le p<\infty ,0<\alpha <\infty .$$

This conjecture has so far been proved only in certain special cases, for example, $p=n/2$, where n is a positive integer. Different statements and proofs can be found in the papers [10], [4] and in the most recent works [16] and [5]. In the classical Fock (or Bargmann-Segal) spaces $F_{\alpha }^p$ of entire functions which are p-integrable with respect to a Gaussian measure in the plane with parameter α, there is also a contractive inequality for the spaces $F_{\alpha }^p$ and $F_{\alpha }^{\infty }$ (cf. [20, p. 40]). Contractive inequalities for generalized Fock spaces in ${\mathbb{C}}^n$ were studied in [11], again in relation to the Wehrl entropy conjecture.

Recently, in [4] a conjecture was formulated that (4) should hold for $c=1/2$; this was already implicitly suggested in [8]. This would have some important consequences. It is well known that, for $0<c<\infty$, the spaces $A_{cp-2}^p$ become larger as p increases. Also, the Hardy space $H^{1/c}$ can be viewed as the limit case of these spaces as $p\to \frac{1}{c}+$, in the sense that

$${\Vert f\Vert }_{H^{1/c}}=\underset{p\to 1/c+}{\lim }{\Vert f\Vert }_{p,cp-2}.$$

Thus, if true, inequality (4) would also readily imply the following contractive inequality mentioned in [17, Problem 2.1, p. 53]:

$${\Vert f\Vert }_{A_{cp-2}^p}\le {\Vert f\Vert }_{H^{1/c}},\frac{1}{c}<p<\infty$$

by taking the limit as $q\to 1/c+$. In [8], a conjecture was posed that

$${\Vert f\Vert }_{A_{\frac{p}{2}-2}^p}\le {\Vert f\Vert }_{H^2},2<p<\infty ,$$

which is a special case of (6) with $c=1/2$. Important related inequalities have been proved in [4]. Proving such inequality would have immediate applications to the theory of Hardy spaces of Dirichlet series and, hence, further direct consequences in Number Theory; cf. [7].

In summary, the case $p<u$ (scenario of Theorem B) seems to be very difficult even in special cases. While some partial results on these cases will be treated elsewhere, in Section 2 this note we devote our effort to characterizing completely the cases when norm inclusions are contractive under the condition of Theorem A: $p\ge u$. It is our belief that this case should also be of some interest and that it seems appropriate to carry out a systematic study of when the inclusions can be contractive in all cases.

We also list some direct consequences that might be of independent interest, such as certain inequalities for the Beta function which we have not been able to find elsewhere in the literature. This is done in Section 3.