Integral means of holomorphic functions as generic log-convex weights

TL;DR

This paper constructs holomorphic functions on the unit ball in complex space whose integral means are equivalent to given log-convex weights, extending understanding of weighted spaces of holomorphic functions.

Contribution

It introduces a method to realize any log-convex weight as the integral mean of a holomorphic function on the unit ball in complex space.

Findings

Constructed functions with prescribed integral mean behavior

Established equivalence between integral means and log-convex weights

Extended results to volume integral means

Abstract

Let $\mathcal{H}ol(B_d)$ denote the space of holomorphic functions on the unit ball $B_d$ of $\mathbb{C}^d$, $d\ge 1$. Given a log-convex strictly positive weight $w(r)$ on $[0,1)$, we construct a function $f\in\mathcal{H}ol(B_d)$ such that the standard integral means $M_p(f, r)$ and $w(r)$ are equivalent for any $0<p\le\infty$. Also, we obtain similar results related to volume integral means.

Equations

$$M_p(f,r)={\left({\int }_{\partial B_d}|f(r\zeta ){|}^{p}d{\sigma }_d(\zeta )\right)}^{\frac{1}{p}},0\le r<1,$$

$$M_{\infty }(f,r)=\sup \{|f(z)|:|z|=r\},0\le r<1.$$

$$C_{1}u(x)\le v(x)\le C_{2}u(x),x\in X.$$

$$M_p(f,r)\asymp w(r),0\le r<1,$$

$${\left(\frac{1}{{\nu }_d(r{B}_d)}{\int }_{r{B}_d}|f(z){|}^{q}d{\nu }_d(z)\right)}^{\frac{1}{q}}\asymp w(r),0<r<1,$$

$$n_k$$

$$\sum _{k=1}^{\infty }{a}_k{r}^{n_k}$$

$$|g_1(r\zeta )|+|g_2(r\zeta )|$$

$$g_1(z)=\sum _{j=1}^{\infty }{a}_{2j-1}{z}^{n_{2j-1}},g_2(z)=\sum _{j=1}^{\infty }{a}_{2j}{z}^{n_{2j}},z\in \mathbb{D}.$$

$$|g_1(r\zeta ){|}^2+|g_2(r\zeta ){|}^2\ge C_3{w}^2(r),r_0\le r<1,\zeta \in \mathbb{T}.$$

$$\sum _{k=1}^{\infty }{a}_k^2{r}^{2n_k}\ge C_3{w}^2(r),r_0\le r<1.$$

$$1+\sum _{k=1}^{\infty }{a}_k^2{r}^{2n_k}\ge C_4{w}^2(r),0\le r<1.$$

$$M_2(f,r)\ge w(r),0\le r<1,$$

$$\sqrt{C_4}f(z)=1+\sum _{k=1}^{\infty }{a}_k{z}^{n_k},z\in \mathbb{D}.$$

$$|f(r\zeta )|\le C_{0}w(r),0\le r<1,\zeta \in \mathbb{T}.$$

$${\sigma }_1\left\{\zeta \in \mathbb{T}:|f(r\zeta )|\ge \frac{w(r)}{2}\right\}\ge \frac{1}{2C_0^2}.$$

$$F(z)=1+\sum _{k=1}^{\infty }{a}_k{W}_k(z),z\in B_d.$$

$$M_p(f,r)\asymp w(r),0\le r<1.$$

$$M_{q,u}(f,r)$$

$$M_{q,u}(f,0)$$

$$M_{q,\frac{1}{u}}(f,r)\asymp w(r),0\le r<1.$$

$$w^q(t)\asymp \sum _{k=0}^{\infty }{a}_k{t}^k,0\le t<1.$$

$${\phi }^q(t)=\sum _{k=0}^{\infty }(k+2d)a_k{t}^k,0\le t<1.$$

$${\int }_{\partial B_d}|f(t\zeta ){|}^{q}d{\sigma }_d(\zeta )\asymp {\phi }^q(t)u(t),0\le t<1,$$

$$\frac{t^{2d-1}}{u(t)}{\int }_{\partial B_d}|f(t\zeta ){|}^{q}d{\sigma }_d(\zeta )\asymp \sum _{k=0}^{\infty }(k+2d)a_k{t}^{k+2d-1},0\le t<1.$$

$$M_{q,\frac{1}{u}}^q(f,r)$$

$$\asymp \sum _{k=0}^{\infty }{a}_k{r}^k,$$

$$\asymp w^q(r),0\le r<1,$$

$${\mathfrak{M}}_q(f,r)$$

$${\mathfrak{M}}_q(f,0)$$

$${\mathfrak{M}}_q(f,r)={\left({\int }_{B_d}|f(rz){|}^{q}d{\nu }_d(z)\right)}^{\frac{1}{q}},0\le r<1.$$

$$M_q^q(f,t)\asymp \sum _{k=0}^{\infty }{a}_k{t}^k,0\le t<1.$$

$${\mathfrak{M}}_q^q(f,r)$$

Full text

The author was supported by the Russian Science Foundation (grant No. 14-41-00010).

1. Introduction

Let $\mathcal{H}ol(B_{d})$ denote the space of holomorphic functions on the unit ball $B_{d}$ of ${\mathbb{C}}^{d}$, $d\geq 1$. For $0<p<\infty$ and $f\in\mathcal{H}ol(B_{d})$, the standard integral means $M_{p}(f,r)$ are defined as

[TABLE]

where $\sigma_{d}$ denotes the normalized Lebesgue measure on the unit sphere $\partial B_{d}$. For $p=\infty$, put

[TABLE]

A function $w:[0,1)\to(0,+\infty)$ is called a weight if $w$ is continuous and non-decreasing. A weight $w$ is said to be log-convex if $\log w(r)$ is a convex function of $\log r$, $0<r<1$. It is known that $M_{p}(f,r)$, $0\leq r<1$, is a log-convex weight for any $f\in\mathcal{H}ol(B_{d})$, $f(0)\neq 0$, $d\geq 1$, $0<p\leq\infty$. In fact, for $d=1$, this result constitutes the classical Hardy convexity theorem (see [2]). The corresponding proofs are extendable to all dimensions $d$, $d\geq 2$ (see, for example [7, Lemma 1]).

In the present paper, for each $0<p\leq\infty$, we show that the functions $M_{p}(f,r)$, $f\in\mathcal{H}ol(B_{d})$, $f(0)\neq 0$, are generic log-convex weights in the sense of the following equivalence:

Let $u,v:X\to(0,+\infty)$. We say that $u$ and $v$ are equivalent ($u\asymp v$, in brief) if there exist constants $C_{1},C_{2}>0$ such that

[TABLE]

Theorem 1.1**.**

Let $d\geq 1$ and let $w:[0,1)\to(0,+\infty)$ be a log-convex weight. There exists $f\in\mathcal{H}ol(B_{d})$ such that

[TABLE]

for each $0<p\leq\infty$.

Also, we consider volume integral means for $0<q<\infty$. The logarithmic convexity properties for such integral means have been recently investigated in a series of papers (see, for example, [5, 6, 7]). Applying Theorem 1.1, we obtain, in particular, the following result.

Corollary 1.2**.**

Let $d\geq 1$, $0<q<\infty$ and let $w:[0,1)\to(0,+\infty)$ be a weight. The following properties are equivalent:

• (i)

$w(r)$* is equivalent to a log-convex weight on $[0,1)$;*

• (ii)

there exists $f\in\mathcal{H}ol(B_{d})$ such that

[TABLE]

where $\nu_{d}$ denotes the normalized volume measure on $B_{d}$.

Organization of the paper

Section 2 is devoted to the proof of Theorem 1.1. Corollary 1.2 and other results related to volume integral means are obtained in Section 3.

2. Proof of Theorem 1.1

Put $\mathbb{D}=B_{1}$ and $\mathbb{T}=\partial\mathbb{D}$. For a log-convex weight $w$ on $[0,1)$, Theorem 1.2 from [1] provides functions $f_{1},f_{2}\in\mathcal{H}ol(\mathbb{D})$ such that $|f_{1}(z)|+|f_{2}(z)|\asymp w(|z|)$, $z\in\mathbb{D}$. These functions are almost sufficient for a proof of Theorem 1.1 with $d=1$. However, we will need additional technical information contained in [1]. Namely, applying Lemma 2.2 from [1] and arguing as in the proof of Theorem 1.2 from [1], we obtain the following lemma.

Lemma 2.1**.**

Let $w$ be a log-convex weight on $[0,1)$. There exist $a_{k}>0$, $n_{k}\in\mathbb{N}$, $k=1,2,\dots$, and constants $r_{0}\in(\frac{9}{10},1)$, $C_{1},C_{2}>0$ with the following properties:

[TABLE]

where

[TABLE]

Proof of Theorem 1.1.

We are given a log-convex weight $w$ on $[0,1)$. First, assume that $d=1$. Let $a_{k}$ and $n_{k}$, $k=1,2,\dots$, $g_{1}$ and $g_{2}$ be those provided by Lemma 2.1. By (2.3),

[TABLE]

Using (2.1) and integrating the above inequality with respect to Lebesgue measure $\sigma_{1}$ on $\mathbb{T}$, we obtain

[TABLE]

Therefore,

[TABLE]

So, by (2.1), we have

[TABLE]

where

[TABLE]

Also, (2.2) guarantees that

[TABLE]

Hence, $M_{2}(f,r)\leq M_{\infty}(f,r)\leq Cw(r)$, $0\leq r<1$. Combining these estimates and (2.4), we conclude that $M_{2}(f,r)\asymp M_{\infty}(f,r)\asymp w(r)$. Thus, $M_{p}(f,r)\asymp w(r)$, $0\leq r<1$, for any $2\leq p\leq\infty$.

Also, we claim that $M_{p}(f,r)\asymp w(r)$ for any $0<p<2$. Indeed, (2.4) and (2.5) guarantee that

[TABLE]

Therefore, $M_{\infty}(f,r)\geq M_{p}(f,r)\geq C_{p}w(r)$, $0\leq r<1$. So, the proof of the theorem is finished for $d=1$.

Now, assume that $d\geq 2$. Let $W_{k}$, $k=1,2,\dots$, be a Ryll–Wojtaszczyk sequence (see [3]). By definition, $W_{k}$ is a holomorphic homogeneous polynomial of degree $k$, $\|W_{k}\|_{L^{\infty}(\partial B_{d})}=1$ and $\|W_{k}\|_{L^{2}(\partial B_{d})}\geq\delta$ for a constant $\delta>0$ which does not depend on $k$. Put

[TABLE]

Clearly, (2.2) guarantees that $|F(r\zeta)|\leq Cw(r)$, $0\leq r<1$, $\zeta\in\partial B_{d}$. Also, the polynomials $W_{k}$, $k=1,2,\dots$, are mutually orthogonal in $L^{2}(\partial B_{d})$; hence, $M_{2}(F,r)\geq C(\delta)w(r)$, $0\leq r<1$. So, arguing as in the case $d=1$, we conclude that $M_{p}(F,r)\asymp w(r)$ for any $0<p\leq\infty$, as required. ∎

As indicated in the introduction, for any $f\in\mathcal{H}ol(B_{d})$, the function $M_{p}(f,r)$ is log-convex; hence, Theorem 1.1 implies the following analog of Corollary 1.2.

Corollary 2.2**.**

Let $d\geq 1$, $0<p\leq\infty$ and let $w:[0,1)\to(0,+\infty)$ be a weight. The following properties are equivalent:

• (i)

$w(r)$* is equivalent to a log-convex weight on $[0,1)$;*

• (ii)

there exists $f\in\mathcal{H}ol(B_{d})$ such that

[TABLE]

3. Volume integral means

In this section, we consider integral means based on volume integrals. Recall that $\nu_{d}$ denotes the normalized volume measure on the unit ball $B_{d}$. For $f\in\mathcal{H}ol(B_{d})$, $0<q<\infty$ and a continuous function $u:[0,1)\to(0,+\infty)$, define

[TABLE]

Proposition 3.1**.**

Let $0<q<\infty$ and let $u,w:[0,1)\to(0,+\infty)$ be log-convex weights. There exists $f\in\mathcal{H}ol(B_{d})$ such that

[TABLE]

Proof.

By Theorem 1.1 with $p=2$, there exist $a_{k}\geq 0$, $k=0,1,\dots$, such that

[TABLE]

Let

[TABLE]

The functions $\varphi^{q}(t)$ and $\varphi(t)$ are correctly defined log-convex weights on $[0,1)$. Hence, $\varphi(t)u^{\frac{1}{q}}(t)$ is a log-convex weight as the product of two log-convex weights. By Theorem 1.1, there exists $f\in\mathcal{H}ol(B_{d})$ such that

[TABLE]

or, equivalently,

[TABLE]

Representing $M^{q}_{q,\frac{1}{u}}(f,r)$ in polar coordinates and integrating the above estimates with respect to $t$, we obtain

[TABLE]

as required. ∎

Clearly, Proposition 3.1 is of special interest if $M_{q,\frac{1}{u}}(f,r)$ is log-convex or equivalent to a log-convex function for any $f\in\mathcal{H}ol(B_{d})$. Also, we have to prove Corollary 1.2. So, assume that $u\equiv 1$ and define

[TABLE]

where $0<q<\infty$.

Proof of Corollary 1.2.

By Proposition 3.1, (i) implies (ii). To prove the reverse implication, assume that $w(t)$ is a weight on $[0,1)$ and $w(r)\asymp\mathfrak{M}_{q}(f,r)$ for some $f\in\mathcal{H}ol(B_{d})$, $f(0)\neq 0$.

If $d=1$ and $0<q<\infty$, then $\mathfrak{M}_{q}(f,r)$ is log-convex by Theorem 1 from [5]. So, (ii) implies (i) for $d=1$. The function $\mathfrak{M}_{q}(f,r)$ is also log-convex if $1\leq q<\infty$ and $d\geq 2$. Indeed, we have

[TABLE]

Thus, Taylor’s Banach space method applies (see [4, Theorem 3.3]).

Now, assume that $d\geq 2$ and $0<q<1$. The function $M^{q}_{q}(f,t)$ is a log-convex weight. Hence, by Theorem 1.1 with $p=2$, there exist $a_{k}\geq 0$, $k=0,1,\dots$, such that

[TABLE]

Thus,

[TABLE]

In other words, $\mathfrak{M}_{q}(f,r)$ is equivalent to a log-convex weight on $[0,1)$. So, (ii) implies (i) for all $d\geq 1$ and $0<q<\infty$. The proof of the corollary is finished. ∎

For $\alpha>0$, Proposition 3.1 also applies to the following integral means:

[TABLE]

However, in general, the above integral means are not log-convex.