Zygmund’s theorem for harmonic quasiregular mappings

Suppose ${\mathbb{D}}\coloneqq\{z\in{\mathbb{C}}:\,|z|<1\}$ is the open unit disk in the complex plane ${\mathbb{C}}$, and $\mathbb{T}\coloneqq\{z\in{\mathbb{C}}:|z|=1\}$ is the unit circle. For a function $f$ analytic in ${\mathbb{D}}$, the integral means of $f$ are defined as

and

The function $f$ is said to be in the Hardy space $H^{p}$ $(0<p\leq\infty)$ if

Every function $f\in H^{p}$ has “nice” boundary behaviour, in the sense that the radial limit

exists in almost every direction, and is of class $L^{p}\coloneqq L^{p}({\mathbb{T}})$. Detailed surveys on Hardy spaces can be found, e.g., in the books of Duren [7] and Koosis [11]. Throughout this paper, we follow notations from [7].

A real-valued function $u(x,y)$, twice continuously differentiable in ${\mathbb{D}}$, is called harmonic if it satisfies the Laplace equation

For a complex-valued function $f=u+iv$, the complex partial derivatives have the form

where $z=x+iy\in{\mathbb{C}}$. In view of this, it is easy to see that the Laplacian operator $\Delta$ can also be written as

A function $f=u+iv$ is harmonic in ${\mathbb{D}}$, if $u$ and $v$ are real-valued harmonic functions in ${\mathbb{D}}$. Every such function has a representation $f=h+\overline{g}$, where $h$ and $g$ are analytic in $\mathbb{D}$. This representation is unique up to an additive constant. In this paper we usually assume $g(0)=0$, unless mentioned otherwise. Similar to the $H^{p}$ spaces, the harmonic Hardy spaces $h^{p}$ are defined as the class of functions $f$, harmonic in ${\mathbb{D}}$, such that

We refer to the books of Duren [6] for the theory of planar harmonic mappings and Pavlović [15] for a concise survey on $h^{p}$ spaces.

Given $K\geq 1$, a sense-preserving harmonic function $f=h+\bar{g}$ is said to be $K$-quasiregular if its complex dilatation $\omega=g^{\prime}/h^{\prime}$ satisfies the inequality

where

We say that $f$ is $K$-quasiconformal if $f$ is $K$-quasiregular and homeomorphic in ${\mathbb{D}}$. One can find the $H^{p}$-theory for quasiregular and quasiconformal mappings, for example, in the survey of Astala and Koskela [2] and the paper of Adamowicz and González [1]. Some recent results on the Hardy space of harmonic quasiconformal mappings are obtained in [18].

Given a real-valued harmonic function $u$ in ${\mathbb{D}}$, let $v$ be its harmonic conjugate, normalized by the condition $v(0)=0$. It is a natural question that if $u$ has a certain property, whether so does $v$. In the context of boundary behaviour, M. Riesz established this principle in a remarkably precise form:

Curiously, the theorem fails for the cases $p=1$ and $p=\infty$, examples are given in [7, p. 56]. Although the harmonic conjugate of an $h^{1}$ function need not be in $h^{1}$, Kolmogorov crucially pointed out that it does belong to $h^{p}$ for all $p<1$. The result, in a stronger form, is as follows.

It follows from the theorems of Riesz and Kolmogorov that the condition $u\in h^{1}$ is not enough to ensure $v\in h^{1}$, but the slightly stronger hypothesis $u\in h^{p}$ (for some $p>1$) is sufficient. Therefore, one naturally asks for the “minimal” growth restriction on $u$ which should imply $v\in h^{1}$. As established by Zygmund, such a condition is the boundedness of the integrals

where $\log^{+}x=\max\{\log x,\,0\}$. The class of such harmonic functions $u$ in ${\mathbb{D}}$ is denoted by ${{\bf h}\log^{+}{\bf h}}$. The following is the precise statement of Zygmund’s theorem.

It should be noted that $h^{p}\subset{{\bf h}\log^{+}{\bf h}}$ for every $p>1$, so that Zygmund’s theorem is indeed best possible. We refer the reader to the paper of Pichorides [17] for the sharp constants in the Riesz, Kolmogorov, and Zygmund theorems.

Suppose $f=u+iv$ is a harmonic function in ${\mathbb{D}}$. If $u\in h^{p}$, $p>1$, then the imaginary part $v$ does not necessarily belong to the space $h^{p}$, i.e., Riesz theorem is not true for harmonic functions. The question arises: Under what additional condition(s) does a harmonic analogue of the Riesz theorem hold? Recently, Liu and Zhu established in their breakthrough paper [12] that such a condition is the quasiregularity of $f$.

It is important to note that the assumption “$f$ is harmonic in ${\mathbb{D}}$” cannot be removed. There exists a quasiconformal mapping for which the first coordinate function belongs to the Hardy space, but the whole function does not (see [2, p. 36]).

Later, Kalaj [10] refined Theorem D by removing the condition $u\geq 0$. Furthermore, he proved a Kolmogorov type theorem for harmonic quasiregular mappings in ${\mathbb{D}}$, thereby making considerable progress in this direction.

Kalaj also points out that the second inequality is new even for analytic functions. Several other Riesz type theorems for harmonic functions can be found in the recent works of Kalaj [8, 9], Chen and Hamada [3], Melentijević [13], and Melentijević and Marković [14]. Furthermore, Chen and Huang [4] have proved Riesz theorem for pluriharmonic functions where the assumption $u\geq 0$ is dropped as well.

In this paper, we make further advance along this line and establish a Zygmund type theorem for harmonic quasiregular mappings. For a harmonic $K$-quasiregular mapping $f=u+iv$ in ${\mathbb{D}}$, we find out the growth condition on $u$ that would imply $f\in h^{1}$ (and $v\in h^{1}$). The following is the central result of this paper.

Theorem 1 is the best possible, in the sense that the growth restriction imposed on $u$ cannot be weakened. This can be seen from the following partial converse, which surprisingly does not require the quasiregularity hypothesis.

The proofs of the main theorems are presented in Section 2. In Section 3, we study the harmonic analogue of a well-known result of Hardy and Littlewood as an application of the classical Riesz and Kolmogorov theorems.

Our technique is based on a method of Green’s theorem, which has also been used in [10] and [12].

First assume that $u\geq 1$, so that $\log^{+}|u(r{e^{i\theta}})|=\log u(r{e^{i\theta}})$. Let us write $f=h+\bar{g}$, and let $\omega(z)=g^{\prime}(z)/h^{\prime}(z)$ be its complex dilatation. An elementary computation gives

Suppose $F=h+g$. Then

so that

This implies

Therefore, we have

as $|\omega(z)|\leq k$. On the other hand, one finds

Thus, it follows that

These inequalities imply

and therefore,

The next step is to apply Green’s theorem in the form

where $z=x+iy$. Then, it follows from (2) that

Now, integrating from $0$ to $r$ and recalling that $v(0)=0$, we get

Therefore

which is the desired inequality.

For the case $u\leq-1$, let us write $F=e^{i\pi}f$. We see that $F$ is harmonic $K$-quasiregular in ${\mathbb{D}}$ with $u_{F}=\real F=-u\geq 1$, and $v_{F}={\operatorname{Im}\,}F=-v$ satisfying $v_{F}(0)=0$. Then, from what we have already proved, it follows that

Since $M_{1}(r,f)=M_{1}(r,F)$ and $u_{F}=|u|$, the proof is complete.∎

The proof of the converse (Theorem 2) relies on the following lemma, which is of some independent interest.

Suppose $f=u+iv$. Without any loss of generality, we may assume that $u\geq 1$, as adding a constant to $u$ does not affect $v$ or the condition $f\in h^{1}$. Since ${\operatorname{Im}\,}h$ is non-vanishing, by Lemma 1 we have $F=h+g\in H^{1}$. Let us write

where $z=r{e^{i\theta}}$, $R=R(r,\theta)$ and $\Phi=\Phi(r,\theta)$. Then

Now, the function $F(z)\log F(z)$ is analytic in ${\mathbb{D}}$, with

We apply mean value theorem to this real part to obtain

since $g(0)=0$. It is to be noted that ${\operatorname{Im}\,}h\neq 0$ implies $h\neq 0$, so that $\log h(0)$ is well-defined. Thus we have

It follows that

which is finite since $F\in H^{1}$. Therefore, $u\in{{\bf h}\log^{+}{\bf h}}$ and the proof is complete.∎

The first-named author thanks Petar Melentijević for helpful discussions on the space $h^{1}$ that led to the classical theory of Zygmund. The research was partially supported by the Natural Science Foundation of Guangdong Province (Grant no. 2024A1515010467) and the Li Ka Shing Foundation STU-GTIIT Joint Research Grant (Grant no. 2024LKSFG06).