Hardy-Littlewood theorems for \(A\)-harmonic tensors
The theory of conjugate harmonic functions plays a central role in such areas of mathematics as potential theory, harmonic analysis and the theory of \(H^p\)-spaces. Conjugate harmonic functions have many analytical properties in common, among which are global \(L^p\)-integrability and Hölder continuity. These discoveries essentially began with the work of Hardy and Littlewood in the 1930's. Here we mention three specific results. Theorem A. For each \(p>0\), there is a constant \(C\) such that \[ \int_\mathbb{D}\bigl|u-u(0) \bigr|^p dx dy\leq C\int_\mathbb{D} \bigl|v-v(0) \bigr|^p dx dy \] for all analytic functions \(u+iv\) in the unit disk \(\mathbb{D}\). Theorem B. For each \(0<k\leq 1\), there is a constant \(C\) such that \[ \|u\|_{\text{Lip}_k, \mathbb{D}} \leq C\|v\|_{\text{Lip}_k,\mathbb{D}} \] for all analytic functions \(u+iv\) in \(\mathbb{D}\). Here \(\|\cdot \|_{\text{Lip}_k, \mathbb{D}}\) is the usual Lipschitz norm over \(\mathbb{D}\). Theorem C. There is a constant \(C\) such that \[ \|u\|_{\text{BMO},\mathbb{D}}\leq C\|v\|_{\text{BMO},\mathbb{D}} \] for all analytic functions \(u+iv\) in \(\mathbb{D}\). Here \(\|\cdot \|_{\text{BMO}, \mathbb{D}}\) is the usual BMO norm over \(\mathbb{D}\). Conjugate \(A\)-harmonic tensors are interesting and important generalizations of conjugate harmonic functions and \(p\)-harmonic functions, \(p>1\). The main results of this paper generalize Theorems A, B and C to conjugate \(A\)-harmonic tensors defined in domains in \(\mathbb{R}^n\) which possess an appropriate geometry. Examples show that in many ways the results are the fest possible. In particular we prove the local integrability result for conjugate \(A\)-harmonic tensors. This is the basic estimate, a result of the conjugacy, which is used throughout the paper to obtain global results. Then we investigate the global \(L^2\)-integrability of conjugate \(A\)-harmonic tensors. Here the global geometry of the domains of integration is important. The integrability exponents and the conjugacy exponents \(p\) and \(q\) directly determine each other. The first main result is a generalization of Theorem A to conjugate \(A\)-harmonic tensors in John domains in \(\mathbb{R}^n\). Next we deal with more general domains than John domains, so-called \(L^s\)-averaging domains and obtain a weaker result. Finally, we treat the local Lipschitz spaces and BMO spaces and obtain generalizations of Theorems B and C. Again the relationship between \(p,q\) and the Lipschitz exponents is seen to be the best possible. A global Lipschitz result holds in the plane for so-called \(\text{Lip}_{k,k'}\)-extension domains.
scientific article; zbMATH DE number 1361048