Boundedness and compactness of Hankel operators on the sphere
Let \(S\) denote the unit sphere in \({\mathbb C}^n\) and \(\sigma\) denote the normalized positive, regular Borel measure on \(S\) invariant under the orthogonal group \(O(2n)\). The Cauchy projection is denoted by \(P\), i.e., \((Pf)(w)=\int_S \frac{f(\xi)}{(1-\langle w,\xi\rangle)^n}\,d\sigma(\xi)\), \(| w| <1\), the multiplication operator by \(M_f\), i.e., \(M_f(g)=fg\), and the Hankel operator by \(H_f\), i.e., \(H_f=(1-P)M_f| _{H^2(S)}\), where \(H^2(S)\) stands for the Hardy space. In this clearly written and interesting paper, the author contributes to the study of the compactness of \(H_f\). Although the compactness result given by \textit{D\,Zheng} [J.~Funct.\ Anal.\ 149, No.\,1, 1--24 (1997; Zbl 0909.47020)] which says that, for \(f\in BMO\), \(H_f\) is compact if and only if \(\lim_{| z| \to 1} \| H_f k_z\| =0\), where \(k_z\) is the normalized Cauchy kernel \(k_z(w)=\frac{(1-| z| ^2)^{n/2}}{(1-\langle w, z\rangle)^n}\), the condition appearing there does not give much information about the symbol \(f\). Due to the connection between \(H_f\) and commutators, from the fundamental result by Coifman, Rochberg and Weiss saying that the commutator \([P,M_f]\) is compact on \(L^2(S)\) if and only if \(\text{VMO}(S)\) and using the fact the \(H_f=H_{f-Pf}\), the author observes that if \(f\in L^\infty(S)\) and \(f-Pf\in \text{VMO}(S)\), then \(H_f\) is compact. In fact, he shows that the converse it true, that is, if \(H_f\) is compact then \(f-Pf\in \text{VMO}(S)\). The main result actually gives a bit more. In fact, denoting the local mean oscillation of \(f\) at the point \(\xi\) by \(\text{LMO}(f)(\xi)= \lim_{\delta\to 0} \sup_{B(\eta,r)\subset B(\xi,\delta)} \frac{1}{\sigma(B(\eta,r))}\int_{B(\eta,r)} | f-f_{B(\eta,r)}| \,d\sigma\), it is proved that if \(f\in \text{BMO}(S)\) and \(\xi\in S\) satisfies that \(\lim_{z\to \xi} \| H_fk_z\| =0\), then the local mean oscillation of \(f-Pf\) at \(\xi\) vanishes, i.e., \(\text{LMO}(f-Pf)(\xi)=0\). He also shows the estimate \(\| f-Pf\|_{\text{BMO}}\leq C\sup_{| z| <1}\| H_fk_z\|\) and the fact that if \(f\in L^2(S)\) and \(\sup_{| z| <1}\| H_fk_z\| <\infty\), then \(f-Pf\in \text{BMO}\). This allows him to conclude that if \(f\in L^2(S)\) and \(\lim_{| z| \to 1}\| H_fk_z\| =0\), then \(f-Pf\in \text{VMO}\) and consequently, \(H_f\) naturally extends to a compact operator mapping \(H^2(S)\) into \(L^2(S) \ominus H^2(S)\).
scientific article; zbMATH DE number 5306458
scientific article