Commutators on half-spaces (Q1423751)

Let \({\mathbb R}_+^n\) be the upper half-space, \(n\geq 2\). The author studies the boundedness and compactness in the Lebesgue spaces \(L^p({\mathbb R}_+^n)\) of the commutator \([M_f,I_k]=M_fI_k-I_kM_f\) where \(M_fg=fg\) and \(I_kg(x)=\int_{{\mathbb R}_+^n}k(x,y)g(y)dy\). Let BMO\(^p\) and VMO\(^p\) be the spaces of functions of bounded and vanishing mean oscillation on \({\mathbb R}_+^n\), respectively (the precise meaning of these terms can be found in the paper under review). From a more general result of \textit{F. Beatrous} and \textit{S.-Y. Li} [J. Funct. Anal. 111, 350--379 (1993; Zbl 0793.47022)] it follows that if \(f\in \text{BMO}^p\) and \(k\) satisfies \(| k(x,y)| \leq c| x-\overline{y}| ^{-n}\) for all \(x,y\in{\mathbb R}_+^n\), then \([M_f,I_k]\) is bounded on \(L^p({\mathbb R}_+^n)\) for \(p\in[2,\infty)\). The author proves that if, moreover, \(f\in \text{VMO}^p\), then \([M_f,I_k]\) is compact on \(L^p({\mathbb R}_+^n)\) for \(p\in[2,\infty)\). Further, he imposes two additional conditions on \(k\) and under those conditions proves that the boundedness (compactness) of \([M_f,I_k]\) on \(L^p({\mathbb R}_+^n)\) implies \(f\in \text{BMO}^p\) (resp., \(f\in \text{VMO}^p\)). The reproducing kernel of the harmonic Bergman space of \({\mathbb R}_+^n\) is shown to satisfy all the required assumptions, so the author's results are applicable in this case.