A necessary condition for Calderón-Zygmund singular integral operators
The author treats the following Calderón-Zygmund singular integral operator \[ T_\Omega (f)(x)=\text{p.v. }\int_{\mathbb R^n}f(x-y)\Omega(y/|y|)|y|^{-n} dy, \] where \(\Omega\) is an integrable function with mean value zero on the unit sphere \(S^{n-1}\). In 1977, W. Connett showed that if \(\Omega\) is in the Hardy space \(H^1(S^{n-1})\), then \(T_\Omega\) is \(L^p(\mathbb R^n)\) bounded for \(1<p<\infty\). The author gives the following partial reverse of this: if \(T_\Omega\) is bounded on the Hardy space \(H^1(\mathbb R^n)\), then the restriction of \(\Omega\) to \(S^{n-1}\) is a function in \(H^1(S^{n-1})\). To prove this, he uses the F. and M. Riesz theorem on \(S^{n-1}\) and spherical harmonics expansion.
scientific article; zbMATH DE number 1340857