Restriction theorems related to atoms
Let \(S^{n-1}\) be the unit sphere in \(\mathbb{R}^{n}\) and \(d\sigma \) be the element of the Lebesgue measure on the unit sphere. Let \(\Omega(x)/|x|^{n}\) be a homogeneous function of degree \(-n\), where \(\Omega(x)\) is assumed to have mean value zero over \(S^{n-1}\) and belong to the Hardy class \(H^{1}(S^{n-1})\). One of the main result is that if the radial function \(b\) for some \(p>1\) satisfies \[ R^{-1}\int_{0}^{R}|b(\rho )|^{p}d\rho \leq A \] for all \(R>0\), then the operator \(f*K\), where \(K= \text{p.v. } b(|x|)\Omega(x)/|x|^{{n}}\), is bounded on \(L^{2}(\mathbb{R}^{n})\). This generalizes the result in \textit{D. Fan} [Can. Math. Bull. 37, No. 2, 197-201 (1994; Zbl 0802.42014)].
scientific article; zbMATH DE number 878725