Marcinkiewicz integrals with rough kernels on \(L^p(1\leq p\leq 2)\)
Let \(n\geq 3\) and \(H\in L^\infty(S^{n-1})\) be a homogeneous function of positive degree \(\leq n-1\) satisfying \(\int_{S^{n-1}}H(x') dx'=0,\) and let \(\phi\) be a characteristic function on a compact set. Define \(\Omega =H\phi\), \[ F_t(x)=\int_{| x-y| \leq t} {\Omega(x-y)\over {| x-y| ^{n-1}}}f(y) dy \] and \[ \mu_\Omega(f)(x)=\left(\int^\infty_0| F_t(x)| ^2 {dt\over t^3}\right)^{1/2}, \] which is called to be the generalized Marcinkiewicz integral. The author proves that \(\mu_\Omega\) is bounded on \(L^p({\mathbf{R}}^n)\) for \(1<p\leq 2\) and of weak type \((1,1)\).
scientific article; zbMATH DE number 1286153
scientific article