\(L^p\) bounds for the function of Marcinkiewicz.
Let \(\mathbb{R}^n\), \(n\geq 2\), be the \(n\)-dimensional Euclidean space and \(S^{n-1}\) be the unit sphere in \(\mathbb{R}^n\) equipped with the normalized Lebesgue measure \(d\sigma\). Let \(\Omega\) be a homogeneous function of degree 0 satisfying \(\Omega\in L^1(S^{n-1})\) and \(\int_{S^{n-1}} \Omega(y')\, d\sigma(y')= 0\), where \(y'= y/| y|\in S^{n-1}\) for any \(y\neq 0\). In this paper the authors study the Marcinkiewicz integral operator defined by \[ {\mathcal M}_\Omega f(x)= \Biggl( \int_0^\infty| K_t* f(x)|^2 t^{-1}\, dt\Biggr)^{1/2}, \] where \(K(x)=| x|^{1-n} \Omega(x) \chi_{(0,1)}(| x|)\), \(K_t(x)= t^{-n} K(x/t)\), and \(\chi_A\) denotes the characteristic function of a set \(A\). Inspired by a 1972 paper of T. Walsh the authors prove the following theorem: Theorem. If \(\Omega\in L(\log L)^{1/2} (S^{n-1})\) and \(p\in (1,\infty)\), then \({\mathcal M}_\Omega\) is bounded on \(L^p(\mathbb{R}^n)\). This result improves some previously known results.
scientific article; zbMATH DE number 1930990