A note on Marcinkiewicz integral operators.
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 \[ {\mathcal M}_\Omega f(z)= \Biggl( \int_0^\infty| K_t* f(x)|^2 t^{-1}\, dt\Biggr)^{1/2}, \] 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 \(A\). Let \(B_q^{\mu,\nu} (S^{n-1})\) be the block space defined in the book by \textit{S. Z. Lu, M. H. Taibleson} and \textit{G. Weiss} [``Spaces generated by blocks'', Beijing Normal University Press, Beijing (1989), per bibl.]. The main Theorem of this paper is the following theorem: (a) If \(\Omega\) belongs to the block space \(B_q^{0,-1/2} (S^{n-1})\), \(q>1\), then \(\|{\mathcal M}_\Omega f\|_p\leq C\| f\|_p\) for any \(p\in (1,\infty)\). (b) There exists an \(\Omega\) which lies in \(B_q^{0,\nu} (S^{n-1})\) for all \(-1< \nu< -1/2\) such that \({\mathfrak M}_\Omega\) is not bounded on \(L^2(\mathbb{R}^n)\).
scientific article; zbMATH DE number 1957243