A generalization for Fourier transforms of a theorem due to Marcinkiewicz

[摘要] It is shown that the maximal operator of the Marcinkiewicz means of a tempered distribution is bounded from H(p)(R(2)) to L(p)(R(2)) for all p(0) < p infinity and, consequently, is of weak type (1, 1), where p(0) < 1. As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkiewicz and Zhizhiashviii, more exactly, the Marcinkiewicz means of a function f L(1)(R(2)) converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz means are uniformly bounded on the spaces H(p)(R(2)) and so they converge in the norm (p(0) < p < infinity). Similar results for the Riesz transforms are also given. (C) 2000 Academic Press.

[发布日期] 2000-12-15

[关键词] Hardy spaces;p-atom;atomic decomposition;interpolation;Marcinkiewicz means