A weighted inequality for the maximal Bochner-Riesz operator on ${\bf R}\sp 2$

Abstract

For/e ?(R2), let (7£/)(£) = (1 - |£|2«2)i/(£). It is a well-known theorem of Carleson and Sjolin that T defines a bounded operator on Z,4 if a > 0. In this paper we obtain an explicit weighted inequality of the form / sup \T%f(x)\2w(x)dxti( \f\2Paw(x)dx, 0 0. This strengthens the above theorem of Carleson and Sjolin. The method gives information on the maximal operator associated to general suitably smooth radial Fourier multipliers of R2.