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

Abstract:

For $f \in \mathcal {S}({{\mathbf {R}}^2})$, let $(T_R^\alpha f)\hat \emptyset (\xi ) = (1 - |\xi {|^2}{R^2})_ + ^\alpha \hat f(\xi )$. It is a well-known theorem of Carleson and Sjölin that $T_1^\alpha$ defines a bounded operator on ${L^4}$ if $\alpha > 0$. In this paper we obtain an explicit weighted inequality of the form \[ \int {\sup \limits _{0 < R < \infty } |T_R^\alpha f(x){|^2}w(x)\;dx \leqslant \int {|f{|^2}{P_\alpha }w(x)\;dx,} } \] with ${P_\alpha }$ bounded on ${L^2}$ if $\alpha > 0$. This strengthens the above theorem of Carleson and Sjölin. The method gives information on the maximal operator associated to general suitably smooth radial Fourier multipliers of ${{\mathbf {R}}^2}$.