Weak type estimates for Bochner-Riesz spherical summation multipliers

Abstract:

We consider the Bochner-Riesz multiplier \[ \widehat {{T_\delta }f}(\xi ) = {(1 - {\left | \xi \right |^2})^\delta } + \hat f(\xi ),\qquad \delta > 0,\] where $\widehat {}$ denotes the Fourier transform. It is shown that the multiplier operator ${T_\delta }$ is weak type $({p_0}, {p_0})$ acting on ${L^{p0}}({{\mathbf {R}}^n})$ radial functions, where ${p_0}$ is the critical value $2n/(n + 1 + 2\delta )$.