Sharp estimates for the Bochner-Riesz operator of negative order in $\mathbf {R}^{2}$

The Bochner-Riesz operator $T^{\alpha }$ on $\mathbf {R}^{n}$ of order $\alpha$ is defined by

where $\;\widehat {}\;$ denotes the Fourier transform and $r_{+}^{\alpha } = r^{\alpha }$ if $r>0$, and $r_{+}^{\alpha }=0$ if $r\leq 0$. We determine all pairs $(p,q)$ such that $T^{\alpha }$ on $\mathbf {R}^{2}$ of negative order is bounded from $L^{p}(\mathbf {R}^{2})$ to $L^{q}(\mathbf {R}^{2})$. To be more precise, we prove that for $0<\delta < 3/2$ the estimate $\Vert T^{-\delta }f \Vert _{L^{q}(\mathbf {R}^{2})} \leq C \Vert f \Vert _{L^{p}(\mathbf {R}^{2})}$ holds if and only if $(1/p,1/q) \in \Delta ^{-\delta }$, where

We also obtain some weak-type results for $T^{\alpha }$.