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

Abstract:

The Bochner-Riesz operator $T^{\alpha }$ on $\mathbf {R}^{n}$ of order $\alpha$ is defined by \begin{equation*}(T^{\alpha } f)\;\widehat {}\;(\xi ) = {\frac {(1-|\xi |^{2})_{+}^{\alpha } }{\Gamma (\alpha +1)}} \hat {f}(\xi ) \end{equation*} 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 \begin{equation*}\Delta ^{-\delta }=\bigg \{ \bigg ({\frac {1}{p}},{\frac {1}{q}} \bigg )\in [0,1]\times [0,1]\colon \;\; {\frac {1}{p}}-{\frac {1}{q}} \geq {\frac {2\delta }{3}}, \;\; {\frac {1}{p}}> {\frac {1}{4}} + {\frac {\delta }{2}} , \;\; {\frac {1}{q}} < {\frac {3}{4}} - {\frac {\delta }{2}} \bigg \} .\end{equation*} We also obtain some weak-type results for $T^{\alpha }$.