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Sharp estimates for the Bochner-Riesz operator of negative order in

Authors:	Jong-Guk Bak

Institution:	Department of Mathematics, Florida State University, Tallahassee, Florida 32306--3027

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 , and $r_{+}^{\alpha }=0$ if $r\leq 0$ . We determine all pairs 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 $T^{\alpha }$ .

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