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

Author: Jong-Guk Bak
Journal: Proc. Amer. Math. Soc. 125 (1997), 1977-1986
MSC (1991): Primary 42B15
MathSciNet review: 1371114
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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 $T^{\alpha }$.

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Additional Information

Jong-Guk Bak
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306–3027
Address at time of publication: Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Korea

Received by editor(s): October 3, 1995
Received by editor(s) in revised form: December 19, 1995
Additional Notes: The author’s research was partially supported by a grant from the Pohang University of Science and Technology
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1997 American Mathematical Society