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Improved bound for the bilinear Bochner-Riesz operator (1711.02425v1)
Published 7 Nov 2017 in math.CA
Abstract: We study $Lp\times Lq\to Lr$ bounds for the bilinear Bochner-Riesz operator $\mathcal{B}\alpha$, $\alpha>0$ in $\mathbb{R}d,$ $d\ge2$, which is defined by [ {\mathcal B}{\alpha}(f,g)=\iint_{\mathbb{R}d\times\mathbb{R}d} e{2\pi i x\cdot(\xi+\eta)} (1-|\xi|2-|\eta|2 ){\alpha}_+ ~ \widehat{f}(\xi)\,\widehat{g}(\eta)\,d\xi d\eta.] We make use of a decomposition which relates the estimates for $\mathcal{B}\alpha$ to those of the square function estimates for the classical Bochner-Riesz operators. In consequence, we significantly improve the previously known bounds.