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arXiv · 2410.17583

On a variant of bilinear spherical maximal function

Ankit Bhojak·Surjeet Singh Choudhary·Saurabh Shrivastava
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Abstract

We obtain $L^p-$estimates for the full and lacunary maximal functions associated to the bilinear spherical averages given by \[\mathfrak{A}_t(f_1,f_2)(x,y)=\int_{\mathbb S^{2d-1}}f_1(x+tz_1,y)f_2(x,y+tz_2)\;d\sigma(z_1,z_2),\;t>0,\] for all dimensions $d\geq1$. We show that the estimates for such operators in dimensions $d\geq2$ essentially rely on the method of slicing. The bounds for the lacunary maximal function in dimension one are more delicate and require a trilinear smoothing inequality, which is based on an appropriate sublevel set estimate in this context.

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Ankit Bhojak, Surjeet Singh Choudhary, Saurabh Shrivastava. 2024-10-23. On a variant of bilinear spherical maximal function. https://arxiv.org/abs/2410.17583

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