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arXiv · 2306.05610

A Comparison of Bessel and Riesz Potentials

Ikemefuna Agbanusi
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Abstract

How large is the Bessel potential, $G_{α,μ}f$, compared to the Riesz potential, $I_αf$? In this paper, we show that if $I_αf\in L^p$ with $0<α<1$ and $p>1$, then the following interpolation bound holds: \[\Vert G_{α,μ}f\Vert_p\leq C(ω(I_αf,1/μ)_p)^α\cdot\Vert I_αf\Vert^{1-α}_p.\] Here $ω(f,t)_p$ is the $L^p$ modulus of continuity. However, if $α=p=1$, we obtain the ``$L\log L$" type result \[\Vert G_{1,μ}f\Vert_1\leq Bω(I_1f,1/μ)_1|\logω(I_1f,1/μ)_1|.\] These and other estimates are obtained by studying the quotient of the two operators, $E_{α,μ}:=\frac{(-Δ)^{α/2}}{(μ^2-Δ)^{α/2}}$. This operator is of independent interest due to its connection to approximation theory.

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Ikemefuna Agbanusi. 2025-06-03. A Comparison of Bessel and Riesz Potentials. https://arxiv.org/abs/2306.05610

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