A Comparison of Bessel and Riesz Potentials

Abstract

How large is the Bessel potential, $G_{\hat{I}\pm ,\hat{I}¼}f$, compared to the Riesz potential, $I_{\hat{I}}\pm f$? In this paper, we show that if $I_{\hat{I}}\pm f\in L^p$ with $0<\hat{I}\pm <1$ 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 $\ddot{I}‰(f,t{)}_p$ is the $L^p$ modulus of continuity. However, if $\hat{I}\pm =p=1$, 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, $E_{\hat{I}\pm ,\hat{I}¼}:=\frac{(-\hat{I}”{)}^{\hat{I}\pm /2}}{(\hat{I}{¼}^2-\hat{I}”{)}^{\hat{I}\pm /2}}$. This operator is of independent interest due to its connection to approximation theory.

11 pages, no figures. The Abstract, Introduction and References have been updated. The exposition has been streamlined considerably and focuses on comparing the Bessel and Riesz potentials