A Comparison of Bessel and Riesz Potentials

How large is the Bessel potential, $G_{\alpha ,\mu }f$, compared to the Riesz potential, $I_{\alpha }f$? In this paper, we show that if $I_{\alpha }f\in L^p$ with $0<\alpha <1$ and $p>1$, then the following interpolation bound holds: $$\mathrm{\parallel }{G}_{\alpha ,\mu }f{\mathrm{\parallel }}_p\le C(\omega (I_{\alpha }f,1\mathrm{/}\mu {)}_p{)}^{\alpha }\cdot \mathrm{\parallel }{I}_{\alpha }f{\mathrm{\parallel }}_p^{1-\alpha }\mathrm{.}$$ Here $\omega (f,t{)}_p$ is the $L^p$ modulus of continuity. However, if $\alpha =p=1$, we obtain the ``$L\log L$" type result $$\mathrm{\parallel }{G}_{1,\mu }f{\mathrm{\parallel }}_1\le B\omega (I_{1}f,1\mathrm{/}\mu {)}_1\mathrm{\mid }\log \omega (I_{1}f,1\mathrm{/}\mu {)}_1\mathrm{\mid }\mathrm{.}$$ These and other estimates are obtained by studying the quotient of the two operators, $E_{\alpha ,\mu }:=\frac{(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}}{({\mu }^2-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}}$. This operator is of independent interest due to its connection to approximation theory.