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: $$\Vert G_{\alpha,\mu}f\Vert_p\leq C(\omega(I_\alpha f,1/\mu)_p)^\alpha\cdot\Vert I_\alpha f\Vert^{1-\alpha}_p.$$ 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 $$\Vert G_{1,\mu}f\Vert_1\leq B\omega(I_1f,1/\mu)_1|\log\omega(I_1f,1/\mu)_1|.$$ These and other estimates are obtained by studying the quotient of the two operators, $E_{\alpha,\mu}:=\frac{(-\Delta)^{\alpha/2}}{(\mu^2-\Delta)^{\alpha/2}}$. This operator is of independent interest due to its connection to approximation theory.