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