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arXiv:2306.05610v2 Announce Type: replace
Abstract: How large is the Bessel potential $G_{1,\mu}f$ compared to the Riesz potential $I_1f$ of a given function? We show that, for certain $f$ and $p$, \[\Vert G_{1,\mu} f\Vert_p\approx \omega(I_1f,1/\mu)_p,\] where $\omega(f,t)_p$ is the $L^p$ modulus of continuity. This estimates quantifies how the Bessel potential "fine tunes" the Riesz potential. It is obtained by studying the quotient of the two operators, $E_\mu=\frac{\sqrt{-\Delta}}{\sqrt{\mu^2-\Delta}}$, and exploiting its approximation theoretic properties. We also derive pointwise estimates for the kernel of $E_\mu$ which imply localization type results.
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