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The Bessel potential is defined by $(I-\Delta)^{-\alpha/2}$ and it has an integral kernel $K$ such that $(I-\Delta)^{-\alpha/2}f = K*f$. Is there a way to generalize this to something like $$ (g-\Delta)^{-\alpha/2}f = \tilde{K}* f$$ where $g$ is some given function? And if so, what are the conditions on $g$?

Jakob Elias
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    Since convolution operators are translation-invariant, heuristics would suggest that the function $g$ has to be translation-invariant for this to work out. But a translation-invariant function is just a constant function, so that would be a pretty restrictive condition. – Jason Jan 07 '21 at 06:42

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