Lévy-type operators with kernels comparable to log(I-Δ) produce strong hypercontractivity and eventual boundedness with differentiability gains, but not supercontractivity, while more singular kernels smooth instantly and less singular ones do not regularize.
W.An introduction to the theory of large deviations
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Gradual smoothing: strong hypercontractivity and logarithmic Sobolev inequalities
Lévy-type operators with kernels comparable to log(I-Δ) produce strong hypercontractivity and eventual boundedness with differentiability gains, but not supercontractivity, while more singular kernels smooth instantly and less singular ones do not regularize.