Assume $A$ is a second order operator, generator of a positive analytic semigroup on the $L^p$-spaces with $p\in (1,\infty)$ and domain $D(A_p)=W^{2,p}$. Do we have regularity estimates $\|T_p(t)f\|_{W^{k;p}}\leq t^{-\alpha} C \|f\|_{W^{l,q}}$ for $k\leq l$,possibly non-integral and $p\leq q$? In the case $p=q$ and $k,l$ integers such a result is obtained from the real characterization of analytic semigroups (more precisely the property $\sup_{t>0} \|tAT(t)\|<\infty$). Also $L^p-L^q$-mapping properties are known. The point is to extend the results to $k,l$ non-integral.
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$\begingroup$ For the record, at time of writing this question has attracted a vote to close as "unclear what you're asking". I strongly disagree with this reason given $\endgroup$– Yemon ChoiCommented May 8, 2017 at 11:05
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