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Let $M$ be a complete manifold. The heat semigroup $e^{-tL}$ is bounded on $L^p(M)$, for any $1 \leq p \leq \infty$; see this for instance.

It seems that we can deduce the time derivative of the heat semigroup $t\partial_t e^{-tL}$ is also bounded on $L^p(M)$, for any $1 \leq p \leq \infty$.

I have seen someone use the conclusion in some references several times. I try to find a reference about it. But I failed. So, does anyone know how to prove it or know a reference which gives proof of it.

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The fact that $t\partial_t e^{-tL}=-tLe^{-tL} $ is bounded in $L^p$ for $1<p<+\infty$ follows from analyticity of the heat semigroup on $L^p$; see the paper Harmonic Analysis on semigroups by Michael Cowling or Page 101 in Klaus-Jochen Engel and Rainer Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000.

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  • $\begingroup$ Do you know of any counterexamples for $p=1$? The only ones I am aware of (Ornstein-Uhlenbeck semigroups) are weighted manifolds, which behave like infinite-dimensional objects in the sense of curvature-dimension. $\endgroup$
    – MaoWao
    Commented Jul 23 at 9:51
  • $\begingroup$ Thanks, sir. So, do you mean that $t\partial_t e^{-tL}$ will not be bounded on $L^p$, when $p=1$ or $p =\infty$. $\endgroup$
    – TianS
    Commented Jul 23 at 13:06

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