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What is the actual inequality that holds for the heat semigroup between fractional Sobolev space $W^{2\alpha,p}$ and classical Lebesgue space $L^q$?

I am trying to derive an inequality $$ \lvert\lvert e^{t\Delta}f \rvert\rvert_{W^{2\alpha,p}} \leq \frac{C}{t^\beta} \lvert\lvert f \rvert\rvert_{L^q} $$ but i cannot manage to find the right value of $\beta$.

I am trying to write the heat semigroup as a convolution $$ e^{t\Delta}f = K_t*f $$ where $K_t$ is the heat kernel, and use Holder inequality for convolutions, but I am stuck when computing $$ \lvert\lvert (I-\Delta)^\alpha K_t \rvert\rvert_{L^r} $$ where $\frac{1}{q}+\frac{1}{r} = \frac{1}{p}+1$.

Any help is appreciated.

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  • $\begingroup$ Do you want an estimate for all $t$ or only for small enough? $\endgroup$
    – Andrew
    Commented Mar 18, 2019 at 7:48

1 Answer 1

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This isn't a complete solution/answer, just a couple estimates that may be useful.

I'm going to try to look into this more, but I do know a couple of estimates that may be of help. I considered posting this as a comment, but it seemed a bit long and perhaps poorly suited.

We have the following estimates for $1\leq r \leq p \leq \infty$ and $t>0$: $$\| e^{t\Delta}f\|_{L_x^p} \lesssim \frac{1}{t^{\frac{d}{2}\left(\frac{1}{7}-\frac{1}{p}\right)}}\|f\|_{L_x^r},$$ $$\| e^{t\Delta}\nabla f\|_{L_x^p} \lesssim \frac{1}{t^{\frac{1}{2} + \frac {d}{2}\left(\frac{1}{r}-\frac{1}{p}\right)}}\|f\|_{L_x^r}.$$ I hope you find this useful. I'll continue looking into/thinking about the desired estimate. If it would be of help, I'm happy to post some info on proofs.

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