# Heat semigroup norm between fractional Sobolev and $L^p$ spaces

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.

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

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.