I have asked this question on MSE, but this is a better place.
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$: $$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&=0\quad\forall x\in\mathbb R. \end{aligned}\right. $$ It is well-known that under suitable hypotheses there exists a unique solution $u$ given by the Duhamel formula (see below). So, we consider the operator $\Gamma:=(\partial_t-\Delta)^{-1} $ which maps $f$ to the solution $u$, that is, $$ u=\Gamma f. $$ Given the heat kernel $$ G(t,x)=\frac{1}{(4\pi t)^{1/2}}e^{-\frac{x^2}{4t}}, $$ one can show that $u$ is given as a space-time convolution between $G$ and $f$: $$ u(t,x)=\left(\Gamma f\right)(t,x)=\int_0^\infty\int_{\mathbb R^d} G(t-s,x-y)f(s,y)\,\,dyds. $$
The problem.
Problem. Given $s\in[0,2]$, find all quadruples $(p,q,r,\sigma)$ such that the following estimate holds for all test functions $f\in\mathcal D((0,\infty)\times\mathbb R)$, for some constant $C>0$: $$ \||\partial_x|^s\Gamma f \|_{L^rL^\sigma}\leq C\|f\|_{L^pL^q}. \qquad (1)$$
The norm in the space $L^pL^q$ is given by $$ \|f\|_{L^pL^q}^p:=\int_0^\infty\|f(s,\cdot)\|^p_{L^q}ds, $$ so it's the classical mixed $L^p-L^q$ norm in which you take first the $L^q$ norm in the $x$ variable and then the $L^p$ norm in the $t$ variable. The fractional derivative $|\partial_x|^s$ is defined via the Fourier transform.
Basically, the problem is to find all the 'Strichartz estimates' of the heat equation with fractional gain of derivatives. Note that this is not about the fractional heat equation: we simply want to prove estimates on the derivatives of the solution to the classical heat equation.
Considerations and known estimates.
By scaling considerations, one immediately finds that the coefficients must satisfy $$ \left(\frac{1}{r}+\frac{1/2}{\sigma}\right)=\left(\frac{1}{p}+\frac{1/2}{q}\right)-1+\frac{s}{2}. \qquad (2)$$ Thus, in what follows I will always assume $(2)$. I will also assume $p\leq r$ and $q\leq\sigma$ (essentially because convolution with a given function maps $L^p$ to $L^q$ with $q\geq p$).
Preliminarly, I state without proof the following estimates for the heat kernel: $$ \||\partial_x|^sG(t,\cdot)\|_{L_x^p}\lesssim |t|^{-\frac{1}{2}(1+s)+\frac{1}{2p}}, \quad s\geq 0,\,\,1\leq p\leq \infty. \qquad (3)$$
What I already know is the following:
The estimate holds in the cases $$ s\in\{0,1\},\quad 1\leq q\leq \sigma\leq \infty,\quad 1<p<r<\infty. $$ This follows from a direct use of the Hardy-Littlewood-Sobolev inequality and the estimates $(3)$ for $s=0$ and $s=1$ (see this book).
This is not in the book, but with the same proof, using the full range of estimates $(3)$, one covers all the cases $s\in[0,2)$ with the same conditions on $p,q,r,\sigma$ as above.
The estimates hold in the case $(r,\sigma)=(\infty,2)$ with $1\leq q\leq 2$. This follows by integrating the heat equation against $u$ and perform standard energy estimates.
What is left is a lot of endpoint cases (most prominently, $s=2$, $p=r$, $p=1$, and $r=\infty$).
Finally, my question.
My question is: what about the case $r=\infty$, $1<\sigma<\infty$ and $s\in(0,2)$? It seems that energy estimates do not help for the full range of exponents $(p,q)$ and work only for certain values of $\sigma$ anyway. Do you know a way to cover all cases, or a reference that covers all cases? I am reasonably sure that one can exchange the roles of $x$ and $t$ and use Hardy-Littlewood-Sobolev, but that would work only for $p\leq q$.
I am especially interested in the case $(r,\sigma)=(\infty,2)$, $q=2$ for $s\in(0,2)$ (which would basically cover all other cases $(p,q)$ by Sobolev embedding), i.e., $$ \||\partial_x|^{2-\frac{2}{p}}\Gamma f\|_{L^\infty L^2}\lesssim \|f\|_{L^pL^2}. $$ If the case $q=2$ is too hard, $q<2$ would be well accepted. Even the case $s=1$ is not clear to me at the moment, except when $(p,q)=(2,2)$.
Besides my specific question, it would be nice to collect some references that address estimate $(1)$ in all known ranges of exponents (unless you can think of a paper which treats all these estimates at once, which would be very appreciated). So, if you have a reference for any of the remaining ranges of exponents, even for higher dimensions, that would be very appreciated.
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