# Holder regularity for the heat potentials

First I apologize for my bad English and for any error: this is my first question.

I need some regularity results for the single and double layer heat potentials. If $\Gamma(t,x)$ is the fundamental solution of the heat equation, the single and double layer potentials are defined as follows:

(SLP) $u(t,x)= \int_{0}^{t}\int_{\partial \Omega}\Gamma(t-\tau,x-y)\varphi(\tau,y)d\sigma(y) d\tau$

(DLP) $v(t,x)= \int_{0}^{t}\int_{\partial \Omega}\frac{\partial\Gamma(t-\tau,x-y)}{\partial\nu(y)}\varphi(\tau,y)d\sigma(y) d\tau$.

In the cylinder $Q=\Omega\times[0,T]$, $\Omega\subset\mathbb{R}^n$, $\Omega$ bounded domain regular enough and $\nu$ in the unit outer normal to $\partial \Omega$

My questions are:

1) It is true that if $\varphi\in C^{1+\alpha/2,2+\alpha}(Q)$ then $v$ extends with continuity to $\bar{\Omega}\times[0,T]$ and this extension stay in $C^{1+\alpha/2,2+\alpha}(\bar{Q})$?

2)It is true that if $\varphi\in C^{(1+\alpha)/2,1+\alpha}(Q)$ then $u$ extends with continuity to $\bar{\Omega}\times[0,T]$ and this extension stay in $C^{1+\alpha/2,2+\alpha}(\bar{Q})$?

Where the functions spaces $C^{1+\alpha/2,2+\alpha},C^{(1+\alpha)/2,1+\alpha}$ are the parabolic holder spaces for example defined in "Lectures on Elliptic and Parabolic Equations in Holder Spaces, N.V. Krylov, pg 117,118"

(Is possible that the parameters of the Holder spaces are different.)

These questions are well known for the armonic potentials and I want to know if similar results hold true for the heat potentials.

• You take $\varphi\in C^{1+\alpha/2,2+\alpha}(Q)$ but densities should be defined on $\partial\Omega\times[0,T]$. Does it means that the trace of $\varphi$ is taken? Sep 23 '14 at 12:33
• You're right, i think that the density $\varphi$ must stay in $C^{1+\alpha/2,2+\alpha}(\partial\Omega\times[0,T])$ Sep 23 '14 at 12:49

The answer is yes for the smooth enough boundary, provided $\varphi|_{t=0}=0$ and for 2) additionally $\partial_t\varphi|_{t=0}=0$. Say, for the SLP from $C^{1+\alpha/2,2+\alpha}(\bar{Q})$ it should hold $u|_{t=0}=\partial_tu|_{t=0}=0$. From here the mentioned conditions on $\varphi$ at $t=0$ follow.

• Thank you.Can you give me some references for these results? Sep 23 '14 at 12:54
• When $\Omega$ is a half-space it is written in O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type". For general case I havn't a reference handy, but smoothness of the heat potentials was studied in the works of Kamynin. Sep 23 '14 at 13:10
• Thank you. I think that the works of Kamynin are what I was looking for. Sep 24 '14 at 13:08
• In the Kamynin's works I think that the SLP and the DLP don't reach the regularity $C^{1+\alpha/2,2+\alpha}$ but $C^{1+\alpha'/2,2+\alpha'}$ for all $\alpha'<\alpha$. If you recall some references for the result with $\alpha$ with holder coefficient, came you give me? Thank you in advance Sep 25 '14 at 11:06
• @foo90 If you can read Russian and would provide some e-mail address, I can send you my article (in Russian) for SLP of a parabolic operator with variable coefficients, which, as a special case, have the required result. Sep 26 '14 at 14:03