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.



Holder regularity for the heat potentials
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. 


