Let $\alpha \in (0, 1)$. Consider $f = f(t, x)$ a bounded measurable function on $[0, 1] \times \mathbb{R}$, such that $f$ is $\alpha$-Hölder with respect to $x$, uniformly in $t$. Let $G(r) := e^{-r^2/2}/\sqrt{2 \pi}$ and $$ v(t, x) := \int_t^1{ \int_{\mathbb{R}} f(u,z) G(\frac{z-x}{\sqrt{u-t}}) \frac{dz}{\sqrt{u-t}}du}.$$ Is $v_{xx}(t, \cdot)$ $\beta$-Hölder for some $\beta \in (0, 1)$ uniformly in $t \in [0, 1]$ ?
Note that $v$ is the solution on $[0, 1] \times \mathbb{R}$ of the PDE: $$v_t + \frac{1}{2} v_{xx} + f = 0 \quad \text{ with } \quad v_{|1} = 0.$$ In addition, we have $$ v_{xx}(t, x) = \int_t^1{\int_{\mathbb{R}} \frac{f(u, z) - f(u, x)}{u-t}G''(\frac{z-x}{\sqrt{u-t}}) \frac{dz}{\sqrt{u-t}}du},$$ so because $f$ is $\alpha$-Hölder and $\frac{1}{(u-t)^{1-\alpha/2}}$ is integrable on $[t, 1]$, it holds that $v_{xx}$ is bounded.
Many thanks!
This integral is the volume potential for the heat equation. The answer is yes. See Ladygenskaja O.A., Solonnikov V.A., Ural'ceva N.N., Linear and quasi-linear equations of parabolic type, ch. 4, $\S2$, estimate (2.8). It is assumed in the formulations that $f$ is Holder with respect to time too, but it's not used in the proof of (2.8).