Smoothing property of the heat kernel on the one-dimensional torus

Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$$G$ is given by \begin{equation}G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \mathbb{Z}} \exp \left ( - \frac{(x+k)^2}{4t} \right),\end{equation}and is smooth, meaning $G \in C^{\infty}(\mathbb{T}^1 \times (0,T])$ and is compactly supported w.r.t to $x,$ since it is defined on the torus.Moreover, let $u \in L^\infty((0,T); L^1(\mathbb{T}^1))$ and $r \in L^\infty((0,T); L^\infty(\mathbb{T}^1)).$ We consider the following convolution:$$ w(x,t) := \int_{0}^{t} \int_{\mathbb{T}^1} G(x-y,t-s)r(y,s)u(y,s) \, dyds.$$My question is, how much regularity can we expect for $w$? I would need $w \in C^{1/2}(\mathbb{T}^1 \times [0,T]).$I suppose it is possible to expect smoothness w.r.t to $x,$ since $G$ is compactly supported on the torus but what about $t$?I would be very grateful for help!