The heat kernel in one dimension for the real line is given by the usual gaussian density function: $$g(t,x,y)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{(x-y)^2}{2t}}\, .$$ In particular, by differentiating this function, one finds that for $|x-y|\leq T$, $$\sup_{t\in [0,T]} g(t,x,y) =\frac{C}{|x-y|}\, ,$$ for some constant $C$. My question is about the heat kernel for a bounded interval. More precisely, in the case of the interval $[0,\pi]$ instead of the real line, the heat kernel is given by $$g(t,x,y)=\frac 2 \pi \sum_{k\geq 1}\sin(kx) \sin(ky) e^{-k^2 t}\, .$$ Then do we have a similar estimate on the suppremum in time of the heat kernel, i.e. do we have an inequality like $$\sup_{t\in [0,T]} g(t,x,y) \geq \frac{C}{|x-y|}\, ,$$ valid for $|x-y|$ small enough?

The heat kernel in one dimension for the real line is given by the usual gaussian density function: $$g(t,x,y)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{(x-y)^2}{2t}}\, .$$ In particular, by differentiating this function, one finds that for $|x-y|\leq \sqrt{T}$, $$\sup_{t\in [0,T]} g(t,x,y) =\frac{C}{|x-y|}\, ,$$ for some constant $C$. My question is about the heat kernel for a bounded interval. More precisely, in the case of the interval $[0,\pi]$ instead of the real line, the heat kernel is given by $$g(t,x,y)=\frac 2 \pi \sum_{k\geq 1}\sin(kx) \sin(ky) e^{-k^2 t}\, .$$ Then do we have a similar estimate on the suppremum in time of the heat kernel, i.e. do we have an inequality like $$\sup_{t\in [0,T]} g(t,x,y) \geq \frac{C}{|x-y|}\, ,$$ valid for $|x-y|$ small enough?