Does $K^{1/2} (t,s)$ inherit the continuity of $K(t,s)$?

Question

Assume that $K(t,s)$ is a (1) symmetric, (2) continuous, and (3) positive definite kernel on $[0,1] \times [0,1]$. The spectral decomposition of $K(t,s)$ is: $$ K (t,s) = \sum_{i=1}^\infty \lambda_i \phi_i (t) \phi_i (s) $$ with $\{ \phi_i (t): i \in \mathbb{N}\}$ orthonormal and $\lambda_1 > \lambda_2 > \cdots > 0$. Then define $$ K^{1/2} (t,s) = \sum_{i=1}^\infty \sqrt{\lambda_i} \phi_i (t) \phi_i (s) . $$ $K(t,s)$ is continuous on $[0,1] \times [0,1]$, is $K^{1/2} (t,s)$ also continuous? Furthermore, if $K \in C^{k}([0,1]^2)$ for some known $k \geq 1$, can we say something about $K^{1/2}$, for example, does $K^{1/2} \in C^{r}([0,1]^2)$ for some $r$ related to $k$?

Answer 1

No.

For example, the Green function for $-\Delta$ (the 1-D Laplace operator) in the interval $(0, 1)$ with Dirichlet boundary condition is continuous (it is $\min(x (1 - y), (1 - x) y)$), but the Green function for the square root of this operator is unbounded: it has logarithmic singularity near the diagonal. This follows, for example, from formula (2.4) in this paper by R. Song and Z. Vondraček. (For further information, see here).