Gradient H\"older regularity in mixed local and nonlocal linear parabolic problem

We prove the local H\"older regularity of weak solutions to the mixed local nonlocal parabolic equation of the form \begin{equation*} u_t-\Delta u+\text{P.V.}\int_{\mathbb{R}^{n}} {\frac{u(x,t)-u(y,t)}{{\left|x-y\right|}^{n+2s}}}dy=0, \end{equation*} where $0<s<1$; for every exponent $\alpha_0\in(0,1)$. Here, $\Delta$ is the usual Laplace operator. Next, we show that the gradients of weak solutions are also $\alpha$-H\"older continuous for some $\alpha\in (0,1)$. Our approach is purely analytic and it is based on perturbation techniques.