Maximal regularity and caloric trace estimates in mixed Lebesgue norms for the heat equation

We study the heat equation in the half-space with nonhomogeneous Dirichlet boundary data. For the caloric extension $v$ of the boundary data $g$, we prove maximal regularity estimates in mixed Lebesgue norms $L_t^p{L}_x^q$ for any order derivative of $v$ in terms of mixed Besov and Lizorkin–Triebel type norms of $g$. We also establish the corresponding reverse inequalities, which are caloric trace estimates recovering the boundary regularity of $g$ from the mixed-norm regularity of $v$. As a model case, our results show that the natural $$\text{dotc}{W}^{1,p}(\mathbb{R};L^q({\mathbb{R}}_{+}^d))\cap L^p(\mathbb{R};\text{dotc}{W}^{2,q}({\mathbb{R}}_{+}^d))$$ regularity norm of $v$ is controlled by the $$\text{dotc}{F}_{p,q}^{1-\frac{1}{2q}}(\mathbb{R};L^q({\mathbb{R}}^{d-1}))\cap L^p(\mathbb{R};\text{dotc}{B}_{q,q}^{2-\frac{1}{q}}({\mathbb{R}}^{d-1}))$$ norm of $g$. The maximal regularity estimate holds for $1\le p,q<\mathrm{\infty }$, while the caloric trace estimate holds for $1<p<\mathrm{\infty }$ and $1\le q\le \mathrm{\infty }$. In particular, the endpoint cases $p=1$ or $q=1$ in the maximal regularity estimate are included and appear to be new. These endpoint estimates may be useful in the analysis of free-boundary Navier–Stokes problems with small initial data, whereas the caloric trace estimates may be relevant to the construction of Stokes or Navier–Stokes flows exhibiting strong boundary singularities.