Vector-valued Lizorkin–Triebel spaces and sharp trace theory for functions in Sobolev spaces with mixed $L_p$-norm for parabolic problems

The trace problem on the hypersurface $y_n=0$ is investigated for a function $u=u(y,t)\in L_q(0,T;W_{\underline p}^{\underline m}(\mathbb R_+^n))$ with $\partial_tu\in L_q(0,T; L_{\underline p}(\mathbb R_+^n))$, that is, Sobolev spaces with mixed Lebesgue norm $L_{\underline p,q}(\mathbb R^n_+\times(0,T)) =L_q(0,T;L_{\underline p}(\mathbb R_+^n))$ are considered; here $\underline p=(p_1,\dots,p_n)$ is a vector and $\mathbb R^n_+=\mathbb R^{n-1}\times (0,\infty)$. Such function spaces are useful in the context of parabolic equations. They allow, in particular, different exponents of summability in space and time. It is shown that the sharp regularity of the trace in the time variable is characterized by the Lizorkin–Triebel space $F_{q,p_n}^{1-1/(p_nm_n)}(0,T;L_{\widetilde{\underline p}}(\mathbb R^{n-1}))$, $\underline p=(\widetilde{\underline p},p_n)$. A similar result is established for first order spatial derivatives of $u$. These results allow one to determine the exact spaces for the data in the inhomogeneous Dirichlet and Neumann problems for parabolic equations of the second order if the solution is in the space $L_q(0,T; W_p^2(\Omega))\cap W_q^1(0,T;L_p(\Omega))$ with $p\leqslant q$.