Hölder, Sobolev, weak-type and $BMO$ estimates in mixed-norm with weights for parabolic equations

Abstract

We prove weighted mixed-norm $L_t^q(W_x^{2,p})$ and $L_t^q(C_x^{2,\alpha })$ estimates for $1<p,q<\infty$ and $0<\alpha <1$, weighted mixed weak-type estimates for $q=1$, $L_t^{\infty }(L_x^p)-BM{O}_t(W_x^{2,p})$, and $L_t^{\infty }(C_x^{\alpha })-BM{O}_t(C_x^{2,\alpha })$, and a.e.~pointwise formulas for derivatives, for solutions $u=u(t,x)$ to parabolic equations of the form $${\partial }_{t}u-a^{ij}(t){\partial }_{ij}u+u=ft\in \mathbb{R},x\in {\mathbb{R}}^n$$ and for the Cauchy problem $$\{\begin{array}{ll}{\partial }_{t}v-a^{ij}(t){\partial }_{ij}v+v=f& fort>0,x\in {\mathbb{R}}^n\\ v(0,x)=g& forx\in {\mathbb{R}}^n.\end{array}$$ The coefficients $a(t)=(a^{ij}(t))$ are just bounded, measurable, symmetric and uniformly elliptic. Furthermore, we show strong, weak type and $BMO$-Sobolev estimates with parabolic Muckenhoupt weights. It is quite remarkable that most of our results are new even for the classical heat equation $${\partial }_{t}u-\Delta u+u=f.$$

27 pages. To appear in Science China Mathematics