H\"older, Sobolev, weak-type and $BMO$ estimates in mixed-norm with weights for parabolic equations

Abstract

We prove weighted mixed-norm $L^q_t(W^{2,p}_x)$ and $L^q_t(C^{2,\alpha}_x)$ estimates for $1<p,q<\infty$ and $0<\alpha<1$, weighted mixed weak-type estimates for $q=1$, $L^\infty_{t}(L^p_x)-BMO_t(W^{2,p}_x)$, and $L^\infty_{t}(C^\alpha_x)-BMO_t(C^{2,\alpha}_x)$, and a.e.~pointwise formulas for derivatives, for solutions $u=u(t,x)$ to parabolic equations of the form $$\partial_tu-a^{ij}(t)\partial_{ij}u+u=f\quad t\in\mathbb{R},~x\in\mathbb{R}^n$$ and for the Cauchy problem $$\begin{cases} \partial_tv-a^{ij}(t)\partial_{ij}v+v=f&\hbox{for}~t >0,~x\in\mathbb{R}^n \\ v(0,x)=g&\hbox{for}~x\in\mathbb{R}^n. \end{cases}$$ 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 $$\partial_tu-\Delta u+u=f.$$