A new maximal regularity for parabolic equations and an application

Abstract

We introduce the Lebesgue--Hölder--Dini and Lebesgue--Hölder spaces $L^p(\mathbb{R};{\mathcal{C}}_{\vartheta ,\ddot{I}‚}^{\hat{I}\pm ,\ddot{I}}({\mathbb{R}}^n))$ ($\vartheta \in \{l,b\},\ddot{I}‚\in \{d,s,c,w\}$, $p\in (1,+\infty ]$ and $\hat{I}\pm \in [0,1)$), and then use a vector-valued Calderón--Zygmund theorem to establish the maximal Lebesgue--Hölder--Dini and Lebesgue--Hölder regularity for a class of parabolic equations. As an application, we obtain the unique strong solvability of the following stochastic differential equation \begin{eqnarray*} X_{s,t}(x)=x+\int\limits_s^tb(r,X_{s,r}(x))dr+W_t-W_{s}, \ \ t\in [s,T], \ x\in \mathbb{R}^n, \ s\in [0,T], \end{eqnarray*} for the low regularity growing drift in critical Lebesgue--Hölder--Dini spaces $L^p([0,T];{\mathcal{C}}_{l,d}^{\frac{2}{p}-1,\ddot{I}}({\mathbb{R}}^n;{\mathbb{R}}^n))$ ($p\in (1,2]$), where $\{W_t{\}}_{0\le t\le T}$ is a $n$-dimensional standard Wiener process. In particular, when $p=2$ we give a partially affirmative answer to a longstanding open problem, which was proposed by Krylov and Röckner for $b\in L^2([0,T];L^{\infty }({\mathbb{R}}^n;{\mathbb{R}}^n))$ based upon their work ({\em Probab. Theory Relat. Fields 131(2): 154--196, 2005}).

45 pages