A new maximal regularity for parabolic equations and an application (2411.13266v1)

Abstract: We introduce the Lebesgue--H\"{o}lder--Dini and Lebesgue--H\"{o}lder spaces $L^p(\mathbb{R};{\mathcal C}{\vartheta,\varsigma}^{\alpha,\rho}({\mathbb R}^n))$ ($\vartheta\in {l,b}, \, \varsigma\in {d,s,c,w}$, $p\in (1,+\infty]$ and $\alpha\in [0,1)$), and then use a vector-valued Calder\'{o}n--Zygmund theorem to establish the maximal Lebesgue--H\"{o}lder--Dini and Lebesgue--H\"{o}lder 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--H\"{o}lder--Dini spaces $L^p([0,T];{\mathcal C}^{\frac{2}{p}-1,\rho}_{l,d}({\mathbb R}^n;{\mathbb R}^n))$ ($p\in (1,2]$), where ${W_t}_{0\leq t\leq T}$ 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 R\"{o}ckner 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}).