A new maximal regularity for parabolic equations and an application

We introduce the Lebesgue–Hölder–Dini and Lebesgue–Hölder spaces $L^p(\mathbb{R};{\mathcal{C}}_{\vartheta ,\varsigma }^{\alpha ,\rho }({\mathbb{R}}^n))$ ($\vartheta \in \{l,b\},\varsigma \in \{d,s,c,w\}$, $p\in (1,+\mathrm{\infty }]$ and $\alpha \in [0,1)$), 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 $L^p([0,T];{\mathcal{C}}_{l,d}^{\frac{2}{p}-1,\rho }({\mathbb{R}}^n;{\mathbb{R}}^n))$ ($p\in (1,2]$), 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 $b\in L^2([0,T];L^{\mathrm{\infty }}({\mathbb{R}}^n;{\mathbb{R}}^n))$ based upon their work ({\em Probab. Theory Relat. Fields 131(2): 154–196, 2005}).