Conical stochastic maximal $L^p$-regularity for $1\le p<\mathrm{\infty }$

Let $A=-\mathrm{d}\mathrm{i}\mathrm{v}a(\cdot )\mathrm{\nabla }$ be a second order divergence form elliptic operator on ${\mathbb{R}}^n$ with bounded measurable real-valued coefficients and let $W$ be a cylindrical Brownian motion in a Hilbert space $H$. Our main result implies that the stochastic convolution process $$u(t)={\int }_0^t{e}^{-(t-s)A}g(s)dW(s),t\ge 0,$$ satisfies, for all $1\le p<\mathrm{\infty }$, a conical maximal $L^p$-regularity estimate $$\text{E}\text{n}\mathrm{\nabla }u{\text{n}}_{T_2^{p,2}({\mathbb{R}}_{+}\times {\mathbb{R}}^n)}^p\le C_p^p\text{E}\text{n}g{\text{n}}_{T_2^{p,2}({\mathbb{R}}_{+}\times {\mathbb{R}}^n;H)}^p\mathrm{.}$$ Here, $T_2^{p,2}({\mathbb{R}}_{+}\times {\mathbb{R}}^n)$ and $T_2^{p,2}({\mathbb{R}}_{+}\times {\mathbb{R}}^n;H)$ are the parabolic tent spaces of real-valued and $H$-valued functions, respectively. This contrasts with Krylov’s maximal $L^p$-regularity estimate $$\text{E}\text{n}\mathrm{\nabla }u{\text{n}}_{L^p({\mathbb{R}}_{+};L^2({\mathbb{R}}^n;{\mathbb{R}}^n))}^p\le C^p\text{E}\text{n}g{\text{n}}_{L^p({\mathbb{R}}_{+};L^2({\mathbb{R}}^n;H))}^p$$ which is known to hold only for $2\le p<\mathrm{\infty }$, even when $A=-\mathrm{\Delta }$ and $H=\mathbb{R}$. The proof is based on an $L^2$-estimate and extrapolation arguments which use the fact that $A$ satisfies suitable off-diagonal bounds. Our results are applied to obtain conical stochastic maximal $L^p$-regularity for a class of nonlinear SPDEs with rough initial data.