We introduce Calderón–Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove $L^p$-extrapolation results under a Hörmander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the $A_2$-conjecture. The results are applied to obtain $p$-independence and weighted bounds for stochastic maximal $L^p$-regularity both in the complex and real interpolation scale. As a consequence we obtain several new regularity results for the stochastic heat equation on ${ℝ}^d$ and smooth and angular domains.

中文翻译：

---

奇异随机积分算子

我们介绍了具有算子值内核的奇异随机积分的 Calderón-Zygmund 理论。特别地，我们证明${升}^{磷}$- 在内核上的 Hörmander 条件下的外推结果。在内核的 D​​ini 条件下获得稀疏支配和尖锐加权边界，从而导致解的随机版本${一}_2$——猜想。结果用于获得$磷$- 随机最大值的独立性和加权界限 ${升}^{磷}$- 复杂和真实插值尺度的正则性。因此，我们获得了随机热方程的几个新的正则性结果${ℝ}^d$ 以及平滑和有角的域。