# Singular stochastic integral operators

## TL;DR

This paper develops Calderón-Zygmund theory for singular stochastic integrals with operator-valued kernels, establishing $L^p$ bounds, sparse domination, and applications to stochastic PDE regularity.

## Contribution

It introduces a novel Calderón-Zygmund framework for stochastic integrals with operator-valued kernels, including $L^p$ extrapolation, sparse bounds, and solutions to the stochastic $A_2$-conjecture.

## Key findings

- Established $L^p$-extrapolation under Hörmander condition.
- Derived sharp weighted bounds via Dini condition.
- Applied results to stochastic heat equation regularity.

## Abstract

In this paper we introduce Calder\'on-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove $L^p$-extrapolation results under a H\"ormander 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 $\mathbb{R}^d$ and smooth and angular domains.