We introduce Calderón–Zygmund theory for singular stochastic
integrals with operator-valued kernel. In particular, we prove
-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
-conjecture. The results are
applied to obtain
-independence
and weighted bounds for stochastic maximal
-regularity
both in the complex and real interpolation scale. As a consequence we
obtain several new regularity results for the stochastic heat equation on
and
smooth and angular domains.
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