Singular stochastic integral operators

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.Comment: typos corrected. Accepted for publication in Analysis & PD

Shape derivatives of boundary integral operators in electromagnetic scattering. Part I: Shape differentiability of pseudo-homogeneous boundary integral operators

In this paper we study the shape differentiability properties of a class of boundary integral operators and of potentials with weakly singular pseudo-homogeneous kernels acting between classical Sobolev spaces, with respect to smooth deformations of the boundary. We prove that the boundary integral operators are infinitely differentiable without loss of regularity. The potential operators are infinitely shape differentiable away from the boundary, whereas their derivatives lose regularity near the boundary. We study the shape differentiability of surface differential operators. The shape differentiability properties of the usual strongly singular or hypersingular boundary integral operators of interest in acoustic, elastodynamic or electromagnetic potential theory can then be established by expressing them in terms of integral operators with weakly singular kernels and of surface differential operators

BCR algorithm and the T(b)T(b) theorem

We show using the Beylkin-Coifman-Rokhlin algorithm in the Haar basis that any singular integral operator can be written as the sum of a bounded operator on $L^p$, $1<p<\infty$, and of a perfect dyadic singular integral operator. This allows to deduce a local $T(b)$ theorem for singular integral operators from the one for perfect dyadic singular integral operators obtained by Hofmann, Muscalu, Thiele, Tao and the first author.Comment: Change of title. New abstract and new introductio

Bounds for singular fractional integrals and related Fourier integral operators

We prove sharp L^p-L^q endpoint bounds for singular fractional integral operators and related Fourier integral operators, under the nonvanishing rotational curvature assumption.Comment: 30 page

Gauge-Invariant Operators for Singular Knots in Chern-Simons Gauge Theory

We construct gauge invariant operators for singular knots in the context of Chern-Simons gauge theory. These new operators provide polynomial invariants and Vassiliev invariants for singular knots. As an application we present the form of the Kontsevich integral for the case of singular knots.Comment: 44 pages, latex, 16 figure