Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients

TL;DR

This paper investigates the boundedness of single layer potentials for divergence form parabolic equations with complex coefficients, reducing the problem to key estimates and establishing a local Tb-theorem for square functions.

Contribution

It introduces a reduction of boundedness to two key estimates and proves a local parabolic Tb-theorem for square functions in this context.

Findings

Boundedness of layer potentials reduces to two crucial estimates.

Established a scale-invariant reverse H{"o}lder inequality for the parabolic Poisson kernel.

Verified key estimates for real, symmetric operators using the Tb-theorem.

Abstract

We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}:=-\mbox{div}\, A(X,t)\nabla,$$ in $\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}$, $n\geq 1$. We assume that $A$ is a $(n+1)\times (n+1)$-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate $x_{n+1}$ as well as of the time coordinate $t$. We prove that the boundedness of associated single layer potentials, with data in $L^2$, can be reduced to two crucial estimates, one being a square function estimate involving the single layer potential. By establishing a local parabolic Tb-theorem for square functions we are then able to verify the two crucial estimates in the case of real, symmetric operators. As part of this argument we establish a scale-invariant reverse H{\"o}lder inequality for the parabolic Poisson kernel. Our results are important when addressing the solvability of the classical Dirichlet, Neumann and Regularity problems for the operator $\partial_t+\mathcal{L}$ in $\mathbb R_+^{n+2}$, with $L^2$-data on $\mathbb R^{n+1}=\partial\mathbb R_+^{n+2}$, and by way of layer potentials.