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

Abstract

We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}:=-\mbox{div}\, A(X,t)\nabla,$$ in ${\mathbb{R}}_{+}^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in {\mathbb{R}}^n\times \mathbb{R}\times \mathbb{R}:x_{n+1}>0\}$, $n\ge 1$. 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{ö}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.

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