Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients
Alejandro J. Castro, Kaj Nyström, Olow Sande · Calculus of Variations and Partial Differential Equations · 2016
We consider parabolic operators of the form $$\begin{aligned} \partial _t+\mathcal {L},\ \mathcal {L}:=-\text{ div }\, A(X,t) abla , \end{aligned}$$ 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 (Theorem 1.1), 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 (Theorem 1.2). As part of this argument we establish a scale-invariant reverse Hölder inequality for the parabolic Poisson kernel (Theorem 1.3). 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.