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

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