### Abstract

We consider parabolic operators of the form ∂ _{t}+L, L:= -div A(X,t) ∇, R^{n+2} _{+}+ := {(X, t) = (x, x_{n+1},t) ∈ Rn × R × R : x_{n+1} > 0}, n≥ 1. We assume that A is a (n+ 1) × (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 ∂_{t}+ L in R+n+2, with L^{2}-data on Rn+1=∂R+n+2, and by way of layer potentials.

Original language | English |
---|---|

Article number | 124 |

Number of pages | 49 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 55 |

Issue number | 5 |

DOIs | |

Publication status | Published - Oct 1 2016 |

Externally published | Yes |

### Keywords

- 31B10
- 35K20

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics