### 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 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 55 |

Issue number | 5 |

DOIs | |

Publication status | Published - Oct 1 2016 |

Externally published | Yes |

### Fingerprint

### Keywords

- 31B10
- 35K20

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Calculus of Variations and Partial Differential Equations*,

*55*(5), [124]. https://doi.org/10.1007/s00526-016-1058-8

**Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients.** / Castro, Alejandro J.; Nyström, Kaj; Sande, Olow.

Research output: Contribution to journal › Article

*Calculus of Variations and Partial Differential Equations*, vol. 55, no. 5, 124. https://doi.org/10.1007/s00526-016-1058-8

}

TY - JOUR

T1 - Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients

AU - Castro, Alejandro J.

AU - Nyström, Kaj

AU - Sande, Olow

PY - 2016/10/1

Y1 - 2016/10/1

N2 - We consider parabolic operators of the form ∂ t+L, L:= -div A(X,t) ∇, Rn+2 ++ := {(X, t) = (x, xn+1,t) ∈ Rn × R × R : xn+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 xn + 1 as well as of the time coordinate t. We prove that the boundedness of associated single layer potentials, with data in L2, 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 L2-data on Rn+1=∂R+n+2, and by way of layer potentials.

AB - We consider parabolic operators of the form ∂ t+L, L:= -div A(X,t) ∇, Rn+2 ++ := {(X, t) = (x, xn+1,t) ∈ Rn × R × R : xn+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 xn + 1 as well as of the time coordinate t. We prove that the boundedness of associated single layer potentials, with data in L2, 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 L2-data on Rn+1=∂R+n+2, and by way of layer potentials.

KW - 31B10

KW - 35K20

UR - http://www.scopus.com/inward/record.url?scp=84988908417&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84988908417&partnerID=8YFLogxK

U2 - 10.1007/s00526-016-1058-8

DO - 10.1007/s00526-016-1058-8

M3 - Article

VL - 55

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 5

M1 - 124

ER -