Abstract. The square root of the heat operator √ ∂ t − ∆, can be realized as the Dirichlet to Neumann map of the heat extension of data on R n+1 to R n+2 + . In this note we obtain similar characterizations for general fractional powers of the heat operator, (∂ t − ∆) s , s ∈ (0, 1). Using the characterizations we derive properties and boundary estimates for parabolic integro-differential equations from purely local arguments in the extension problem.2000 Mathematics Subject Classification.

We consider parabolic operators of the formWe 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+ , and by way of layer potentials. Mathematics Subject Classification 35K20 · 31B10Communicated by Y. Giga.

Let Ω ⊂ R n be a bounded NTA-domain and let Ω T = Ω × (0, T ) for some T > 0. We study the boundary behaviour of non-negative solutions to the equationWe assume that A(x, t) = {a ij (x, t)} is measurable, real, symmetric and thatfor some constant β ≥ 1 and for some non-negative and real-valued function λ = λ(x) belonging to the Muckenhoupt class A 1+2/n (R n ). Our main results include the doubling property of the associated parabolic measure and the Hölder continuity up to the boundary of quotients of non-negative solutions which vanish continuously on a portion of the boundary. Our results generalize previous results of Fabes, Kenig, Jerison, Serapioni, see [18], [19], [20], to a parabolic setting.

Boundary estimates for non-negative solutions to non-linear parabolic equations.Access to the published version may require subscription. N.B. When citing this work, cite the original published paper. Abstract This paper is mainly devoted to the boundary behavior of non-negative solutions to the equationwhere Ω ⊂ R n is a bounded non-tangentially accessible (NTA) domain and T > 0. The assumptions we impose on A imply that H is a non-linear parabolic operator with linear growth. Our main results include a backward Harnack inequality, and the Hölder continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary. Furthermore, to each such solution one can associate a natural Riesz measure supported on the lateral boundary and one of our main result is a proof of the doubling property for this measure. Our results generalize, to the setting of non-linear equations with linear growth, previous results concerning the boundary behaviour, in Lipschitz cylinders and time-independent NTA-cylinders, established for non-negative solutions to equations of the type ∂ t u − ∇ · (A(x, t)∇u) = 0, where A is a measurable, bounded and uniformly positive definite matrixvalued function. In the latter case the measure referred to above is essentially the caloric or parabolic measure associated to the operator and related to Green's function. At the end of the paper we also remark that our arguments are general enough to allow us to generalize parts of our results to general fully non-linear parabolic partial differential equations of second order.

No abstract