Parabolic mean values and maximal estimates for gradients of temperatures

Abstract

We aim to prove inequalities of the form |δ k−λ (x, t)∇ k u (x, t)is the parabolic distance of (x, t) to parabolic boundary of Ω, M − R + is the one-sided Hardy-Littlewood maximal operator in the time variable on R + , M #,λ,k D is a Calderón-Scott type d-dimensional elliptic maximal operator in the space variable on the domain D in R d , and 0 < λ < k < λ + d. As a consequence, when D is a bounded Lipschitz domain, we obtain estimates for the L p (Ω) norm of δ 2n−λ (∇ 2,1 ) n u in terms of some mixed norm ∞ 0 u(·, t) p B λ,p p (D) dt for the space L p (R + , B λ,p p (D)) with · B λ,p p (D) denotes the Besov norm in the space variable x and where ∇ 2,1 = (∇ 2 , ∂ ∂t ).

On Besov Regularity of Temperatures

“…In fact, for λ large enough, and ε small, the functions in B λ−ε p ( ) are continuous in . Let us point out that the result in Theorem 1.1 can be regarded as a second step for the program explicitly stated in the introduction of [1]. In particular we expect that the result proved in this note, could be some help to obtain improvements of Besov regularity for temperatures of the type of those in [5] for harmonic functions.…”

“…We point out that (3.3) follows from the mean value formula for solutions of the heat equation as in the proof of Theorem 5.1 in [1]. The fact that u is a temperature in Theorem 3.1 is used in the required mean value representation and in order to obtain estimates for the partial derivative with respect to time in terms of second order space derivatives.…”

“…Let us start by stating a local version in time of Corollary 6.2 in [1]. Given a bounded Lipschitz domain D in R d and T > 0 we shall write δ = δ(x; t) to denote the parabolic distance of (x; t) ∈ = D × (0, T ) to the parabolic boundary of ,…”

“…Section 2 is devoted to briefly introduce the spaces of regularity considered here. Section 3 contains localizations of the main result in [1] which provide L p ( ) estimates for gradients of temperatures in terms of mixed Lebesgue-Besov norms in space-time. In Sect.…”

Parabolic Besov Regularity for the Heat Equation

“…For our first result, we shall use the parabolic analog of Theorem 3.1 in [8], essentially contained in Corollary 5.2 of [2]. Our second result is a corollary of the first one if we use Theorem 1.1 in [3].…”