Parabolic mean values and maximal estimates for gradients of temperatures

We aim to prove inequalities of the form |δk−λ(x, t)∇ku(x, t)| CM−R+M #,λ,k D u(x, t) for solutions of ∂u ∂t = u on a domain Ω = D ×R+, where δ(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 Rd, and 0 < λ < k < λ + d. As a consequence, when D is a bounded Lipschitz domain, we obtain estimates for the Lp(Ω) norm of δ2n−λ(∇2,1)nu in terms of some mixed norm ∞0 u(·, t) p B λ,p p (D) dt for the space Lp(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 ). © 2008 Elsevier Inc. All rights reserved.