Parabolic mean values and maximal estimates for gradients of temperatures

Abstract

AbstractWe aim to prove inequalities of the form |δk−λ(x,t)∇ku(x,t)|⩽CMR+−MD#,λ,ku(x,t) for solutions of ∂u∂t=Δu on a domain Ω=D×R+, where δ(x,t) is the parabolic distance of (x,t) to parabolic boundary of Ω, MR+− is the one-sided Hardy–Littlewood maximal operator in the time variable on R+, MD#,λ,k 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)‖Bpλ,p(D)pdt for the space Lp(R+,Bpλ,p(D)) with ‖⋅‖Bpλ,p(D) denotes the Besov norm in the space variable x and where ∇2,1=(∇2,∂∂t)