Parabolic mean values and maximal estimates for gradients of temperatures

We aim to prove inequalities of the form for solutions of on a domain Ω=D×R⁺, where δ(x,t) is the parabolic distance of (x,t) to parabolic boundary of Ω, is the one-sided Hardy-Littlewood maximal operator in the time variable on R⁺, 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−λn(∇2,1)u in terms of some mixed norm for the space with denotes the Besov norm in the space variable x and where .