Parabolic mean values and maximal estimates for gradients of temperatures

Abstract

We aim to prove inequalities of the form | δk - λ (x, t) ∇k u (x, t) | ≤ C MR+- MD#, λ, k u (x, t) for solutions of frac(∂ u, ∂ t) = Δ u 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 δ2 n - λ (∇2, 1)n u in terms of some mixed norm ∫0∞ {norm of matrix} u (ṡ, t) {norm of matrix}Bpλ, p (D)p d t for the space Lp (R+, Bpλ, p (D)) with {norm of matrix} ṡ {norm of matrix}Bpλ, p (D) denotes the Besov norm in the space variable x and where ∇2, 1 = (∇2, frac(∂, ∂ t)).