Parabolic mean values and maximal estimates for gradients of temperatures

AIMAR, HUGO; GÓMEZ, IVANA; IAFFEI, BIBIANA

JOURNAL OF FUNCTIONAL ANALYSIS

Elsevier

Año: 2008 vol. 255 p. 1939 - 1956

0022-1236

We aim to prove inequalities of the form$abs{delta^{k-lambda}(x,t), abla^k, u(x,t)}leq CM^{-}_{Real^{+}}M^{#,lambda,k}_{D} u(x,t)$ for solutions of$ frac{partial u}{partial t}=Delta u$ on a domain$Omega=D imes Real^{+}$, where $delta(x,t)$ is the parabolicdistance of $(x,t)$ to parabolic boundary of $Omega$,$M^{-}_{Real^{+}}$ is the one-sided Hardy-Littlewood maximaloperator in the time variable on $Real^{+}$,$M^{#,lambda,k}_{D}$ is a Calder´on-Scott type$d$-dimensional elliptic maximal operator in the space variableon the  domain $D$ in $Real^{d}$, and $0<lambda<k<lambda +d$.As a consequence, when $D$ is a bounded Lipschitz domain, weobtain estimates for the $L^p(Omega)$ norm of$delta^{2n-lambda}( abla^{2,1})^n u$ in terms of some mixednorm$int_{0}^{infty} orm{u(cdot,t)}^p_{B^{lambda,p}_{p}(D)}dt$for the space $L^p(Real^{+},B^{lambda,p}_{p}(D))$ with$ orm{cdot}_{B^{lambda,p}_{p}(D)}$ denotes the Besov norm inthe space variable $x$ and where$ abla^{2,1}=( abla^2, frac{partial}{partial t})$.