The classical Whitney extension theorem states that every function in Lip$(\beta ,F)$, $F\subset {\mathbf{R}}^n$, $F$ closed, $k\<\beta \le k+1$, $k$ a non-negative integer, can be extended to a function in Lip$(\beta ,{\mathbf{R}}^n)$. Her Lip$(\beta ,F)$ stands for the class of functions which on $F$ have continuous partial derivatives up to order $k$ satisfying certain Lipschitz conditions in the supremum norm. We formulate and prove a similar theorem in the $L^p$-norm. The restrictions to ${\mathbf{R}}^d$, $d\<n$, of the Bessel potential spaces in ${\mathbf{R}}^n$ and the Besov or generalized Lipschitz...

On démontre plusieurs théorèmes concernant le balayage des mesures sur un espace harmonique satisfaisant aux axiomes de Bauer, parmi lesquels nous indiquons les suivants : a) la balayée ${\mu }_{A\cup B}$ d’une mesure $\mu$ sur la réunion ${\mu }^A\vee {\mu }^B$ (dans l’espace de Riesz de mesure) ; b) ${ϵ}_x^A\ne {ϵ}_x$ caractérise l’effilement de $A$ en $x$ ; c) il existe un potentiel fini et continue $p$ tel que pour tout ensemble $A\left\{x\right|{\widehat{R}}_p\left(x\right)\<p\left(x\right)\}$ est exactement l’ensemble des points où $A$ est effilé ; d) ${\mu }^A$ est portée par la fermeture fine de $A$ ; e) si $A$ et $B$ sont...

Suppose that μ is a Radon measure on ${ℝ}^d$, which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀ > 0 such that for all x ∈ supp(μ) and r > 0, μ(B(x,r)) ≤ C₀rⁿ, where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spaces ${Ḟ}_{pq}^s\left(\mu \right)$ for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C₀, n and d. The method does not use the vector-valued maximal function...

We prove unique continuation for solutions of the inequality $\left|\Delta u\right(x\left)\right|\le V\left(x\right)\left|\nabla u\right(x\left)\right|$, $x\in \Omega$ a connected set contained in ${\mathbf{R}}^n$ and $V$ is in the Morrey spaces $F^{\alpha ,p}$, with $p\ge (n-2)/2(1-\alpha )$ and $\alpha \&lt;1$. These spaces include $L^q$ for $q\ge (3n-2)/2$ (see [H], [BKRS]). If $p=(n-2)/2(1-\alpha )$, the extra assumption of $V$ being small enough is needed.

Let $u$ be a $\delta$-subharmonic function with associated measure $\mu$, and let $v$ be a superharmonic function with associated measure $\nu$, on an open set $E$. For any closed ball $B(x,r)$, of centre $x$ and radius $r$, contained in $E$, let $ℳ(u,x,r)$ denote the mean value of $u$ over the surface of the ball. We prove that the upper and lower limits as $s,t\to 0$ with $0<s<t$ of the quotient $\left(ℳ\right(u,x,s)-ℳ(u,x,t\left)\right)/\left(ℳ\right(v,x,s)-ℳ(v,x,t\left)\right)$, lie between the upper and lower limits as $r\to 0+$ of the quotient $\mu \left(B\right(x,r\left)\right)/\nu \left(B\right(x,r\left)\right)$. This enables us to use some well-known measure-theoretic results to prove new variants...

In this paper we consider the regularity problem for the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ where $b$ is a locally integrable function and $(R_j{)}_{1\le j\le n}$ are the Riesz transforms in the $n$-dimensional euclidean space ${ℝ}^n$. More precisely, we prove that these commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded from $L^p$ into the Besov space ${\dot{B}}_p^{s,p}$ for $1\&lt;p\&lt;+\infty$ and $0\&lt;s\&lt;1$ if and only if $b$ is in the $BMO$-Triebel-Lizorkin space ${\dot{F}}_{\infty }^{s,p}$. The reduction of our result to the case $p=2$ gives in particular that the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded form $L^2$ into the Sobolev space ${\dot{H}}^s$ if and only if $b$...