Displaying similar documents to “A Whitney extension theorem in L p and Besov spaces”

Thin sets in nonlinear potential theory

Lars-Inge Hedberg, Thomas H. Wolff (1983)

Annales de l'institut Fourier

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Let L α q ( R D ) , α > 0 , 1 < q < , denote the space of Bessel potentials f = G α * g , g L q , with norm f α , q = g q . For α integer L α q can be identified with the Sobolev space H α , q . One can associate a potential theory to these spaces much in the same way as classical potential theory is associated to the space H 1 ; 2 , and a considerable part of the theory was carried over to this more general context around 1970. There were difficulties extending the theory of thin sets, however. By means of a new inequality, which characterizes the...

Unique continuation for the solutions of the laplacian plus a drift

Alberto Ruiz, Luis Vega (1991)

Annales de l'institut Fourier

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We prove unique continuation for solutions of the inequality | Δ u ( x ) | V ( x ) | u ( x ) | , x Ω a connected set contained in R n and V is in the Morrey spaces F α , p , with p ( n - 2 ) / 2 ( 1 - α ) and α < 1 . These spaces include L q for q ( 3 n - 2 ) / 2 (see [H], [BKRS]). If p = ( n - 2 ) / 2 ( 1 - α ) , the extra assumption of V being small enough is needed.

Regularity properties of commutators and B M O -Triebel-Lizorkin spaces

Abdellah Youssfi (1995)

Annales de l'institut Fourier

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In this paper we consider the regularity problem for the commutators ( [ b , R k ] ) 1 k n where b is a locally integrable function and ( R j ) 1 j n are the Riesz transforms in the n -dimensional euclidean space n . More precisely, we prove that these commutators ( [ b , R k ] ) 1 k n are bounded from L p into the Besov space B ˙ p s , p for 1 < p < + and 0 < s < 1 if and only if b is in the B M O -Triebel-Lizorkin space F ˙ s , p . The reduction of our result to the case p = 2 gives in particular that the commutators ( [ b , R k ] ) 1 k n are bounded form L 2 into the Sobolev space H ˙ s if and only if b ...

Regularity properties of the equilibrium distribution

Hans Wallin (1965)

Annales de l'institut Fourier

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Soit F un sous-ensemble compact de R m ayant des points intérieurs et soit μ α F la distribution d’équilibre sur F de masse totale 1 par rapport au noyau r α - m avec 0 < α < 2 pour m 2 , et 0 < α < 1 pour m = 1 . La restriction de μ α F à l’intérieur de F est absolument continue et a pour densité f α F . On donne une formule explicite pour f α F et, pour une classe générale d’ensembles F , on démontre que f α F , définie en réalité sur un ensemble de mesure de Lebesgue nulle, croît comme la distance à la frontière F de F élevée à la puissance...

The density of the area integral in + n + 1

Richard F. Gundy, Martin L. Silverstein (1985)

Annales de l'institut Fourier

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Let u ( x , y ) be a harmonic function in the half-plane R + n + 1 , n 2 . We define a family of functionals D ( u ; r ) , - > r > , that are analogs of the family of local times associated to the process u ( x t , y t ) where ( x t , y t ) is Brownian motion in R + n + 1 . We show that D ( u ) = sup r D ( u ; r ) is bounded in L p if and only if u ( x , y ) belongs to H p , an equivalence already proved by Barlow and Yor for the supremum of the local times. Our proof relies on the theory of singular integrals due to Caldéron and Zygmund, rather than the stochastic calculus.