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Regularity properties of singular integral operators

Abdellah Youssfi — 1996

Studia Mathematica

For s>0, we consider bounded linear operators from D ( n ) into D ' ( n ) whose kernels K satisfy the conditions | x γ K ( x , y ) | C γ | x - y | - n + s - | γ | for x≠y, |γ|≤ [s]+1, | y x γ K ( x , y ) | C γ | x - y | - n + s - | γ | - 1 for |γ|=[s], x≠y. We establish a new criterion for the boundedness of these operators from L 2 ( n ) into the homogeneous Sobolev space s ( n ) . This is an extension of the well-known T(1) Theorem due to David and Journé. Our arguments make use of the function T(1) and the BMO-Sobolev space. We give some applications to the Besov and Triebel-Lizorkin spaces as well as some other potential...

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

Abdellah Youssfi — 1995

Annales de l'institut Fourier

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 is in the B M O -Sobolev...

Continuité-Sobolev de certains opérateurs paradifférentiels.

Abdellah Youssfi — 1990

Revista Matemática Iberoamericana

L'objet de ce travail est l'étude de la continuité des opérateurs d'intégrales singulières (au sens de Calderón-Zygmund) sur les espaces de Sobolev H. Il complète le travail fondamental de David-Journé [6], concernant le cas s = 0, et ceux de P. G. Lemarié [10] et M. Meyer [11] concernant le cas 0 < s < 1.

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