Regularity properties of singular integral operators

Abdellah Youssfi

Studia Mathematica (1996)

  • Volume: 119, Issue: 3, page 199-217
  • ISSN: 0039-3223

Abstract

top
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 spaces.

How to cite

top

Youssfi, Abdellah. "Regularity properties of singular integral operators." Studia Mathematica 119.3 (1996): 199-217. <http://eudml.org/doc/216296>.

@article{Youssfi1996,
abstract = {For s>0, we consider bounded linear operators from $D(ℝ^n)$ into $D^\{\prime \}(ℝ^n)$ whose kernels K satisfy the conditions $|∂^\{γ\}_\{x\}K(x,y)| ≤ C_\{γ\}|x-y|^\{-n+s-|γ|\}$ for x≠y, |γ|≤ [s]+1, $|∇_\{y\} ∂^\{γ\}_\{x\} K(x,y)| ≤ C_\{γ\}|x-y|^\{-n+s-|γ|-1\}$ for |γ|=[s], x≠y. We establish a new criterion for the boundedness of these operators from $L^2(ℝ^n)$ into the homogeneous Sobolev space $Ḣ^s(ℝ^n)$. 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 spaces.},
author = {Youssfi, Abdellah},
journal = {Studia Mathematica},
keywords = {Besov spaces; regularity; singular integral operators; Theorem; BMO-Sobolev space; Triebel-Lizorkin spaces},
language = {eng},
number = {3},
pages = {199-217},
title = {Regularity properties of singular integral operators},
url = {http://eudml.org/doc/216296},
volume = {119},
year = {1996},
}

TY - JOUR
AU - Youssfi, Abdellah
TI - Regularity properties of singular integral operators
JO - Studia Mathematica
PY - 1996
VL - 119
IS - 3
SP - 199
EP - 217
AB - For s>0, we consider bounded linear operators from $D(ℝ^n)$ into $D^{\prime }(ℝ^n)$ whose kernels K satisfy the conditions $|∂^{γ}_{x}K(x,y)| ≤ C_{γ}|x-y|^{-n+s-|γ|}$ for x≠y, |γ|≤ [s]+1, $|∇_{y} ∂^{γ}_{x} K(x,y)| ≤ C_{γ}|x-y|^{-n+s-|γ|-1}$ for |γ|=[s], x≠y. We establish a new criterion for the boundedness of these operators from $L^2(ℝ^n)$ into the homogeneous Sobolev space $Ḣ^s(ℝ^n)$. 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 spaces.
LA - eng
KW - Besov spaces; regularity; singular integral operators; Theorem; BMO-Sobolev space; Triebel-Lizorkin spaces
UR - http://eudml.org/doc/216296
ER -

References

top
  1. [1] G. Bourdaud, Analyse fonctionnelle dans l'espace Euclidien, Publ. Math. Paris VII 23, 1987. Zbl0627.46048
  2. [2] G. Bourdaud, Réalisation des espaces de Besov homogènes, Ark. Mat. 26 (1988), 41-54. 
  3. [3] G. Bourdaud, Une algèbre maximale d'opérateurs pseudo-différentiels, Comm. Partial Differential Equations 13 (1988), 1059-1083. Zbl0659.35115
  4. [4] A. P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092-1099. Zbl0151.16901
  5. [5] R. Coifman et Y. Meyer, Au-delà des opérateurs pseudo-différentiels, Astérisque 57 (1978). Zbl0483.35082
  6. [6] G. David and J.-L. Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. 120 (1984), 371-397. Zbl0567.47025
  7. [7] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799. Zbl0551.46018
  8. [8] M. Frazier and B. Jawerth, A discrete transform and applications to distribution spaces, J. Funct. Anal. 93 (1990), 34-170. 
  9. [9] M. Frazier, R. Torres and G. Weiss, The boundedness of Calderón-Zygmund operators on the spaces α , q p , Rev. Math. Iberoamericana 4 (1988), 41-72. Zbl0694.42023
  10. [10] L. Hörmander, Pseudo-differential operators of type 1,1, Comm. Partial Differential Equations 13 (1988), 1085-1111. Zbl0667.35078
  11. [11] L. Hörmander, Continuity of pseudo-differential operators of type 1,1, ibid. 14 (1989), 231-243. Zbl0688.35107
  12. [12] M. Meyer, Une classe d'espace fonctionnels de type BMO. Application aux intégrales singulières, Ark. Mat. 27 (1989), 305-318. Zbl0698.42007
  13. [13] Y. Meyer, Régularité des solutions des équations aux dérivées partielles non linéaires, in: Lecture Notes in Math. 842, Springer, 1980, 293-302. 
  14. [14] Y. Meyer, Ondelettes et Opérateurs, I, II, Hermann, 1990. Zbl0694.41037
  15. [15] L. Päivärinta, Pseudo-differential operators in Hardy-Triebel spaces, Z. Anal. Anwendungen 2 (1983), 235-242. Zbl0544.47046
  16. [16] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Ser. 1, Durham, N.C., 1976. Zbl0356.46038
  17. [17] T. Runst, Pseudo-differential operators of the “exotic” class L 1 , 1 0 in spaces of Besov and Triebel-Lizorkin type, Ann. Global Anal. Geom. 3 (1985), 13-28. Zbl0549.46020
  18. [18] R. S. Strichartz, Bounded mean oscillation and Sobolev spaces, Indiana Univ. Math. J. 29 (1980), 539-558. Zbl0437.46028
  19. [19] M. S. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, 1991. 
  20. [20] R. Torres, Continuity properties of pseudodifferential operators of type 1,1, Comm. Partial Differential Equations 15 (1990), 1313-1328. Zbl0737.35170
  21. [21] H. Triebel, Theory of Function Spaces, Geest & Portig, Leipzig, and Birkhäuser, 1983. 
  22. [22] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992. 
  23. [23] A. Youssfi, Continuité-Besov des opérateurs définis par des intégrales singulières, Manuscripta Math. 65 (1989), 289-310. Zbl0677.42012
  24. [24] A. Youssfi, Commutators on Besov spaces and factorization of the paraproduct, Bull. Sci. Math. 119 (1995), 157-186. Zbl0827.46031
  25. [25] A. Youssfi, Regularity properties of commutators and BMO-Triebel-Lizorkin spaces, Ann. Inst. Fourier (Grenoble) 43 (1995), 795-807. Zbl0827.46030

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.