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Some Aspects of Vector-Valued Singular Integrals

  • Oscar Blasco
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

Let A, B be Banach spaces and \(1 < p < \infty. \; T\) is said to be a (p, A, B)- CalderóLon–Zygmund type operator if it is of weak type (p, p), and there exist a Banach space E, a bounded bilinear map \(u: E \times A \rightarrow B,\) and a locally integrable function k from \(\mathbb{R}^n \times \mathbb{R}^n \backslash \{(x, x): x \in \mathbb{R}^n\}\) into E such that
$$T\;f(x) = \int u(k(x, y), f(y))dy$$
for every A-valued simple function f and \(x \notin \; supp \; f.\)

The study of the boundedness of such operators between spaces of vector-valued functions under the classical assumption of the kernel is established.

Given a bounded operator T from \(L_A^p (\mathbb{R}^n) \; {\rm to} \; L^p_B(\mathbb{R}^n), \; {\rm and} \; \mathcal{L}(A) \; {\rm and} \; \mathcal{L}(B)\)-valued functions b 1 and b 2, we define the commutator
$$T_{b_1,b_2}(f) = b_2T(f) - T(b_1 f)$$
for any A-valued simple function f. The boundedness of commutators of (p, A, B)- CalderöLon–Zygmund type operators and operator-valued functions in a space of bounded mean oscillation (BMO), under certain commuting properties on the couple (b 1, b 2), are also analyzed.

Keywords

Banach Space Hardy Space Singular Integral Weak Type Commute Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [BCP]
    Benedek, A.; Calderón, A.P.; Panzone, R. Convolution operators on Banach space valued functions. Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356–365.CrossRefMATHMathSciNetGoogle Scholar
  2. [Bl1]
    Blasco, O. Hardy spaces of vector-valued functions: Duality. Trans. Amer. Math. Soc. 308 (1988), 495–507.CrossRefMATHMathSciNetGoogle Scholar
  3. [Bl2]
    Blasco, O. Boundary values of functions in vector-valued Hardy spaces and geometry of Banach spaces. J. Funct. Anal. 78 (1988), 346–364.CrossRefMATHMathSciNetGoogle Scholar
  4. [Bl3]
    Blasco, O. Interpolation between \(H_{B_0}^p\) and \(L_{B_1}^p\). Studia Math. 92 (1989), no. 3, 205–210.MATHMathSciNetGoogle Scholar
  5. [BX]
    Blasco, O.; Xu, Q. Interpolation between vector-valued Hardy spaces. J. Funct. Anal. 102 (1991), no. 2, 331–359.CrossRefMATHMathSciNetGoogle Scholar
  6. [B]
    Bloom, S. A commutator theorem and weighted BMO. Trans. Amer. Math. Soc. 292 (1985), 103–122.CrossRefMATHMathSciNetGoogle Scholar
  7. [Bo1]
    Bourgain, J. Some remarks on Banach spaces in which martingale differences are unconditional. Ark. Mat. 21 (1983), 163–168.CrossRefMATHMathSciNetGoogle Scholar
  8. [Bo2]
    Bourgain, J. Extension of a result of Benedek, Calderón and Panzone. Ark. Mat. 22 (1984), 91–95.CrossRefMATHMathSciNetGoogle Scholar
  9. [Bu1]
    Burkholder, D.L. A geometrical characterization of the Banach spaces in which martingale differences are unconditional. Ann. of Prob. 9 (1981), 997–1011.CrossRefMATHMathSciNetGoogle Scholar
  10. [Bu2]
    Burkholder, D.L. A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. Conf. Harmonic Analysis in honor of A. Zygmund, (W. Beckner, A.P. Calderón, R. Fefferman and P.W. Jones editors). Wadsworth Inc., 1981, 270–286.Google Scholar
  11. [CZ]
    Calderón, A; Zygmund, A. On the existence of certain singular integrals. Acta Math. 88 (1952), 85–139.CrossRefMATHMathSciNetGoogle Scholar
  12. [C]
    Coifman R.,A real variable characterization of Hp. Studia Math. 51 (1974), 269–274.MATHMathSciNetGoogle Scholar
  13. [CM]
    Coifman, R.; Meyer, Y. Au-dèlá des opérateurs pseudodiffer‘entiels. Astérisque 57 (1978), Soc. Math. France, Paris.Google Scholar
  14. [CRW]
    Coifman, R.; Rochberg, R.; Weiss, G. Factorization theorems for Hardy spaces in several variables. Ann. of Math. 103 (1976), no. 2, 611–635.CrossRefMATHMathSciNetGoogle Scholar
  15. [CP]
    Cruz-Uribe, D.; Pérez, C. Two-weight, weak type norm inequalities for fractional integrals, Calderón–Zygmund operators and commutators. Indiana Univ. Math. J. 49 (2000), no. 2, 697–721.CrossRefMATHMathSciNetGoogle Scholar
  16. [D]
    Duoandikoetxea, J. Fourier Analysis. Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, 2001.Google Scholar
  17. [FS]
    Fefferman C.; Stein E. M.,Hp spaces of several variables. Acta Math. 129 (1972), 137–193.CrossRefMATHMathSciNetGoogle Scholar
  18. [GMT]
    García–Cuerva, J.; Macías, R.; Torrea, J.L. The Hardy–Littlewood property of Banach lattices. Israel J. Math. 83 (1993), no. 1–2, 177–201.CrossRefMATHMathSciNetGoogle Scholar
  19. [GMT2]
    García–Cuerva, J.; Macías, R. A.; Torrea, J.L. Maximal operators and B.M.O. for Banach lattices. Proc. Edinburgh Math. Soc. (2) 41 (1998), no. 3, 585–609.CrossRefMATHMathSciNetGoogle Scholar
  20. [GR]
    García-Cuerva, J.; Rubio de Francia, J.L. Weighted norm inequalities and related topics. North-Holland, Amsterdam, 1985.MATHGoogle Scholar
  21. [HMST]
    Harboure, E.; Macías, R. A.; Segovia, C.; Torrea, J. L. Some estimates for maximal functions on Köthe function spaces. Israel J. Math. 90 (1995), no. 1–3, 349–371.CrossRefMATHMathSciNetGoogle Scholar
  22. [HST]
    Harboure, E., Segovia, C., Torrea, J.L. Boundedness of commutators of fractional and singular integrals for the extreme values of p, Illinois J. Math. 41 no. 4 (1997), 676–700.MathSciNetGoogle Scholar
  23. [H]
    Hörmander, L.Estimates for translation invariant operators in Lp spaces. Acta Math. 104 (1960), 93–140.CrossRefMATHMathSciNetGoogle Scholar
  24. [J]
    Journé, J.L. Calderón–Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón. Lecture Notes 994 Springer-Verlag, Berlin, 1983.Google Scholar
  25. [P1]
    Pérez, C. Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 128 (1995), 163–185.CrossRefMATHMathSciNetGoogle Scholar
  26. [PP]
    Pérez, C.; Pradolini, G. Sharp weighted endpoint estimates for commutators of singular integrals. Michigan Math. J. 49 (2001), no. 1, 23–37.CrossRefMATHMathSciNetGoogle Scholar
  27. [PT2]
    Pérez, C.; Trujillo-González, R. Sharp weighted estimates for vector-valued singular integral operators and commutators. Tohoku Math. J. (2) 55 (2003), no. 1, 109–129.CrossRefMATHMathSciNetGoogle Scholar
  28. [RRT]
    Rubio de Francia, J.L.; Ruiz, F.; Torrea, J.L. Calderó–Zygmund theory for vector-valued functions. Adv. in Math. 62 (1986), 7–48.CrossRefMATHMathSciNetGoogle Scholar
  29. [ST1]
    Segovia, C.; Torrea, J.L. Vector-valued commutators and applications. Indiana Univ. Math. J. 38 (1989), no. 4, 959–971.CrossRefMATHMathSciNetGoogle Scholar
  30. [ST2]
    Segovia, C.; Torrea, J.L. A note on the commutator of the Hilbert transform. Rev. Un. Mat. Argentina 35 (1989), 259–264.MATHMathSciNetGoogle Scholar
  31. [ST3]
    Segovia, C.; Torrea, J.L. Weighted inequalities for commutators of fractional and singular integrals. Conference on Mathematical Analysis (El Escorial, 1989). Publ. Mat. 35 (1991), no. 1, 209–235.CrossRefMATHMathSciNetGoogle Scholar
  32. [ST4]
    Segovia, C.; Torrea, J.L. Commutators of Littlewood–Paley sums. Ark. Mat. 31 (1993), no. 1, 117–136.CrossRefMATHMathSciNetGoogle Scholar
  33. [ST5]
    Segovia, C.; Torrea, J.L. Higher order commutators for vector-valued Calderón–Zygmund operators. Trans. Amer. Math. Soc. 336 (1993), no. 2, 537–556.MATHMathSciNetGoogle Scholar
  34. [S]
    Stein, E.M. Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, (1970).Google Scholar
  35. [SW]
    Stein E.M. and Weiss G., Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press, (1971).Google Scholar

Copyright information

© Birkhäuser Boston 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversidad de ValenciaValenciaSpain

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