# Some Aspects of Vector-Valued Singular Integrals

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

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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.
2. [Bl1]
Blasco, O. Hardy spaces of vector-valued functions: Duality. Trans. Amer. Math. Soc. 308 (1988), 495–507.
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.
4. [Bl3]
Blasco, O. Interpolation between $$H_{B_0}^p$$ and $$L_{B_1}^p$$. Studia Math. 92 (1989), no. 3, 205–210.
5. [BX]
Blasco, O.; Xu, Q. Interpolation between vector-valued Hardy spaces. J. Funct. Anal. 102 (1991), no. 2, 331–359.
6. [B]
Bloom, S. A commutator theorem and weighted BMO. Trans. Amer. Math. Soc. 292 (1985), 103–122.
7. [Bo1]
Bourgain, J. Some remarks on Banach spaces in which martingale differences are unconditional. Ark. Mat. 21 (1983), 163–168.
8. [Bo2]
Bourgain, J. Extension of a result of Benedek, Calderón and Panzone. Ark. Mat. 22 (1984), 91–95.
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.
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.
12. [C]
Coifman R.,A real variable characterization of Hp. Studia Math. 51 (1974), 269–274.
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.
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.
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.
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.
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.
20. [GR]
García-Cuerva, J.; Rubio de Francia, J.L. Weighted norm inequalities and related topics. North-Holland, Amsterdam, 1985.
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.
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.
23. [H]
Hörmander, L.Estimates for translation invariant operators in Lp spaces. Acta Math. 104 (1960), 93–140.
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.
26. [PP]
Pérez, C.; Pradolini, G. Sharp weighted endpoint estimates for commutators of singular integrals. Michigan Math. J. 49 (2001), no. 1, 23–37.
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.
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.
29. [ST1]
Segovia, C.; Torrea, J.L. Vector-valued commutators and applications. Indiana Univ. Math. J. 38 (1989), no. 4, 959–971.
30. [ST2]
Segovia, C.; Torrea, J.L. A note on the commutator of the Hilbert transform. Rev. Un. Mat. Argentina 35 (1989), 259–264.
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.
32. [ST4]
Segovia, C.; Torrea, J.L. Commutators of Littlewood–Paley sums. Ark. Mat. 31 (1993), no. 1, 117–136.
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.
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