Calderón-Zygmund operators and unconditional bases of weighted Hardy spaces

J. García-Cuerva; K. Kazarian

ArticleOriginal scientific text

Title

Calderón-Zygmund operators and unconditional bases of weighted Hardy spaces

Authors ¹, ¹

Affiliations

Departamento de Matemáticas, C-XV Universidad Autónoma, 28049 Madrid, Spain

Abstract

We study sufficient conditions on the weight w, in terms of membership in the

A_{p}

classes, for the spline wavelet systems to be unconditional bases of the weighted space

H^{p} (w)

. The main tool to obtain these results is a very simple theory of regular Calderón-Zygmund operators.

Keywords

ENG

wavelets, splines,

H^{p}

spaces,

A_{p}

weights, Schauder and unconditional bases

[B] S. Banach, Théorie des opérations linéaires, Warszawa, 1932; English transl.: Elsevier, 1987.
[Bo] S. V. Bochkarev, Existence of bases in the space of analytic functions and some properties of the Franklin system, Mat. Sb. 98 (1974), 3-18.
[Ca] L. Carleson, An explicit unconditional basis in $H^{1}$ , Bull. Sci. Math. 104 (1980), 405-416.
[C-C] A. Chang and Z. Ciesielski, Spline characterizations of $H^{1}$ , Studia Math. 75 (1983), 183-192.
[C1] Z. Ciesielski, Properties of the orthonormal Franklin system, ibid. 23 (1963), 141-157.
[C2] Z. Ciesielski, Properties of the orthonormal Franklin system II, ibid. 27 (1966), 289-323.
[C-F] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, ibid. 51 (1974), 241-250.
[D] G. David, Wavelets and Singular Integrals on Curves and Surfaces, Lecture Notes in Math. 1465, Springer, 1991.
[G] J. García-Cuerva, Weighted $H^{p}$ spaces, Dissertationes Math. 162 (1979).
[G1] J. García-Cuerva, Extrapolation of weighted norm inequalities from endpoint spaces to Banach lattices, J. London Math. Soc. (2) 46 (1992), 280-294.
[G-R] J. García-Cuerva and J.-L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 114, 1985.
[H-M-W] R. A. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-252.
[J-N] R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_{p}$ , Rev. Mat. Iberoamericana 3 (1987), 249-273.
[K] K. S. Kazarian, On bases and unconditional bases in the spaces $L^{p} (d μ)$ , 1 ≤ p < ∞ , Studia Math. 71 (1982), 227-249.
[Ma] S. G. Mallat, Multiresolution approximation and wavelet orthonormal bases of $L^{2} (ℝ)$ , Trans. Amer. Math. Soc. 315 (1989), 69-87.
[Mau] B. Maurey, Isomorphismes entre espaces $H^{1}$ , Acta Math. 145 (1980), 79-120.
[Me] Y. Meyer, Ondelettes et Opérateurs, Vols. I and II, Hermann, Paris, 1990.
[Mu] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226.
[S-S] P. Sjölin and J. O. Strömberg, Basis properties of Hardy spaces, Ark. Mat. 21 (1983), 111-125.
[St] J. O. Strömberg, A modified Franklin system and higher order spline systems on $ℝ^{n}$ as unconditional bases for Hardy spaces, in: Proc. Conf. in Honor of Antoni Zygmund, W. Beckner, A. P. Calderón, R. Fefferman and P. W. Jones (eds.), Wadsworth, 1981, 475-493.
[S-T] J. O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, 1989.
[S-W] J. O. Strömberg and R. Wheeden, Fractional integrals on weighted $H^{p}$ and $L^{p}$ spaces, Trans. Amer. Math. Soc. 287 (1985), 293-321.
[W] P. Wojtaszczyk, The Franklin system is an unconditional basis in $H^{1}$ , Ark. Mat. 20 (1982), 293-300.
[W1] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Stud. Adv. Math. 25, 1991.

Title

Calderón-Zygmund operators and unconditional bases of weighted Hardy spaces

Affiliations

Abstract

Keywords

Bibliography