The concept of t-basis and vector-valued Hardy classes

Authors

BILAL BILALOVFollow
SABINA RAHIB SADIGOVAFollow

Author ORCID Identifier

BILALBİLAL BILALOV: 0000-0003-0750-9339

SABINA SADIGOVA: 0000-0003-4654-0494

Abstract

This paper introduces the concept of a t-basis generated by some bilinear mapping t(·;·). It is considered the vector-valued class L_p(X) := L_p(J; X), 1 ≤ p < +∞, where J = [−π, π] and X is a Banach space with the UMD property, and it is proven that the classical system of exponents {e^int}_n∈ℤ forms a t-basis for L_p(X), 1 < p < +∞. Using this fact, the Hardy vector classes _nH_p^±(X), 1 < p < +∞, different from the classical ones, are defined, and an equivalent definition of these classes is given and some of their properties are studied. In addition, the concept of t-Riesz property of a system of exponentials is introduced in L_p(X), 1 < p < +∞, and it is proved that this system has the t-Riesz property. A new method is given for establishing the Plemelj–Sokhotski formulas for X-valued Cauchy type integrals when X has the UMD property. An abstract analogue of the “1/4-Kadets” theorem is obtained for L₂(H), where H is a Hilbert space.

DOI

10.55730/1300-0098.3588

Keywords

t-basis, UMD space, vector Hardy classes, Plemelj-Sokhotski formulas

First Page

261

Last Page

286

Publisher

The Scientific and Technological Research Council of Türkiye (TÜBİTAK)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Recommended Citation

BILALOV, BILAL and SADIGOVA, SABINA RAHIB (2025) "The concept of t-basis and vector-valued Hardy classes," Turkish Journal of Mathematics: Vol. 49: No. 3, Article 3. https://doi.org/10.55730/1300-0098.3588
Available at: https://journals.tubitak.gov.tr/math/vol49/iss3/3