Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 20 (2025), 133 -- 148

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ON LIPSCHITZ (p,σ)-DOMINATED OPERATORS

Athmane Ferradi

Abstract. This paper focuses on the study of a new class of Lipschitz operators known as Lipschitz (p,σ)-dominated operators, which serve as an interpolating class positioned between Lipschitz p-dominated and Lipschitz mappings. We present a proof of a nonlinear Pietsch domination theorem tailored to this specific class of operators. Additionally, we provide a new characterization of Lipschitz p-dominated operators within the framework of an injective Lipschitz tensor product. As an application, we demonstrate the utility of these results by showcasing new domination-factorization theorems associated with this concept.

2020 Mathematics Subject Classification: 46B28; 47L20; 47B10; 46M05
Keywords: Lipschitz p-dominated; Lipschitz (p,G)-summing; Injective Lipschitz tensor product; Pietsch domination-factorization theorem

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References

  1. D. Achour, P. Rueda, and R. Yahi, (p,σ )-Absolutely Lipschitz operators, Ann. Funct. Anal. 8(2017), 38-50. MR3566889. Zbl 06667768.

  2. O. Blasco and T. Signes, Remarks on (q,p,Y)-summing operators, Quaestiones Mathematicae. 25(2002), 97-103. MR1974408. Zbl 1042.47012.

  3. A. Bougoutaia, A. Belacel and R. Macedo, Strongly (p, σ)-Lipschitz operators, Advances in Operator Theory, 8(2023), 1-17. MR4544658. Zbl 1525.47128.

  4. G. Botelho, D. Pellegrino and P. Rueda, A Unified Pietsch Domination Theorem, J. Math. Anal. Appl. 365(2010), 269-276. MR2585098. Zbl 1193.46026.

  5. G. Botelho, M. Maia, D. Pellegrino and J. Santos, A unified factorization theorem for Lipschitz summing operators. The Quarterly Journal of Mathematics. 70(2019), 1521-1533. MR4045112. Zbl 1545.47047.

  6. M. G. Cabrera-Padilla, J. A. Chávez-Domínguez, A. Jiménez-Vargas, and M. Villegas-Vallecillos, Maximal Banach ideals of Lipschitz maps. Annals of Functional Analysis. 7(2016), 593-608. MR3550938. Zbl 1365.46016.

  7. M. G. Cabrera-Padilla, J. A. Chàvez-Domínguez, A. Jiménez-Vargas and M. Villegas-Vallecillos, Lipschitz Tensor Product, Khayyam J. Math. 1(2015), 185-218. MR3460078. Zbl 1356.46015.

  8. D. Chen and B. Zheng, Lipschitz p-integral operators and Lipschitz p-nuclear operators, Nonlinear Analysis. 75(2012), 5270-5282. MR2927588. Zbl 1263.47025.

  9. E. Dahia, On the tensorial representation of multi-linear ideals, Doctoral Thesis, M’sila University, 2014.

  10. A. Defant and K. Floret, Tensor Norms and Operator Ideals. 176, Amsterdam: North-Holland Mathematics Studies; 1993. MR1209438. Zbl 0774.46018.

  11. J. D. Farmer and W. B. Johnson, Lipschitz p-summing operators, Proc. Amer. Math. Soc. 137(2009), 2989-2995. MR2506457. Zbl 1183.46020.

  12. A. Ferradi and L. Mezrag, Dominated multilinear operators defined on tensor products of Banach spaces. Advances in Operator Theory, 7(2022), 1-19. MR4392391. Zbl 1490.46070.

  13. H. Jarchow and U. Matter, Interpolative constructions for operator ideals. Note. Mat. 8(1988), 45-56. MR1050508. Zbl 0712.47039.

  14. S. V. Kislyakov, Absolutely summing operators on disk algebra, St. Petrsburg Math. J. 4(1992), 705-774. MR1152601. Zbl 0791.47021.

  15. U. Matter, Absolutely continuous operators and super-reflexivity. Math. Nachr. 130(1987), 193-216. MR0885628. Zbl 0622.47045.

  16. B. Maurey, Rappels sur les opérateurs sommants et radonifiants, Séminaire d'analyse fonctionnelle (Polytechnique). I(1973-1974), 1-9. MR0425660. Zbl 0297.47016.

  17. S. Montgomery-Smith and P. Saab, p-summing operators on injective tensor products of spaces, Proc. Roy. Soc. Edinburgh Sect. A 120(1992) 283-296. MR1159186. Zbl 0793.47017.

  18. D. Pellegrino and J. Santos, A General Pietsch Domination Theorem, J. Math. Anal. Appl., 375(2011), 371-374. MR2735722 . Zbl 1210.47047.

  19. D. Pellegrino, J. Santos and J. B. Seoane-Sepúlveda, Some techniques on nonlinear analysis and applications. Adv. Math. 229(2012), 1235-1265. MR2855092. Zbl 1248.47024.

  20. K. Saadi, On the composition ideals of Lipschitz mappings. Banach Journal of Mathematical Analysis. 11(2017), 825-840. MR3708531 . Zbl 1489.47039.

  21. E. A. Sanchez Perez, Ideales de operadores absolutamente continuos y normas tensoriales asociadas, PhD Thesis, Universidad Politecnica de Valencia, Spain, 1997.

  22. A. Tallab, Lipschitz (q, p, E)-summing operators on injective Lipschitz tensor products of spaces. Moroccan Journal of Pure and Applied Analysis. 6(2020), 127-142. Zbl 1549.46087.

  23. N. Weaver, Lipschitz Algebras, World Scientific, Singapore 2018. MR3792558. Zbl 1419.46001.



Athmane Ferradi
Ecole Normale Supèrieure de Bousaada, 28001 Bousaada, Algeria.
Department of Mathematics,
Laboratoire d’Analyse Fonctionnelle et Géométrie des Espaces,
University of M’sila, 28000 M’Sila, Algeria.
e-mail: ferradi.athmane@ens-bousaada.dz , athmane.ferradi@univ-msila.dz

https://www.utgjiu.ro/math/sma