Optimal constants for a class of Hausdorff operators on Lebesgue spaces

Abstract

As the extension of classical Hardy operator and Cesàro operator, Hausdorff operator plays an important role in the harmonic analysis, so it is significant to discuss the boundedness of this kind of operator on various function spaces. The article explores the boundedness of a kind of Hausdorff operators on Lebesgue spaces and calculates the optimal constants for the operators to be bounded on such spaces. In addition, the paper also obtains the necessary and sufficient for a kind of multilinear Hausdorff operators to be bounded on Lebesgue spaces and their optimal constants.

Key words： Hausdorff operators; multilinear operators; Lebesgue spaces; optimal constants

References

1 Bennett G. An inequality for Hausdorff means. Houston J Math 1999; 25(4): 709–744

2 Brown G, Móricz F. Multivariate Hausdorff operators on the spaces Lp(Rn). J Math6 Anal Appl 2002; 271(2): 443–454

3 Chen J C, Fan D S, Li J. Hausdorff operators on function spaces. Chin Ann Math Ser B 2012; 33(4): 537–556

4 Chen J C, Fan D S, Wang S L. Hausdorff operators on Euclidean spaces. Appl Math J Chin Univ 2013; 28(4): 548–564

5 Chen J C, Fan D S, Zhang C J. Boundedness of Hausdorff operators on some product Hardy type spaces. Appl Math J Chin Univ 2012; 27(1): 114–126

6 Chen J C, Fan D S, Zhang C J. Multilinear Hausdorff operators and their best constants. Acta Math Sin Engl Ser 2012; 28(8): 1521–1530

7 Fan D S, Zhao F Y. Multilinear fractional Hausdorff operators. Acta Math Sin Engl Ser 2014; 30(8): 1407–1421

8 Georgakis C. The Hausdorff mean of a Fourier-Stieltjes transform. Proc Amer Math Soc 1992; 116(2): 465–471

9 Hardy G H. Note on a theorem of Hilbert. Math Z 1920; 6(3/4): 314–317

10 Hardy G H. An inequality for Hausdorff means. J London Math Soc Ser (1) 1943; 18: 46–50

11 Hausdorff F. Summationsmethoden und Momentfolgen, I. Math Z 1921; 9(1/2): 74–109

12 LiflyandE. Open problems on Hausdorff operators, In: Complex Analysis and Potential Theory: Proceedings of the Conference, Istanbul, Turkey (Azero TA, Tamrazov PM eds). New Jersey: World Scientific, 2006, 280–285

13 Liflyand E, Miyachi A. Boundedness of the Hausdorff operators in Hp spaces, 0<p<1. Studia Math 2009; 194(3): 279–292

14 Ruan J M, Fan D S. Hausdorff operators on the power weighted Hardy spaces. J Math Anal Appl 2016; 433(1): 31–48

15 Wu X M, Chen J C. Best constants for Hausdorff operators on n-dimensional product spaces. Sci China Math 2014; 57(3): 569–578

16 Zhu X R, Chen J C. Integrability of the general product Hardy operators on the product Hardy spaces. Appl Math J Chin Univ 2012; 27(2): 225–233