Abstract
In this paper, we state as a conjecture a vector-valued Hopf–Dunford–Schwartz lemma and give a partial answer to it. As an application of this powerful result, we prove some Fefferman–Stein inequalities in the setting of Dunkl analysis where covering methods are not available.
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Asmar, N., Berkson, E., Gillespie, T.A.: Transference of strong type maximal inequalities by separation preserving representation. Am. J. Math. 113, 47–74 (1991)
Chacon, R.V.: On the ergodic theorem without assumption of positivity. Bull. Am. Math. Soc. 67(2), 186–190 (1961)
de Jeu, M.: The Dunkl transform. Invent. Math. 113(1), 147–162 (1993)
Deleaval, L.: Fefferman–Stein inequalities for the \({\mathbb{Z}}^{d}_{2}\) Dunkl maximal operator. J. Math. Anal. Appl. 360(2), 711–726 (2009)
Deleaval, L.: Two results on the Dunkl maximal operator. Studia Math. 203(1), 47–68 (2011)
Dunford, N., Schwartz, J.T.: Linear operators part I: general theory. In: Pure and Applied Mathematics. Interscience Publishers, Inc., New York (1958)
Dunford, N., Schwartz, J.T.: Convergence almost everywhere of operator averages. J. Math. Mech. 5, 129–178 (1959)
Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311(1), 167–183 (1989)
Dunkl, C.F.: Hankel transforms associated to finite reflection groups. Contemp. Math. 138, 123–138 (1992)
Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math. 93(1), 107–115 (1971)
Garciá-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics. In: North-Holland Math. Studies, vol. 116. North-Holland, Amsterdam (1985)
Grafakos, L.: Classical fourier analysis. In: Graduate Texts in Mathematics. Springer, Berlin (2008)
Hardy, G.H., Littlewood, J.E.: A maximal theorem with function-theoretic applications. Acta Math. 54(1), 81–116 (1930)
Hopf, E.: The general temporally discrete Markoff process. J. Math. Mech. 3, 13–45 (1954)
Krengel, U.: Ergodic theorems. In: de Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter & Co., Berlin (1985)
Opdam, E.M.: Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group. Compositio Math. 85(3), 333–373 (1993)
Rösler, M.: Bessel-type signed hypergroups on R. Probability Measures on Groups and Related Structures, vol. XI, 292–304 (Oberwolfach, 1994) (1995)
Rösler, M.: Dunkl operators: theory and applications, Orthogonal polynomials and special functions (Leuven, 2002). Lecture Notes in Mathematics, vol. 1817, pp. 93–135. Springer, Berlin (2003) & Co., Berlin (1985)
Rösler, M.: Generalized Hermite polynomials and the heat equation for Dunkl operators. Commun. Math. Phys. 192(3), 519–542 (1998)
Rubio de Francia, J.L.: Factorization theory and \(A_p\) weights. Am. J. Math. 106(1), 533–547 (1984)
Schwartz, J.T.: Another proof of E. Hopf’s ergodic lemma. Commun. Pure Appl. Math. 12(1), 399–401 (1959)
Stein, E.M.: Topics in harmonic analysis related to the Littlewood–Paley theory. In: Annals of Mathematics Studies, vol. 63. Princeton University Press, Princeton (1970)
Strömberg, J.-O.: Weak type \(L^{1}\) estimates for maximal functions on noncompact symmetric spaces. Ann. Math. (2) 114(1), 115–126 (1981)
Taggart, R.J.: Pointwise convergence for semigroups in vector-valued Lp spaces. Math. Z. 261, 933–949 (2009)
Thangavelu, S., Xu, Y.: Convolution operator and maximal function for the Dunkl transform. J. Anal. Math. 97, 25–55 (2005)
Yoshimoto, T.: Vector-valued ergodic theorems for operators satisfying norm conditions. Pac. J. Math. 85(2), 485–499 (1979)
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Charpentier, S., Deleaval, L. On a vector-valued Hopf–Dunford–Schwartz lemma. Positivity 17, 899–910 (2013). https://doi.org/10.1007/s11117-012-0211-7
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DOI: https://doi.org/10.1007/s11117-012-0211-7