Equivalence of operator norm for Hardy-Littlewood maximal operators and their truncated operators on Morrey spaces

Abstract

We will prove that for and , the central Morrey norm of the truncated centered Hardy-Littlewood maximal operator {{\displaystyle M}}_{\gamma }^c equals that of the centered Hardy-Littlewood maximal operator for all . When p = 1 and , it turns out that the weak central Morrey norm of the truncated centered Hardy-Littlewood maximal operator {{\displaystyle M}}_{\gamma }^c equals that of the centered Hardy-Littlewood maximal operator for all . Moreover, the same results are true for the truncated uncentered Hardy-Littlewood maximal operator. Our work extends the previous results of Lebesgue spaces to Morrey spaces.

Keywords

Hardy-Littlewood maximal function / truncated Hardy-Littlewood maximal function / Morrey norms / weak Morrey norms

References

[1] Alvarez J, Lakey J, Guzmán-Partida M. Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures. Collect Math, 2000, 50(1): 1–47

[2] Chiarenza F, Frasca M. Morrey spaces and Hardy-Littlewood maximal function. Rend Mat Appl (7), 1987, 7(3-4): 273–279

[3] Christ M, Grafakos L. Best constants for two nonconvolution inequalities. Proc Amer Math Soc, 1995, 123(6): 1687–1693

[4] Duoandikoetxea J. Fourier Analysis. Grad Stud Math, Vol 29. Providence: Amer Math Soc, 2001

[5] Grafakos L. Classical Fourier Analysis. 3rd ed. Grad Texts in Math, Vol 249. New York: Springer, 2014

[6] Lu S Z, Yan D Y. L^p-boundedness of multilinear oscillatory singular integrals with Calderón-Zygmund kernel. Sci China Ser A, 2002, 45(2): 196–213

[7] Morrey C B Jr. On the solutions of quasi-linear elliptic partial differential equations. Trans Amer Math Soc, 1938, 43(1): 126–166

[8] Peetre J. On the theory of Lp,λ spaces. J Funct Anal, 1969, 4: 71–87

[9] Shi Z S H, Yan D Y. Criterion on Lp1×Lp2→Lq-boundedness for oscillatory bilinear Hilbert transform. Abstr Appl Anal, 2014, Art. ID 712051 (11 pp)

[10] Tang L. Endpoint estimates for multilinear fractional integrals. J Aust Math Soc, 2008, 84(3): 419–429

[11] Wei M Q, Nie X D, Wu D, Yan D Y. A note on Hardy-Littlewood maximal operators. J Inequal Appl, 2016, Paper No 21 (13 pp)

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature