Oscillation and variation inequalities for singular integrals and commutators on weighted Morrey spaces

Abstract

This paper is devoted to investigating the bounded behaviors of the oscillation and variation operators for Calderón-Zygmund singular integrals and the corresponding commutators on the weighted Morrey spaces. We establish several criterions of boundedness, which are applied to obtain the corresponding bounds for the oscillation and variation operators of Hilbert transform, Hermitian Riesz transform and their commutators with BMO functions, or Lipschitz functions on weighted Morrey spaces.

Keywords

Oscillation / variation / singular integrals / commutators / Morrey spaces / weights

References

[1] Adams D R. A note on Riesz potentials. Duke Math J, 1975, 42: 765–778

[2] Akcoglu M, Jones R L, Schwartz P. Variation in probability, ergodic theory and analysis. Illinois J Math, 1998, 42(1): 154–177

[3] Bougain J. Pointwise ergodic theorem for arithmetic sets. Publ Math Inst Hautes Études Sci, 1989, 69: 5–45

[4] Campbell J T, Jones R L, Reinhold K, Wierdl M. Oscillation and variation for Hilbert transform. Duke Math J, 2000, 105: 59–83<?Pub Caret?>

[5] Campbell J T, Jones R L, Reinhold K, Wierdl M. Oscillation and variation for singular integrals in higher dimension. Trans Amer Math Soc, 2003, 355: 2115–2137

[6] Chen S, Wu H, Xue Q. A note on multilinear Muckenhoupt classes for multiple weights. Studia Math, 2014, 223(1): 1–18

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

[8] Crescimbeni R, Martin-Reyes F J, Torre A L, Torrea J L. The ρ-variation of the Hermitian Riesz transform. Acta Math Sin (Engl Ser), 2010, 26: 1827–1838

[9] Garćıa-Cuerva J. Weighted H^p space. Dissertations Math, 1979, 162: 1–63

[10] Gillespie T A, Torrea J L. Dimension free estimates for the oscillation of Riesz transforms. Israel J Math, 2004, 141: 125–144

[11] Grafakos L. Classical and Modern Fourier Analysis. Upper Saddle River: Pearson Education, Inc, 2004

[12] Jones R L. Ergodic theory and connections with analysis and probability. New York J Math, 1997, 3A: 31–67

[13] Jones R L. Variation inequalities for singular integrals and related operators. Contemp Math, 2006, 411: 89–121

[14] Jones R L, Reinhold K. Oscillation and variation inequalities for convolution powers. Ergodic Theory Dynam Systems, 2001, 21: 1809–1829

[15] Komori Y, Shirai S. Weighted Morrey spaces and a singular integral operator. Math Nachr, 2009, 282(2): 219–231

[16] Liu F, Wu H. A criterion on osillation and variation for the commutators of singular integral operators. Forum Math, 2015, 27: 77–97

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

[18] Muckenhoupt B, Wheeden R. Weighted bounded mean oscillation and the Hilbert transform. Studia Math, 1976, 54: 221–237

[19] Stempark K, Torrea J L. Poisson integrals and Riesz transforms for Hermite function expansions with weights. J Funct Anal, 2003, 202: 443–472

[20] Thangavelu S. Lectures on Hermite and Laguerre Expansions. Math Notes 42. Princeton: Princeton Univ Press, 1993

[21] Torchinsky A. Real Variable Methods in Harmonic Analysis. New York: Academic Press, 1986

[22] Zhang J, Wu H. Oscillation and variation inequalities for the commutators of singular integrals with Lipschitz functions. J Inequal Appl, 2015, 214, DOI: 10.1186/s13660-015-0737-x

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg