Limiting weak-type behaviors for Riesz transforms and maximal operators in Bessel setting

Abstract

We establish the limiting weak type behaviors of Riesz transforms associated to the Bessel operators on ℝ⁺; which are closely related to the best constants of the weak type (1; 1) estimates for such operators. Meanwhile, the corresponding results for Hardy-Littlewood maximal operator and fractional maximal operator in Bessel setting are also obtained.

Keywords

Limiting weak-type behaviors / Bessel operators / Riesz transforms / maximal operators

References

[1] Betancor J, Chicco Ruiz A, Fariña J, Rodríuez-Mesa L. Maximal operators, Riesz transforms and Littlewood-Paley functions associated with Bessel operators on BMO. J Math Anal Appl, 2010, 363(1): 310–326

[2] Betancor J,Fariña J, Buraczewski D, Martínez T, Torrea J. Riesz transform related to Bessel operators. Proc Roy Soc Edinburgh Sect A, 2007, 137(4): 701–725

[3] Betancor J, Harboure E, Nowak A, Viviani B. Mapping properties of functional operators in harmonic analysis related to Bessel operators. Studia Math, 2010, 197(2): 101–140

[4] Coifman R, Weiss G. Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes. Lecture Notes in Math, Vol 242. Berlin: Springer-Verlag, 1971

[5] Davis B. On the weak type (1; 1) inequality for conjugate functions. Proc Amer Math Soc, 1974, 44(2): 307–311

[6] Ding Y, Lai X. Behavior of bounds of singular integrals for large dimension. Forum Math, 2016, 28(6): 1015–1030

[7] Ding Y, Lai X. L¹-Dini conditions and limiting behavior of weak type estimates for singular integrals. Rev Mat Iberoam, 2017, 33(4): 1267–1284

[8] Ding Y, Lai X.Weak type (1; 1) behavior for the maximal operator with L¹-Dini kernel. Potential Anal, 2017, 47(2): 169–187

[9] Grafakos L, Kinnunen J. Sharp inequalities for maximal functions associated with general measures. Proc Roy Soc Edinburgh Sect A, 1998, 128(4): 717–723

[10] Hou X, Wu H. On the limiting weak-type behaviors for maximal operators associated to power weighted measure. Canad Math Bull, 2019, 62(2): 313–326

[11] Hu J, Huang X. A note on the limiting weak-type behavior for maximal operators. Proc Amer Math Soc, 2008, 136(5): 1599–1607

[12] Janakiraman P. Weak-type estimates for singular integral and the Riesz transform. Indiana Univ Math J, 2004, 53(2): 533–555

[13] Janakiraman P. Limiting weak-type behavior for singular integral and maximal operators. Trans Amer Math Soc, 2006, 358(5): 1937–1952

[14] Melas A. The best constant for the centered Hardy-Littlewood maximal inequality. Ann of Math, 2003, 157(2): 647–688

[15] Muckenhoupt B, Stein E. Classical expansions and their relation to conjugate harmonic functions. Trans Amer Math Soc, 1965, 118: 17–92

[16] Pan W. Fractional integrals on spaces of homogeneous type. Approx Theory Appl, 1992, 8(1): 1–15

[17] Stein E, Strömberg J. Behavior of maximal functions in ℝⁿ for large n: Ark Mat, 1983, 21(2): 259–269

[18] Villani M. Riesz transforms associated to Bessel operators. Illinois J Math, 2008, 52(1): 77–89

[19] Wu H, Yang D, Zhang J. Oscillation and variation for semigroups associated with Bessel operators. J Math Anal Appl, 2016, 443(2): 848–867

[20] Wu H, Yang D, Zhang J. Oscillation and variation for Riesz transform associated with Bessel operators. Proc Roy Soc Edinburgh Sect A, 2019, 149(1): 169–190

[21] Yang D C, Yang D Y. Real-variable characterizations of Hardy spaces associated with Bessel operators. Anal Appl, 2011, 9(3): 345–368

RIGHTS & PERMISSIONS

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