Commutators of Hardy operators. (English) Zbl 1022.42011

Long, Shunchao; Wang, Jian

J. Math. Anal. Appl. 274, No. 2, 626-644 (2002).

Hardy’s integral inequalities for commutators generating Hardy operators and fractional order Hardy operators with onesided dyadic central mean oscillation functions are established.

Reviewer: James Adedayo Oguntuase (Abeokuta)

MSC:

42B25	Maximal functions, Littlewood-Paley theory
26D10	Inequalities involving derivatives and differential and integral operators
42B30	\(H^p\)-spaces
47B38	Linear operators on function spaces (general)
47B47	Commutators, derivations, elementary operators, etc.

Keywords:

Hardy’s integral inequality; Hardy operator; commutator; CMO; onesided CMO; dyadic CMO; fractional order Hardy operators

References:

[1]	Bicheng, Y.; Zhuohua, Z.; Debnath, L., Note on new generalizations of Hardy’s integral inequality, J. math. anal. appl., 217, 2, 321-327, (1998) · Zbl 0893.26008
[2]	Bicheng, Y.; Zhuohua, Z.; Debnath, L., Generalizations of Hardy integral inequality, Internat. J. math. math. sci., 22, 3, 535-542, (1999) · Zbl 0971.26012
[3]	Bloom, S., A commutator theorem and weighted BMO, Trans. amer. math. soc., 292, 103-122, (1985) · Zbl 0578.42012
[4]	Broadbent, T.A.A., A proof of Hardy’s convergence theorem, J. London math. soc., 3, 242-243, (1928) · JFM 54.0226.01
[5]	Chen, Y.Z.; Lau, K.S., Some new classes of Hardy spaces, J. functional anal., 84, 255-278, (1989) · Zbl 0677.30030
[6]	Coifman, R.R.; Rochberg, R.; Weiss, G., Factorization theorems for Hardy spaces in several variables, Ann. math., 103, 611-635, (1976) · Zbl 0326.32011
[7]	Copson, E., Some integral inequality, Proc. roy. soc. Edinburgh sect. A, 75, 157-164, (1975-76) · Zbl 0331.26015
[8]	Elliott, E.B., A simple exposition of some recently proved facts as to convergency, J. London math. soc., 1, 93-96, (1926) · JFM 52.0207.02
[9]	Garcia-Cuerva, J., Hardy spaces and Beurling algebras, J. London math. soc., 39, 499-513, (1989) · Zbl 0681.42014
[10]	Garcia-Cuerva, J.; Herrero, M.J.L., A theory of Hardy spaces associated to the Herz spaces, Proc. London math. soc., 69, 3, 605-628, (1994) · Zbl 0831.42012
[11]	Hardy, G.H., Note on a theorem of Hilbert, Math. Z., 6, 314-317, (1920) · JFM 47.0207.01
[12]	Hardy, G.H., Note on some points in the integral calculus, Messenger math., 57, 12-16, (1928)
[13]	Hardy, G.H.; Littlewood, J.E.; Polya, G., Inequalities, (1959), Cambridge University Press Cambridge, UK · Zbl 0634.26008
[14]	Izumi, M.; Izumi, S., On some inequalities for Fourier series, J. anal. math., 21, 277-291, (1968) · Zbl 0164.36803
[15]	Lakey, J.D., Constructive decomposition of functions of finite central Mean oscillation, Proc. amer. math. soc., 127, 8, 2375-2384, (1999) · Zbl 0922.42008
[16]	Landau, E., A note on a theorem concerning series of positive terms, J. London math. soc., 1, 38-39, (1926) · JFM 52.0207.01
[17]	Levinson, N., Generalizations of an inequality Hardy, Duke math. J., 31, 389-394, (1964) · Zbl 0126.28101
[18]	Love, E.R., Generalizations of Hardy’s inequality, Proc. roy. soc. Edinburgh sect. A, 100, 237-262, (1985) · Zbl 0584.26011
[19]	Oguntuase, J.A.; Imoru, C.O., New generalizations of Hardy’s integral inequality, J. math. anal. appl., 241, 73-82, (2000) · Zbl 0944.26021
[20]	Pachpatte, B.G., On a new class of Hardy type inequality, Proc. roy. soc. Edinburgh sect. A, 105, 265-274, (1987) · Zbl 0624.26014
[21]	Pachpatte, B.G., On some integral inequality similar to Hardy’s integral inequality, J. math. anal. appl., 129, 596-606, (1988) · Zbl 0638.26013
[22]	Pachpatte, B.G., A note on certain inequality related to Hardy’s inequality, Indian J. pure appl. math., 23, 773-776, (1992) · Zbl 0766.26015
[23]	Pachpatte, B.G., On some generalizations of Hardy’s integral inequality, J. math. anal. appl., 234, 1, 15-30, (1999) · Zbl 0937.26010
[24]	Pachpatte, B.G.; Love, E.R., On some new inequality related to Hardy’s integral inequality, J. math. anal. appl., 149, 17-25, (1990) · Zbl 0706.26017
[25]	Pecaric, J.E.; Love, E.R., Still more generalizations of Hardy’s integral inequality, J. austral. math. soc. ser. A, 58, 1-11, (1995) · Zbl 0849.26011
[26]	Segovia, C.; Torrea, J.L., Higher order commutators for vector-valued calderon – zygmund operators, Trans. amer. math. soc., 336, 537-556, (1993) · Zbl 0799.42009