Mapping properties of integral averaging operators
Heinig, H. ; Sinnamon, G.
Studia Mathematica, Tome 129 (1998), p. 157-177 / Harvested from The Polish Digital Mathematics Library
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Characterizations are obtained for those pairs of weight functions u and v for which the operators Tf(x)=ʃa(x)b(x)f(t)dt with a and b certain non-negative functions are bounded from Lup(0,∞) to Lvq(0,∞), 0 < p,q < ∞, p≥ 1. Sufficient conditions are given for T to be bounded on the cones of monotone functions. The results are applied to give a weighted inequality comparing differences and derivatives as well as a weight characterization for the Steklov operator.

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:216496 
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Heinig, H.; Sinnamon, G. Mapping properties of integral averaging operators. Studia Mathematica, Tome 129 (1998) pp. 157-177. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv129i2p157bwm/

[00000] [1] M. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for non-increasing functions, Trans. Amer. Math. Soc. 320 (1990), 727-735. | Zbl 0716.42016

[00001] [2] E. N. Batuev and V. D. Stepanov, Weighted inequalities of Hardy type, Siberian Math. J. 30 (1989), 8-16. | Zbl 0729.42007

[00002] [3] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408. | Zbl 0402.26006

[00003] [4] V. Burenkov and W. D. Evans, Hardy inequalities for differences and the extension problem for spaces with generalized smoothness, to appear. | Zbl 0922.46033

[00004] [5] M. J. Carro and J. Soria, Boundedness of some integral operators, Canad. J. Math. 45 (1993), 1155-1166. | Zbl 0798.42010

[00005] [6] P. Grisvard, Espaces intermédiaires entre espaces de Sobolev avec poids, Ann. Scoula Norm. Sup. Pisa 23 (1969), 373-386.

[00006] [7] H. P. Heinig, A. Kufner and L.-E. Persson, On some fractional order Hardy inequalities, J. Inequalities Appl. 1 (1997), 25-46. | Zbl 0880.26021

[00007] [8] G. N. Jakovlev, Boundary properties of functions from the space Wp(l) on domains with angular points, Dokl. Akad. Nauk SSSR 140 (1961), 73-76 (in Russian).

[00008] [9] L. V. Kantorovitch and G. P. Akilov, Functional Analysis, 2nd ed., Pergamon Press, Oxford, 1982.

[00009] [10] B. Opic and A. Kufner, Hardy-Type Inequalities, Longman Sci. Tech., Harlow, 1990.

[00010] [11] E. T. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145-158. | Zbl 0705.42014

[00011] [12] E. T. Sawyer, personal communication, ~1985.

[00012] [13] G. Sinnamon and V. Stepanov, The weighted Hardy inequality: New proofs and the case p=1, J. London Math. Soc. (2) 54 (1996), 89-101. | Zbl 0856.26012

[00013] [14] V. D. Stepanov, Integral operators on the cone of monotone functions, ibid. 48 (1993), 465-487. | Zbl 0837.26011