# Mapping properties of integral averaging operators

Studia Mathematica (1998)

- Volume: 129, Issue: 2, page 157-177
- ISSN: 0039-3223

## Access Full Article

top## Abstract

top## How to cite

topHeinig, H., and Sinnamon, G.. "Mapping properties of integral averaging operators." Studia Mathematica 129.2 (1998): 157-177. <http://eudml.org/doc/216496>.

@article{Heinig1998,

abstract = {Characterizations are obtained for those pairs of weight functions u and v for which the operators $Tf(x) = ʃ_\{a(x)\}^\{b(x)\} f(t)dt$ with a and b certain non-negative functions are bounded from $L^p_u(0,∞)$ to $L^q_v(0,∞)$, 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.},

author = {Heinig, H., Sinnamon, G.},

journal = {Studia Mathematica},

keywords = {integral averaging operator; weight characterizations; Hardy inequalities; Steklov operator; differences; weighted inequalities; derivatives},

language = {eng},

number = {2},

pages = {157-177},

title = {Mapping properties of integral averaging operators},

url = {http://eudml.org/doc/216496},

volume = {129},

year = {1998},

}

TY - JOUR

AU - Heinig, H.

AU - Sinnamon, G.

TI - Mapping properties of integral averaging operators

JO - Studia Mathematica

PY - 1998

VL - 129

IS - 2

SP - 157

EP - 177

AB - Characterizations are obtained for those pairs of weight functions u and v for which the operators $Tf(x) = ʃ_{a(x)}^{b(x)} f(t)dt$ with a and b certain non-negative functions are bounded from $L^p_u(0,∞)$ to $L^q_v(0,∞)$, 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.

LA - eng

KW - integral averaging operator; weight characterizations; Hardy inequalities; Steklov operator; differences; weighted inequalities; derivatives

UR - http://eudml.org/doc/216496

ER -

## References

top- [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. Zbl0716.42016
- [2] E. N. Batuev and V. D. Stepanov, Weighted inequalities of Hardy type, Siberian Math. J. 30 (1989), 8-16. Zbl0729.42007
- [3] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408. Zbl0402.26006
- [4] V. Burenkov and W. D. Evans, Hardy inequalities for differences and the extension problem for spaces with generalized smoothness, to appear. Zbl0922.46033
- [5] M. J. Carro and J. Soria, Boundedness of some integral operators, Canad. J. Math. 45 (1993), 1155-1166. Zbl0798.42010
- [6] P. Grisvard, Espaces intermédiaires entre espaces de Sobolev avec poids, Ann. Scoula Norm. Sup. Pisa 23 (1969), 373-386.
- [7] H. P. Heinig, A. Kufner and L.-E. Persson, On some fractional order Hardy inequalities, J. Inequalities Appl. 1 (1997), 25-46. Zbl0880.26021
- [8] G. N. Jakovlev, Boundary properties of functions from the space ${W}_{p}^{(}l)$ on domains with angular points, Dokl. Akad. Nauk SSSR 140 (1961), 73-76 (in Russian).
- [9] L. V. Kantorovitch and G. P. Akilov, Functional Analysis, 2nd ed., Pergamon Press, Oxford, 1982.
- [10] B. Opic and A. Kufner, Hardy-Type Inequalities, Longman Sci. Tech., Harlow, 1990.
- [11] E. T. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145-158. Zbl0705.42014
- [12] E. T. Sawyer, personal communication, ~1985.
- [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. Zbl0856.26012
- [14] V. D. Stepanov, Integral operators on the cone of monotone functions, ibid. 48 (1993), 465-487. Zbl0837.26011

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.