Mapping properties of integral averaging operators

H. Heinig; G. Sinnamon

Studia Mathematica (1998)

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

Abstract

top
Characterizations are obtained for those pairs of weight functions u and v for which the operators with a and b certain non-negative functions are bounded from to , 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.

How to cite

top

Heinig, 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. [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. [2] E. N. Batuev and V. D. Stepanov, Weighted inequalities of Hardy type, Siberian Math. J. 30 (1989), 8-16. Zbl0729.42007
  3. [3] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408. Zbl0402.26006
  4. [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. [5] M. J. Carro and J. Soria, Boundedness of some integral operators, Canad. J. Math. 45 (1993), 1155-1166. Zbl0798.42010
  6. [6] P. Grisvard, Espaces intermédiaires entre espaces de Sobolev avec poids, Ann. Scoula Norm. Sup. Pisa 23 (1969), 373-386. 
  7. [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. [8] G. N. Jakovlev, Boundary properties of functions from the space on domains with angular points, Dokl. Akad. Nauk SSSR 140 (1961), 73-76 (in Russian). 
  9. [9] L. V. Kantorovitch and G. P. Akilov, Functional Analysis, 2nd ed., Pergamon Press, Oxford, 1982. 
  10. [10] B. Opic and A. Kufner, Hardy-Type Inequalities, Longman Sci. Tech., Harlow, 1990. 
  11. [11] E. T. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145-158. Zbl0705.42014
  12. [12] E. T. Sawyer, personal communication, ~1985. 
  13. [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. [14] V. D. Stepanov, Integral operators on the cone of monotone functions, ibid. 48 (1993), 465-487. Zbl0837.26011

NotesEmbed ?

top

You must be logged in to post comments.