Mapping properties of integral averaging operators

H. Heinig; G. Sinnamon

Studia Mathematica (1998)

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

Abstract

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Characterizations are obtained for those pairs of weight functions u and v for which the operators T f ( x ) = ʃ a ( x ) b ( x ) f ( t ) d t with a and b certain non-negative functions are bounded from L u p ( 0 , ∞ ) to L v q ( 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.

How to cite

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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

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  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

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