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Mapping properties of integral averaging operators

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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.
  • Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada
  • Department of Mathematics, University of Western Ontario, London, Ontario, N6A 5B7, Canada
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