On the Geometric Mean Operator
1994; Elsevier BV; Volume: 183; Issue: 3 Linguagem: Inglês
10.1006/jmaa.1994.1172
ISSN1096-0813
Autores Tópico(s)Analytic and geometric function theory
ResumoWe give a characterization of pairs of weights (u, v) such that the geometric mean operator Gf(x) = exp((1/x) ∫x0 log ƒ(t) dt), defined for ƒ > 0 almost everywhere on (0, ∞), is bounded from Lp,v (0, ∞) to Lq,u (0, ∞), where 0 < q < p ≤ ∞. Our proofs are based on the rather surprising but simple observation that in the case v ≡ 1 and p > 1 the good weights for G coincide with those good for the averaging operator Af(x) = (1/x) ∫x0 ƒ(t) dt. Our result applies to a certain independence on p, q of weighted Lp − Lq inequalities involving the operator A.
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