Weighted inequalities through factorization

• Autores: Eugenio Hernández Rodríguez
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 35, Nº 1, 1991, págs. 141-153
• Idioma: inglés
• Títulos paralelos:
  • Desigualdades ponderadas mediante factorización
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• Resumen

  • In [4] P. Jones solved the question posed by B. Muckenhoupt in [7] concerning the factorization of Ap weights. We recall that a non-negative measurable function w on Rn is in the class Ap, 1 < p < 8 if and only if the Hardy-Littlewood maximal operator is bounded on Lp(Rn, w). In what follows, Lp(X, w) denotes the class of all measurable functions f defined on X for which ||fw1/p||Lp(X) < 8, where X is a measure space and w is a non-negative measurable function on X.

    It has recently been proved that the factorization of Ap weights is a particular case of a general factorization theorem concerning positive sublinear operators. The case in which the operator is bounded from Lp(X, v) to Lp(Y, u), 1 < p < 8, for u and v non-negative measurable functions on X and Y respectively, is treated in [8]. The case in which the operator is bounded from Lp(X, v) to Lq(X, u), 1 < p < q < 8 is treated in [3].

    Our first result is a factorization theorem for weights u and v associated to operators bounded from Lp(X, v) to Lq(Y, u) where X and Y are two, possibly different, measure spaces and p and q are any index between 1 and 8.