Weighted generalized weak type inequalities for modified Hardy operators

• Autores: Pedro Ortega Salvador
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 51, Fasc. 2, 2000, págs. 149-156
• Idioma: inglés
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• Resumen

  • We consider the operator $T_gf(x)=g(x) \int^x_0 f$, where $g$ is a positive nonincreasing function, and characterize the pairs of positive measurable functions $(u,v)$ such that the generalized weak type inequality \begin{center}$\Phi^{-1}_2\Bigg(\Phi_2(\lambda)\int_{\{x\in(0,\infty);\vert T_gf(x)\vert >\lambda\}}u\Bigg)\leq\Phi^{-1}_1\bigg(\int^\infty_0\Phi_1(K\vert f\vert)v\bigg)$\end{center} holds, where either $\Phi_1$ $N$-function and $\Phi_2$ is a positive increasing function such that $\Phi_1\circ\Phi^{-1}_2$ is countably subadditive or $\Phi_1 (t)= t$ and $\Phi_2$ is a positive increasing function whose inverse is countably subadditive.

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