Journal of Operator Theory
Volume 59, Issue 2, Spring 2008 pp. 309-332.
Weighted inequalities involving two Hardy operators with applications to embeddings of function spacesAuthors: Maria Carro (1), Amiran Gogatishvili (2), Joaquim Martin (3), and Lubos Pick (4)
Author institution: (1) Departament de Matematica Aplicada i Analisi, Universitat de Barcelona, 08071 Barcelona, Spain
(2) Mathematical Institute, Academy of Sciences of the Czech Republic, Zitna 25, 115 67 Praha 1, Czech Republic
Summary: We find necessary and sufficient conditions for the two-operator weighted inequality $ \Big(\int\limits_0\sp\infty\!\Big(\frac1t\!\int\limits_0\sp t f(s)\mathrm{d}s\Big)\sp q\!w(t)\dd t\Big)\sp{1/q} \!\!\!\leqslant\!\! C \Big(\int\limits_0\sp\infty\!\Big(\int\limits_t\sp{\infty}\!\frac{f(s)}s\dd s\Big)\sp p\!v(t)\dd t\Big)\sp{1/p}. $ We use this inequality to study embedding properties between the function spaces $S\sp p(u)$ equipped with the norm $ \|f\|_{S\sp p(u)}\!=\! \!\Big(\!\int\limits_{0}^{\infty}[f\sp{**}\!(t)\!-\!f\sp*\!(t)]\sp pu(t)\dd t\Big)\sp{1/p} $ and the classical Lorentz spaces $\Lambda\sp p(v)$ and $\Gamma\sp q(w)$. Moreover, we solve the only missing open case of the embedding $\Lambda\sp p(v)\hra\Gamma\sp q(w)$, where $0<q<p\leqslant 1$.
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