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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 spaces

Authors 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 (3) Departartement de Matematiques, Universitat Autonoma de Barcelona, Edifici C 08193 Bellaterra, Barcelona, Spain (4) Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Praha 8, 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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