Journal of Mathematics of Kyoto University

On the inclusion of some Lorentz spaces

A. Turan Gürkanli

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Let $(X,\Sigma ,\mu )$ be a measure space. It is well known that $l^{p}(X) \subseteq l^{q}(X)$ whenever $0 < p \leq q \leq \infty$. Subramanian [12] characterized all positive measures $\mu$ on $(X,\Sigma )$ for which $L^{p}(\mu ) \subseteq L^{q}(\mu )$ whenever $0 < p \leq q \leq \infty$ and Romero [10] completed and improved some results of Subramanian [12]. Miamee [6] considered the more general inclusion $L^{p}(\mu ) \subseteq L^{q}(\nu )$ where $\mu$ and $\nu$ are two measures on $(X,\Sigma )$.

Let $L(p_{1}, q_{1})(X,\mu )$ and $L(p_{2},q_{2})(X,\nu )$ be two Lorentz spaces,where $0 < p_{1}, p_{2} < \infty$ and $0 < q_{1}, q_{2} \leq \infty$. In this work we generalized these results to the Lorentz spaces and investigated that under what conditions $L(p_{1}, q_{1})(X,\mu ) \subseteq L(p_{2},q_{2})(X,\nu )$ for two different measures $\mu$ and $\nu$ on $(X,\Sigma )$.

Article information

J. Math. Kyoto Univ., Volume 44, Number 2 (2004), 441-450.

First available in Project Euclid: 14 August 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)


Gürkanli, A. Turan. On the inclusion of some Lorentz spaces. J. Math. Kyoto Univ. 44 (2004), no. 2, 441--450. doi:10.1215/kjm/1250283559.

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