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Embeddings and Duality Theorem for Weak Classical Lorentz Spaces |
Canad. Math. Bull. 49(2006), 82-95
Published:2006-03-01
Printed: Mar 2006
Printed: Mar 2006
- Amiran Gogatishvili
- Luboš Pick
Abstract
We characterize the weight functions
$u,v,w$ on $(0,\infty)$ such that
$$
\left(\int_0^\infty f^{*}(t)^
qw(t)\,dt\right)^{1/q}
\leq
C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t),
$$
where
$$
f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{-1}
\int_{0}^{t}f^*(s)u(s)\,ds.
$$
As an application we present a~new simple characterization of
the associate space to the space $\Gamma^ \infty(v)$, determined by the
norm
$$
\|f\|_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t),
$$
where
$$
f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds.
$$
| Keywords: | Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequality |
| MSC Classifications: |
26D10 - Inequalities involving derivatives and differential and integral operators 46E20 - Hilbert spaces of continuous, differentiable or analytic functions |