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Embeddings and duality theorem for weak classical Lorentz spaces. (English) Zbl 1106.26018
The authors introduce and investigate four types of the so-called classical Lorentz spaces. Let \((X,\mu)\) be a totally \(\sigma\)-finite measure space with a non-atomic measure \(\mu\) and \(\mathcal M(X,\mu)\) the set of all extended complex-valued \(\mu\)-measurable functions on \(X\). Let \(u, v, w\) be weights on \((0,\infty)\) and set \[ U(t):=\int_0^t u(s)\,ds,\quad V(t):=\int_0^t v(s)\,ds,\quad W(t):=\int_0^t w(s)\,ds. \] For \(f\in\mathcal M(X,\mu)\), let
\[ \begin{alignedat}{2} f_*(t)&:=\mu(\{x\in X: |f(x)|>t\}),&\quad &t>0,\\ f^*(t)&:=\inf\{s>0: f_*(s)\leq t\},&\quad &t\in[0,\mu(X)),\\ f_u^{**}(t)&:=\frac{1}{U(t)} \int_0^tf^*(s)u(s)\,ds,&\quad &t\in (0,\infty).\end{alignedat} \] Let \(q\in (0,\infty)\). Assume that \(u\) is such that \(U\) is admissible. Let \[ \varphi(t):=\text{ess\,sup}_{s\in(0,t)} U(s)\text{ ess sup}_{\tau\in(s,\infty)}\frac{v(\tau)}{U(\tau)},\quad t\in(0,\infty), \] be non-degenerate with respect to \(U\). Let \(\lambda\) be the representation measure of \(U^q/\varphi^q\) with respect to \(U^q\). The authors show that
(i) If \(1\leq q < \infty\), then \[ \left(\int_0^\infty f^*(t)^q w(t)\,dt\right)^{1/q}\lesssim \text{ess sup}_{t\in(0,\infty)}f_u^{**}(t)v(t)\tag{1} \] holds for all \(f\in\mathcal M(X,\mu)\) if and only if \[ \left(\int_0^\infty \sup_{s\in(t,\infty)}\frac{W(s)}{U(s)^q} d\lambda(t)\right)^{1/q}<\infty. \] (ii) If \(0 < q < 1\), then (1) holds for all \(f\in\mathcal M(X,\mu)\) if and only if
\[ \left(\int_0^\infty \frac{\zeta(t)}{U(t)^q}d\lambda(t)\right)^{1/q}<\infty, \] where \[ \zeta(t):=W(t)+U(t)^q\left(\int_t^\infty\left(\frac{W(s)}{U(s)}\right) ^{q/(1-q)} w(s)ds\right)^{1-q},\quad t\in(0,\infty). \] The proof is presented in a precise and elegant way.

26D10 Inequalities involving derivatives and differential and integral operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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