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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.

##### MSC:
 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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