Embeddings between generalized weighted Lorentz spaces

We give a new characterization of a continuous embedding between two function spaces of type $G\mathrm{\Gamma }$. Such spaces are governed by functionals of type \begin{equation*} |f|{GΓ(r,q;w,δ)} := \left(\int{0}^{L} \left( \frac1{Δ(t)} \int_0^t f^(s)^r δ(s) ds \right)^{\frac{q}{r}} w(t) dt \right)^\frac1{q}, \end{equation} in which $f^{\ast }$ is the nonincreasing rearrangement of $f$, $L\in (0,\mathrm{\infty }]$, $r,q\in (0,\mathrm{\infty })$, $w,\delta$ are weights on $(0,L)$ and $\mathrm{\Delta }(t)={\int }_0^t\delta (s)ds$ for $t\in (0,L)$. To characterize the embedding of such a space, say $G\mathrm{\Gamma }(r_1,q_1;w_1,{\delta }_1)$, into another, $G\mathrm{\Gamma }(r_2,q_2;w_2,{\delta }_2)$, means to find a balance condition on the four positive real parameters and the four weights in order that an appropriate inequality holds for every admissible function. We develop a new discretization technique which will enable us to get rid of restrictions on parameters imposed in earlier work such as the non-degeneracy conditions or certain relations between the $r$’s and $q$’s. Such restrictions were caused mainly by the use of duality techniques, which we avoid in this paper. On the other hand we consider here only the case when $q_1\le q_2$, leaving the reverse case to future work.