Embeddings between generalized weighted Lorentz spaces

We give a new characterization of a continuous embedding between two function spaces of type $G\Gamma$. Such spaces are governed by functionals of type \begin{equation*} \|f\|_{G\Gamma(r,q;w,\delta)} := \left(\int_{0}^{L} \left( \frac1{\Delta(t)} \int_0^t f^*(s)^r \delta(s) ds \right)^{\frac{q}{r}} w(t) dt \right)^\frac1{q}, \end{equation*} in which $f^*$ is the nonincreasing rearrangement of $f$, $L\in(0,\infty]$, $r,q \in (0, \infty)$, $w, \delta$ are weights on $(0,L)$ and $\Delta(t)=\int_{0}^{t}\delta(s)\,ds$ for $t\in(0,L)$. To characterize the embedding of such a space, say $G\Gamma(r_1,q_1;w_1,\delta_1)$, into another, $G\Gamma(r_2,q_2;w_2,\delta_2)$, 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.