Embeddings between generalized weighted Lorentz spaces

We give a new characterization of a continuous embedding between two function spaces of type GΓ. Such spaces are governed by functionals of type ∥f∥GΓ(r,q;w,δ):=⎛⎝∫L0(1Δ(t)∫t0f∗(s)rδ(s)ds)q/rw(t)dt⎞⎠1/q, where f∗ is the nonincreasing rearrangement of f, L∈(0,∞], r,q∈(0,∞), w,δ are weights on (0,L) and Δ(t)=∫t0δ(s)ds for t∈(0,L). To characterize the embedding of such a space, say GΓ(r1,q1;w1,δ1), into another, GΓ(r2,q2;w2,δ2), 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 enables us to get rid of restrictions on parameters imposed in earlier work such as the nondegeneracy conditions or certain relations between the r’s and the 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 q1≤q2, leaving the reverse case to future work.