Embeddings of Lorentz-type spaces involving weighted integral means

We solve the problem of characterizing weights on (0,∞) for which the inequality involving two possibly different general inner weighted means(∫0∞(∫0tf⁎(s)m2u2(s)ds)p2m2w2(t)dt)1p2≤C(∫0∞(∫0tf⁎(s)m1u1(s)ds)p1m1w1(t)dt)1p1 holds, where p1,p2,m1,m2∈(0,∞) and p2>m2. The proof is based on a new approach combining duality techniques with sharp weighted estimates for iterated integral and supremum operators.