Embedding theorems for Lipschitz and Lorentz spaces on lower Ahlfors regular sets
We prove norm inequalities between Lorentz and Besov–Lipschitz spaces of fractional smoothness. 1. Main results. In what follows we let (F, ρ) be a metric space with a positive σ-finite Borel measure μ. By B(x, r) we denote the open ball centred at x with radius r. We always assume that there exist d > 0 and C1 > 0 such that (1) μ(B(x, r)) ≥ C1r for all 0 < r ≤ 1 and x ∈ F , i.e., the lower Ahlfors d-regularity of F . In particular, F may be a d-set in Rn and μ the d-dimensional Hausdorff measure, or F may be an h-set with h(r) ≥ rd for 0 < r ≤ 1 and μ an h-measure [10, 11, 5, 6, 18]. We denote Lp = Lp(F, μ). We obtain the following inequality of Sobolev type. Theorem 1. If 0 < p < ∞ and 0 < α < d/p, then there exists a constant c = c(d,C1, p, α) such that (2) ‖u‖Lpd/(d−αp) ≤ c ( ‖u‖Lp + ( ρ(x,y)<1 |u(x)− u(y)|p ρ(x, y)d+αp μ(dy)μ(dx) )1/p) for all u ∈ Lp. Under certain additional assumptions we can get rid of the Lp norm on the right hand side. 2010Mathematics Subject Classification: Primary 46E35; Secondary 46E30, 26D15, 60J75.