Characterization of the critical Sobolev space on the optimal singularity at the origin

概要

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In the present paper, we investigate the optimal singularity at the origin for the functions belonging to the critical Sobolev space H^[n/p, p] (ℝ^[n]), 1 < p < ∞. With this purpose, we shall show the weighted Gagliardo-Nirenberg type inequality: ||u|| _[Lq(ℝ^[n] ; dx/|x|s)] ≤ C(1/[n-s])^[1/q + 1/p'] q^[1/p'] ||u|| ^[(n-s)p/nq] _[Lp(ℝ^[n]] ||(-Δ)^[n/2p] u||^[1-[(n-s)p]/nq] _[Lp(ℝ^[n]], (GN) where C depends only on n and p. Here, 0 ≤ s < n and p~ ≤ q < ∞ with some p~ ∈ (p, ∞) determined only by n and p. Additionally, in the case n ≥ 2 and n/[n-1] ≤ p < ∞, we can prove the growth orders for s as s ↑ n and for q as q → ∞ are both optimal. (GN) allows us to prove the Trudinger type estimate with the homogeneous weight. Furthermore, it is obvious that (GN) can not hold with the weight |x|^[n] itself. However, with a help of the logarithmic weight of the type (log 1/|x|)^[r] |x|^[n] at the origin, we cover this critical weight. Simultaneously, we shall give the minimal exponent r = [q+p']/p' so that the continuous embedding can hold.
2010-06-01