Characterization of the critical Sobolev space on the optimal singularity at the origin
Resource
Journal of Functional Analysis, 258(11), 3725-3757
Journal
Journal of Functional Analysis
Pages
3725-3757
Date Issued
2010
Date
2010
Author(s)
Nagayasu, Sei
Wadade, Hidemitsu
Abstract
In the present paper, we investigate the optimal singularity at the origin for the functions belonging to the critical Sobolev space Hfrac(n, p), p (Rn), 1 < p < ∞. With this purpose, we shall show the weighted Gagliardo-Nirenberg type inequality:(GN){norm of matrix} u {norm of matrix}Lq (Rn ; frac(d x, | x |s)) ≤ C (frac(1, n - s))frac(1, q) + frac(1, p′) qfrac(1, p′) {norm of matrix} u {norm of matrix}Lp (Rn)frac((n - s) p, n q) {norm of matrix} (- Δ)frac(n, 2 p) u {norm of matrix}Lp (Rn)1 - frac((n - s) p, n q), where C depends only on n and p. Here, 0 ≤ s < n and over(p, ̃) ≤ q < ∞ with some over(p, ̃) ∈ (p, ∞) determined only by n and p. Additionally, in the case n ≥ 2 and frac(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) cannot hold with the weight | x |n itself. However, with a help of the logarithmic weight of the type (log frac(1, | x |))r | x |n at the origin, we cover this critical weight. Simultaneously, we shall give the minimal exponent r = frac(q + p′, p′) so that the continuous embedding can hold. © 2010 Elsevier Inc. All rights reserved.
Type
journal article
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