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Characterization of the critical Sobolev space on the optimal singularity at the origin
| Title: | Characterization of the critical Sobolev space on the optimal singularity at the origin |
| Authors: | Nagayasu, Sei Browse this author | | Wadade, Hidemitsu Browse this author |
| Keywords: | Sobolev embedding theorem | | Gagliardo-Nirenberg type inequality | | Trudinger type inequality | | Caffarelli-Kohn-Nirenberg inequality |
| Issue Date: | 1-Jun-2010 |
| Publisher: | Elsevier |
| Journal Title: | Journal of Functional Analysis |
| Volume: | 258 |
| Issue: | 11 |
| Start Page: | 3725 |
| End Page: | 3757 |
| Publisher DOI: | 10.1016/j.jfa.2010.02.015 |
| Abstract: | 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. |
| Type: | article (author version) |
| URI: | http://hdl.handle.net/2115/43080 |
| Appears in Collections: | 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)
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Submitter: 永安 聖
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