Abstract
Our main purpose in this article is to establish a Gagliardo-Nirenberg type inequality in the critical Sobolev–Morrey space \(H\mathcal{M}^{\frac{n}{p}}_{p,q}(\mathbb{R}^{n})\) with n∈ℕ and 1<q≤p<∞, which coincides with the usual critical Sobolev space \(H^{\frac{n}{p}}_{p}(\mathbb{R}^{n})\) in the case of q=p. Indeed, we shall show the following interpolation inequality. If q<p, there exists a positive constant C p,q depending only on p and q such that
for all \(u\in H\mathcal{M}^{\frac{n}{p}}_{p,q}( \mathbb{R}^{n})\) and for all p≤r<∞. In the case of q=p, that is, the case of the critical Sobolev space \(H^{\frac{n}{p}}_{p}(\mathbb{R}^{n})\), the corresponding inequality was obtained in Ogawa (Nonlinear Anal. 14:765–769, 1990), Ogawa-Ozawa (J. Math. Anal. Appl. 155:531–540, 1991) and Ozawa (J. Func. Anal. 127:259–269, 1995) with the growth order \(r^{1-\frac{1}{p}}\) as r→∞. The inequality (GN) implies that the growth order as r→∞ is linear, which might look worse compared to the case of the critical Sobolev space. However, we investigate the optimality of the growth order and prove that this linear order is best-possible. Furthermore, as several applications of the inequality (GN), we shall obtain a Trudinger-Moser type inequality and a Brézis-Gallouët-Wainger type inequality in the critical Sobolev-Morrey space.
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Communicated by Hans Triebel.
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Sawano, Y., Wadade, H. On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space. J Fourier Anal Appl 19, 20–47 (2013). https://doi.org/10.1007/s00041-012-9223-8
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DOI: https://doi.org/10.1007/s00041-012-9223-8
Keywords
- Gagliardo-Nirenberg’s inequality
- Trudinger-Moser’s inequality
- Sobolev-Morrey space
- Triebel-Lizorkin-Morrey space