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Bilinear forms on non-homogeneous Sobolev spaces.
- Source :
- Forum Mathematicum; Jul2020, Vol. 32 Issue 4, p995-1026, 32p
- Publication Year :
- 2020
-
Abstract
- In this paper, we show that if b ∈ L 2 (ℝ n) {b\in L^{2}(\mathbb{R}^{n})} , then the bilinear form defined on the product of the non-homogeneous Sobolev spaces H s 2 (ℝ n) × H s 2 (ℝ n) {H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})} , 0 < s < 1 {0<s<1} , by (f , g) ∈ H s 2 (ℝ n) × H s 2 (ℝ n) → ∫ ℝ n (Id - Δ) s 2 (f g) (𝐱) b (𝐱) d 𝐱 (f,g)\in H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})\to\int_{% \mathbb{R}^{n}}(\mathrm{Id}-\Delta)^{\frac{s}{2}}(fg)(\mathbf{x})b(\mathbf{x})% \mathop{}\!d\mathbf{x} is continuous if and only if the positive measure | b (𝐱) | 2 d 𝐱 {\lvert b(\mathbf{x})\rvert^{2}\mathop{}\!d\mathbf{x}} is a trace measure for H s 2 (ℝ n) {H_{s}^{2}(\mathbb{R}^{n})}. [ABSTRACT FROM AUTHOR]
- Subjects :
- SOBOLEV spaces
BILINEAR forms
Subjects
Details
- Language :
- English
- ISSN :
- 09337741
- Volume :
- 32
- Issue :
- 4
- Database :
- Complementary Index
- Journal :
- Forum Mathematicum
- Publication Type :
- Academic Journal
- Accession number :
- 144584195
- Full Text :
- https://doi.org/10.1515/forum-2019-0311