Bilinear forms on non-homogeneous Sobolev spaces.

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]