Sobolev inequality

In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev. ∫ | x | ≥ ρ | u ^ ( x ) | 2 d x ≤ ∫ | x | ≥ ρ | x | 2 ρ 2 | u ^ ( x ) | 2 d x ≤ ρ − 2 ∫ R n | D u | 2 d x {displaystyle int _{|x|geq ho }left|{hat {u}}(x) ight|^{2},dxleq int _{|x|geq ho }{frac {|x|^{2}}{ ho ^{2}}}left|{hat {u}}(x) ight|^{2},dxleq ho ^{-2}int _{mathbf {R} ^{n}}|Du|^{2},dx}     (1) ∫ | x | ≤ ρ | u ^ ( x ) | 2 d x ≤ ρ n ω n ‖ u ‖ L 1 2 {displaystyle int _{|x|leq ho }|{hat {u}}(x)|^{2},dxleq ho ^{n}omega _{n}|u|_{L^{1}}^{2}}     (2) In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev. Let W k,p(Rn) denote the Sobolev space consisting of all real-valued functions on Rn whose first k weak derivatives are functions in Lp. Here k is a non-negative integer and 1 ≤ p < ∞. The first part of the Sobolev embedding theorem states that if k > ℓ and 1 ≤ p < q < ∞ are two real numbers such that