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Take $u\in W_0^{1,p}(U)$ and extend it by zero outside $U$. The zero extension of W01 belongs to $W^{1,p}({\mathbb{R}}^n)$ and the Sobolev inequality givesBecause $U$ has finite measure, the Holder inequality givesThusThe reverse estimate follows directly from $\Vert Du{\Vert }_{L^p}\le \Vert u{\Vert }_{W^{1,p}}$. After adjusting constants,This is the Poincare inequality on $W_0^{1,p}(U)$.

For $1\le p<n$, define the Sobolev conjugate exponentThe Sobolev inequality, also called the Gagliardo--Nirenberg--Sobolev inequality, states that there is a constant $C=C(n,p)$ such thatfor every $u\in C_c^{\infty }({\mathbb{R}}^n)$, and hence by completion for every $u\in W^{1,p}({\mathbb{R}}^n)$ for which the right formulation applies.

Interpolation between $L^2$ and $L^6$, followed by the three-dimensional Sobolev inequality, givesApply this with $v=P_{N}w$ and use part i:Orthogonal projection is contractive in $H$, soThe Sobolev constant is dimensionless after using the periodic-domain normalization, so the estimate is scale invariant.

For $1\le p<n$, the Sobolev conjugate exponent $p^{\ast }$ is defined byThe first-order Sobolev inequality maps one derivative in $L^p({\mathbb{R}}^n)$ to the function in $L^{p^{\ast }}({\mathbb{R}}^n)$.