Sobolev embedding into bounded continuous derivatives

Statement

For an integer $k\ge0$ and $s>n/2+k$ , each $u\in H^s(\mathbb R^n)$ has a representative in $C^k$ whose derivatives through order $k$ are bounded and continuous, with

\max_{|\alpha|\le k}\|\partial^\alpha u\|_\infty\le C_{n,s,k}\|u\|_{H^s}.

The space $H^s$ is the inhomogeneous Fourier Sobolev space.

Fourier proof

Cauchy–Schwarz bounds $\int |\xi|^{|\alpha|}|\widehat u(\xi)|\,d\xi$ by a constant times $\|u\|_{H^s}$ , since $\int |\xi|^{2|\alpha|}(1+4\pi^2|\xi|^2)^{-s}\,d\xi<\infty$ . The inverse Fourier integrals for these derivatives therefore converge absolutely and define bounded continuous functions. Approximation, or distributional Fourier inversion, identifies them with the derivatives of $u$ . For example, in three dimensions $H^3$ controls both the function and its first derivatives in $L^\infty$ .