For a compactly supported function $f: \mathbb{R}^n \to \mathbb{C}$, the Sobolev norm is defined by $$\|f\|_s^2 = \int |\hat{f}(y)|^2(1+|y|^2)^sdy.$$ Here $\hat{f}$ is the Fourier transform of $f$, i.e. $\hat{f}(y) = (2\pi)^{-n}\int e^{-i\langle x,y\rangle} f(x)dx$.
According to Wells (GTM 65), the Sobolev norm can extends to a $\mathbb{C}^n$-valued functions by taking the $s$-norm of the Euclidean norm of the vector. But I can not understand this explanation.
Can you tell me the explicit definition of the Sobolev norm for vector-valued functions?
You can abstract away the particulars of the norm by thinking in terms of direct sum $$H^s(\mathbb R^n,\mathbb C^m) = \underset{m \text{ times}}{\underbrace{H^s(\mathbb R^n,\mathbb C)\oplus\dots \oplus H^s(\mathbb R^n,\mathbb C)}} $$ where the direct sum of Hilbert spaces has a canonical Hilbert space structure, given by the sum of inner products on the components. As a consequence, $$\|(f_1,\dots,f_m)\|_{H^s}^2 = \sum_{k=1}^m \|f_k\|_{H^s}^2 = \int \sum_{k=1}^m |\widehat {f_k}(y)|^2 (1+|y|^2)^s\,dy$$