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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?

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It is written in exactly the same way, except that $\lvert \hat{f}(y) \rvert$ now stands for the norm in $\mathbb{C}^n$. –  Giuseppe Negro Jan 20 '13 at 15:43
    
If so, there is no need to divide the definition into scalar-valued functions and vector-valued functions. (We use the norm in $\mathbb{C}^n$ in the definition of the Sobolev norm.) –  H. Shindoh Jan 20 '13 at 15:55
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Yes. I agree. The same holds for $L^p$ spaces. The differences between scalar-valued and vector-valued functions start to appear when it comes to multiplication, since you cannot multiply two vectors. –  Giuseppe Negro Jan 20 '13 at 16:59
    
I see. I think that I should forget the explanation of the book. –  H. Shindoh Jan 21 '13 at 11:35
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1 Answer

up vote 2 down vote accepted

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, $$\|(f_1,\dots,f_m)\|_{H^s}^2 = \sum_{k=1}^m \|f_k\|_{H^s}^2 = \int \sum_{k=1}^m |\widehat {f_k}(y)|^2 (1+|y|^2)^s\,dy$$

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