Let $f$ be in the Sobolev space $H^s(\Omega)$. Then the standard definition of the norm is \begin{equation} ||f||_{H^s}^2 = \sum_{|\alpha| \leq s} ||D^{\alpha} f||_{L^2}^2 \qquad\qquad\qquad(1) \end{equation} when $s$ is a positive integer. With Fourier transform, an equivalent norm is formulated as \begin{equation} \|f\|_{H^s}^2=\int|\hat f(\xi)|^2(1+|\xi|^2)^sd\xi\qquad\qquad\qquad(2) \end{equation} where $s$ can be extended to any positive real numbers. My question is whether $(2)$ is equivalent to the following norm \begin{equation} \|f\|_{H^s}^2=\int|\hat f(\xi)|^2(1+|\xi|^{2s})d\xi\qquad\qquad\qquad(3) \end{equation} The following is my attempt for proving $(1+|\xi|^2)^s\approx1+|\xi|^{2s}$. Notice that for any positive number $a$, if $0<s<1$, then $(1+a)^s\leq 1+a^s$ and if $1\leq s$, then $(1+a)^s\leq 2^{s-1}(1+a^s)$. When $0<s<1$, we have $$ (1+a)^{1/s}\leq 2^{\frac{1}{s}-1}(1+a^{1/s}) $$ which leads to $$ 1+b^s\leq 2^{1-s}(1+b)^s \quad \ $$ for any positive number $b$. So in the case of $0<s<1$, we have $$ 1+|\xi|^{2s}\lesssim(1+|\xi|^2)^s\leq 1+|\xi|^{2s}. $$ The case of $1\leq s$ is similar. Is there anything wrong here?
And I also see an equivalent norm defined by (for example see equivalent norm) $$ ||f||_{H^s}^2 = ||f||_{L^2}^2 + \sum_{|\alpha| = s} ||D^{\alpha} f||_{L^2}^2. \qquad\qquad\qquad(4)$$ By $\|f\|_2=\|\hat f\|_2$ and $(\hat{f^{(s)}})(\xi)=(2\pi i\xi)^s\hat f(\xi)$, (4) is truely equivalent to $(3)$. So are $(1),(2),(3),(4)$ equivalent for $\Omega=\mathbb{R}$ or $\Omega$ being a interval?
