I was reading about Sobolev spaces and came across the notation $\dot{H}^1, \dot{H}^{-1}, \dot{H}^t$. I'm familiar with $H^1, H^{-1}, H^t$, but not the dot, and I can't find these spaces defined anywhere. Is this notation common, and could you explain it to me or point me to a reference?
I have more or less the same question about the spaces $L_t^2,H_x^2$, and the norm $||\cdot||_{L_t^2H_x^2}$.
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I have seen another meaning of $\dot{H}^s(\mathbb{R}^d)$ in Fourier analysis and nonlinear PDEs by Bahouri-Chemin-Danchin. In this book, this is called a homogenous Sobolev space defined as the set of all tempered distributions such that $\hat{u} \in L^1_{loc}$ and $$ ||u||_{\dot{H}^s}^2 := \int_{\mathbb{R}^d} |\xi|^{2s} |\hat{u}(\xi)|^2 d \xi < \infty. $$ This can be compared with the Slobodecki seminorm, and in case $s \in \mathbb{N}$ with the usual Sobolev seminorm. For the definition of $H^s$ without a dot, you would add $1$, i.e. replace $|\xi|^{2s}$ with $(1+|\xi|^{2})^s$.
I have seen the dot notation to mean trace-free: $\dot{H}^1 \equiv H^1_0$. The space $L^2_t,H_x^2$ should mean that the function $u=u(x,t)$ is $L^2$ in time and $H^2$ in space. The norm would be
$$\|u\|^2_{L^2_t,H^2_x} = \int_0^T \|u(\cdot,t)\|^2_{H^2(\Omega)} dt.$$
In general, a common notation is $L^p(0,T; X)$, where $X$ is a Banach space, and the norm is defined as
$$\|u\|_{L^p(0,T;X)} = \left( \int_0^T \|u\|^p_{X} dt\right)^{1/p}.$$
In your case we consider $L^2(0,T;H^2(\Omega))$.