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Let $\Omega$ be a bounded (at least) Lipschitz domain in $\mathbb{R}^{N}$. Its boundary $\partial\Omega$, if I'm right, is $(N-1)$-dimensional object embedded in $\mathbb{R}^N$.

We can define Sobolev spaces $H^1(\partial\Omega)$. My question is what dimension should I think of for $\partial\Omega$? I want to use Sobolev inequalities on the boundary (= manifolds) where I need to specify the dimension obviously. So should I say that Sobolev spaces over $\partial\Omega$ are spaces on a $(N-1)$-dimensional space or an $N$-dimensional space? The former seems right, but I don't know how to justify this. Thanks.

Edit: my main issue is that to define $H^1(\partial\Omega)$, you need to define the tangential gradient, which is defined as $\nabla_T f = \nabla \tilde f - \nabla \tilde f\cdot \nu$ where $\nu$ is the unit normal and $\tilde f$ is an extension of $f$ to an open set surrounding $\partial\Omega$. This tangential gradient has $N$ components, not $N-1$. SO how can $H^1(\partial\Omega)$ be on a $N-1$-dimensional set?

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The number of components is not the dimension. Here, because there is 1 "constraint" (that the normal component of the tangential derivative is zero), the boundary is indeed a codimension 1 submanifold.

You probably don't actually believe the number of components is the dimension, but this particular scenario has confused you. For example, in $\mathbb{R}^n$ you could have the constraint $x_1 + x_2 = 0$. The set of all points satisfying the constraint is a hyperplane of course.

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