Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space $$X = \left\{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{n+1}} < \infty\right\}$$ with norm $\lVert u \rVert_X^2 := \lVert u \rVert_{L^2}^2 + |u|_X^2$.

We can also define $Y=(L^2(\Gamma), H^1(\Gamma))_{\frac 12, J}$ as the interpolation space using the J method. It is known that these spaces are equivalent with an equivalence of norms: $$C_1\lVert u \rVert_X \leq \lVert u \rVert_Y \leq C_2\lVert u \rVert_X$$

My question is, do the constants in the equivalence of norms depend on $\Gamma$ only in a nice way i.e. on the diameter of $\Gamma$, eigenvalues of the Laplacian, curvature, etc? Is there some explicit expression for these constants?

I tried Demengel, Adams and Triebel without much success.


1 Answer 1


Tartar's An Introduction to Sobolev Spaces and Interpolation Spaces Chapter 36 (and the chapters before) might be helpful. His computations seem quite explicit.


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