Restriction estimates

Question

What is the defining property of what someone in the harmonic analysis community would call a "restriction estimate?" I see sobolev norms, Fourier transforms, and inequalities relating these. The one thing that I don't always see, but would expect given the name, is estimating the "size" of $f$ restricted to a hyperplane vs. $f$ on its entire domain.

Answer 1

To me*, restriction estimate means a bound on the norm of an operator $T:X\to Y$ where

1. $X$ is some space of functions in a domain $\Omega$ (often the entire Euclidean space)
2. $Y$ is some space of functions on a subset $E\subset \Omega$ of strictly smaller dimension than $\Omega$ itself
3. For smooth functions, the operator agrees with the restriction $f\mapsto f_{|E}$.

The set $E$ is not always a hyperplane; it could be a manifold of some sort, or a more general set. A quick look at randomly selected papers with this keyword did not produce counterexamples.

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^{(*) a non-harmonic analyst}