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Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.
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Boundary value of Sobolev space
This is not true. Take a sequence of points $(x_n) \subset D$ which converge to a point on the boundary (or even worse: whose accumulation points are $\partial D$).
Then, for every point $x_n$ we can …