Global automorphic Sobolev spaces

The goal is legitimization of term-wise differentiation of L spectral expansions, so that computations producing a classical outcome are correct. We are fond of L expansions because they are what Plancherel gives. Typically, L expansions are not continuous, much less differentiable, so the issue cannot be proving classical differentiability, which does not hold. To say that L spectral expansions are term-wise differentiable in a distributional sense is often valid, but too weak, since it is difficult to return from the large world of distributions to the smaller world of L functions. Further, already for Fourier transforms on R, the integral expressing Fourier inversion is not a superposition of L functions, since the exponentials are not in L(R). Notions of L Sobolev spaces are a balance of the simplicity of Hilbert space structures with extensions of notions of differentiability, insofar as solving elliptic partial differential equations of sufficiently high degree can move back to L. That is, Sobolev spaces are within finite distance of L, in terms of basic processes of analysis.