@IgorKhavkine The counter-example presented in the answer to your question concerns the lift of the fundamental representation of the (special) Lorentz group to the space of symmetric contravariant 2-tensors, but your question is about whether closed orbits of the representation can be separated by invariant polynomials. I'm missing the connection (due to my own ignorance, sorry) of that with the failure of Schwarz's theorem. Do invariant smooth functions necessarily separate closed orbits even if the group is non-compact? If so, then I can see why Schwarz's theorem must fail in that case.

@ThomasRichard Thanks. Interesting, I had no idea that a counter-example to this could be that simple... If non-properness is key to the failure of Schwarz's theorem, that could potentially happen to the fundamental representation of the Lorentz group, as the isotropy group of a nonzero lightlike vector is non-compact and hence this action is non-proper.

@ThomasRichard could you please provide at least some of these counterexamples when $G$ is non compact? Do they include, for instance, some pseudo-orthogonal group?

Sorry for the delay in getting back to you. The Bernstein inequality in the OP essentially follows in the case of $\mathbb{R}^d$ from the second inequality in Proposition 1.3.1 in the case $N=2^n$ if we remove the first $n$ terms from the sum in the lhs, up to normalization terms. In the case of $\mathbb{T}^d$, the argument is exactly the same but one then replaces the Littlewood-Paley projections $\Delta_q u$ with its Fourier components $\langle e_k,u\rangle e_k=\hat{u}_k e_k$. Prop. 1.3.1 in that case then follows from integration by parts and the Parseval formula.

(continued) Removing the interaction cutoff (the so-called "adiabatic limit") can only be done for the matrix elements of the S matrix, and even so can only be done in the presence of a mass gap, as shown already by Epstein and Glaser. This is consistent with the underlying hypotheses of the LSZ reduction formulae. In the case of QED, one needs to compute inclusive cross sections directly and show that soft photon contributions cancel out in their adiabatic limit, as done e.g. by the Bloch-Nordsieck approximation. This is pointed in Scharf's book as well.

Scharf doesn't address the LSZ reduction formulae directly, but the Bogolyubov S matrix has them somewhat built in. More precisely, the space-time interaction cutoff ("adiabatic switching") evades Haag's theorem, so one can construct the S matrix at the operator level using the interaction picture as long as the cutoff is not removed. Moreover, the fact that the interaction has compact space-time support makes it easier to relate it to time-ordered products. In fact, Bogolyubov's formula states that the cutoff S matrix is the generating function of the (cutoff) time-ordered products.

One can upgrade the LSZ reduction formulae to the strong sense (i.e. for Hilbert space vectors) in the case of non-collinear momenta for the multiparticle "in" and "out" arrangements using the Haag-Ruelle QFT scattering theory, provided the mass gap hypothesis still holds. If the latter fails, the S matrix no longer exists in the usual sense due to the presence of soft massless particles (i.e. with arbitrarily small 4-momenta), and formulating a proper QFT scattering theory in this scenario in a rigorous way still is a major open problem.

No, LSZ doesn't violate Haag's theorem if formulated correctly. Namely, if the energy-momentum spectum has a mass gap, the LSZ reduction formulae hold true in the weak sense, that is, for the matrix elements of the S matrix. That's why the time-ordered products only enter there as their vacuum expectation values. Haag's theorem prevents these formulae from holding true in the operator sense, that's why so-called "derivations" of LSZ from the interaction picture (as still done e.g. by QFT books like Peskin-Schroeder) are fundamentally flawed.

Answer to 3. - have a look at A. Pietsch, Nuclear Locally Convex Spaces (Springer-Verlag, 1972). H. Jarchow's Locally Convex Vector Spaces (B.G. Teubner, 1981), the second volume of G. Köthe's book on topological vector spaces and the fourth volume of Gel'fand's "Generalized Functions" (with N. Ya. Vilenkin as the second author) should also have information on that.

The first two volumes of Gel'fand-Shilov's Generalized Functions consider some of these intermediate test function spaces and their corresponding topological duals = "spaces of generalized functions", as far as I remember. It's the first place I'd look for that, at least.

More generally, there is no loss of information to restrict oneself to vacuum expectation values thanks to the Wightman-GNS and Osterwalder-Schrader reconstruction theorems, which allow us to recover the field operators and their products.

(continued) ...and from there one can recover the time-ordered field operator products through its matrix elements (at least as sesquilinear forms). These matrix elements are obtained by time ordering subsets of the space-time arguments of the Wightman functions. Also, the main use for time-ordered products is for the S-matrix elements through the LSZ reduction formulae, and there the former only enter as their vacuum expectation values = time-ordered Green functions, so the latter are the objects that really matter in practice.

Moreover, as I said in the edit, it's not a trivial problem to define time-ordered Green functions because of the renormalization problem. A similar problem happens with time-ordered field products, only worse because you're dealing with (possibly unbounded) operator-valued distributions and hence have even less analytic control over the objects you're trying to construct. The renormalization problem for time-ordered products is easier to manage for the Green functions...