We consider weighted ray-transforms PW (weighted Radon transforms along oriented straight lines) in Rd,d≥2, with strictly positive weights W. We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions on Rd. In addition, the constructed weight W is rotation-invariant continuous and is infinitely smooth almost everywhere on Rd×Sd−1. In particular, by this construction we give counterexamples to some well-known injectivity results for weighted ray transforms for the case when the regularity of W is slightly relaxed. We also give examples of continous strictly positive W such that dimkerPW≥n in the space of infinitely smooth compactly supported functions on Rd for arbitrary n∈N∪{∞}, where W are infinitely smooth for d=2 and infinitely smooth almost everywhere for d≥3.

This paper deals with the moment problem on a (not necessarily finitely generated) commutative unital real algebra A. We define moment functionals on A as linear functionals which can be written as integrals over characters of A with respect to cylinder measures. Our main results provide such integral representations for A+–positive linear functionals (generalized Haviland theorem) and for positive functionals fulfilling Carleman conditions. As an application, we solve the moment problem for the symmetric algebra S(V) of a real vector space V. As a byproduct, we obtain new approaches to the moment problem on S(V) for a nuclear space V and to the integral decomposition of continuous positive functionals on a barrelled nuclear topological algebra A.

We prove that if f is a distribution on RN with N>1 and if ∂jf∈Lpj,σj∩LN,1uloc with 1≤pj≤N and σj=1 when pj=1 or N, then f is bounded, continuous and has a finite constant radial limit at infinity. Here, Lp,σ is the classical Lorentz space and Lp,σuloc is a “uniformly local” subspace of Lp,σloc larger than Lp,σ when p<∞. We also show that f∈BUC if, in addition, ∂jf∈Lpj,σj∩Lquloc with q>N whenever pj<N and that, if so, the limit of f at infinity is uniform if the pj are suitably distributed. Only a few special cases have been considered in the literature, under much more restrictive assumptions that do not involve uniformly local spaces (pj=N and f vanishing at infinity, or ∂jf∈Lp∩Lq with p<N<q). Various similar results hold under integrability conditions on the higher order derivatives of f. All of them are applicable to g∗f with g∈L1 and f as above, or under weaker assumptions on f and stronger ones on g. When g is a Bessel kernel, the results are provably optimal in some cases.

It is shown for the non-centered Hardy–Littlewood maximal operator M that ∥DMf∥1≤Cn∥Df∥1 for all radial functions in W1,1(Rn).

We prove that every lattice in a product of higher-rank simple Lie groups or higher-rank simple algebraic groups over local fields has Vincent Lafforgue’s strong property (T). Over non-Archimedean local fields, we also prove that they have strong Banach property (T) with respect to all Banach spaces with non-trivial type, whereas in general we obtain such a result with additional hypotheses on the Banach spaces. The novelty is that we deal with non-co-compact lattices, such as SLn(Z) for n⩾3. To do so, we introduce a stronger form of strong property (T) which allows us to deal with more general objects than group representations on Banach spaces that we call two-step representations, namely families indexed by a group of operators between different Banach spaces that we can compose only once. We prove that higher-rank groups have this property and that this property passes to undistorted lattices. Full Text (PDF format)