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)

It is proved that the natural embedding of a separable Banach space$X$into the corresponding Bourgain–Pisier space extends $\mathcal{L}_{\infty}$ -valued operators.

Moduli of path families are widely used to mark curves which may be neglected for some applications. We introduce ordinary and approximation modulus with respect to Banach function spaces. While these moduli lead to the same result in reflexive spaces, we show that there are important path families (like paths tangent to a given set) which can be labeled as negligible by the approximation modulus with respect to the Lorentz Lp,1-space for an appropriate p, in particular, to the ordinary L1-space if p=1, but not by the ordinary modulus with respect to the same space.

We consider a multidimensional version of an inequality due to Leray as a substitute for Hardy’s inequality in the case $p = n ≥ 2$. In this paper we provide an optimal Sobolev-type improvement of this substitute, analogous to the corresponding improvements obtained for $p = 2 < n$ in S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (1) (2002) 186–233, and for $p > n ≥ 1$ in G. Psaradakis, An optimal Hardy-Morrey inequality, Calc. Var. Partial Differential Equations 45 (3-4) (2012) 421–441.

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.