Positivity properties in noncommutative convolution algebras with applications in pseudo-differential calculus

We study , of all such that for every ?∈C^∞₀, where denotes the twisted convolution. We prove that certain boundedness for are completely determined of the behaviour for a at origin, for example that , and that if a(0)<∞, then a∈L²∩L^∞. We use the results in order to determine wether positive pseudo-differential operators belong to certain Schatten-casses or not.     

A spectral radius formula for the Fourier transform on compact groups and applications to random walks

Let G be a compact group, not necessarily abelian, let ? be its unitary dual, and for f∈L¹(G), let fn?f∗?∗f denote n-fold convolution of f with itself and f? the Fourier transform of f. In this paper, we derive the following spectral radius formula

     

Moment conditions and support theorems for Radon transforms on affine Grassmann manifolds

Let G(p,n) and G(q,n) be the affine Grassmann manifolds of p- and q-planes in Rⁿ, respectively, and let be the Radon transform from smooth functions on G(p,n) to smooth functions on G(q,n) arising from the inclusion incidence relation. When p<q and dimG(p,n)=dimG(p,n), we present a range characterization theorem for via moment conditions. We then use this range result to prove a support theorem for . This complements a previous range characterization theorem for via differential equations when dimG(p,n)<dimG(p,n). We also present a support theorem in this latter case.     

Monoids with sub-log-exponential free spectra

A classic result from the 1960s states that the asymptotic growth of the free spectrum of a finite group is sub-log-exponential if and only if is nilpotent. Thus a monoid is sub-log-exponential implies , the pseudovariety of semigroups with nilpotent subgroups. Unfortunately, little more is known about the boundary between the sub-log-exponential and log-exponential monoids.The pseudovariety consists of those finite semigroups satisfying (x^ωy^ω)^ω(y^ωx^ω)^ω(x^ωy^ω)^ω≈(x^ωy^ω)^ω. Here it is shown that a monoid is sub-log-exponential implies . A quick application: a regular sub-log-exponential monoid is orthodox. It is conjectured that a finite monoid is sub-log-exponential if and only if it is , the finite monoids in having nilpotent subgroups. The forward direction of the conjecture is proved; moreover, the conjecture is proved for when is completely (0)-simple. In particular, the six-element Brandt monoid (the Perkins semigroup) is sub-log-exponential.     

Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator

Let γ be the Gauss measure on and the Ornstein-Uhlenbeck operator. For every p in [1,∞)?{2}, set , and consider the sector . The main results of this paper are the following. If p is in (1,∞)?{2}, and , i.e., if M is an Lp(γ)uniform spectral multiplier of in our terminology, and M is continuous on , then M extends to a bounded holomorphic function on the sector . Furthermore, if p=1 a spectral multiplier M, continuous on , satisfies the condition if and only if M extends to a bounded holomorphic function on the right half-plane, and its boundary value M(i·) on the imaginary axis is the Euclidean Fourier transform of a finite Borel measure on the real line. We prove similar results for uniform spectral multipliers of second order elliptic differential operators in divergence form on belonging to a wide class, which contains . From these results we deduce that operators in this class do not admit an H^∞ functional calculus in sectors smaller than .     

The Whitney embedding theorem for tropical torsion modules: Classification of tropical modules

We prove here a tropical version of the well-known Whitney embedding theorem [32] stating that a smooth connected m-dimensional compact differential manifold can be embedded into R^2m+1.The tropical version of this theorem states that a tropical torsion module with m generators can always be embedded into the free tropical module , where p (equals to 2 for m=2, and otherwise) is the number of rows supporting the torsion, when the generators are given by the (independent) columns of a matrix of size n×m.As a corollary, we get that tropical m-dimensional torsion modules are classified by a (m-1)(m(m-1)-1)-parameter family.     

On maximal functions for Mikhlin-Hörmander multipliers

Given Mikhlin-Hörmander multipliers , with uniform estimates we prove an optimal bound in L^p for the maximal function and related bounds for maximal functions generated by dilations. These improve the results in [M. Christ, L. Grafakos, P. Honzík, A. Seeger, Maximal functions associated with multipliers of Mikhlin-Hörmander type, Math. Z. 249 (2005) 223-240].