Helgason-Marchaud inversion formulas for Radon transforms

Let be either the hyperbolic space or the unit sphere , and let be the set of all -dimensional totally geodesic submanifolds of . For and , the totally geodesic Radon transform is studied. By averaging over all at a distance from , and applying Riemann-Liouville fractional differentiation in , S. Helgason has recovered . We show that in the hyperbolic case this method blows up if does not decrease sufficiently fast. The situation can be saved if one employs Marchaud's fractional derivatives instead of the Riemann-Liouville ones. New inversion formulas for , are obtained.

     

Radon transforms and lamplighter random walks

We compute the spectrum of the lamplighter random walk in the case where the underlying graph is a path, representing the state space as a product of two rooted q-ary trees and using suitable Radon transforms. We analyze with the same techniques two additional examples in which the action of the automorphism group on the state space has orbits and the restriction on each orbit is not multiplicity free. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.     

Radon transforms and finite type conditions

We prove regularity of Radon type integral operators in -Sobolev spaces.

     

Topological Radon transforms and degree formulas for dual varieties

We give a simpler and purely topological proof of Ernström's class formula (1997) for the degree of dual varieties. Our new proof also allows us to obtain a formula describing the degrees of the associated varieties studied by Gelfand, Kapranov and Zelevinsky (1994).

     

Mixed norm estimates for certain generalized Radon transforms

We obtain essentially optimal local mixed norm inequalities for certain generalized Radon transforms defined by integration over curves.

     

Martingale transforms and Hardy spaces

Summary Burkholder's martingale transforms are especially useful in studying predictable martingale Hardy spaces. Characterizations of such spaces via martingale transforms are provided. In particular, it is shown that for 0<p<, a martingale inh ^p , defined by the conditioned square function, is the martingale transform of a bmo₂ martingale with a multiplier sequence whose maximal function is inL ^p .     

Isometric embed-dings between classical Banach spaces,cubature formulas,and spherical designs

If an isometric embeddingl _p ^m →l _q ⁿ with finitep, q>1 exists, thenp=2 andq is an even integer. Under these conditions such an embedding exists if and only ifn?N(m, q) where $$\left( {\begin{array}{*{20}c} {m + q/2 - 1} \\ {m - 1} \\ \end{array} } \right) \leqslant N(m,q) \leqslant \left( {\begin{array}{*{20}c} {m + q - 1} \\ {m - 1} \\ \end{array} } \right).$$ To construct some concrete embeddings, one can use orbits of orthogonal representations of finite groups. This yields:N(2,q)=q/2+1 (by regular (q+2)-gon),N(3, 4)=6 (by icosahedron),N(3, 6)?11 (by octahedron), etc. Another approach is based on relations between embeddings, Euclidean or spherical designs and cubature formulas. This allows us to sharpen the above lower bound forN(m, q) and obtain a series of concrete values, e.g.N(3, 8)=16 andN(7, 4)=28. In the cases (m, n, q)=(3, 6, 10) and (3, 8, 15) some ε-embeddings with ε ～ 0.03 are constructed by the orbit method. The rigidness of spherical designs in Bannai's sense and a similar property for the embeddings are considered, and a conjecture of [7] is proved for any fixed (m, n, q).     

New inversion formulas for Radon transforms on affine Grassmannians

We obtain new inversion formulas for the Radon transform and the corresponding dual transform acting on affine Grassmann manifolds of planes in ${\mathbb{R}}^n$. The consideration is performed in full generality on continuous functions and functions belonging to $L^p$ spaces.     

Discrete Radon transforms and applications to ergodic theory

We prove L ^p boundedness of certain non-translation-invariant discrete maximal Radon transforms and discrete singular Radon transforms. We also prove maximal, pointwise, and L ^p ergodic theorems for certain families of non-commuting operators.     

On Efimov spaces and Radon measures

We give a construction under CH of an infinite Hausdorff compact space having no converging sequences and carrying no Radon measure of uncountable type. Under ? we obtain another example of a compact space with no convergent sequences, which in addition has the stronger property that every nonatomic Radon measure on it is uniformly regular. This example refutes a conjecture of Mercourakis from 1996 stating that if every measure on a compact space K is uniformly regular then K is necessarily sequentially compact.     

Metrics and smoothing of translation-invariant Radon transforms along curves

In this paper new L_α^p→L_β^q estimates are proved for translation-invariant Radon transforms along curves for α?β and p<q. For a fixed α and β, if p is sufficiently close to 2 the best possible q is obtained, up to ε. The method is related to that of Greenblatt (Ph.D. Thesis, Princeton University, 1998).     

Null sets and games in Banach spaces

The notion of Aronszajn-null sets generalizes the notion of Lebesgue measure zero in the Euclidean space to infinite dimensional Banach spaces. We present a game-theoretic approach to Aronszajn-null sets, establish its basic properties, and discuss some ensuing open problems.     

Null sets and games in Banach spaces

The notion of Aronszajn-null sets generalizes the notion of Lebesgue measure zero in the Euclidean space to infinite dimensional Banach spaces. We present a game-theoretic approach to Aronszajn-null sets, establish its basic properties, and discuss some ensuing open problems.     

Pfaffian systems and Radon transforms on affine Grassmann manifolds

Mathematische Annalen -