Elliptic operators and their symbols

Elliptic symbols, elliptic operators and Poincaré duality on conical pseudomanifolds

In [7], a notion of noncommutative tangent space is associated with a conical pseudomanifold and the Poincaré duality in K-theory is proved between this space and the pseudomanifold. The present paper continues this work. We show that an appropriate presentation of the notion of symbols on a manifold generalizes right away to conical pseudomanifolds and that it enables us to interpret the Poinc...

Modular Symbols and Hecke Operators

We survey techniques to compute the action of the Hecke operators on the cohomology of arithmetic groups. These techniques can be seen as generalizations in different directions of the classical modular symbol algorithm, due to Manin and Ash-Rudolph. Most of the work is contained in papers of the author and the author with Mark McConnell. Some results are unpublished work of Mark McConnell and ...

Pseudodifferential operators with homogeneous symbols

We prove boundedness of pseudodifferential operators with symbols satisfying the conditions |∂ ξ ∂ xa(x, ξ)| ≤ Cβ,γ |ξ|m−|β|+|γ| on homogeneous Besov-Lipschitz and Triebel-Lizorkin spaces

Pseudodifferential Operators with Rough Symbols

In this work, we develop L boundedness theory for pseudodifferential operators with rough (not even continuous in general) symbols in the x variable. Moreover, the B(L) operator norms are estimated explicitly in terms of scale invariant quantities involving the symbols. All the estimates are shown to be sharp with respect to the required smoothness in the ξ variable. As a corollary, we obtain L...

Toeplitz operators with special symbols