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On convolution operators with small support which are far from being convolution by a bounded measure

Edmond Granirer (1994)

Colloquium Mathematicae

Let C V p ( F ) be the left convolution operators on L p ( G ) with support included in F and M p ( F ) denote those which are norm limits of convolution by bounded measures in M(F). Conditions on F are given which insure that C V p ( F ) , C V p ( F ) / M p ( F ) and C V p ( F ) / W are as big as they can be, namely have l as a quotient, where the ergodic space W contains, and at times is very big relative to M p ( F ) . Other subspaces of C V p ( F ) are considered. These improve results of Cowling and Fournier, Price and Edwards, Lust-Piquard, and others.

On the Cauchy problem for convolution equations

(2013)

Colloquium Mathematicae

We consider one-parameter (C₀)-semigroups of operators in the space ' ( ; m ) with infinitesimal generator of the form ( G * ) | ' ( ; m ) where G is an M m × m -valued rapidly decreasing distribution on ℝⁿ. It is proved that the Petrovskiĭ condition for forward evolution ensures not only the existence and uniqueness of the above semigroup but also its nice behaviour after restriction to whichever of the function spaces ( ; m ) , L p ( ; m ) , p ∈ [1,∞], ( a ) ( ; m ) , a ∈ ]0,∞[, or the spaces L q ' ( ; m ) , q ∈ ]1,∞], of bounded distributions.

On the Fourier cosine—Kontorovich-Lebedev generalized convolution transforms

Nguyen Thanh Hong, Trinh Tuan, Nguyen Xuan Thao (2013)

Applications of Mathematics

We deal with several classes of integral transformations of the form f ( x ) D + 2 1 u ( e - u cosh ( x + v ) + e - u cosh ( x - v ) ) h ( u ) f ( v ) d u d v , where D is an operator. In case D is the identity operator, we obtain several operator properties on L p ( + ) with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on L 2 ( + ) and define the inversion formula. Further, for an other class of differential operators of finite...

On the Fourier transform, Boehmians, and distributions

Dragu Atanasiu, Piotr Mikusiński (2007)

Colloquium Mathematicae

We introduce some spaces of generalized functions that are defined as generalized quotients and Boehmians. The spaces provide simple and natural frameworks for extensions of the Fourier transform.

On the Haagerup inequality and groups acting on A ˜ n -buildings

Alain Valette (1997)

Annales de l'institut Fourier

Let Γ be a group endowed with a length function L , and let E be a linear subspace of C Γ . We say that E satisfies the Haagerup inequality if there exists constants C , s > 0 such that, for any f E , the convolutor norm of f on 2 ( Γ ) is dominated by C times the 2 norm of f ( 1 + L ) s . We show that, for E = C Γ , the Haagerup inequality can be expressed in terms of decay of random walks associated with finitely supported symmetric probability measures on Γ . If L is a word length function on a finitely generated group Γ , we show that,...

On the hessian of the optimal transport potential

Stefán Ingi Valdimarsson (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study the optimal solution of the Monge-Kantorovich mass transport problem between measures whose density functions are convolution with a gaussian measure and a log-concave perturbation of a different gaussian measure. Under certain conditions we prove bounds for the Hessian of the optimal transport potential. This extends and generalises a result of Caffarelli. We also show how this result fits into the scheme of Barthe to prove Brascamp-Lieb inequalities and thus prove a new generalised Reverse...

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