56 research outputs found
Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups
Let G be a compact Hausdorff group and let H be a closed subgroup of G. We introduce pseudo-differential operators with symbols on the homogeneous space G/H. We present a necessary and sufficient condition on symbols for which these operators are in the class of Hilbert-Schmidt operators. We also give a characterization of and a trace formula for the trace class pseudo-differential operators on the homogeneous space G/H
-boundedness and -nuclearity of multilinear pseudo-differential operators on and the torus
In this article, we begin a systematic study of the boundedness and the
nuclearity properties of multilinear periodic pseudo-differential operators and
multilinear discrete pseudo-differential operators on -spaces. First, we
prove analogues of known multilinear Fourier multipliers theorems (proved by
Coifman and Meyer, Grafakos, Tomita, Torres, Kenig, Stein, Fujita, Tao, etc.)
in the context of periodic and discrete multilinear pseudo-differential
operators. For this, we use the periodic analysis of pseudo-differential
operators developed by Ruzhansky and Turunen. Later, we investigate the
-nuclearity, of periodic and discrete pseudo-differential
operators. To accomplish this, we classify those -nuclear multilinear
integral operators on arbitrary Lebesgue spaces defined on -finite
measures spaces. We also study similar properties for periodic Fourier integral
operators. Finally, we present some applications of our study to deduce the
periodic Kato-Ponce inequality and to examine the -nuclearity of multilinear
Bessel potentials as well as the -nuclearity of periodic Fourier integral
operators admitting suitable types of singularities.Comment: 40 pages, This version is a revised version based on reviewer's
comments. Final version appeared in JFA
Multilinear analysis for discrete and periodic pseudo-differential operators in Lp-spaces
In this note we announce our investigation on the Lp properties for periodic and discrete multilinear pseudo-differential operators. First, we review the periodic analysis of multilinear pseudo-differential operators byshowing classical multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Tomita, Miyachi, Fujita, Grafakos, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. The s-nuclearity, 0 < s ≤ 1, for the discrete and periodic multilinear pseudo-differential operators will be investigated. To do so, we classify those s-nuclear, 0 < s ≤ 1, multilinear integral operators on arbitrary Lebesgue spaces defined on σ-finite measures spaces. Finally, we present some applications of our analysis to deduce the periodic Kato-Ponce inequality and to examine the s-nuclearity of multilinear Bessel potentialsas well as the s-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.En esta nota anunciamos los resultados de nuestra investigación sobre las propiedades Lp de operadores pseudodiferenciales multilineales periódicos y/o discretos. Primero, revisaremos el análisis multilineal de tales operadores mostrando versiones análogas de los teoremas clásicos disponibles en el análisis multilineal euclidiano (debidos a Coifman y Meyer, Tomita, Miyachi, Fujita, Grafakos, Tao, etc.), pero, en el contexto de operadores periódicos y/o discretos. Se caracterizará la s-nuclearidad, 0 < s ≤ 1, para operadores multilineales pseudodiferenciales periódicos y/o discretos. Para cumplir este objetivo se clasificarán aquellos operadores lineales s-nucleares, 0 < s ≤ 1, multilineales con núcleo, sobre espacios de Lebesgue arbitrarios definidos en espacios de medida σ-finitos. Finalmente, como aplicación de los resultados presentados se obtiene la versión periódica de la desigualdad de Kato-Ponce, y se examina la s-nuclearidad de potenciales de Bessel lineales y multilineales, como también la s-nuclearidad de operadores integrales de Fourier periódicos admitiendo sÃmbolos con tipos adecuados de singularidad
Hilbert-Schmidt and Trace Class Pseudo-differential Operators on the Abstract Heisenberg Group
In this paper we introduce and study pseudo-differential operators with
operator valued symbols on the abstract Heisenberg group where a locally compact abelian
group with its dual group . We obtain a necessary and sufficient
condition on symbols for which these operators are in the class of
Hilbert-Schmidt operators. As a key step in proving this we derive a trace
formula for the trace class -Weyl transform, with
symbols in We go on to present a characterization
of the trace class pseudo-differential operators on . Finally,
we also give a trace formula for these trace class operators.Comment: 16 page
The Hausdorff-Young inequality for Orlicz spaces on compact hypergroups
We prove the classical Hausdorff-Young inequality for the Lebesgue spaces on a compact hypergroup using interpolation of sublinear operators. We use this result to prove the Hausdorff-Young inequality for Orlicz spaces on a compact hypergroup
Hypergroup Deformations of Semigroups
We view the well-known example of the dual of a countable compact hypergroup,
motivated by the orbit space of p-adic integers by Dunkl and Ramirez (1975), as
hypergroup deformation of the max semigroup structure on the linearly ordered
set of the non-negative integers along the diagonal. This works
as motivation for us to study hypergroups or semi convolution spaces arising
from "max" semigroups or general commutative semigroups via hypergroup
deformation on idempotents.Comment: 28 pages, 1 Table, This version is a truncated version with fourth
section deleted from version 3, which is being developed into a separate
paper. The title and abstract have been changed accordingl
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