Search results for "Integral operators"
showing 5 items of 5 documents
Mapping properties of weakly singular periodic volume potentials in Roumieu classes
2020
The analysis of the dependence of integral operators on perturbations plays an important role in the study of inverse problems and of perturbed boundary value problems. In this paper, we focus on the mapping properties of the volume potentials with weakly singular periodic kernels. Our main result is to prove that the map which takes a density function and a periodic kernel to a (suitable restriction of the) volume potential is bilinear and continuous with values in a Roumieu class of analytic functions. This result extends to the periodic case of some previous results obtained by the authors for nonperiodic potentials, and it is motivated by the study of perturbation problems for the solut…
On Fourier integral operators with Hölder-continuous phase
2018
We study continuity properties in Lebesgue spaces for a class of Fourier integral operators arising in the study of the Boltzmann equation. The phase has a H\"older-type singularity at the origin. We prove boundedness in $L^1$ with a precise loss of decay depending on the H\"older exponent, and we show by counterexamples that a loss occurs even in the case of smooth phases. The results can be seen as a quantitative version of the Beurling-Helson theorem for changes of variables with a H\"older singularity at the origin. The continuity in $L^2$ is studied as well by providing sufficient conditions and relevant counterexamples. The proofs rely on techniques from Time-frequency Analysis.
A characterization of riesz operators
1987
Boundedness and compactness of operators related to time-frequency analysis
2018
En esta tesis, estudiamos diferentes aspectos de los operadores relacionados con el análisis tiempo-frecuencia. Cada operador lineal y continuo de la clase de Schwartz en su dual, el espacio de distribuciones temperadas, se puede escribir como un operador integral con núcleo K, o también como un operador integral de Fourier (de hecho, pseudodiferencial). Las diferentes condiciones en el núcleo o el símbolo y la fase (en el caso de los operadores integrales de Fourier) permiten extender el operador a varios espacios de funciones y distribuciones. A continuación detallamos los contenidos de la memoria. En el primer capítulo presentamos la notación, las definiciones de algunos espacios, de suc…
Two examples related to conical energies
2022
In a recent article we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and M\"{o}rters, the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.