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EnglishFor kernels nu which are positive and integrable we show that the operator g-->J_nu g=int_0^x u(x-s)g(s) ds on a finite time interval enjoys a regularizing effect when applied to Hölder continuous and Lebesgue functions and a "contractive” effect when applied to Sobolev functions. For Hölder continuous functions, we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel, namely by the factor N(x)=int_0^x nu(s) ds. For functions in Lebesgue spaces, we prove that an improvement always exists, and it can be expressed in terms of Orlicz integrability. Finally, for functions in Sobolev spaces, we show that the operator J_nu “shrinks” the norm of the argument by a factor that, as in the Hölder case, depends on the function N (whereas no regularization result can be obtained). These results can be applied, for instance, to Abel kernels and to the Volterra function I(x)=mu(x,0,-1)=int_0^infty x^{s-1}/Gamma(s) ds, the latter being relevant for instance in the analysis of the Schrödinger equation with concentrated nonlinearities in R^2.
The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions / Carlone, Raffaele; Fiorenza, Alberto; Tentarelli, Lorenzo. - In: JOURNAL OF FUNCTIONAL ANALYSIS. - ISSN 0022-1236. - 273:3(2017), pp. 1258-1294.
| Titolo: | The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions | |
| Autori: | ||
| Data di pubblicazione: | 2017 | |
| Rivista: | ||
| Digital Object Identifier (DOI): | http://dx.doi.org/10.1016/j.jfa.2017.04.013 | |
| Appare nelle tipologie: | 1.1 Articolo in rivista |
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| File | Descrizione | Tipologia | Licenza | |
|---|---|---|---|---|
| Carlone R., Fiorenza A., Tentarelli L., The action of Volterra integral operators with highly singular kernels on Holder continuous, Lebesgue and Sobolev functions, 2017.pdf | 2a Post-print versione editoriale / Version of Record | Non Pubblico - Accesso privato/ristretto | Administrator Richiedi una copia | |
| Carlone_Fiorenza_TentarelliRevised.pdf | 2. Post-print / Author's Accepted Manuscript |  | Visibile a tuttiVisualizza/Apri |
http://hdl.handle.net/11583/2771913
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