2016/11/30 by Raffaele Carlone, Alberto Fıorenza, Alberto Fiorenza +1 · 20 citations
Mathematics · Physics and Astronomy · #Banach space #Differential Equations and Boundary Problems #Fractional Differential Equations Solutions #Integrable system #Kernel (algebra) #Lebesgue integration #Lebesgue measure #Lp space #Mathematical analysis #Mathematics #Maximal function #Nonlinear Differential Equations Analysis #Operator (biology) #Pure mathematics #Sobolev space #Standard probability space #math-ph #math.AP #math.FA #math.MP #msc:26A33 #msc:44A99 #msc:45E99 #msc:46E30 #msc:47G10
paper · pdf · doi:10.1016/j.jfa.2017.04.013
published in Journal of Functional Analysis 273(3), 1258-1294 (Elsevier BV) · 27 pages, 3 figures
openalex publication_date 2017/05/03 · arxiv created 2017/05/23 · arxiv updated 2019/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
For kernels ν which are positive and integrable we show that the operator g↦ Jνg=∫0x ν(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)=∫0x ν(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 Jν "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) = μ(x,0,-1) = ∫0∞xs-1/Γ(s) ds, the latter being relevant for instance in the analysis of the Schrödinger equation with concentrated nonlinearities in ℝ2.