Actis, Marcelo; Aimar, Hugo Pointwise convergence to the initial data for nonlocal dyadic diffusions. (English) Zbl 1389.42054 Czech. Math. J. 66, No. 1, 193-204 (2016). Summary: We solve the initial value problem for the diffusion induced by dyadic fractional derivative \(s\) in \(\mathbb{R}^+\). First we obtain the spectral analysis of the dyadic fractional derivative operator in terms of the Haar system, which unveils a structure for the underlying “heat kernel”. We show that this kernel admits an integrable and decreasing majorant that involves the dyadic distance. This allows us to provide an estimate of the maximal operator of the diffusion by the Hardy-Littlewood dyadic maximal operator. As a consequence we obtain the pointwise convergence to the initial data. Cited in 2 Documents MSC: 42B37 Harmonic analysis and PDEs 26A33 Fractional derivatives and integrals 35K57 Reaction-diffusion equations Keywords:pointwise convergence; nonlocal diffusion; dyadic fractional derivatives; Haar base PDFBibTeX XMLCite \textit{M. Actis} and \textit{H. Aimar}, Czech. Math. J. 66, No. 1, 193--204 (2016; Zbl 1389.42054) Full Text: DOI Link Link References: [1] H. Aimar, B. Bongioanni, I. Gómez: On dyadic nonlocal Schrödinger equations with Besov initial data. J. Math. Anal. Appl. 407 (2013), 23–34. · Zbl 1306.35106 · doi:10.1016/j.jmaa.2013.05.001 [2] R. M. Blumenthal, R. K. Getoor: Some theorems on stable processes. Trans. Am. Math. Soc. 95 (1960), 263–273. · Zbl 0107.12401 · doi:10.1090/S0002-9947-1960-0119247-6 [3] L. Caffarelli, L. Silvestre: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equations 32 (2007), 1245–1260. · Zbl 1143.26002 · doi:10.1080/03605300600987306 [4] I. Daubechies: Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61, SIAM, Philadelphia, 1992. · Zbl 0776.42018 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.