# Article

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Keywords:
pointwise convergence; nonlocal diffusion; dyadic fractional derivatives; Haar base
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.
References:
[1] Aimar, H., Bongioanni, B., Gómez, I.: On dyadic nonlocal Schrödinger equations with Besov initial data. J. Math. Anal. Appl. 407 (2013), 23-34. DOI 10.1016/j.jmaa.2013.05.001 | MR 3063102 | Zbl 1306.35106
[2] Blumenthal, R. M., Getoor, R. K.: Some theorems on stable processes. Trans. Am. Math. Soc. 95 (1960), 263-273. DOI 10.1090/S0002-9947-1960-0119247-6 | MR 0119247
[3] Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equations 32 (2007), 1245-1260. DOI 10.1080/03605300600987306 | MR 2354493 | Zbl 1143.26002
[4] Daubechies, I.: Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61 SIAM, Philadelphia (1992). MR 1162107

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