pointwise convergence; nonlocal diffusion; dyadic fractional derivatives; Haar base
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.
 Daubechies, I.: Ten Lectures on Wavelets
. CBMS-NSF Regional Conference Series in Applied Mathematics 61 SIAM, Philadelphia (1992). MR 1162107