Pointwise convergence to the initial data for nonlocal dyadic diffusions

Marcelo Actis; Hugo Aimar

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 1, page 193-204
  • ISSN: 0011-4642

Abstract

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We solve the initial value problem for the diffusion induced by dyadic fractional derivative s in + . 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.

How to cite

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Actis, Marcelo, and Aimar, Hugo. "Pointwise convergence to the initial data for nonlocal dyadic diffusions." Czechoslovak Mathematical Journal 66.1 (2016): 193-204. <http://eudml.org/doc/276803>.

@article{Actis2016,
abstract = {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.},
author = {Actis, Marcelo, Aimar, Hugo},
journal = {Czechoslovak Mathematical Journal},
keywords = {pointwise convergence; nonlocal diffusion; dyadic fractional derivatives; Haar base},
language = {eng},
number = {1},
pages = {193-204},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Pointwise convergence to the initial data for nonlocal dyadic diffusions},
url = {http://eudml.org/doc/276803},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Actis, Marcelo
AU - Aimar, Hugo
TI - Pointwise convergence to the initial data for nonlocal dyadic diffusions
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 193
EP - 204
AB - 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.
LA - eng
KW - pointwise convergence; nonlocal diffusion; dyadic fractional derivatives; Haar base
UR - http://eudml.org/doc/276803
ER -

References

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  1. Aimar, H., Bongioanni, B., Gómez, I., 10.1016/j.jmaa.2013.05.001, J. Math. Anal. Appl. 407 (2013), 23-34. (2013) MR3063102DOI10.1016/j.jmaa.2013.05.001
  2. Blumenthal, R. M., Getoor, R. K., 10.1090/S0002-9947-1960-0119247-6, Trans. Am. Math. Soc. 95 (1960), 263-273. (1960) MR0119247DOI10.1090/S0002-9947-1960-0119247-6
  3. Caffarelli, L., Silvestre, L., 10.1080/03605300600987306, Commun. Partial Differ. Equations 32 (2007), 1245-1260. (2007) MR2354493DOI10.1080/03605300600987306
  4. Daubechies, I., Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics 61 SIAM, Philadelphia (1992). (1992) MR1162107

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