Pointwise regularity associated with function spaces and multifractal analysis

Stéphane Jaffard

Banach Center Publications, Tome 72 (2006), p. 93-100 / Harvested from The Polish Digital Mathematics Library

Résumé

The purpose of multifractal analysis of functions is to determine the Hausdorff dimensions of the sets of points where a function (or a distribution) f has a given pointwise regularity exponent H. This notion has many variants depending on the global hypotheses made on f; if f locally belongs to a Banach space E, then a family of pointwise regularity spaces $C_{E}^{α} (x ₀)$ are constructed, leading to a notion of pointwise regularity with respect to E; the case $E = L^{\infty}$ corresponds to the usual Hölder regularity, and $E = L^{p}$ corresponds to the $T_{α}^{p} (x ₀)$ regularity of Calderón and Zygmund. We focus on the study of the spaces $T_{α}^{p} (x ₀)$ ; in particular, we give their characterization in terms of a wavelet basis and show their invariance under standard pseudodifferential operators of order 0.

Publié le : 2006-01-01

Zbl 1119.26007

EUDML-ID : urn:eudml:doc:282152

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     author = {St\'ephane Jaffard},
     title = {Pointwise regularity associated with function spaces and multifractal analysis},
     journal = {Banach Center Publications},
     volume = {72},
     year = {2006},
     pages = {93-100},
     zbl = {1119.26007},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc72-0-7}
}

Stéphane Jaffard. Pointwise regularity associated with function spaces and multifractal analysis. Banach Center Publications, Tome 72 (2006) pp. 93-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc72-0-7/