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Logarithmic Dimension Bounds for the Maximal Function Along a Polynomial Curve
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2010 (English)In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 20, no 3, 771-785 p.Article in journal (Refereed) Published
Abstract [en]

Let M denote the maximal function along the polynomial curve (gamma(1)t,..., gamma(d)t(d)): M(f)(x) = sup(r>0) 1/2r integral(vertical bar t vertical bar <= r) vertical bar f(x(1) - gamma(1)t,..., x(d) - gamma(d)t(d))vertical bar dt. We show that the L(2) norm of this operator grows at most logarithmically with the parameter d: parallel to Mf parallel to(L2(Rd)) <= c log d parallel to f parallel to(L2(Rd)), where c > 0 is an absolute constant. The proof depends on the explicit construction of a "parabolic" semigroup of operators which is a mixture of stable semigroups.

Place, publisher, year, edition, pages
2010. Vol. 20, no 3, 771-785 p.
Keyword [en]
Maximal function, Polynomial curve, Parabolic dilations, Semigroup of operators
National Category
URN: urn:nbn:se:kth:diva-46647DOI: 10.1007/s12220-010-9127-2ISI: 000276881900012ScopusID: 2-s2.0-77956395280OAI: diva2:454774
QC 20111108Available from: 2011-11-08 Created: 2011-11-04 Last updated: 2011-11-15Bibliographically approved

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Parissis, Ioannis
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