79 research outputs found
On generalized sequence spaces defined by modulus functions
[This abstract contains special characters that cannot correctly be shown in standard HTML. Please refer to the full text.
On a Biparameter Maximal Multilinear Operator
It is well-known that estimates for maximal operators and questions of
pointwise convergence are strongly connected. In recent years, convergence
properties of so-called `non-conventional ergodic averages' have been studied
by a number of authors, including Assani, Austin, Host, Kra, Tao, and so on. In
particular, much is known regarding convergence in of these averages, but
little is known about pointwise convergence. In this spirit, we consider the
pointwise convergence of a particular ergodic average and study the
corresponding maximal trilinear operator (over , thanks to a
transference principle). Lacey and Demeter, Tao, and Thiele have studied
maximal multilinear operators previously; however, the maximal operator we
develop has a novel bi-parameter structure which has not been previously
encountered and cannot be estimated using their techniques. We will carve this
bi-parameter maximal multilinear operator using a certain Taylor series and
produce non-trivial H\"{o}lder-type estimates for one of the two "main" terms
by treating it as a singular integrals whose symbol's singular set is similar
to that of the Biest operator studied by Muscalu, Tao, and Thiele.Comment: 32 pages, 1 figur
Fourier interpolation with zeros of zeta and -functions
We construct a large family of Fourier interpolation bases for functions
analytic in a strip symmetric about the real line. Interesting examples involve
the nontrivial zeros of the Riemann zeta function and other -functions. We
establish a duality principle for Fourier interpolation bases in terms of
certain kernels of general Dirichlet series with variable coefficients. Such
kernels admit meromorphic continuation, with poles at a sequence dual to the
sequence of frequencies of the Dirichlet series, and they satisfy a functional
equation. Our construction of concrete bases relies on a strengthening of
Knopp's abundance principle for Dirichlet series with functional equations and
a careful analysis of the associated Dirichlet series kernel, with coefficients
arising from certain modular integrals for the theta group.Comment: We have made an update and addition in the bibliograph
Rigidity sequences, Kazhdan sets and group topologies on the integers
International audienceWe study the relationships between three different classes of sequences (or sets) of integers, namely rigidity sequences, Kazhdan sequences (or sets) and nullpotent sequences. We prove that rigidity sequences are non-Kazhdan and nullpotent, and that all other implications are false. In particular, we show by probabilistic means that there exist sequences of integers which are both nullpotent and Kazhdan. Moreover, using Baire category methods, we provide general criteria for a sequence of integers to be a rigidity sequence. Finally, we give a new proof of the existence of rigidity sequences which are dense in Z for the Bohr topology, a result originally due to Griesmer
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