147 research outputs found
Szemeredi's theorem, frequent hypercyclicity and multiple recurrence
Let T be a bounded linear operator acting on a complex Banach space X and
(\lambda_n) a sequence of complex numbers. Our main result is that if
|\lambda_n|/|\lambda_{n+1}| \to 1 and the sequence (\lambda_n T^n) is
frequently universal then T is topologically multiply recurrent. To achieve
such a result one has to carefully apply Szemer\'edi's theorem in arithmetic
progressions. We show that the previous assumption on the sequence (\lambda_n)
is optimal among sequences such that |\lambda_n|/|\lambda_{n+1}| converges in
[0,+\infty]. In the case of bilateral weighted shifts and adjoints of
multiplication operators we provide characterizations of topological multiple
recurrence in terms of the weight sequence and the symbol of the multiplication
operator respectively.Comment: 18 pages; to appear in Math. Scand., this second version of the paper
is significantly revised to deal with the more general case of a sequence of
operators (\lambda_n T^n). The hypothesis of the theorem has been weakened.
The numbering has changed, the main theorem now being Th. 3.8 (in place of
Proposition 3.3). The changes incorporate the suggestions and corrections of
the anonymous refere
A sharp estimate for the Hilbert transform along finite order lacunary sets of directions
Let be a nonnegative integer and be a
lacunary set of directions of order . We show that the norms,
, of the maximal directional Hilbert transform in the plane are comparable to
. For vector fields with
range in a lacunary set of of order and generated using suitable
combinations of truncations of Lipschitz functions, we prove that the truncated
Hilbert transform along the vector field , is -bounded for all . These
results extend previous bounds of the first author with Demeter, and of Guo and
Thiele.Comment: 20 pages, 2 figures. Submitted. Changes: clarified the definition of
D-lacunary set and streamlined the notatio
Solyanik estimates and local H\"older continuity of halo functions of geometric maximal operators
Let be a homothecy invariant basis consisting of convex sets in
, and define the associated geometric maximal operator
by and the halo function
on by It is shown that if
satisfies the Solyanik estimate for
sufficiently close to 1 then lies in the H\"older class . As a consequence we obtain that the halo functions associated
with the Hardy-Littlewood maximal operator and the strong maximal operator on
lie in the H\"older class .Comment: 19 pages, 1 figure, minor typos corrected, incorporates referee's
report, to appear in Adv. Mat
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