74 research outputs found
Dimensions of sums with self-similar sets
For some self-similar sets K in d-dimensional Euclidean space we obtain
certain lower bounds for the lower Minkowski dimension of K+E in terms of the
lower Minkowski dimension of E.Comment: 12 page
Spherical Means and Pinned Distance Sets
We use mixed norm estimates for the spherical averaging operator to obtain
some results concerning pinned distance sets
Estimates for compositions of maximal operators with singular integrals
We prove weak-type (1,1) estimates for compositions of maximal operators with
singular integrals. Our main object of interest is the operator
where is Bourgain's maximal multiplier operator and is the
sum of several modulated singular integrals; here our method yields a
significantly improved bound for the operator norm when . We
also consider associated variation-norm estimates
Variation-norm and fluctuation estimates for ergodic bilinear averages
For any dynamical system, we show that higher variation-norms for the
sequence of ergodic bilinear averages of two functions satisfy a large range of
bilinear Lp estimates. It follows that, with probability one, the number of
fluctuations along this sequence may grow at most polynomially with respect to
(the growth of) the underlying scale. These results strengthen previous works
of Lacey and Bourgain where almost surely convergence of the sequence was
proved (which is equivalent to the qualitative statement that the number of
fluctuations is finite at each scale). Via transference, the proof reduces to
establishing new bilinear Lp bounds for variation-norms of truncated bilinear
operators on R, and the main ingredient of the proof of these bounds is a
variation-norm extension of maximal Bessel inequalities of Lacey and
Demeter--Tao--Thiele.Comment: 37 pages, new version fixed some references not displaying correctl
A variation norm Carleson theorem
We strengthen the Carleson-Hunt theorem by proving estimates for the
-variation of the partial sum operators for Fourier series and integrals,
for . Four appendices are concerned with transference, a
variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear
Fourier transforms and ergodic theory.Comment: 41 page
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