You are currently browsing the tag archive for the ‘maximal inequalities’ tag.

Ciprian Demeter, Michael Lacey, Christoph Thiele and I have just uploaded our joint paper, “The Walsh model for $M_2^*$ Carleson” to the arXiv. This paper (which was recently accepted for publication in Revista Iberoamericana) establishes a simplified model for the key estimate (the “$M_2^*$ Carleson estimate”) in another (much longer) paper of ours on the return times theorem of Bourgain, in which the Fourier transform is replaced by its dyadic analogue, the Walsh-Fourier transform. This model estimate is established by the now-standard techniques of time-frequency analysis: one decomposes the expression to be estimated into a sum over tiles, and then uses combinatorial stopping time arguments into group the tiles into trees, and the trees into forests. One then uses (phase-space localised, and frequency-modulated) versions of classical Calderòn-Zygmund theory (or in this particular case, a certain maximal Fourier inequality of Bourgain) to control individual trees and forests, and sums up over the trees and forests using orthogonality methods (excluding an exceptional set if necessary).

Rather than discuss time-frequency analysis in detail here, I thought I would dwell instead on the return times theorem, and sketch how it is connected to the $M_2^*$ Carleson estimate; this is a more complicated version of the “$M_2$ Carleson estimate”, which is an estimate which is logically equivalent to Carleson’s famous theorem (and its extension by Hunt) on the almost everywhere convergence of Fourier series.

### Recent Comments

 Anonymous on Open question: scarring for th… Various Items | Not… on IMU Graduate Breakout Fellowsh… Anonymous on Open question: scarring for th… louigiaddario on IMU Graduate Breakout Fellowsh… gninrepoli on P=NP, relativisation, and mult… Wolfgang Arendt on A quick application of the clo… Terence Tao on Ultrafilters, nonstandard anal… sagar on Ultrafilters, nonstandard anal… Anonymous on On time management Terence Tao on An elementary non-commutative… Anonymous on An elementary non-commutative… Nathan on 254A, Notes 1: Concentration o… Terence Tao on Distinguished Lecture Series I… Terence Tao on 254A, Notes 1: Concentration o… Anonymous on Distinguished Lecture Series I…