Results 1  10
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12
Breaking duality in the Return Times Theorem
, 2006
"... We prove Bourgain’s Return Times Theorem for a range of exponents p and q that are outside the duality range. An oscillation result is used to prove hitherto unknown almost everywhere convergence for the signed average analog of Bourgain’s averages. As an immediate corollary we obtain a WienerWin ..."
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Cited by 19 (9 self)
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We prove Bourgain’s Return Times Theorem for a range of exponents p and q that are outside the duality range. An oscillation result is used to prove hitherto unknown almost everywhere convergence for the signed average analog of Bourgain’s averages. As an immediate corollary we obtain a WienerWintner type of result for the ergodic Hilbert series.
On the two dimensional Bilinear Hilbert Transform
, 2008
"... We investigate the Bilinear Hilbert Transform in the plane and the pointwise convergence of bilinear averages in Ergodic theory, arising from Z 2 actions. Our techniques combine novel one and a halfdimensional phasespace analysis with more standard onedimensional theory. ..."
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Cited by 18 (4 self)
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We investigate the Bilinear Hilbert Transform in the plane and the pointwise convergence of bilinear averages in Ergodic theory, arising from Z 2 actions. Our techniques combine novel one and a halfdimensional phasespace analysis with more standard onedimensional theory.
Oscillation and the mean ergodic theorem for uniformly convex Banach spaces
, 2013
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RANDOM SEQUENCES AND POINTWISE CONVERGENCE OF MULTIPLE ERGODIC AVERAGES
"... n=1 f(T nx) ·g(Sanx), where T and S are commuting measure preserving transformations, and an is a random version of the sequence [n c] for some appropriate c> 1. We also prove similar mean convergence results for averages of the form 1 N ∑N n=1 f(T anx) · g(Sanx), as well as pointwise results wh ..."
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Cited by 4 (4 self)
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n=1 f(T nx) ·g(Sanx), where T and S are commuting measure preserving transformations, and an is a random version of the sequence [n c] for some appropriate c> 1. We also prove similar mean convergence results for averages of the form 1 N ∑N n=1 f(T anx) · g(Sanx), as well as pointwise results when T and S are powers of the same transformation. The deterministic versions of these results, where one replaces an with [n c], remain open, and we hope that our method will indicate a fruitful way to approach these problems as well. 1.
Variational bounds for a dyadic model of the bilinear Hilbert transform
 the Illinois J. Math
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Spaces of infinite measure and the pointwise convergence of the bilinear Hilbert and ergodic averages defined by Lpisometries
, 2008
"... We generalize the respective “double recurrence” results of Bourgain and of the second author, which established for pairs of L∞ functions on a finite measure space the a.e. convergence of the discrete bilinear ergodic averages and of the discrete bilinear Hilbert averages defined by invertible meas ..."
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Cited by 1 (1 self)
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We generalize the respective “double recurrence” results of Bourgain and of the second author, which established for pairs of L∞ functions on a finite measure space the a.e. convergence of the discrete bilinear ergodic averages and of the discrete bilinear Hilbert averages defined by invertible measurepreserving point transformations. Our generalizations are set in the context of arbitrary sigmafinite measure spaces and take the form of a.e. convergence of such discrete averages, as well as of their continuous variable counterparts, when these averages are defined by Lebesgue space + p−1 2 < 3/2). In the setting of an arbitrary measure space, this yields the a.e. convergence of these discrete bilinear averages when they act on L p1 p2 × L and are defined by an invertible measurepreserving point transformation. isometries and act on L p1 × L p2 (1 < p1, p2 < ∞, p −1 1
ON SOME MAXIMAL MULTIPLIERS IN L p
, 2009
"... We extend an L 2 maximal multiplier result of Bourgain to all L p spaces, 1 < p < ∞. ..."
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We extend an L 2 maximal multiplier result of Bourgain to all L p spaces, 1 < p < ∞.
VARIATIONNORM AND FLUCTUATION ESTIMATES FOR ERGODIC BILINEAR AVERAGES
, 2015
"... For any dynamical system, we show that higher variationnorms 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 respe ..."
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For any dynamical system, we show that higher variationnorms 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 variationnorms of truncated bilinear operators on R, and the main new ingredient of the proof of these bounds is a variationnorm extension of maximal Bessel inequalities of Lacey and Demeter–Tao–Thiele.