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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.
IMPROVED RANGE IN THE RETURN TIMES THEOREM
, 2009
"... We prove that the Return Times Theorem holds true for pairs of L p − L q functions, whenever 1/p + 1/q < 3/2. ..."
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Cited by 3 (0 self)
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We prove that the Return Times Theorem holds true for pairs of L p − L q functions, whenever 1/p + 1/q < 3/2.
A Survey of the Return Times Theorem
, 2014
"... The goal of this paper is to survey the history, development and current status of the Return Times Theorem and its many extensions and variations. Let (X,F, µ) be a finite measure space and let T: X → X be a measure preserving transformation. Perhaps the oldest result in ergodic theory is that of ..."
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Cited by 2 (2 self)
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The goal of this paper is to survey the history, development and current status of the Return Times Theorem and its many extensions and variations. Let (X,F, µ) be a finite measure space and let T: X → X be a measure preserving transformation. Perhaps the oldest result in ergodic theory is that of Poincaré’s Recurrence Principle [Poi87] which states: Theorem 0.1. For any set A ∈ F, the set of points x of A such that Tnx / ∈ A for all n> 0 has zero measure. This says that almost every point of A returns to A. In fact, almost every point of A returns to A infinitely often. The return time for a given element x ∈ A, rA(x) = inf{k ≥ 1: T kx ∈ A}, is the first time that the element x returns to the set A. This is visualized in the figure below. By Theorem 0.1, there is set of full measure in A such that all elements of this set have a finite return time. Our study of the return times theorem asks how we can further generalize this notion. 1 ar
Plinio: Logarithmic Lp Bounds for Maximal Directional Singular Integrals in the Plane
 J. Geom. Anal
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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 < ∞.
Linear Sequences and Weighted Ergodic Theorems
"... We present a simple way to produce good weights for several types of ergodic theorem including the WienerWintner type multiple return time theorem and the multiple polynomial ergodic theorem. These weights are deterministic and come from orbits of certain bounded linear operators on Banach spaces. ..."
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We present a simple way to produce good weights for several types of ergodic theorem including the WienerWintner type multiple return time theorem and the multiple polynomial ergodic theorem. These weights are deterministic and come from orbits of certain bounded linear operators on Banach spaces. This extends the known results for nilsequences and return time sequences of the form (g(S y)) for a measure preserving system (Y, S) and ∈ ∞ ( ), avoiding in the latter case the problem of finding the full measure set of appropriate points y.
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"... Nikolskiitype inequalities for shift invariant function spaces. (English summary) ..."
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Nikolskiitype inequalities for shift invariant function spaces. (English summary)