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17
Use of abstract Hardy spaces, Real interpolation and Applications to bilinear operators.
, 2008
"... This paper can be considered as the sequel of [6], where the authors have proposed an abstract construction of Hardy spaces H 1. They shew an interpolation result for these Hardy spaces with the Lebesgue spaces. Here we describe a more precise result using the real interpolation theory and we clarif ..."
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Cited by 11 (8 self)
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This paper can be considered as the sequel of [6], where the authors have proposed an abstract construction of Hardy spaces H 1. They shew an interpolation result for these Hardy spaces with the Lebesgue spaces. Here we describe a more precise result using the real interpolation theory and we clarify the use of Hardy spaces. Then with the help of the bilinear interpolation theory, we then give applications to study bilinear operators on Lebesgue spaces. These ideas permit us to study singular operators with singularities similar to those of bilinear CalderónZygmund operators in a far more abstract framework as in the euclidean case.
Distributional Estimates for the Bilinear Hilbert Transform
"... ABSTRACT. We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power −2/3 (modulo some logarithmic factors). These results ..."
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Cited by 5 (2 self)
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ABSTRACT. We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power −2/3 (modulo some logarithmic factors). These results yield all known Lp bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on Lp1 × Lp2 when either 1/p1 + 1/p2 = 3/2 or one of p1, p2 is equal to 1. As a consequence of this work we also obtain that the square root of the bilinear Hilbert transform of two characteristic functions is exponentially integrable over any compact set. 1.
A NEW WAY OF LOOKING AT DISTRIBUTIONAL ESTIMATES; APPLICATIONS FOR THE BILINEAR HILBERT TRANSFORM.
"... Abstract. Distributional estimates for the Carleson operator acting on characteristic functions of measurable sets of finite measure were obtained by Hunt [12]. In this article we describe a simple method that yields such estimates for general operators acting on one or more functions. As an applica ..."
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Cited by 4 (1 self)
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Abstract. Distributional estimates for the Carleson operator acting on characteristic functions of measurable sets of finite measure were obtained by Hunt [12]. In this article we describe a simple method that yields such estimates for general operators acting on one or more functions. As an application we discuss how distributional estimates are obtained for the linear and bilinear Hilbert transform. These distributional estimates show that the square root of the bilinear Hilbert transform is exponentially integrable over compact sets. They also provide restricted type endpoint results on products of Lebesgue spaces where one exponent is 1 or the sum of the reciprocal of the exponents is 3/2. The proof of the distributional estimates for the bilinear Hilbert transform rely on an improved energy estimate for characteristic functions with respect to sets of tiles from which appropriate exceptional subsets have been removed. 1.
THE MULTILINEAR MARCINKIEWICZ INTERPOLATION THEOREM REVISITED: THE BEHAVIOR OF THE CONSTANT
"... Abstract. We provide a selfcontained proof of the multilinear extension of the Marcinkiewicz real method interpolation theorem with initial assumptions a set of restricted weak type estimates, considering possible degenerate situations that may arise. The advantage of this proof is that it yields a ..."
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Cited by 3 (3 self)
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Abstract. We provide a selfcontained proof of the multilinear extension of the Marcinkiewicz real method interpolation theorem with initial assumptions a set of restricted weak type estimates, considering possible degenerate situations that may arise. The advantage of this proof is that it yields a logarithmically convex bound for the norm of the operator on the intermediate spaces in terms of the initial restricted weak type bounds; it also provides an explicit estimate in terms of the exponents of the initial estimates: the constant blows up like a negative power of the distance from the intermediate point to the boundary of the convex hull of the initial points. In memory of Nigel Kalton 1.
Boundedness of smooth bilinear square functions and applications to some bilinear pseudodifferential operators
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Interplay between distributional estimates and boundedness of operators, submitted
"... Abstract. We prove that certain boundedness properties of operators yield distributional estimates that have exponential decay at infinity. Such distributional estimates imply local exponential integrability and apply to many operators such as mlinear CalderónZygmund operators and their maximal co ..."
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Cited by 2 (2 self)
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Abstract. We prove that certain boundedness properties of operators yield distributional estimates that have exponential decay at infinity. Such distributional estimates imply local exponential integrability and apply to many operators such as mlinear CalderónZygmund operators and their maximal counterparts.
Lp estimates for multilinear and multiparameter pseudodifferential operators
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