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Problems on averages and lacunary maximal functions

100%
Seeger A., Wright J.
Banach Center Publications
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2011
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tom 95
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nr 1
235-250
EN
We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an H¹ to $L^{1,∞}$ bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an $L^p$ regularity bound for some p > 1. Secondly, we obtain a necessary and sufficient condition for L² boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an $L^p$ regularity result for such averages. We formulate various open problems.
2
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Weak-type (1,1) bounds for oscillatory singular integrals with rational phases

100%
Folch-Gabayet M., Wright J.
Studia Mathematica
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2012
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tom 210
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nr 1
57-76
EN
We consider singular integral operators on ℝ given by convolution with a principal value distribution defined by integrating against oscillating kernels of the form $e^{iR(x)}/x$ where R(x) = P(x)/Q(x) is a general rational function with real coefficients. We establish weak-type (1,1) bounds for such operators which are uniform in the coefficients, depending only on the degrees of P and Q. It is not always the case that these operators map the Hardy space H¹(ℝ) to L¹(ℝ) and we will characterise those rational phases R(x) = P(x)/Q(x) which do map H¹ to L¹ (and even H¹ to H¹).
3
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An estimation for a family of oscillatory integrals

100%
Folch-Gabayet M., Wright J.
Studia Mathematica
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2003
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tom 154
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nr 1
89-97
EN
Let K be a Calderón-Zygmund kernel and P a real polynomial defined on ℝⁿ with P(0) = 0. We prove that convolution with Kexp(i/P) is continuous on L²(ℝⁿ) with bounds depending only on K, n and the degree of P, but not on the coefficients of P.
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