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Oscillatory and Fourier integral operators with degenerate canonical relations.

Allan GreenleafAndreas Seeger — 2002

Publicacions Matemàtiques

We survey results concerning the L2 boundedness of oscillatory and Fourier integral operators and discuss applications. The article does not intend to give a broad overview; it mainly focuses on topics related to the work of the authors. [Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].

Problems on averages and lacunary maximal functions

Andreas SeegerJames Wright — 2011

Banach Center Publications

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...

Two problems associated with convex finite type domains.

Alexander IosevichEric SawyerAndreas Seeger — 2002

Publicacions Matemàtiques

We use scaling properties of convex surfaces of finite line type to derive new estimates for two problems arising in harmonic analysis. For Riesz means associated to such surfaces we obtain sharp L estimates for p > 4, generalizing the Carleson-Sjölin theorem. Moreover we obtain estimates for the remainder term in the lattice point problem associated to convex bodies; these estimates are sharp in some instances involving sufficiently flat boundaries.

A variation norm Carleson theorem

Richard OberlinAndreas SeegerTerence TaoChristoph ThieleJames Wright — 2012

Journal of the European Mathematical Society

We strengthen the Carleson-Hunt theorem by proving L p estimates for the r -variation of the partial sum operators for Fourier series and integrals, for r > 𝚖𝚊𝚡 { p ' , 2 } . Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

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