In this paper, we present a new proof of a theorem of Carleson and Hunt: The Fourier series of an LP function on [0, 2J] converges almost everywhere (p > 1). (See [1], [51.) Our proof is very much in the spirit of the classical theorem of Kolmogoroff-Seliverstoff-Plessner [8]. Unlike Carleson's proof, which makes a careful analysis of the structure of an L2 function f, our arguments essentially ignore f, and concentrate instead on building up a basic "partial sum" operator from simpler pieces. Our methods are (almost) entirely L2. Sections 1-7 of this paper contain a proof of pointwise convergence for L2 functions; Section 8 contains the modifications necessary to handle LP, and includes various further comments.

https://www.jstor.org/stable/pdfplus/1970917.pdf

Pointwise Convergence of Fourier Series

This two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and will be useful to graduate students and researchers in both pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. This first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional Calderon–Zygmund and Littlewood–Paley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. The second volume goes beyond the classical to the highly contemporary and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; Coifman–Meyer theory; Carleson's resolution of the Lusin conjecture; Calderon's commutators and the Cauchy integral on Lipschitz curves. The material in this volume has not previously appeared together in book form.

/pdf/classical-and-multilinear-harmonic-analysis-1s81cq3r8w.pdf

Classical and Multilinear Harmonic Analysis

T s Ts = o ( l o g l o g I~]), t~]-+ 0% for almost eve ry x in T,. I n Sections 1 to 3 in this pape r we prove among o ther things the following theorem, which generalizes the L p es t imate of the opera tor M* in [7]. TEEORE~. Assume that k is a Calderdn -Zygmund kernel defined in R', s ~_ 2, which has continuous derivatives of order ~ s 41 outside the origin. Let the operator M be defined by

/pdf/convergence-almost-everywhere-of-certain-singular-integrals-vsfm97sw9m.pdf

Convergence almost everywhere of certain singular integrals and multiple Fourier series

Baire’s category theorem and trigonometric series

This is a survey on transformation of fractal type sets and measures under orthogonal projections and more general mappings.

/pdf/hausdorff-dimension-projections-and-the-fourier-transform-4yq5qa70wk.pdf

Hausdorff dimension, projections, and the Fourier transform

/pdf/an-inequality-of-paley-and-convergence-a-e-of-walsh-fourier-c3pm8zpzf3.pdf

An inequality of Paley and convergence a.e. of Walsh-Fourier series

Estimates of spherial averages of Fourier transforms and dimensions of sets

A note on Littlewood-Paley decompositions with arbitrary intervals

/pdf/regularity-of-solutions-to-nonlinear-equations-of-1nhtyv4mxj.pdf

Regularity of solutions to nonlinear equations of schrödinger type

In P. Sjogren and P. Sjolin we studied the regularity of solutions to the Schrodinger equation i∂u/∂t=Pu+Vu in a half-space {(x,t)∈R n ×R + }. Here P is an elliptic self-adjoint constant-coefficient operator in x of order m≥2 and V=V(x) a real-valued potential. We assumed that V∈C ∞ (R n ) and that DαV is bounded for every α, where D=(D 1 , ..., D n ) and D k =i∂/∂x k

Per Sjölin

Papers

An inequality of Paley and convergence a.e. of Walsh-Fourier series

Estimates of spherial averages of Fourier transforms and dimensions of sets

A note on Littlewood-Paley decompositions with arbitrary intervals

Regularity of solutions to nonlinear equations of schrödinger type

Local regularity of solutions to nonlinear Schrödinger equations