8/12/2019 On the Convergence of Multiple Fourier Series - Charles Fefferman (AMS 11-01-1971)

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B U L L E T I N O F T H EA M E R I C A N M A T H E M A T I C A L S O C I E T YVolume 77, Number 5, September 1971

O N T H E C O N V E R G E N C E O F M U L T I P L EFOURIER SERIES

B Y C H A R L E S F E F F E R M A NCommunicated by M. H. Protter , January 11, 1971

We cont inue f rom [2] .T H E O R E M . Let P be an open polygonal region in R 2, containing theorigin. Set XP = {(Xx,A;y) (x, y) P } fo r X > 0 . Then for

~ ]T) a mn exp[i(mx + ny) ]m,n~

in L2 ( [ 0 , 2 T T ] X [ 0 , 2 T T ] ) , we havef(x, y) = limx->oo 12(tn,n)e\p a mnexp [i(mx+ny) ]

almost everywhere.Surpr i s ing ly , th i s is an easy co nsequen ce of Ca r leson ' s theo rem [ l ]on convergence of Fourier ser ies of one var iable .P R O O F . I t i s enough to prove the maximal inequa l i ty

( i ) sup 7 j a y)-t

8/12/2019 On the Convergence of Multiple Fourier Series - Charles Fefferman (AMS 11-01-1971)

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ON THE CONVERGENCE OF MULTIPLE FOURIER SERIES 7453)

Here,sup X) a>m'n> exp[i(mV + n'y')](m tn )ES ,m mnexp[i(tnx + ny)]almost everywhere.

Compare with the counterexamples of [2].R E F E R E N C E S

1. L. Carleson,On convergenceand growthofpartial sumsofFourier series,ActaMath . 116(1966), 135-157.MR33 7774.2. C.Fefferman, Onthe divergenceofmultiple Fourier series, Bull. Amer. Math.Soc.77(1971), 191-195.3. R.A. Hunt, On the convergenceofFourier series,Proc.Conf.on Orthogonal Expansions and their C ontinuous Analogues (Edwardsville, 111.,1967), Southe rn IllinoisUniv. Press, Carbondale, 111., 1968, pp. 235-255. MR38 6296.4. P.Sjlin, On the convergence almost everywhereofcertain singular integrals andmultiple Fourier series, Ark. Mat.9 (1971) (toappear).5. N.Tevzadze, On the convergence of double Fourier seriesofquadratic summablefunctions, SoobS. Akad. Nauk Gruzin.SSR1970, 277-279.UNIVERSITY OF CHICAGO, CHICAGO, ILLINOIS60637