By V.P. Khavin, N.K. Nikol'skii, J. Peetre, Sh.A. Alimov, R.R. Ashurov, E.M. Dyn'kin, S.V. Kislyakov, A.K. Pulatov

With the foundation laid within the first quantity (EMS 15) of the Commutative Harmonic research subseries of the Encyclopaedia, the current quantity takes up 4 complicated issues within the topic: Littlewood-Paley conception for singular integrals, unprecedented units, a number of Fourier sequence and a number of Fourier integrals. The authors think that the reader understands the basics of harmonic research and with simple sensible research. The exposition begins with the fundamentals for every subject, additionally taking account of the ancient improvement, and ends via bringing the topic to the extent of present examine. desk of Contents I. a number of Fourier sequence and Fourier Integrals. Sh.A.Alimov, R.R.Ashurov, A.K.Pulatov II. equipment of the speculation of Singular Integrals. II: Littlewood Paley idea and its purposes E.M.Dyn'kin III.Exceptional units in Harmonic research S.V.Kislyakov

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Theorem (Carleson-Sjölin (1972)). 23) is a multiplier in > O. L p (JR 2 ) for pE 12 = [~, 4J and any s The proof of this theorem is hard. Let us just remark that it suffices to prove it for p = 4. The result for p = ~ then follows by duality + = 1) and the remaining cases follow by interpolation. If p ~ 12 , i. e. 23) is a Fourier multiplier in L p (JR2) provided (! i ! s> 21~ - ~I-~, p ~h Thus, the theorems of Fefferman and Carleson-Sjölin give an exhaustive solution to the question of the convergence in Lp (lR2 ) of spherical Riesz means of Fourier integral expansions.

11) 1. Multiple Fourier Series and Fourier Integrals let us fix our attention to the fact that the operator f integral operator (T~(X, f) = f D~(x jTN with kernel D~(x 33 I--t (T~(x, f) is an y)f(y)dy, - y) (the Dirichlet kernei) given by D~(x-y)=(27r)-N L Inl<#' (1_ln~2)8ein(x_y). L) at the point { = n. L). 13) constitutes one of the main tools with the aid of which one studies spherical partial sums of multiple Fourier series. L). 7), we obtain If s ]RN: > N ;1 the series to the right is uniformly convergent for x E {x E IXjl :::; 27r - ö}.

Ashurov, A. K. Pulatov in an arbitrary small neighborhood of this point. Let us indicate the plan of the proof of this theorem. As is well-known, the Dirichlet kernel Dn(x) = Le ikx Ikl$n in the one-dimensional case has the form ( ) _ sin(n + ~)x Dn x 2' sm 2x . Let us fix a number 6,0< 6 < 71", and set Qs = {x E Tl,lxl > 6}. It follows then from the summability of f that the function f (x + y) . (sin ~)-l is summable in the domain Qs. Therefore we have [ f(x + y)Dn(y)dy - t 0, ins n - t 00, uniformly in x E [-71",71"].