Number Theory

Real-Variable Methods in Harmonic Analysis (Pure and Applied by Alberto Torchinsky

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By Alberto Torchinsky

An exploration of the harmony of numerous parts in harmonic research, this article emphasizes real-variable tools. Discusses classical Fourier sequence, summability, norm convergence, and conjugate function. Examines the Hardy-Littlewood maximal functionality, the Calder?n-Zygmund decomposition, the Hilbert rework and homes of harmonic services, the Littlewood-Paley idea, extra. 1986 variation.

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Example text

Proof. Since CnlA,IZ < co,A, # 0 only for countably many a’s. 4. 8. Let {x,},,~ be an ONS in H. Each of the following conditions implies the others: (i) (Completeness){xa},EAis a maximal orthonormal set in H. 2. s A,x,} is dense in H. (iii) For every x in H, llx112 = C,lx(a)12. (iv) (Parseval's Identity) For x, y in H ( x , y ) = C, x ( a ) y ( a ) . Proof. (i) implies (ii). Let M denote the closure of S; M is also a subspace of H. If M f H, then by well-known properties of Hilbert spaces there is a nonzero vector x orthogonal to M.

We then take A = 1 and A = i to obtain the desired conclusion. (iv) implies (i). By (iv), with x = y there, we get that lly112 = 0, or y = 0, which is a contradiction. + How do these results apply to L2(T)? 8 above holds. Therefore, (ii)-(iv) of that theorem, as well as the rest of the results in this section also hold. 6) and 2. 6), for the case p = 2. On the other hand, the case p = 00 was ruled out, even with L"( T) replaced by C ( T), by the results in Section 2 of Chapter I. Moreover, since for 1 =zp < 00 the Lp(T) spaces are Hilbert only when p = 2, a different approach is needed here.

3) We let now n -+ 00. 3) converges to c j , whereas the right-hand side converges, on account of the uniform convergence of the a n ’ s , to C j ( f ) . 3) and Minkowski’s inequality, we obtain that ) bounded in Lp( T) (by the Lp norm In other words, the sequence a n ( f is of f). 1) be the Fourier series of an Lp function requires some knowledge about LP-bounded sequences of functions. 2. } be a bounded sequence in Lp(T), 1 < p S 00, with bound M. Then there exist a subsequence nk and an Lp functionf, Ilfll, S M, such that for each g E Lp’(T), l/p + l/p’ = 1 (here p’ = 1 when p = 00).

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