Read e-book online Classical Fourier Analysis PDF

By Loukas Grafakos

ISBN-10: 0387094318

ISBN-13: 9780387094311

The fundamental objective of those volumes is to provide the theoretical origin of the sector of Euclidean Harmonic research. the unique version was once released as a unmarried quantity, yet as a result of its measurement, scope, and the addition of latest fabric, the second one version contains volumes. the current version incorporates a new bankruptcy on time-frequency research and the Carleson-Hunt theorem. the 1st quantity includes the classical subject matters similar to Interpolation, Fourier sequence, the Fourier rework, Maximal capabilities, Singular Integrals, and Littlewood-Paley thought. the second one quantity comprises more moderen themes similar to functionality areas, Atomic Decompositions, Singular Integrals of Nonconvolution variety, and Weighted Inequalities.

These volumes are often addressed to graduate scholars in arithmetic and are designed for a two-course series at the topic with extra fabric integrated for reference. the must haves for the 1st quantity are passable final touch of classes in actual and intricate variables. the second one quantity assumes fabric from the 1st. This publication is meant to provide the chosen themes intensive and stimulate extra examine. even supposing the emphasis falls on genuine variable equipment in Euclidean areas, a bankruptcy is dedicated to the basics of research at the torus. This fabric is integrated for historic purposes, because the genesis of Fourier research are available in trigonometric expansions of periodic features in numerous variables.

About the 1st edition:

"Grafakos's ebook is especially ordinary with quite a few examples illustrating the definitions and ideas... The therapy is carefully glossy with loose use of operators and sensible research. Morever, in contrast to many authors, Grafakos has in actual fact spent loads of time getting ready the exercises."

- Kenneth Ross, MAA Online

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7). 2 Complex Method: The Riesz–Thorin Interpolation Theorem The next interpolation theorem assumes stronger endpoint estimates, but yields a more natural bound on the norm of the operator on the intermediate spaces. Unfortunately, it is mostly applicable for linear operators and in some cases for sublinear operators (often via a linearization process). It does not apply to quasilinear operators without some loss in the constant. A short history of this theorem is discussed at the end of this chapter.

15. Property (iii) follows from the expression giving FN in terms of sines, while property (i) follows from the expression giving FN in terms of exponentials. Property (ii) is identical to property (i), since FN is nonnegative. Next comes the basic theorem concerning approximate identities. 19. Let kε be an approximate identity on a locally compact group G with left Haar measure λ . (1) If f ∈ L p (G) for 1 ≤ p < ∞, then kε ∗ f − f L p (G) → 0 as ε → 0. (2) When p = ∞, the following is valid: If f is continuous in a neighborhood of a a compact subset K of G, then kε ∗ f − f L∞ (K) → 0 as ε → 0.

7. (Yano [294] ) Let (X, µ) and (Y, ν) be two measure spaces with µ(X) < ∞ and ν(Y ) < ∞. Let T be a sublinear operator that maps L p (X) to L p (Y ) for every 1 < p ≤ 2 with norm T L p →L p ≤ A(p − 1)−α for some fixed A, α > 0. Prove that for all f measurable on X we have 1 L p Spaces and Interpolation 44 1 1 |T ( f )| dν ≤ 6A(1 + ν(Y )) 2 Y X α 2 , | f |(log+ 2 | f |) dµ +Cα + µ(X) α k where Cα = ∑∞ k=1 k (2/3) . This result provides an example of extrapolation. Hint: Write ∞ f= ∑ f χSk , k=0 where Sk = {2k ≤ | f | < 2k+1 } when k ≥ 1 and S0 = {| f | < 2}.

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