Download Classical Fourier Analysis by Loukas Grafakos PDF

By Loukas Grafakos

The basic objective of those volumes is to provide the theoretical starting place of the sector of Euclidean Harmonic research. the unique version was once released as a unmarried quantity, yet as a result of its dimension, scope, and the addition of recent fabric, the second one version contains volumes. the current version encompasses a new bankruptcy on time-frequency research and the Carleson-Hunt theorem. the 1st quantity includes the classical issues comparable to Interpolation, Fourier sequence, the Fourier remodel, Maximal capabilities, Singular Integrals, and Littlewood-Paley concept. the second one quantity includes more moderen themes corresponding to functionality areas, Atomic Decompositions, Singular Integrals of Nonconvolution sort, and Weighted Inequalities.

These volumes are generally addressed to graduate scholars in arithmetic and are designed for a two-course series at the topic with extra fabric integrated for reference. the necessities for the 1st quantity are passable crowning glory of classes in actual and intricate variables. the second one quantity assumes fabric from the 1st. This e-book is meant to offer the chosen themes extensive and stimulate additional examine. even if the emphasis falls on actual variable tools in Euclidean areas, a bankruptcy is dedicated to the basics of study at the torus. This fabric is integrated for historic purposes, because the genesis of Fourier research are available in trigonometric expansions of periodic services in different variables.

About the 1st edition:

"Grafakos's booklet is especially straightforward with various examples illustrating the definitions and ideas... The remedy is punctiliously sleek with unfastened use of operators and sensible research. Morever, not like many authors, Grafakos has basically spent loads of time getting ready the exercises."

- Kenneth Ross, MAA Online

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13). 4 can now be extended to the case in which the interpolated operators are allowed to vary. In particular, if a family of operators depends analytically on a parameter z, then the proof of this theorem can be adapted to work in this setting. We now describe the setup for this theorem. Let (X, µ) and (Y, ν) be measure spaces. 21) whenever f and g are simple functions on X and Y , respectively. 22) is analytic in the open strip S = {z ∈ C : 0 < Re z < 1} and continuous on its closure. 23) for all z satisfying 0 ≤ Re z ≤ 1.

28) G as h → e. Now approximate a given f in L p (G) by a continuous function with compact support g to deduce that | f (h−1 x) − f (x)| p dλ (x) → 0 h → e. 15 (i). 31) ( f (y−1 x) − f (x))kε (y) dλ (y) + Vc ( f (y−1 x) − f (x))kε (y) dλ (y) . 31). 33) δ ≤ 2 f L p (G) |kε (y)| dλ (y) < , 2 Vc provided we have that Vc |kε (x)| dλ (x) < δ 4 f . 34) is valid for ε < ε0 by property (iii). 33) imply the required conclusion. 2 Convolution and Approximate Identities 27 The case p = ∞ follows similarly.

Hint: On the multiplicative group (R+ , dtt ) consider the convolution of the function 1 | f (x)|x p with the function x − p1 1 χ[1,∞) and the convolution of the function | f (x)|x1+ p 1 with x p χ(0,1] . 8. (G. H. Hardy ) Let 0 < b < ∞ and 1 ≤ p < ∞. Prove that | f (t)| dt 0 1 p p x ∞ x −b−1 dx 0 ∞ | f (t)| dt 0 1 p p ∞ x b−1 x dx p ≤ b p ≤ b ∞ p p−b−1 | f (t)| t 1 p , dt 0 ∞ p p+b−1 | f (t)| t 1 p dt . 0 Hint: On the multiplicative group (R+ , dtt ) consider the convolution of the function b b b b | f (x)|x1− p with x− p χ[1,∞) and of the function | f (x)|x1+ p with x p χ(0,1] .

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