By Loukas Grafakos
ISBN-10: 0387094334
ISBN-13: 9780387094335
The basic objective of those volumes is to give the theoretical beginning of the sector of Euclidean Harmonic research. the unique variation was once released as a unmarried quantity, yet because of its dimension, scope, and the addition of latest fabric, the second one version involves volumes. the current version features a new bankruptcy on time-frequency research and the Carleson-Hunt theorem. the 1st quantity comprises the classical themes reminiscent of Interpolation, Fourier sequence, the Fourier rework, Maximal services, Singular Integrals, and Littlewood-Paley thought. the second one quantity comprises newer subject matters corresponding to functionality areas, Atomic Decompositions, Singular Integrals of Nonconvolution style, and Weighted Inequalities.
These volumes are in general addressed to graduate scholars in arithmetic and are designed for a two-course series at the topic with extra fabric incorporated for reference. the must haves for the 1st quantity are passable of completion of classes in genuine and complicated variables. the second one quantity assumes fabric from the 1st. This booklet is meant to offer the chosen themes intensive and stimulate extra examine. even though the emphasis falls on genuine variable tools in Euclidean areas, a bankruptcy is dedicated to the basics of research at the torus. This fabric is incorporated for historic purposes, because the genesis of Fourier research are available in trigonometric expansions of periodic features in different variables.
About the 1st edition:
"Grafakos's e-book is particularly ordinary with quite a few examples illustrating the definitions and ideas... The remedy is carefully smooth 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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Example text
4. 20) where Ps is the Poisson kernel. The Fourier transform Φ is Φ (ξ ) = ∞ 1 ψ (s)Ps (ξ ) ds = ∞ 1 ψ (s)e−2π s|ξ | ds (cf. 11), which is easily seen to be rapidly decreasing as |ξ | → ∞. The same is true for all the derivatives of Φ . The function Φ is clearly smooth on Rn \ {0}. Moreover, 44 6 Smoothness and Function Spaces ∂ j Φ (ξ ) = L−1 ∑ (−2π )k+1 k=0 |ξ |k ξ j k! |ξ | ∞ 1 sk+1 ψ (s) ds + O(|ξ |L ) = O(|ξ |L ) as |ξ | → 0, which implies that the distributional derivative ∂ j Φ is continuous at the origin.
1. Fix k ∈ Z+ . Show that Dkh ( f )(x) = 0 for all x, h in Rn if and only if f is a polynomial of degree at most k − 1. Hint: One direction may be proved by direct verification. 1. 2. 1 to the case γ = 0 and show that for all continuous functions f we have f L∞ ≤ f Λ ≤ 3 f L∞ ; 0 can be identified with L∞ (Rn ) ∩C(Rn ). hence the space Λ0 (b) Given a measurable function f on Rn we define (Rn ) f L˙ ∞ = inf f +c L∞ : c∈C . Let L˙ ∞ (Rn ) be the space of equivalent classes of bounded functions whose difference is a constant, equipped with this norm.
6. 10). Fix s ∈ R and all 1 < p < ∞. Then there exists a constant C1 that depends only on n, s, p, Φ , and Ψ such that for all f ∈ Lsp we have ∞ S0 ( f ) Lp + ∑ (2 js |Δ j ( f )|)2 j=1 1 2 Lp ≤ C1 f p Ls . 11) 18 6 Smoothness and Function Spaces Conversely, there exists a constant C2 that depends on the parameters n, s, p, Φ , and Ψ such that every tempered distribution f that satisfies ∞ S0 ( f ) Lp 1 2 ∑ (2 js|Δ j ( f )|)2 + Lp j=1 <∞ is an element of the Sobolev space Lsp with norm f p Ls ∞ ≤ C2 S0 ( f ) 1 2 ∑ (2 js|Δ j ( f )|)2 + Lp Lp j=1 .