By Loukas Grafakos

The fundamental objective of those volumes is to provide the theoretical starting place of the sphere 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 variation contains volumes. the current version encompasses a new bankruptcy on time-frequency research and the Carleson-Hunt theorem. the 1st quantity comprises the classical themes akin to Interpolation, Fourier sequence, the Fourier rework, Maximal features, Singular Integrals, and Littlewood-Paley concept. the second one quantity includes newer subject matters equivalent to functionality areas, Atomic Decompositions, Singular Integrals of Nonconvolution kind, and Weighted Inequalities.

These volumes are more often than not 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 of completion 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 intensive and stimulate extra learn. even if the emphasis falls on genuine variable tools in Euclidean areas, a bankruptcy is dedicated to the basics of study at the torus. This fabric is incorporated for old purposes, because the genesis of Fourier research are available in trigonometric expansions of periodic services in different variables.

About the 1st edition:

"Grafakos's publication is particularly ordinary with a number of examples illustrating the definitions and ideas... The remedy is punctiliously glossy with unfastened use of operators and sensible research. Morever, in contrast to many authors, Grafakos has essentially spent loads of time getting ready the exercises."

- Kenneth Ross, MAA Online

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**Example text**

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. The extension of the Riesz–Thorin interpolation theorem is now stated. 7. Let Tz be an analytic family of linear operators of admissible growth. 24) 1 L p Spaces and Interpolation 38 for j = 0, 1 and some b < π. Let 0 < θ < 1 satisfy 1 1−θ θ = + p p0 p1 and 1 1−θ θ = + .

5) Proof. 5, and we may therefore concentrate on the case q < ∞. 2, then N d f (s) = ∑ B j χ[a j+1 ,a j ) (s) j=1 with the understanding that aN+1 = 0. 5) for simple functions. e. 5 (8)). 5). Since L p,p ⊆ L p,∞ , one may wonder whether these spaces are nested. The next result shows that for any fixed p, the Lorentz spaces L p,q increase as the exponent q increases. 10. Suppose 0 < p ≤ ∞ and 0 < q < r ≤ ∞. Then there exists a constant c p,q,r (which depends on p, q, and r) such that f L p,r ≤ c p,q,r f L p,q .

Let f be a complex-valued function defined on X. 4 Lorentz Spaces 45 f ∗ (t) = inf{s > 0 : d f (s) ≤ t} . 2) We adopt the convention inf 0/ = ∞, thus having f ∗ (t) = ∞ whenever d f (α) > t for all α ≥ 0. Observe that f ∗ is decreasing and supported in [0, µ(X)]. Before we proceed with properties of the function f ∗ , we work out three examples. f(x) f*(t) a1 a1 a2 a2 a3 a3 0 . E3 E1 E2 . 0 B1 x . B2 B3 t Fig. 3 The graph of a simple function f (x) and its decreasing rearrangement f ∗ (t). 2. 2, N f (x) = ∑ a j χE j (x) , j=1 where the sets E j have finite measure and are pairwise disjoint and a1 > · · · > aN .