By Loukas Grafakos

The basic aim of those volumes is to offer the theoretical origin of the sector of Euclidean Harmonic research. the unique variation was once released as a unmarried quantity, yet as a result of its measurement, scope, and the addition of recent fabric, the second one version involves volumes. the current version encompasses a new bankruptcy on time-frequency research and the Carleson-Hunt theorem. the 1st quantity comprises the classical issues corresponding to Interpolation, Fourier sequence, the Fourier rework, Maximal services, Singular Integrals, and Littlewood-Paley concept. the second one quantity includes more moderen themes similar to functionality areas, Atomic Decompositions, Singular Integrals of Nonconvolution sort, and Weighted Inequalities.

These volumes are quite 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 necessities for the 1st quantity are passable crowning glory of classes in actual and complicated variables. the second one quantity assumes fabric from the 1st. This publication is meant to give the chosen issues intensive and stimulate additional research. 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 incorporated for old purposes, because the genesis of Fourier research are available in trigonometric expansions of periodic features in different variables.

About the 1st edition:

"Grafakos's booklet is especially easy with a variety of examples illustrating the definitions and ideas... The therapy is carefully glossy with loose use of operators and practical research. Morever, in contrast to many authors, Grafakos has truly spent loads of time getting ready the exercises."

- Kenneth Ross, MAA Online

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**Example text**

Implies that g∗ ≤ f ∗ and | f |∗ = f ∗ . (5) (k f )∗ = |k| f ∗ . (6) ( f + g)∗ (t1 + t2 ) ≤ f ∗ (t1 ) + g∗ (t2 ). (7) ( f g)∗ (t1 + t2 ) ≤ f ∗ (t1 )g∗ (t2 ). e. implies fn∗ ↑ f ∗ . e. implies f ∗ ≤ lim inf fn∗ . n→∞ n→∞ (10) f∗ (11) t ≤ µ({| f | ≥ f ∗ (t)}) if µ({| f | ≥ f ∗ (t) − c}) < ∞ for some c > 0. (12) df = df∗. (13) (| f | p )∗ = ( f ∗ ) p when 0 < p < ∞ . is right continuous on [0, ∞). ∞ | f | p dµ = (14) X 0 f ∗ (t) p dt when 0 < p < ∞ . 4 Lorentz Spaces (15) (16) f L∞ 47 = f ∗ (0). sup t q f ∗ (t) = sup α d f (α) t>0 q for 0 < q < ∞ .

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 . 2 that N d f (α) = ∑ B j χ[a j+1 ,a j ) (α) , j=0 where j B j = ∑ µ(Ei ) i=1 and aN+1 = B0 = 0 and a0 = ∞. Observe that for B0 ≤ t < B1 , the smallest s > 0 with d f (s) ≤ t is a1 . Similarly, for B1 ≤ t < B2 , the smallest s > 0 with d f (s) ≤ t is a2 . Arguing this way, it is not difficult to see that f ∗ (t) = N ∑ a j χ[B j−1 ,B j ) (t) .

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}.