By Loukas Grafakos
The basic aim of this article is to offer the theoretical origin of the sector of Fourier research. This publication is principally addressed to graduate scholars in arithmetic and is designed to serve for a three-course series at the topic. the single prerequisite for figuring out the textual content is passable finishing touch of a path in degree concept, Lebesgue integration, and complicated variables. This ebook is meant to offer the chosen subject matters in a few intensity and stimulate extra research. even supposing the emphasis falls on actual variable equipment in Euclidean areas, a bankruptcy is dedicated to the basics of research at the torus. This fabric is incorporated for ancient purposes, because the genesis of Fourier research are available in trigonometric expansions of periodic services in numerous variables.
While the first variation was once released as a unmarried quantity, the hot version will comprise one hundred twenty pp of latest fabric, with an extra bankruptcy on time-frequency research and different glossy subject matters. therefore, the e-book is now being released in 2 separate volumes, the 1st quantity containing the classical issues (Lp areas, Littlewood-Paley thought, Smoothness, etc...), the second one quantity containing the trendy subject matters (weighted inequalities, wavelets, atomic decomposition, etc...).
From a evaluate of the 1st edition:
“Grafakos’s e-book is particularly straightforward with quite a few examples illustrating the definitions and ideas. it truly is superior for readers who are looking to get a think for present study. The therapy is carefully sleek with unfastened use of operators and sensible research. Morever, in contrast to many authors, Grafakos has sincerely spent loads of time getting ready the exercises.” - Ken Ross, MAA Online
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This quantity will serve numerous reasons: to supply an creation for graduate scholars now not formerly conversant in the fabric, to function a reference for mathematical physicists already operating within the box, and to supply an advent to varied complicated subject matters that are obscure within the literature.
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Extra resources for Classical Fourier Analysis
It is not difficult to verify that h(ζ ) = 1 1+ζ log i πi 1−ζ is a conformal map from D onto the strip S = (0, 1) × R. Indeed, i(1 + ζ )/(1 − ζ ) lies in the upper half-plane and the preceding complex logarithm is a well defined holomorphic function that takes the upper half-plane onto the strip R × (0, π). Since F ◦ h is a holomorphic function on D, log |F ◦ h| is a subharmonic function on D. 36) R2 − 2ρR cos(θ − ϕ) + ρ 2 when z = ρeiϕ and |z| = ρ < R. Observe that when |ζ | = 1 and ζ = ±1, h(ζ ) has real part zero or one.
3. Similarly, on the multiplicative group G = R+ , a Haar measure is dx/x. 4. Counting measure is a Haar measure on the group Zn with group operation the usual addition. 5. The Heisenberg group Hn is the set Cn ×R with the group operation n (z1 , . . , zn ,t)(w1 , . . , wn , s) = z1 + w1 , . . , zn + wn ,t + s + 2 Im ∑ z j w j . j=1 It can easily be seen that the identity element e of this group is 0 ∈ Cn × R and (z1 , . . , zn ,t)−1 = (−z1 , . . , −zn , −t). Topologically the Heisenberg group is identified with Cn × R, and both left and right Haar measure on Hn is Lebesgue measure.
Operators that map L p to Lq are called of strong type (p, q) and operators that map L p to Lq,∞ are called weak type (p, q). 1. Let T be an operator defined on a linear space of complex-valued measurable functions on a measure space (X, µ) and taking values in the set of all complex-valued finite almost everywhere measurable functions on a measure space (Y, ν). Then T is called linear if for all f , g and all λ ∈ C, we have T ( f + g) = T ( f ) + T (g) and T (λ f ) = λ T ( f ). 2) T is called sublinear if for all f , g and all λ ∈ C, we have |T ( f + g)| ≤ |T ( f )| + |T (g)| and |T (λ f )| = |λ ||T ( f )|.
Classical Fourier Analysis by Loukas Grafakos