By Loukas Grafakos
This textual content is addressed to graduate scholars in arithmetic and to researchers who desire to collect a close figuring out of Euclidean Harmonic research. The textual content covers modern subject matters and methods in functionality areas, atomic decompositions, singular integrals of nonconvolution kind and the boundedness and convergence of Fourier sequence and integrals. The exposition and elegance are designed to stimulate additional research and advertise study. historic info and references are integrated on the finish of every chapter.
This 3rd variation incorporates a new bankruptcy entitled "Multilinear Harmonic research" which makes a speciality of subject matters on the topic of multilinear operators and their functions. Sections 1.1 and 1.2 also are new during this variation. a variety of corrections were made to the textual content from the former versions and a number of other advancements were integrated, akin to the adoption of transparent and stylish statements. a number of extra workouts were further with appropriate tricks whilst necessary.
Reviews from the second one Edition:
“The books hide a large number of arithmetic. they're definitely a important and beneficial addition to the present literature and will function textbooks or as reference books. scholars will particularly delight in the huge number of exercises.”
—Andreas Seeger, Mathematical stories
“The routines on the finish of every part complement the fabric of the part well and supply an excellent chance to strengthen extra instinct and deeper comprehension. The ancient notes in each one bankruptcy are meant to supply an account of prior examine in addition to to indicate instructions for additional research. the amount is especially addressed to graduate scholars who desire to research harmonic analysis.”
—Leonid Golinskii, zbMATH
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Additional info for Modern Fourier Analysis
Example text
7) Notice that the support properties of the operators Δ Ψ j yield the simple identity Ψ Ψ Ψ Ψ ΔΨ j = Δ j−1 + Δ j + Δ j+1 Δ j for all j ∈ Z. We also define a Schwartz function Φ such that Φ (ξ ) = ∑ j≤0 Ψ (2− j ξ ) 1 when ξ = 0, when ξ = 0. 9) j=1 for all ξ in Rn . 10) for f ∈ S (Rn ). 6(b). Having introduced the relevant background, we are now ready to state and prove the following result. 6. 10), respectively. Fix s ∈ R and 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 S0Φ ( f ) Lp ∞ 1 2 ∑ (2 js |Δ Ψj ( f )|)2 + Lp j=1 ≤ C1 f p Ls .
24) Proof. The proof of the theorem is similar to . 6. 23), we start with f ∈ Lsp and note that 2 js Δ j ( f ) = 2 js |ξ |s |ξ |−sΨ (2− j ξ ) f ∨ = σ (2− j ξ ) fs ∨ = Δ σj ( fs ) , where σ (ξ ) = Ψ (ξ )|ξ |−s and Δ σj is the Littlewood–Paley operator given on the Fourier transform side by multiplication with the function σ (2− j ξ ). We have ∑ |2 js Δ Ψj ( f )|2 j∈Z 1 2 Lp = ∑ |Δ σj ( fs )|2 j∈Z 1 2 Lp ≤ C fs L. p = C f L. 2 in [156]. 23). 24) is finite, n n then the . p distribution f in S (R )/P(R ) must lie in the homogeneous Sobolev space Ls with norm controlled by a multiple of this expression.
5. Let f be a C m function on Rn for some m ∈ Z+ . Then for all h = (h1 , . . , hn ) and x ∈ Rn the following identity holds: Dh ( f )(x) = 1 n ∑ h j (∂ j f )(x + sh) ds . 4) 0 j=1 More generally, we have that Dm h ( f )(x) = 1 0 ··· 1 n ∑ ··· 0 j =1 1 n ∑ h j1 · · ·h jm (∂ j1 · · ·∂ jm f )(x + (s1 +· · ·+sm )h) ds1 · · · dsm . 5) jm =1 . Consequently, if, for some γ ∈ (0, 1), ∂ α f lies in Λγ for all multi-indices |α | = m, . then f lies in Λm+γ . Proof. 4) by induction.. Now suppose that ∂ α f lie in Λγ for all multi-indices |α | = m.