By Camil Muscalu, Wilhelm Schlag

This two-volume textual content in harmonic research introduces a wealth of analytical effects and strategies. it truly is principally self-contained and should be invaluable to graduate scholars and researchers in either natural and utilized research. various workouts and difficulties make the textual content appropriate for self-study and the study room alike. this primary quantity starts off with classical one-dimensional subject matters: Fourier sequence; harmonic services; Hilbert remodel. Then the higher-dimensional Calderón-Zygmund and Littlewood-Paley theories are constructed. Probabilistic tools and their purposes are mentioned, as are functions of harmonic research to partial differential equations. the amount concludes with an advent to the Weyl calculus. the second one quantity is going past the classical to the hugely modern and makes a speciality of multilinear points of harmonic research: the bilinear Hilbert remodel; Coifman-Meyer conception; Carleson's answer of the Lusin conjecture; Calderón's commutators and the Cauchy necessary on Lipschitz curves. the cloth during this quantity has now not formerly seemed jointly in ebook shape.

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**Classical and Multilinear Harmonic Analysis**

This two-volume textual content in harmonic research introduces a wealth of analytical effects and strategies. it's mostly self-contained and may be worthwhile to graduate scholars and researchers in either natural and utilized research. a variety of routines and difficulties make the textual content appropriate for self-study and the study room alike.

**Additional resources for Classical and Multilinear Harmonic Analysis**

**Example text**

Does this generalize to H s (T) and, if so, for which values of s? 7 Use the previous two problems to prove the following theorem of Pal and Bohr: for any real function f ∈ C(T) there exists a homeomorphism ϕ : T → T such that Sn (f ◦ ϕ) −→ f ◦ ϕ uniformly. Hint: Without loss of generality let f > 0. Consider the domain defined in terms of polar coordinates by means of r(θ) = f (θ/2π). Then apply the Riemann mapping theorem to the unit disk. 8 Suppose that f ∈ L1 (T) satisfies fˆ(j ) = 0 for all j with |j | < n.

We now apply Wiener’s covering lemma to pass to a more convenient sub-cover. Select an arc of maximal length from {Ij }; call it J1 . Observe that any Ij such that Ij ∩ J1 = ∅ satisfies Ij ⊂ 3J1 where 3J1 is the arc with the same center as J1 and three times the length (if 3J1 has length larger than 1 then set 3J1 = T). Now remove all arcs from {Ij }N j =1 that intersect J1 . Let J2 be a remaining are having maximal length. Continuing in this fashion we obtain arcs {J }L=1 that are pairwise disjoint and such that N L Ij ⊂ j =1 3J .

1) π r 2 z+rD T Proof By the divergence theorem, with dm the Lebesgue measure in the plane, ˆ ˆ ˆ d 1 1 u(z + re(θ )) dθ = ∂n u(w) σ (dw) = u dm = 0. dr T 2π r z+r∂D 2π r z+rD Thus, the circular means (with σ the arc length) ˆ ˆ 1 M(r) := u(z + re(θ )) dθ = u(w) σ (dw) 2π r |w−z|=r T are constant and, since M(r) → u(z) as r → 0, we see that u(z) = M(r) as claimed. Integrating su(z) = sM(s) over 0 < s < r implies the mean-value property over solid disks. The special role of circles here is no coincidence.