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Carleson's Theorem: proof, complements, variations.

Michael T. Lacey (2004)

Publicacions Matemàtiques

Carleson's Theorem from 1965 states that the partial Fourier sums of a square integrable function converge pointwise. We prove an equivalent statement on the real line, following the method developed by the author and C. Thiele. This theorem, and the proof presented, is at the center of an emerging theory which complements the statement and proof of Carleson's theorem. An outline of these variations is also given.

Certain lacunary cosine series are recurrent

D. Grubb, Charles Moore (1994)

Studia Mathematica

Let the coefficients of a lacunary cosine series be bounded and not square-summable. Then the partial sums of the series are recurrent.

Cesàro summability of one- and two-dimensional trigonometric-Fourier series

Ferenc Weisz (1997)

Colloquium Mathematicae

We introduce p-quasilocal operators and prove that if a sublinear operator T is p-quasilocal and bounded from L ∞ to L ∞ then it is also bounded from the classical Hardy space H p ( T ) to L p (0 < p ≤ 1). As an application it is shown that the maximal operator of the one-parameter Cesàro means of a distribution is bounded from H p ( T ) to L p (3/4 < p ≤ ∞) and is of weak type ( L 1 , L 1 ) . We define the two-dimensional dyadic hybrid Hardy space H 1 ♯ ( T 2 ) and verify that the maximal operator of the Cesàro means of a two-dimensional...

Character sums in complex half-planes

Sergei V. Konyagin, Vsevolod F. Lev (2004)

Journal de Théorie des Nombres de Bordeaux

Let A be a finite subset of an abelian group G and let P be a closed half-plane of the complex plane, containing zero. We show that (unless A possesses a special, explicitly indicated structure) there exists a non-trivial Fourier coefficient of the indicator function of A which belongs to P . In other words, there exists a non-trivial character χ ∈ G ^ such that ∑ a ∈ A χ ( a ) ∈ P .

Characterization of the convolution operators on quasianalytic classes of Beurling type that admit a continuous linear right inverse

José Bonet, Reinhold Meise (2008)

Studia Mathematica

Extending previous work by Meise and Vogt, we characterize those convolution operators, defined on the space ( ω ) ( ℝ ) of (ω)-quasianalytic functions of Beurling type of one variable, which admit a continuous linear right inverse. Also, we characterize those (ω)-ultradifferential operators which admit a continuous linear right inverse on ( ω ) [ a , b ] for each compact interval [a,b] and we show that this property is in fact weaker than the existence of a continuous linear right inverse on ( ω ) ( ℝ ) .

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