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Fourier coefficients of continuous functions and a class of multipliers

Serguei V. Kislyakov (1988)

Annales de l'institut Fourier

If x is a bounded function on Z , the multiplier with symbol x (denoted by M x ) is defined by ( M x f ) ^ = x f ^ , f ∈ L 2 ( T ) . We give some conditions on x ensuring the “interpolation inequality” ∥ M x f ∥ L p ≤ C ∥ f ∥ L 1 α ∥ M x f ∥ L q 1 - α (here 1 < p < q and α = α ( p , q , x ) is between 0 and 1). In most cases considered M x fails to have stronger L 1 -regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets E ⊂ Z every positive sequence in ℓ 2 ( E ) can be majorized by the sequence { ...

Fourier inversion of distributions on projective spaces

Francisco Javier González Vieli (2006)

Commentationes Mathematicae Universitatis Carolinae

We show that the Fourier-Laplace series of a distribution on the real, complex or quarternionic projective space is uniformly Cesàro-summable to zero on a neighbourhood of a point if and only if this point does not belong to the support of the distribution.

Fourier multipliers for Hölder continuous functions and maximal regularity

Wolfgang Arendt, Charles Batty, Shangquan Bu (2004)

Studia Mathematica

Two operator-valued Fourier multiplier theorems for Hölder spaces are proved, one periodic, the other on the line. In contrast to the L p -situation they hold for arbitrary Banach spaces. As a consequence, maximal regularity in the sense of Hölder can be characterized by simple resolvent estimates of the underlying operator.

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