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A Marcinkiewicz type multiplier theorem for H¹ spaces on product domains

Michał Wojciechowski — 2000

Studia Mathematica

It is proved that if m : d satisfies a suitable integral condition of Marcinkiewicz type then m is a Fourier multiplier on the H 1 space on the product domain d 1 × . . . × d k . This implies an estimate of the norm N ( m , L p ( d ) of the multiplier transformation of m on L p ( d ) as p→1. Precisely we get N ( m , L p ( d ) ) ( p - 1 ) - k . This bound is the best possible in general.

On the relationships between Fourier-Stieltjes coefficients and spectra of measures

Przemysław OhryskoMichał Wojciechowski — 2014

Studia Mathematica

We construct examples of uncountable compact subsets of complex numbers with the property that any Borel measure on the circle group with Fourier coefficients taking values in this set has a natural spectrum. For measures with Fourier coefficients tending to 0 we construct an open set with this property. We also give an example of a singular measure whose spectrum is contained in our set.

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