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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...

Estimates with global range for oscillatory integrals with concave phase

Bjorn Gabriel Walther (2002)

Colloquium Mathematicae

We consider the maximal function | | ( S a f ) [ x ] | | L [ - 1 , 1 ] where ( S a f ) ( t ) ( ξ ) = e i t | ξ | a f ̂ ( ξ ) and 0 < a < 1. We prove the global estimate | | S a f | | L ² ( , L [ - 1 , 1 ] ) C | | f | | H s ( ) , s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.

Ideal norms and trigonometric orthonormal systems

Jörg Wenzel (1994)

Studia Mathematica

We characterize the UMD-property of a Banach space X by sequences of ideal norms associated with trigonometric orthonormal systems. The asymptotic behavior of those numerical parameters can be used to decide whether X is a UMD-space. Moreover, if this is not the case, we obtain a measure that shows how far X is from being a UMD-space. The main result is that all described sequences are not only simultaneously bounded but are also asymptotically equivalent.

Lebesgue type points in strong (C,α) approximation of Fourier series

Włodzimierz Łenski, Bogdan Roszak (2011)

Banach Center Publications

We present an estimation of the H k , k r q , α f and H λ , u ϕ , α f means as approximation versions of the Totik type generalization (see [5], [6]) of the result of G. H. Hardy, J. E. Littlewood. Some corollaries on the norm approximation are also given.

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