Weak type inequalities for the Walsh and bounded Ciesielski systems

Weisz, Ferenc

doi:10.1023/b:anam.0000033225.98714.4b

Cited by 22 publications

(13 citation statements)

References 18 publications

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“…We may suppose that 0 < < 1 since the case = 1 was proved in Weisz [18]. As the operators * and , * are bounded on L ∞ (see [16,17]), we have to prove (1).…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Weak inequalities for Cesàro and Riesz summability of Walsh–Fourier series

Šimon

Weisz

2008

Journal of Approximation Theory

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The maximal operators for Cesàro or (C, ) and Riesz summability with respect to Walsh-Fourier series are investigated as mappings between dyadic Hardy and Lebesgue spaces. It is well known that they are bounded from H p to L p for all 1/( + 1) < p < ∞. In this work we prove that this boundedness result does not hold anymore if p 1/( + 1). However, for p = 1/( + 1) the maximal operators are bounded from H 1/( +1) to the weak L 1/( +1) space. To the proof some known estimations for the Cesàro and Riesz kernels have to be sharpened.

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“…We may suppose that 0 < < 1 since the case = 1 was proved in Weisz [18]. As the operators * and , * are bounded on L ∞ (see [16,17]), we have to prove (1).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…For the endpoint case p = 1/( + 1) Weisz obtained in [18] for the Fejér means that 1 * is bounded from H 1/2 to the weak L 1/2 . Simon [10] gave a counterexample which shows that 1 * is not bounded from H p to L p if 0 < p < 1 2 .…”

Section: Introductionmentioning

confidence: 95%

Weak inequalities for Cesàro and Riesz summability of Walsh–Fourier series

Šimon

Weisz

2008

Journal of Approximation Theory

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“…The counterexample for p = 1/2 due to Goginava [6], (see also [3] and [10]). Weisz [25] proved that σ * is bounded from the Hardy space…”

Section: The Concept Of Modulus Of Continuity In Hmentioning

confidence: 99%

A note on the norm convergence by Vilenkin–Fejér means

Tephnadze

2014

Georgian Mathematical Journal

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“…The counterexample for p = 1/2 due to Goginava [7], (see also [1] and [20]). Weisz [28] proved that σ * is bounded from the Hardy space H 1/2 to the space L 1/2,∞ . The second author [21,22] proved that the following is true: Theorem T1.…”

Section: This Theorem Implies Thatmentioning

confidence: 99%