Walsh Multipliers for Dyadic Hardy Spaces

Daly, James E.; Fridli, S.

doi:10.1080/0003681031000148564

Cited by 7 publications

(9 citation statements)

References 9 publications

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“…. The authors have studied these conditions extensively in the context of Hardy spaces on the dyadic group [2], Vilenkin groups [5], dyadic field [4], and the classical case [3]. In two dimensions, set…”

Section: Resultsmentioning

confidence: 99%

Hörmander multipliers on two-dimensional dyadic Hardy spaces

Daly¹,

Fridli²

2008

Journal of Mathematical Analysis and Applications

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In this paper we are interested in conditions on the coefficients of a two-dimensional Walsh multiplier operator that imply the operator is bounded on certain of the Hardy type spaces H p , 0 < p < ∞. We consider the classical coefficient conditions, the MarcinkiewiczHörmander-Mihlin conditions. They are known to be sufficient for the trigonometric system in the one and two-dimensional cases for the spaces L p , 1 < p < ∞. This can be found in the original papers of Marcinkiewicz [J. Marcinkiewicz, Sur les multiplicateurs des series de Fourier, Studia Math. 8 (1939) 78-91], Hörmander [L. Hörmander, Estimates for translation invariant operators in L p spaces, Acta Math. 104 (1960) 93-140], and Mihlin [S.G. Mihlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR 109 (1956) 701-703; S.G. Mihlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, 1965]. In this paper we extend these results to the two-dimensional dyadic Hardy spaces.

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Section: Resultsmentioning

confidence: 99%

Hörmander multipliers on two-dimensional dyadic Hardy spaces

Daly¹,

Fridli²

2008

Journal of Mathematical Analysis and Applications

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“…Discrete analogs of certain spaces (namely, the dyadic Hardy space and the dyadic space of BM O(D) functions of bounded mean oscillation) are applied in recent years (see, e.g., [40] as well as [2,28,42,93] and others).…”

Section: Nikol'skiȋ-besov Dyadic Spaces and Their Propertiesmentioning

confidence: 99%

Piecewise Polynomial Approximation Methods in the Theory of Nikol’Skiĭ–Besov Spaces

Irodova

2015

J Math Sci

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“…Note that with the same conditions T (m,k) is not bounded from H 1 to L 1 in general (see [7,8]). Under slightly stronger conditions the Marcinkiewicz multiplier theorem will be extended to Hardy spaces in the next section.…”

Section: Then T (Mk) F ∈ L P Andmentioning

confidence: 99%

“…Hörmander [16] generalized the Marcinkiewicz condition and theorem. Under some Hörmander-type conditions the boundedness of the multiplier operator was proved also on the Hardy spaces H p (for trigonometric Fourier series see [1,8,20], for Walsh-and Vilenkin-Fourier series see [7,[17][18][19]). …”

Section: Introductionmentioning

confidence: 99%