The Hausdorff Mean of a Fourier-Stieltjes Transform

Georgakis, Constantine

doi:10.2307/2159753

Cited by 7 publications

(5 citation statements)

References 6 publications

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“…For the real line case, Goldberg [5] investigated the properties of the operator H φ on the spaces L p (R) with 1 < p ≤ 2. Georgakis [3] studied the Fourier analytic properties of H φ on the space of complex bounded regular Borel measures on R, and as a special case he showed that if φ ∈ L 1 (R), then H φ is a bounded operator on L 1 (R). Giang and Móricz [4] and Liflyand and Móricz [8] followed as stated above.…”

Section: Giang and Móricz [4] Proved The Following Resultmentioning

confidence: 99%

The Hausdorff operators on the real Hardy spaces H^p(R)

Kanjin

2001

Studia Math.

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Abstract. We prove that the Hausdorff operator generated by a function φ is bounded on the real Hardy space H p (R), 0 < p ≤ 1, if the Fourier transform φ of φ satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Cesàro operator of order α on H p (R), 2/(2α + 1) < p ≤ 1. Our proof is based on the atomic decomposition and molecular characterization of H p (R).

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Section: Giang and Móricz [4] Proved The Following Resultmentioning

confidence: 99%

The Hausdorff operators on the real Hardy spaces H^p(R)

Kanjin

2001

Studia Math.

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“…But, the Fourier integral of f is also convergent to f (x) a.e. by (3). Hence, F (0) − F (0−) = 0 in (1), and so (1) reduces to equation (4).…”

Section: B) If the Fourier Integral Of A Unimodal Distribution Is Conmentioning

confidence: 97%

The convergence of the Fourier integral of a unimodal distribution

Georgakis

1998

Proc. Amer. Math. Soc.

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Abstract. We show that the characteristic function of a unimodal probability distribution function can be inverted by the Fourier transform a.e. if and only if the distribution is absolutely continuous. The result complements Khintchine's criterion for unimodal distributions.The distribution function F (x) = Pr(X ≤ x) of a random variable X is unimodal with vertex at the origin if F (x) is convex for x < 0 and concave for x > 0. Some examples of unimodal distributions are: i) the standard normal, exponential and Cauchy probability densities; ii) the class of stable distributions; iii) the class of symmetric infinitely divisible distributions. According to Khintchine's criterion ( for 0 < x ≤ t < ∞, −t −1 for − ∞ < t ≤ x < 0, 0 otherwise.

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“…It can go back to the Hausdorff summability method which was introduced in 1917 in connecting with summability of number series. The modern (continuous) versions of Hausdorff operators were initiated with the work of Siskakas in complex analysis setting [16] and with the work of Georgakis [13] and Liflyand-Móricz in the Fourier transform setting [15]. Recently, an increasing attention has been acquired on the boundedness of Hausdorff operators and their commutators on various spaces and their sharp bounds.…”

Section: Introductionmentioning

confidence: 99%

Necessary and sufficient conditions for generalized Hausdorff operators and commutators

Wu¹

2015

Ann. Funct. Anal.

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In this paper, we introduce a type of generalized Hausdorff operators and characterize the boundedness of these operators on Lebesgue spaces and central Morrey spaces. Moreover, we obtain the operator norms on these spaces. We also obtain sufficient and necessary conditions which ensure the boundedness of their commutators on Lebesgue spaces and central Morrey spaces with symbols in central BMO spaces. As applications, we give a new method to obtain sharp bounds for weighted Hardy operators and weighted Cesàro operators on Lebesgue spaces and central Morrey spaces.

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The Hausdorff Mean of a Fourier-Stieltjes Transform

Cited by 7 publications

References 6 publications

The Hausdorff operators on the real Hardy spaces H^p(R)

The Hausdorff operators on the real Hardy spaces H^p(R)

The convergence of the Fourier integral of a unimodal distribution

Necessary and sufficient conditions for generalized Hausdorff operators and commutators

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The Hausdorff Mean of a Fourier-Stieltjes Transform

Cited by 7 publications

References 6 publications

The Hausdorff operators on the real Hardy spaces Hp(R)

The Hausdorff operators on the real Hardy spaces Hp(R)

The convergence of the Fourier integral of a unimodal distribution

Necessary and sufficient conditions for generalized Hausdorff operators and commutators

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Product

Resources

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The Hausdorff operators on the real Hardy spaces H^p(R)

The Hausdorff operators on the real Hardy spaces H^p(R)