Marcinkiewicz multiplier theorem and the Sunouchi operator for Ciesielski–Fourier series

Weisz, Ferenc

doi:10.1016/j.jat.2004.12.017

Cited by 4 publications

(6 citation statements)

References 39 publications

(57 reference statements)

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“…For more general multipliers we refer to [69]. It is easy to see that λ (1) satisfies the conditions of Theorems 21 and 22 and, moreover, λ (2) fulfills the conditions in Theorem 21.…”

Section: 4mentioning

confidence: 99%

“…It is easy to see that λ (1) satisfies the conditions of Theorems 21 and 22 and, moreover, λ (2) fulfills the conditions in Theorem 21. These two multipliers are used in [69] to prove some inequalities for the Sunouchi operators.…”

Section: 4mentioning

confidence: 99%

“…The Marcinkiewicz multiplier theorem is generalized for Ciesielski systems in the next theorem (see Weisz [69]). …”

Section: Spline-fourier and Ciesielski-fourier Seriesmentioning

confidence: 99%

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Results on spline-Fourier and Ciesielski-Fourier series

Weisz

2006

Approximation and Probability

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Abstract. Some recent results on spline-Fourier and Ciesielski-Fourier series are summarized. The convergence of spline Fourier series and the basis properties of the spline systems are considered. Some classical topics, that are well known for trigonometric and Walsh-Fourier series, are investigated for Ciesielski-Fourier series, such as inequalities for the Fourier coefficients, convergence a.e. and in norm, Fejér and θ-summability, strong summability and multipliers. The connection between Fourier series and Hardy spaces is studied.1. Introduction. It is known that the spline Fourier series of f ∈ L p converges a.e. and in L p norm to f , whenever 1 ≤ p < ∞. The maximal operator of the partial sums with respect to the spline (or unbounded Ciesielski) systems of order (m, k) is bounded from the Hardy space H p to L p (1/(m − k + 2) < p < ∞) and is of weak type (1, 1). If 1 < p < ∞ then H p is equivalent to L p . Moreover, the spline systems are unconditional and equivalent bases to the Haar system in H p (1/(m − k + 2) < p < ∞).We investigate also the bounded Ciesielski systems, which can be obtained from the spline systems in the same way as the Walsh system from the Haar. Some results, that are well known for trigonometric and Walsh-Fourier series, are extended to CiesielskiFourier series. Paley and Hardy-Littlewood type inequalities are shown for CiesielskiFourier coefficients. If f ∈ L p (1 < p < ∞) then the Ciesielski-Fourier series of f converges

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“…For more general multipliers we refer to [69]. It is easy to see that λ (1) satisfies the conditions of Theorems 21 and 22 and, moreover, λ (2) fulfills the conditions in Theorem 21.…”

Section: 4mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

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Results on spline-Fourier and Ciesielski-Fourier series

Weisz

2006

Approximation and Probability

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“…The general M-H-M condition does not give as precise results as those obtained for this pair of multipliers by Simon, Daly and Phillips, but this is not unexpected. Concerning the properties of multiplier operators, in particular the Marcinkiewicz and the Sunouchi multipliers, with respect to the Ciesielski system we call the attention to two recent papers by Weisz [21,22]. …”

Section: Corollary 22mentioning

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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“…In the special case σ = δ = 1 this model was considered more in detail in [21]. The author obtained a potential decay estimate for solutions localized to low frequencies, whereas the high frequency part decays exponentially under the requirement of a suitable regularity for the data by application of the Marcinkiewicz theorem (see, for example, [13,24]) to related Fourier multipliers. The case of semilinear visco-elastic damped wave models (1) and (2) with σ = δ = 1 was studied in several recent papers such as [3] and [19].…”

mentioning

confidence: 99%

<inline-formula><tex-math id="M1">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula> estimates for oscillating integrals and their applications to semi-linear models with <inline-formula><tex-math id="M2">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-evolution like structural damping

Dao¹,

Reissig²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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The present paper is a continuation of our recent paper [4]. We will consider the following Cauchy problem for semi-linear structurally damped σ-evolution models:Our aim is to study two main models including σ-evolution models with structural damping δ ∈ ( σ 2 , σ) and those with visco-elastic damping δ = σ. Here the function f (u, ut) stands for power nonlinearities |u| p and |ut| p with a given number p > 1. We are interested in investigating the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on L q spaces by assuming additional L m regularity for the initial data, with q ∈ (1, ∞) and m ∈ [1, q).

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Marcinkiewicz multiplier theorem and the Sunouchi operator for Ciesielski–Fourier series

Cited by 4 publications

References 39 publications

Results on spline-Fourier and Ciesielski-Fourier series

Results on spline-Fourier and Ciesielski-Fourier series

Hörmander multipliers on two-dimensional dyadic Hardy spaces

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