Martingale transforms and fractional integrals on rearrangement-invariant martingale Hardy spaces

Periodica Mathematica Hungarica - We establish an interpolation result for the rearrangement-invariant martingale Hardy spaces. By using this interpolation result, we extend the mapping properties...     

Martingale Musielak-Orlicz Hardy spaces

In this article,we introduce the martingale Musielak-Orlicz Hardy spaces H_φ~*(?),Pφ(?),H_φ~S(?),Qφ(?)and H_φ~s(?),respectively,via the maximal function,the quadratic variation and the conditional quadratic variation of martingales.We then establish the atomic characterizations of H_φ~s(?),Pφ(?)and Qφ(?).As applications,we obtain the dual space of H_φ~s(?)and several martingale inequalities which further clarify the relations among H_φ~*(?),Pφ(?),H_φ~S(?),Qφ(?)and H_φ~s(?).Especially,as special cases,the results on atomic characterizations of H_φ~s(?),Pφ(?)and Qφ(?)as well as on the dual space of H_φ~s(?)in the weighted case are also new.     

Martingale Hardy spaces and the dyadic derivative

It is shown that the maximal operator of the one-dimensional dyadic derivative of the dyadic integral is bounded from the dyadic Hardy-Lorentz spaceH _p,q toL _p,q (1/2<p<∞, 0<q≤∞) and is of weak type (L ₁,L ₁). We define the twodimensional dyadic hybrid Hardy spaceH ₁ ^‖ and verify that the corresponding maximal operator of a two-dimensional function is of weak type (H ₁ ^‖ ,L ₁). As a consequence, we obtain that the dyadic integral of a two-dimensional functionfεH ₁ ^‖ ?LlogL is dyadically differentiable and its derivative is a.e.f.     

Local Hardy spaces and summability of Fourier transforms

New Wiener amalgam spaces are introduced for local Hardy spaces. A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from the amalgam space to W(L_p,ℓ_∞). This implies the almost everywhere convergence of the θ-means for all fW(L₁,ℓ_∞)L₁.     

Martingale Hardy spaces,BMO and VMO spaces with nonlinearly ordered stochastic basis

VMO. , ⁺, «» VMO . « » .     

Restricted summability of Fourier transforms and local Hardy spaces

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.     

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R ², and acting on the real Hardy space H ¹(R), or the product Hardy space H ¹¹(R×R), or one of the hybrid Hardy spaces H ¹⁰(R ²) and H ⁰¹(R ²), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform. This revised version was published online in June 2006 with corrections to the Cover Date.     

Riesz transforms associated to Schrödinger operators on weighted Hardy spaces

Let w be some Ap weight and enjoy reverse Hölder inequality, and let L=−Δ+V be a Schrödinger operator on Rn, where is a non-negative function on Rn. In this article we introduce weighted Hardy spaces associated to L in terms of the area function characterization, and prove their atomic characters. We show that the Riesz transform ∇L−1/2 associated to L is bounded on for 1<p<2, and bounded from to the classical weighted Hardy space .     

The group Fourier transforms and multipiers of the Hardy spaces on the Heisenberg group

We give the estimate for the growth of the group Fourier transforms of the Hardy spaces and establish and H^p-multipleir theorem satisfying Michlin condition on the Heisenberg group.     

On Hardy and Bellman transforms of the Fourier coefficients of functions in symmetric spaces

The author finds sufficient conditions for invariance of a symmetric function space under the Hardy and Bellman transforms in terms of the fundamental function of the space. Under some additional assumptions about the space, these conditions are proved to be necessary.Translated from Matematicheskie Zametki, Vol. 53, No. 4, pp. 3–12, April, 1993.In conclusion, we use the opportunity to express our deep gratitude to B.I.Golubov for his constant attention to this work and also to E.M.Semënov for valuable remarks on the original version of this article.     

Martingale transforms and uniformly convergent orthogonal series

S. A. Vinogradov's method is adapted to prove for certain orthogonal product systems the analogue of his inequality concerning the trigonometric system. For example, for the Walsh system W={w_n} the following holds. Let U(W) be the space of functions with a uniformly convergent Walsh-Fourier series. Then, for every functional F on U(W) we have the inequality .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 18–38, 1985.     

弱Hardy正规鞅空间与原子分解

本文证明了弱Hardy正规鞅空间wHp和wH^sp上的原子分解定理.利用鞅的原子分解给出了弱Hardy正规鞅空间上的次线性算子有界的一个充分条件.利用这个条件得到了关于正规鞅的一些弱Lp范数不等式和弱（p,p）型不等式.这些结果是经典Hp鞅论中一些重要结果的弱型对应.     

双参数B值鞅Hardy空间的实内插

利用原子分解方法,研究了双参数Ｂ值鞅Ｈａｒｄｙ空间之间的实内插。     

Hardy spaces via distribution spaces

Let ℐ(ℝⁿ) be the Schwartz class on ℝⁿ and ℐ_∞(ℝⁿ) be the collection of functions ϕ ∊ ℐ(ℝⁿ) with additional property that

Martingale transforms with non-atomic limits and stochastic approximation

Summary A sufficient condition for the limit of a martingale transform to posses a continuous distribution is given. The result is used to show that for a stochastic approximation procedure, if the adjustment rate is too small then it would not converge to the target value a.s. Furthermore, if the adjustment rate is taken to be 1/n as usual but the derivative of the regression function at the target value is 0, then the convergence rate is shown to be logn instead of , the rate obtained when the derivative is non-zero.Research supported by the National Science Council, R.O.C.     

Generalized Hardy spaces

Hardy spaces with generalized parameter are introduced following the maximal characterization approach. As particular cases, they include the classical Hp spaces and the Hardy-Lorentz spaces H^p,q. Real interpolation results with function parameter are obtained, Based on them, the behavior of some classical operators is studied in this generalized setting.     

Atomic Hardy spaces

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This paper is a part of the author's Ph.D. thesis written under the supervision of Prof. F. Schipp, Eötvös L. University, Budapest.