On Rearrangements of the Haar System in L p

In this paper we study operators rearranging the Haar system in each bundle. It is proved that the norm of any nonidentical rearrangement admits a nontrivial lower bound in L ^p spaces, .     

On rearrangements of the double Haar system

In this paper estimates of the norm of the operator connected with the rearrangement of the double Haar system in dyadic Hardy and BMO spaces are given. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 385–389, September, 2000.     

Carleson measures and the BMO space on the p ‐adic vector space

For a prime number p, let Q _p be the p ‐adic field and let Q _p ^d denote a vector space over Q _p which consists of all d ‐tuples of Q _p . Then we study the p ‐adic version of the Calderón–Zygmund decomposition, Carleson measures on the vector space Q _p ^{d +1} and the space BMO ( Q _p ^d ) of functions of bounded mean oscillation on Q _p ^d . In particular, it turns out that the operator norms of various oncoming operators are independent of the dimension d and the prime number p, which is one of the big differences from that of the Euclidean case. Interestingly, the independence of the dimension d and p makes it possible to develop Harmonic Analysis on the infinite dimensional p ‐adic vector space as the importance had already been pointed out in the Euclidean case (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Blow-up Criterion of Smooth Solutions to the MHD System in BMO Space

In this paper we study the blow-up criterion of smooth solutions to the incompressible magnetohydrodynamics system in BMO space. Let （u（x,t）,b（x,t）） be smooth solutions in （0, T）. It is shown that the solution （u（x, t）, b（x, t）） can be extended beyond t = T if （u（x,t）, b（x, t）） ∈ L^1 （0, T; BMO） or the vorticity （rot u（x, t）, rot b（x, t）） ∈ L^1 （0, T; BMO） or the deformation （Def u（x, t）, Def b（x, t）） ∈ L^1 （0, T; BMO）.     

On series by Haar system

The paper considers the series by Haar system \(\sum\limits_{n = 1}^\infty {a_n \chi _n (x)} \), satisfying the conditions \(\sum\limits_{n = 1}^\infty {a_n^2 \chi _n^2 (x)} = \infty \) and a _n χ _n (x) → 0 almost everywhere. Some theorems about correcting a function on sets of arbitrarily small measures are proved.     

Rearrangements of the Haar system

It is proved that any fixed rearrangement of the Haar system either is or is not a system of convergence almost everywhere simultaneously for all classes L^p[0, 1] (1 p ).Translated from Matematicheskie Zametki, Vol. 15, No. 1, pp. 63–71, January, 1974The author gratefully acknowledges A. M. Olevskii for stating the problem and offering a number of comments in a discussion of the work.     

广义Littlewood-Paley函数的BMO有界性

本文证明了BMO函数在广义Littlewood-Paley函数下的象或者几乎处处等于无穷或者属于BMO。     

The Haar wavelet characterization of weighted Herz spaces and greediness of the Haar wavelet basis

In this paper we consider the Haar wavelet on weighted Herz spaces. Our weight class, whose name is Ap-dyadic local, is the one defined by the first author (2007). We shall investigate the class of Ap-dyadic weights in connection with the maximal inequalities. After obtaining the properties of weights in the first half of the present paper, we consider the Haar wavelet on weighted Herz spaces in the latter half. We shall show that the Haar wavelet basis is an unconditional basis. We also show that the Haar wavelet is not greedy except for the trivial case, that is, the Haar wavelet is greedy if and only if the Herz space under consideration is a weighted Lp space.     

BMO空间中f与f_N的范数比较

ＢＭＯ空间中ｆ与ｆ_Ｎ的范数比较刘为铨（安徽芜湖师专）设ｆ（ｘ）为定义于Ｒ上的局部可积函数，Ｑ为Ｒ中的任一立方体。｜Ｑ｜为Ｑ的Ｌｅｂｅｓｅｕｅ测度，ｆ（Ｘ）在Ｑ上的平均值记作若ｆ（Ｘ）满足条件则称人工）为有界平均振动函数，一切有界平均振动函数所构成的?..     

Weighted BMO and Carleson measures on spaces of homogeneous type

In this paper we characterize the weighted BMO(ω)(X), with X a space of homogeneous type, through an adequate weighted Carleson measure. As a byproduct we can define the weighted Triebel-Lizorkin space and obtain the identification with the above space.     

Duality of Hardy and BMO spaces associated with operators with heat kernel bounds

Let be the infinitesimal generator of an analytic semigroup on with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space by means of an area integral function associated with the operator . By using a variant of the maximal function associated with the semigroup , a space of functions of BMO type was defined by Duong and Yan and it generalizes the classical BMO space. In this paper, we show that if has a bounded holomorphic functional calculus on , then the dual space of is where is the adjoint operator of . We then obtain a characterization of the space in terms of the Carleson measure. We also discuss the dimensions of the kernel spaces of BMO when is a second-order elliptic operator of divergence form and when is a Schrödinger operator, and study the inclusion between the classical BMO space and spaces associated with operators.

     

Shift Generated Haar Spaces on Compact Domains in the Complex Plane

Haar spaces are certain finite-dimensional subspaces of $\cc(K)$, where $K$ is a compact set and $\cc(K)$ is the Banach space of continuous functions defined on $K$ having values in $\C$. We characterize those Haar spaces which are generated by shifts applied to a single, analytic function for $K\subset\C$. This means that an arbitrary finite number of shifts generates Haar spaces by forming linear hulls. We have to distinguish two cases: (a) $K\not=\overline{K^\circ}$; (b) $K=\overline{K^\circ}$. It turns out that, in case (a), an analytic Haar space generator for dimensions one and two is already a universal Haar space generator for all dimensions. The geometrically simplest case that, in case (b), $K$ is convex with smooth boundary turns out to be the most difficult case. There is one numerical example in which the entire function $f:=1/\Gamma$ is interpolated in a shift generated Haar space of dimension four.     

On rearrangements of the Haar system in the space BMO

, .     

Perturbations of the Haar wavelet by convolution

In this note we show that the standard convolution regularization of the Haar system generates Riesz bases of smooth functions for , providing in this way an alternative to the approach given by Govil and Zalik [Proc. Amer. Math. Soc. 125 (1997), 3363-3370].