Walsh位相平面中的j-尺寸与树上的BMO型空间

建立了Walsh位相平面中树上的BMO型空间,并得到了j-尺寸的两个等价定义.     

Variable Lebesgue norm estimates for BMO functions

In this paper, we are going to characterize the space BMO(? ⁿ ) through variable Lebesgue spaces and Morrey spaces. There have been many attempts to characterize the space BMO(? ⁿ ) by using various function spaces. For example, Ho obtained a characterization of BMO(? ⁿ ) with respect to rearrangement invariant spaces. However, variable Lebesgue spaces and Morrey spaces do not appear in the characterization. One of the reasons is that these spaces are not rearrangement invariant. We also obtain an analogue of the well-known John-Nirenberg inequality which can be seen as an extension to the variable Lebesgue spaces.     

Wavelet expansions for weighted,vector-valued BMO functions

We introduce a scale of weighted Carleson norms, which depend on an integrability parameter p, where p = 2 corresponds to the classical Carleson measure condition. Relations between the weighted BMO norm of a vector-valued function f: ℝ → X, and the Carleson norm of the sequence of its wavelet coefficients, are established. These extend the results of Harboure-Salinas-Viviani, also in the scalar-valued case when p ≠ 2.     

On global integrability of BMO functions on general domains

Extending results of Staples and Smith-Stegenga, we characterize measurable subsets of a given domainD ⊃R ⁿ on which BMO(D) functions areL ^p integrable or exponentially integrable. In particular, we characterize uniform domains by the integrability of BMO functions. We also remark on the boundedness of domains satisfying a certain integrability condition for the quasihyperbolic metric.     

Representation of measurable functions by series in Walsh subsystems

For any sequence {ω(n)} _n∈ℕ tending to infinity we construct a “quasiquadratic” representation spectrum Λ = {n ² + o(ω(n))} _n∈ℕ: for any almost everywhere (a. e.) finite measurable function f(x) there exists a series in the form $ mathop sum limits_{k in Lambda } $ mathop sum limits_{k in Lambda } α _k ω _k (x) that converges a. e. to this function, where {w _k (x)} _k∈ℕ is the Walsh system. We find representation spectra in the form {n ^l + o(n ^l )} _n∈ℕ, where l ∈ {2 ^k } _k∈ℕ.     

Sequences of semicontinuous functions accompanying continuous functions

A space X is said to have property (USC) (resp. (LSC)) if whenever is a sequence of upper (resp. lower) semicontinuous functions from X into the closed unit interval [0,1] converging pointwise to the constant function 0 with the value 0, there is a sequence of continuous functions from X into [0,1] such that fn?gn (n∈ω) and converges pointwise to 0. In this paper, we study spaces having these properties and related ones. In particular, we show that (a) for a subset X of the real line, X has property (USC) if and only if it is a σ-set; (b) if X is a space of non-measurable cardinal and has property (LSC), then it is discrete. Our research comes of Scheepers' conjecture on properties S₁(Γ,Γ) and wQN.     

Variable Lebesgue norm estimates for BMO functions. II

The paper concerns characterization of BMO in terms of Banach function spaces. In particular, we are interested in characterizing BMO by using the variable Lebesgue norm.     

On the arithmetic Walsh coefficients of Boolean functions

We generalize to the arithmetic Walsh transform (AWT) some results which were previously known for the Walsh–Hadamard transform of Boolean functions. We first generalize the classical Poisson summation formula to the AWT. We then define a generalized notion of resilience with respect to an arbitrary statistical measure of Boolean functions. We apply the Poisson summation formula to obtain a condition equivalent to resilience for one such statistical measure. Last, we show that the AWT of a large class of Boolean functions can be expressed in terms of the AWT of a Boolean function of algebraic degree at most three in a larger number of variables.     

Removable singularities for analytic functions in BMO and locally Lipschitz spaces

In this paper we study removable singularities for holomorphic functions such that sup_z|f⁽ⁿ⁾(z)|dist(z,)^s<. Spaces of this type include spaces of holomorphic functions in Campanato classes, BMO and locally Lipschitz classes. Dolzhenko (1963), Král (1976) and Nguyen (1979) characterized removable singularities for some of these spaces. However, they used a different removability concept than in this paper. They assumed the functions to belong to the function space on and be holomorphic on \ E, whereas we only assume that the functions belong to the function space on \ E, and are holomorphic there. Koskela (1993) obtained some results for our type of removability, in particular he showed the usefulness of the Minkowski dimension. Kaufman (1982) obtained some results for s=0.In this paper we obtain a number of examples with certain important properties. Similar examples have earlier been obtained for Hardy H^p classes and weighted Bergman spaces, mainly by the author. Because of the similarities in these three cases, an axiomatic approach is used to obtain some results that hold in all three cases with the same proofs. Supported by the Swedish Research Council and Gustaf Sigurd Magnusons fund of the Royal Swedish Academy of Sciences.Mathematics Subject Classification (2000):30B40, 30D45, 30D55, 46E15.     

Variational inequalities for the commutators of rough operators with BMO functions

Starting with the relatively simple observation that the variational estimates of the commutators of the standard Calderón-Zygmund operators with the bounded mean oscillation(BMO) functions can be obtained from their weighted variational estimates, we establish the similar variational estimates for the commutators of the BMO functions with rough singular integrals, which do not admit any weighted variational estimates. The proof involves several Littlewood-Paley-type inequalities with the commutators as well as Bony decomposition and related para-product estimates.     

Sequences of functions of binomial type

The theory of binomial enumeration leads to sequences of functions of binomial type which are not polynomials. The results of Mullin-Rota for these sequences are developed and a ring structure on the set of sequences is studied.     

Idempotents in the neighbourhood of Patterson-Wiedemann functions having Walsh spectra zeros

In this paper we study the neighbourhood of 15-variable Patterson-Wiedemann (PW) functions, i.e., the functions that differ by a small Hamming distance from the PW functions in terms of truth table representation. We exploit the idempotent structure of the PW functions and interpret them as Rotation Symmetric Boolean Functions (RSBFs). We present techniques to modify these RSBFs to introduce zeros in the Walsh spectra of the modified functions with minimum reduction in nonlinearity. Our technique demonstrates 15-variable balanced and 1-resilient functions with currently best known nonlinearities 16272 and 16264 respectively. In the process, we find functions for which the autocorrelation spectra and algebraic immunity parameters are best known till date.     

Frequency domain Walsh functions and sequences: An introduction

In this letter, a new set of orthogonal band-limited basis functions is introduced. This set of basis functions is derived from the inverse Fourier transform of the frequency domain Walsh functions. The Fourier transforms of the Walsh functions were calculated by Siemens and Kitai in 1973 but they have been overlooked in the literature. Some of the properties of these functions are studied in this paper. Moreover, the orthogonal discrete version of these functions is obtained by truncation, sampling and orthogonalization utilizing the orthogonal Procrustes problem.