Dyadic Hardy Research Articles

Two-weight dyadic Hardy inequalities

We present various results concerning the two-weight Hardy inequality on infinite trees. Our main aim is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization in terms of a reverse Hölder inequality for trace measures, and one based on the well-known Muckenhoupt–Wheeden–Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so-called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so-called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy inequality, characterize the compactness of the Hardy operator, provide a list of open problems, and suggest some possible lines of future research.

Maximal operators of Walsh–Nörlund means on the dyadic Hardy spaces

We establish necessary and sufficient conditions for the maximal operator of Walsh–Nörlund means with non-increasing weights to be bounded from the dyadic Hardy space \(H_{p}(\mathbb{I})\) to the space \( L_{p}(\mathbb{I})\).

On the maximal operators of weighted Marcinkiewicz type means of two-dimensional Walsh–Fourier series

Abstract Goginava proved that the maximal operator σ α , * {\sigma^{\alpha,*}} ( 0 < α < 1 {0<\alpha<1} ) of two-dimensional Marcinkiewicz type ( C , α ) {(C,\alpha)} means is bounded from the two-dimensional dyadic martingale Hardy space H p ⁢ ( G 2 ) {H_{p}(G^{2})} to the space L p ⁢ ( G 2 ) {L^{p}(G^{2})} for p > 2 2 + α {p>\frac{2}{2+\alpha}} . Moreover, he showed that assumption p > 2 2 + α {p>\frac{2}{2+\alpha}} is essential for the boundedness of the maximal operator σ α , * {\sigma^{\alpha,*}} . It was shown that at the point p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}} the maximal operator σ α , * {\sigma^{\alpha,*}} is bounded from the dyadic Hardy space H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} to the space weak- L 2 / ( 2 + α ) ⁢ ( G 2 ) {L^{2/(2+\alpha)}(G^{2})} . The main aim of this paper is to investigate the behaviour of the maximal operators of weighted Marcinkiewicz type σ α , * {\sigma^{\alpha,*}} means ( 0 < α < 1 {0<\alpha<1} ) in the endpoint case p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}} . In particular, the optimal condition on the weights is given which provides the boundedness from H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} to L 2 / ( 2 + α ) ⁢ ( G 2 ) {L^{2/(2+\alpha)}(G^{2})} . Furthermore, a strong summation theorem is stated for functions in the dyadic martingale Hardy space H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} .

Approximation by Θ-Means of Walsh—Fourier Series in Dyadic Hardy Spaces and Dyadic Homogeneous Banach Spaces

Approximation by Θ-Means of Walsh—Fourier Series in Dyadic Hardy Spaces and Dyadic Homogeneous Banach Spaces

Maximal summability operators on the dyadic hardy spaces

It is proved that the maximal operators of subsequences of N?rlund logarithmic means and Ces?ro means with varying parameters of Walsh-Fourier series is bounded from the dyadic Hardy spaces Hp to Lp. This implies an almost everywhere convergence for the subsequences of the summability means.

Dimension dependence of factorization problems: Hardy spaces and $$SL_n^\infty$$

Given $1 \leq p < \infty$, let $W_n$ denote the finite-dimensional dyadic Hardy space $H_n^p$, its dual or $SL_n^\infty$. We prove the following quantitative result: The identity operator on $W_n$ factors through any operator $T : W_N\to W_N$ which has large diagonal with respect to the Haar system, where $N$ depends \emph{linearly} on $n$.

Image Restoration Models Based on Dyadic Hardy Space and Dyadic Bounded Mean Oscillation Space

Texture is widely existed in various images and plays an important role in many area such as medical image diagnosis, remote sensing, etc. However, the image in texture regions is tend to be deteriorated during restoration process. In this paper, we apply the dyadic Hardy space $H_{d}^{1}$ and dyadic Bounded Mean Oscillation (BMO) space in the texture preserving image restoration model. We propose a $H_{d}^{1}$ regularized minimization model to extract texture from noisy data. In this model, $H^{1}_{d}$ norm is taken as regularizer to enforce the prior that the local variance of the noise is below certain level depending on the regularization parameter. We also analyze the mathematical properties of this model which indicate the mechanism of $H^{1}_{d}$ regularizer to control the local variance. For the numerical solution of the model, we transform it into wavelet domain based on the wavelet characterization of dyadic Hardy space and dyadic BMO space, and solve it by the fixed iteration algorithm. Combing the total variation (TV) regularization method and frame based regularization method, a two-layers regularization model is proposed for edge and texture preserving, and then analyzed and solved in the frame of split Bregman method. Finally, we present various numerical results on images to demonstrate the potential of our methods.

Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type

We give an explicit construction of Haar functions associated to a system of dyadic cubes in a geometrically doubling quasi-metric space equipped with a positive Borel measure, and show that these Haar functions form a basis for Lp. Next we focus on spaces X of homogeneous type in the sense of Coifman and Weiss, where we use these Haar functions to define a discrete square function, and hence to define dyadic versions of the function spaces H1(X) and BMO(X). In the setting of product spaces X˜=X1×⋯×Xn of homogeneous type, we show that the space BMO(X˜) of functions of bounded mean oscillation on X˜ can be written as the intersection of finitely many dyadic BMO spaces on X˜, and similarly for Ap(X˜), reverse-Hölder weights on X˜, and doubling weights on X˜. We also establish that the Hardy space H1(X˜) is a sum of finitely many dyadic Hardy spaces on X˜, and that the strong maximal function on X˜ is pointwise comparable to the sum of finitely many dyadic strong maximal functions. These dyadic structure theorems generalize, to product spaces of homogeneous type, the earlier Euclidean analogues for BMO and H1 due to Mei, and Li, Pipher and Ward.

The Walsh–Kaczmarz–Marcinkiewicz means and Hardy spaces

It is known that the maximal operator \({\sigma^{\kappa,*}(f)} := sup_{n \in \mathbf{P}}{|{\sigma}_{n}^{\kappa} (f)|}\) is bounded from the dyadic Hardy space \({H_{p}}\) into the space \({L_{p}}\) for \({p > 2/3}\) [6]. Moreover, Goginava and Nagy showed that \({\sigma^{\kappa,*}}\) is not bounded from the Hardy space \({H_{2/3}}\) to the space \({L_{2/3}}\) [9]. The main aim of this paper is to investigate the case \({0 < p < 2/3}\). We show that the weighted maximal operator \({\tilde{\sigma}^{\kappa,*,p}(f) :=sup_{n\in \mathbf{P}} \frac{|{\sigma}_{n}^\kappa (f)|}{n^{2/p-3}}}\), is bounded from the Hardy space \({H_{p}}\) into the space \({L_{p}}\) for any \({0 < p < 2/3}\). With its aid we provide a necessary and sufficient condition for the convergence of Walsh–Kaczmarz–Marcinkiewicz means in terms of modulus of continuity on the Hardy space \({H_p}\), and prove a strong convergence theorem for this means.

The maximal operator of Marcinkiewicz-Fejér means with respect to Walsh-Kaczmarz-Fourier series

In the paper [4, Theorem 1] Gat, Goginava and the author proved that the maximal operator σκ,∗ of Marcinkiewicz-Fejer means of Walsh-Kaczmarz-Fourier series, is bounded from the dyadic Hardy space Hp into the space Lp for p > 2/3 . Moreover, Goginava and the author showed that σκ,∗ is not bounded from the Hardy space H2/3 to the space L2/3 [6, Theorem 1]. The main aim of this paper is to show that the maximal operator σκ,∗ f := supn∈P |σκ n f | log3/2(n+1) , is bounded from the Hardy space H2/3 into the space L2/3. Moreover, we prove that the order of deviant behavior of the n th Walsh-Kacmarz-Marcinkiewicz-Fejer mean is exactly log3/2(n+1) in the endpoint p = 2/3 . Mathematics subject classification (2010): 42C10.

Renorming spaces with greedy bases

We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given ε>0, so that the basis becomes (1+ε)-democratic, and hence (2+ε)-greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is (1+ε)-greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applications include the Haar systems in Lp[0,1], 1<p<∞, and in dyadic Hardy space H1, as well as the unit vector basis of Tsirelson space.

The one-sided dyadic Hardy—Littlewood maximal operator

The main aim of this paper is to introduce an appropriate dyadic one-sided maximal operator , smaller than the one-sided Hardy–Littlewood maximal operator M+ but such that it controls M+ in a similar way to how the usual dyadic maximal operator controls the Hardy-Littlewood maximal operator. We characterize the weighted inequalities for this dyadic one-sided maximal operator.

Фейеровская суммируемость по треугольникам двойных рядов Фурье

It is proved that the operators σ Δ n of the triangular-Fejér-means of a two-dimensional Walsh-Fourier series are uniformly bounded from the dyadic Hardy space H p to L p for all 4/5 < p≤∞.

On dyadic nonlocal Schrödinger equations with Besov initial data

In this paper we consider the pointwise convergence to the initial data for the Schrödinger–Dirac equation i∂u∂t=Dβu with u(x,0)=u0 in a dyadic Besov space. Here Dβ denotes the fractional derivative of order β associated to the dyadic distance δ on R+. The main tools are a summability formula for the kernel of Dβ and pointwise estimates of the corresponding maximal operator in terms of the dyadic Hardy–Littlewood function and the Calderón sharp maximal operator.

Logarithmic mean oscillation on the polydisc, endpoint results for multi-parameter paraproducts, and commutators on BMO

We study boundedness properties of a class ofmultiparameter paraproducts on the dual space of the dyadic Hardy space H 1 d (T N), the dyadic product BMO space BMOd (T N). For this, we introduce a notion of logarithmic mean oscillation on the polydisc. We also obtain a result on the boundedness of iterated commutators on BMO [0, 1]N).

Maximal operator of the Fejér means of triangular partial sums of two-dimensional Walsh–Fourier series

It is proved that the maximal operator of the triangular-Fejér-means of a two-dimensional Walsh–Fourier series is bounded from the dyadic Hardy space to for all and, consequently, is of weak type (1,1). As a consequence we obtain that the triangular-Fejér-means of a function converge a.e. to . The maximal operator is bounded from the Hardy space to the space weak- and is not bounded from the Hardy space to the space .

Renormings and symmetry properties of 1-greedy bases

We continue the study of 1 -greedy bases initiated by Albiac and Wojtaszczyk (2006) [1]. We answer several open problems that they raised concerning symmetry properties of 1 -greedy bases and the improving of the greedy constant by renorming. We show that 1 -greedy bases need not be symmetric or subsymmetric. We also prove that one cannot in general make a greedy basis 1 -greedy as demonstrated for the Haar basis of the dyadic Hardy space H 1 ( R ) and for the unit vector basis of Tsirelson space. On the other hand, we give a renorming of L p ( 1 < p < ∞ ) that makes the Haar basis 1 -unconditional and 1 -democratic. Other results in this paper clarify the relationship between various basis constants that arise in the context of greedy bases.

Dyadic Fefferman–Stein inequalities and the equivalence of Haar bases on weighted Lebesgue spaces

We give sufficient conditions on two dyadic systems to obtain the equivalence of corresponding Haar systems on dyadic weighted Lebesgue spaces on spaces of homogeneous type. In order to obtain these results, we prove a Fefferman–Stein weighted inequality for vector-valued dyadic Hardy–Littlewood maximal operators with dyadic weights in this general setting.

Weak type inequality for the one-dimensional 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 space H1/2 to the space weak-L1/2 . Mathematics subject classification (2010): Primary 42C10, 43A75, Secondary 42B30, 60G42.

Dyadic sets, maximal functions and applications on ax + b-groups

Let S be the Lie group \({{\mathbb R}^n\ltimes {\mathbb R}}\), where \({{\mathbb R}}\) acts on \({{\mathbb R}^n}\) by dilations, endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure ρ, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245: 37–61, 2003] proved that any integrable function on (S, ρ) admits a Calderón–Zygmund decomposition which involves a particular family of sets, called Calderón–Zygmund sets. In this paper, we show the existence of a dyadic grid in the group S, which has nice properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid, we prove a Fefferman–Stein type inequality, involving the dyadic Hardy–Littlewood maximal function and the dyadic sharp function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space H 1 and the space BMO introduced in [Collect. Math. 60: 277–295, 2009].