Maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a dimension n. For α∈ (0, ∞) denote by Hαp(X ), Hdp(X ), and H?,p(X ) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calder′on reproducing formula, it is shown that all these Hardy spaces coincide with Lp(X ) when p ∈ (1, ∞] a...     

Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space Hp (X1 × X2 ) for 0 p0 p ≤ 1 for some p0 and all 1 q ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (X1 × X2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp (X1 × X2 ). The similar boundedness criterion from Hp (X1 × X2 ) to Hp (X1 × X2 ) is also obtained.     

Some generalizations on the boundedness of bilinear operators

For 0<p<∞, let H^p(R ⁿ) denote the Lebesgue space for p>1 and the Hardy space for p ≤1. In this paper, the authors study H^p(R ⁿ)×H^q(R ⁿ)→H^r(R ⁿ) mapping properties of bilinear operators given by finite sums of the products of the standard fractional integrals or the standard fractional integral with the Calderón-Zygmund operator. The authors prove that such mapping properties hold if and only if these operators satisfy certain cancellation conditions. Supported by the NNSF and the National Education Comittee of China.     

Topology of random clique complexes

In a seminal paper, Erd?s and Rényi identified a sharp threshold for connectivity of the random graph G(n,p). In particular, they showed that if p?logn/n then G(n,p) is almost always connected, and if p?logn/n then G(n,p) is almost always disconnected, as n→∞.The clique complexX(H) of a graph H is the simplicial complex with all complete subgraphs of H as its faces. In contrast to the zeroth homology group of X(H), which measures the number of connected components of H, the higher dimensional homology groups of X(H) do not correspond to monotone graph properties. There are nevertheless higher dimensional analogues of the Erd?s-Rényi Theorem.We study here the higher homology groups of X(G(n,p)). For k>0 we show the following. If p=n^α, with α<−1/k or α>−1/(2k+1), then the kth homology group of X(G(n,p)) is almost always vanishing, and if −1/k<α<−1/(k+1), then it is almost always nonvanishing.We also give estimates for the expected rank of homology, and exhibit explicit nontrivial classes in the nonvanishing regime. These estimates suggest that almost all d-dimensional clique complexes have only one nonvanishing dimension of homology, and we cannot rule out the possibility that they are homotopy equivalent to wedges of a spheres.     

Orlicz-Hardy spaces and their duals

We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions.The class will be wider than the class of all the N-functions.In particular,we consider the non-smooth atomic decomposition.The relation between Orlicz-Hardy spaces and their duals is also studied.As an application,duality of Hardy spaces with variable exponents is revisited.This work is different from earlier works about Orlicz-Hardy spaces H(Rn)in that the class of admissible functions is largely widened.We can deal with,for example,(r)≡(rp1(log(e+1/r))q1,0r 6 1,rp2(log(e+r))q2,r1,with p1,p2∈(0,∞)and q1,q2∈(.∞,∞),where we shall establish the boundedness of the Riesz transforms on H(Rn).In particular,is neither convex nor concave when 0p11p2∞,0p21p1∞or p1=p2=1 and q1,q20.If(r)≡r(log(e+r))q,then H(Rn)=H(logH)q(Rn).We shall also establish the boundedness of the fractional integral operators I of order∈(0,∞).For example,I is shown to be bounded from H(logH)1./n(Rn)to Ln/(n.)(log L)(Rn)for 0n.     

Blocking Sets in PG(r, q n )

Let $\mathcal S$ be a Desarguesian (n – 1)-spread of a hyperplane Σ of PG(rn, q). Let Ω and ${\bar B}$ be, respectively, an (n – 2)-dimensional subspace of an element of $\mathcal S $ and a minimal blocking set of an ((r – 1)n + 1)-dimensional subspace of PG(rn, q) skew to Ω. Denote by K the cone with vertex Ω and base ${\bar B}$ , and consider the point set B defined by $$B=\left(K\setminus\Sigma\right)\cup \{X\in \mathcal S\, : \, X\cap K\neq \emptyset\}$$ in the Barlotti–Cofman representation of PG(r, q ⁿ ) in PG(rn, q) associated to the (n – 1)-spread $\mathcal S$ . Generalizing the constructions of Mazzocca and Polverino (J Algebraic Combin, 24(1):61–81, 2006), under suitable assumptions on ${\bar B}$ , we prove that B is a minimal blocking set in PG(r, q ⁿ ). In this way, we achieve new classes of minimal blocking sets and we find new sizes of minimal blocking sets in finite projective spaces of non-prime order. In particular, for q a power of 3, we exhibit examples of r-dimensional minimal blocking sets of size q ⁿ⁺² + 1 in PG(r, q ⁿ ), 3 ≤ r ≤ 6 and n ≥ 3, and of size q ⁴ + 1 in PG(r, q ²), 4 ≤ r ≤ 6; actually, in the second case, these blocking sets turn out to be the union of q ³ Baer sublines through a point. Moreover, for q an even power of 3, we construct examples of minimal blocking sets of PG(4, q) of size at least q ² + 2. From these constructions, we also get maximal partial ovoids of the hermitian variety H(4, q ²) of size q ⁴ + 1, for any q a power of 3.     

Hardy Spaces, Regularized BMO Spaces and the Boundedness of Calderón–Zygmund Operators on Non-homogeneous Spaces

One defines a non-homogeneous space (X,μ) as a metric space equipped with a non-doubling measure μ so that the volume of the ball with center x, radius r has an upper bound of the form r ⁿ for some n>0. The aim of this paper is to study the boundedness of Calderón–Zygmund singular integral operators T on various function spaces on (X,μ) such as the Hardy spaces, the L ^p spaces, and the regularized BMO spaces. This article thus extends the work of X. Tolsa (Math. Ann. 319:89–149, 2011) on the non-homogeneous space (? ⁿ ,μ) to the setting of a general non-homogeneous space (X,μ). Our framework of the non-homogeneous space (X,μ) is similar to that of Hytönen (2011) and we are able to obtain quite a few properties similar to those of Calderón–Zygmund operators on doubling spaces such as the weak type (1,1) estimate, boundedness from Hardy space into L ¹, boundedness from L ^∞ into the regularized BMO, and an interpolation theorem. Furthermore, we prove that the dual space of the Hardy space is the regularized BMO space, obtain a Calderón–Zygmund decomposition on the non-homogeneous space (X,μ), and use this decomposition to show the boundedness of the maximal operators in the form of a Cotlar inequality as well as the boundedness of commutators of Calderón–Zygmund operators and BMO functions.     

More coverings by rook domains

The set V_kⁿ of all n-tuples (x₁, x₂,…, x_n) with x_i?, $\text{Z}$_k is considered. The problem treated in this paper is determining σ(n, k), the minimum size of a set W ? V_kⁿ such that for each x in V_kⁿ, there is an element in W that differs from x in at most one coordinate. By using a new constructive method, it is shown that σ(n, p) ? (p ? t + 1)p^n?r, where p is a prime and $\text{n = 1 +}\text{t(p}{}^{\mathrm{r?1}}\text{? 1)}\text{(p ? 1)}$ for some integers t and r. The same method also gives σ(7, 3) ? 216. Another construction gives the inequality σ(n, kt) ? σ(n, k)t^n?1 which implies that σ(q + 1, qt) = q^q?1t^q when q is a prime power. By proving another inequality σ(np + 1, p) ? σ(n, p)p^n(p?1), σ(10, 3) ? 5 · 3⁶ and σ(16, 5) ? 13 · 5¹² are obtained.     

Atomic decomposition characterizations of weighted multiparameter Hardy spaces

Let w ?? A _??. In this paper, we introduce weighted-(p, q) atomic Hardy spaces H _w ^p,q (? ⁿ ×? ^m ) for 0 < p ? 1, q >q _w and show that the weighted Hardy space H _w ^p (? ⁿ × ? ^m ) defined via Littlewood-Paley square functions coincides with H _w ^p,q (? ⁿ × ? ^m ) for 0 < p ? 1, q > q _w . As applications, we get a general principle on the H _w ^p (? ⁿ × ? ^m ) to L _w ^p (? ⁿ ×? ^m ) boundedness and a boundedness criterion for two parameter singular integrals on the weighted Hardy spaces.     

On the space Ext for the group SL(2, q)

We consider the space Ext ^r (A,B) = Ext _KG ^r (A, B), where G = SL(2, q), q = p ⁿ , K is an algebraically closed field of characteristic p, A and B are irreducible KG-modules, and r ? 1. Carlson [6] described a basis of Ext ^r (A, B) in arithmetical terms. However, there are certain difficulties concerning the dimension of such a space. In the present article, we find the dimension of Ext ^r (A, B) for r = 1, 2 (in the above-mentioned article, Carlson presents the corresponding assertions without proofs; moreover, there are errors in their formulations). As a corollary, we find the dimension of the space H ^r (G, A), where A is an irreducible KG-module. This result can be used in studying nonsplit extensions of L ₂(q).     

Some Maximal Function Fields and Additive Polynomials

We derive explicit equations for the maximal function fields F over 𝔽 _{q ²ⁿ} given by F = 𝔽 _{q ²ⁿ} (X, Y) with the relation A(Y) = f(X), where A(Y) and f(X) are polynomials with coefficients in the finite field 𝔽 _{q ²ⁿ} , and where A(Y) is q-additive and deg(f) = q ⁿ  + 1. We prove in particular that such maximal function fields F are Galois subfields of the Hermitian function field H over 𝔽 _{q ²ⁿ} (i.e., the extension H/F is Galois).     

Boundedness of Marcinkiewicz integrals and their commutators in H 1 (ℝ n × ℝ m )

The authors establish the boundedness of Marcinkiewicz integrals from the Hardy space H ¹ (? ⁿ × ? ^m ) to the Lebesgue space L ¹(? ⁿ × ? ^m ) and their commutators with Lipschitz functions from the Hardy space H ¹ (? ⁿ × ? ^m ) to the Lebesgue space L ^q (? ⁿ × ? ^m ) for some q > 1.     

Discrete Calderón’s Identity,Atomic Decomposition and Boundedness Criterion of Operators on Multiparameter Hardy Spaces

In this paper we establish a discrete Calderón’s identity which converges in both L ^q (ℝ ^n+m ) (1<q<∞) and Hardy space H ^p (ℝ ⁿ ×ℝ ^m ) (0<p≤1). Based on this identity, we derive a new atomic decomposition into (p,q)-atoms (1<q<∞) on H ^p (ℝ ⁿ ×ℝ ^m ) for 0<p≤1. As an application, we prove that an operator T, which is bounded on L ^q (ℝ ^n+m ) for some 1<q<∞, is bounded from H ^p (ℝ ⁿ ×ℝ ^m ) to L ^p (ℝ ^n+m ) if and only if T is bounded uniformly on all (p,q)-product atoms in L ^p (ℝ ^n+m ). The similar result from H ^p (ℝ ⁿ ×ℝ ^m ) to H ^p (ℝ ⁿ ×ℝ ^m ) is also obtained.     

On the nonvanishing of homogeneous product sums

For integer n ≥ 1 let H_n = H_n(x, y, z) = Σ_{p + q + r = n}x^py^qz^r be the homogeneous product sum of weight n on three letters x, y, z. Morgan Ward conjectured that H_n ≠ 0 for all integers n, x, y, z with n > 1 and xyz ≠ 0. In support of this conjecture he proved that H_n ≠ 0 if n is even or if n + 2 is a prime number greater than 3. This paper adds considerably more evidence in support of Ward's conjecture by showing that in many cases H_n(a, b, c)¬=0 modulo 2, 4, or 16. The parity of H_n(a, b, c) is determined in all cases and, when H_n(a, b, c) is even, further congruences are given modulo 4 or 16.     

Classifying Hilbert bundles

A Hilbert bundle (p, B, X) is a type of fibre space p:B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, G_m($\text{C}$ⁿ)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A.     

Boundedness of Marcinkiewicz integrals and their commutators in H1 (?n × ?m)

The authors establish the boundedness of Marcinkiewicz integrals from the Hardy space H ¹ (ℝ ⁿ × ℝ ^m ) to the Lebesgue space L ¹(ℝ ⁿ × ℝ ^m ) and their commutators with Lipschitz functions from the Hardy space H ¹ (ℝ ⁿ × ℝ ^m ) to the Lebesgue space L ^q (ℝ ⁿ × ℝ ^m ) for some q > 1.     

Generalizations of Bohr inequality for Hilbert space operators

Let B(H) be the space of all bounded linear operators on a complex separable Hilbert space H. Bohr inequality for Hilbert space operators asserts that for A,B∈B(H) and p,q>1 real numbers such that 1/p+1/q=1,

²|A+B|?p²|A|+q²|B|     

Non-linear Approximations Using Sets of Finite Cardinality or Finite Pseudo-dimension

We investigate optimal non-linear approximations of multivariate periodic functions with mixed smoothness. In particular, we study optimal approximation using sets of finite cardinality (as measured by the classical entropy number), as well as sets of finite pseudo-dimension (as measured by the non-linear widths introduced by Ratsaby and Maiorov). Approximation error is measured in the L_q(T^d)-sense, where T^d is the d-dimensional torus. The functions to be approximated are in the unit ball SB^r_p, θ of the mixed smoothness Besov space or in the unit ball SW^r_p of the mixed smoothness Sobolev space. For 1<p, q<∞, 0<θ⩽∞ and r>0 satisfying some restrictions, we establish asymptotic orders of these quantities, as well as construct asymptotically optimal approximation algorithms. We particularly prove that for either r>1/p and θ⩾p or r>(1/p−1/q)₊ and θ⩾min{q, 2}, the asymptotic orders of these quantities for the Besov class SB^r_p, θ are both n^−r(log n)^{(d−1)(r+1/2−1/θ)}.     

Approximation in weighted Hardy spaces

This paper is concerned with several approximation problems in the weighted Hardy spacesH ^p(Ω) of analytic functions in the open unit disc D of the complex plane ℂ. We prove that ifX is a relatively closed subset of D, the class of uniform limits onX of functions inH ^p(Ω) coincides, moduloH ^p(Ω), with the space of uniformly continuous functions on a certain hull ofX which are holomorphic on its interior. We also solve the simultaneous approximation problems of describing Farrell and Mergelyan sets forH ^p(Ω), giving geometric characterizations for them. By replacing approximating polynomials by polynomial multipliers of outer functions, our results lead to characterizations of the same sets with respect to cyclic vectors in the classical Hardy spacesH ^p(D), 1 ⪯p < ∞. Dedicated to Professor Nácere Hayek on the occasion of his 75th birthday.     

Explicit constants for Hardy’s inequality with power weight on n-dimensional product spaces

In this paper, Hardy operator H on n-dimensional product spaces G = (0, ∞)n and its adjoint operator H* are investigated. We use novel methods to obtain two main results. One is that we characterize the sufficient and necessary conditions for the operators H and H* being bounded from Lp(G, xα) to Lq(G, xβ), and the bounds of the operators H and H* are explicitly worked out. The other is that when 1 < p = q < +∞, norms of the operators H and H* are obtained.