A sharp embedding relation between local Hardy spaces and modulation spaces is given.

One-sided versions of maximal functions for suitable defined distributions are considered. Weighted norm equivalences of these maximal functions for weights in the Sawyer's Aq+ classes are obtained.

For any n ∈ ℕ, we obtain a bound for oscillatory singular integral operators with polynomial phases on the Hardy space H¹(ℝⁿ). Our estimate, expressed in terms of the coefficients of the phase polynomial, establishes the H¹ boundedness of such operators in all dimensions when the degree of the phase polynomial is greater than one. It also subsumes a uniform boundedness result of Hu and Pan (1992) for phase polynomials which do not contain any linear terms. Furthermore, the bound is shown to be valid...

The purpose of this paper is to obtain a discrete version for the Hardy spaces $H^p\left(\mathbb{Z}\right)$ of the weak factorization results obtained for the real Hardy spaces $H^p\left(ℝⁿ\right)$ by Coifman, Rochberg and Weiss for p > n/(n+1), and by Miyachi for p ≤ n/(n+1). It represents an extension, in the one-dimensional case, of the corresponding result by A. Uchiyama who obtained a factorization theorem in the general context of spaces X of homogeneous type, but with some restrictions on the measure that exclude the case of points...

We extend the well known factorization theorems on the unit disk to product Hardy spaces, which generalizes the previous results obtained by Coifman, Rochberg and Weiss. The basic tools are the boundedness of a certain bilinear form on ℝ²₊ × ℝ²₊ and the characterization of BMO(ℝ²₊ × ℝ²₊) recently obtained by Ferguson, Lacey and Sadosky.

The two-dimensional classical Hardy spaces $H_p\left(ℝ\times ℝ\right)$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from $H_p\left(ℝ\times ℝ\right)$ to $L_p\left({ℝ}^2\right)$ (1/2 < p ≤ ∞) and is of weak type $(H_1^{}\left(ℝ\times ℝ\right),L_1\left({ℝ}^2\right))$ where the Hardy space $H_1\left(ℝ\times ℝ\right)$ is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ $H_1^{(}ℝ\times ℝ)$ ⊃ $LlogL\left({ℝ}^2\right)$ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on $H_p\left(ℝ\times ℝ\right)$ whenever 1/2 < p < ∞. Thus, in case f ∈ $H_p\left(ℝ\times ℝ\right)$, the Fejér means...

Clearly, one of the most basic contributions to the fields of real variables, partial differential equations and Fourier analysis in recent times has been the celebrated theorem of Calderón and Zygmund on the boundedness of singular integrals on Rn [1].

We define a class of spaces $H_{\mu }^p$, 0 < p < ∞, of holomorphic functions on the tube, with a norm of Hardy type: ${\left|\right|F\left|\right|}_{H_{\mu }^p}^p=su{p}_{y\in \Omega }{\int }_{\Omega ̅}{\int }_{ℝⁿ}{\left|F(x+i(y+t))\right|}^{p}dxd\mu \left(t\right)$. We allow μ to be any quasi-invariant measure with respect to a group acting simply transitively on the cone. We show the existence of boundary limits for functions in $H_{\mu }^p$, and when p ≥ 1, characterize the boundary values as the functions in $L_{\mu }^p$ satisfying the tangential CR equations. A careful description of the measures μ when their supports lie on the boundary of the cone is also provided....

On graphs satisfying the doubling property and the Poincaré inequality, we prove that the space $H_{max}^1$ is equal to $H_{at}^1$, and therefore that its dual is BMO. We also prove the atomic decomposition for $H_{max}^p$ for p ≤ 1 close enough to 1.

Let A = -Δ + V be a Schrödinger operator on ${ℝ}^d$, d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of $H_A^p$ if the maximal function $su{p}_{t>0}\left|T_{t}f\left(x\right)\right|$ belongs to $L^p\left({ℝ}^d\right)$, where ${T_t}_{t>0}$ is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space $H_A^p$ admits a special atomic decomposition.

In a previous paper the authors developed an H¹-BMO theory for unbounded metric measure spaces (M,ρ,μ) of infinite measure that are locally doubling and satisfy two geometric properties, called “approximate midpoint” property and “isoperimetric” property. In this paper we develop a similar theory for spaces of finite measure. We prove that all the results that hold in the infinite measure case have their counterparts in the finite measure case. Finally, we show that the theory applies to a class...