In this paper we discuss a weighted version of Journé's covering lemma, a substitution for Whitney decomposition of an open set in R2 where squares are replaced by rectangles. We then apply this result to obtain a sharp version of the atomic decomposition of the weighted Hardy spaces Hu'p (R+2 x R+2) and a description of their duals when p is close to 1.

A non-homogeneous Hardy-like inequality has recently been found to be closely related to the knowledge of the lowest eigenvalue of a large class of Dirac operators in the gap of their continuous spectrum.

Hardy spaces consisting of adapted function sequences and generated by the q-variation and by the conditional q-variation are considered. Their dual spaces are characterized and an inequality due to Stein and Lepingle is extended.

We prove the continuity of an oscillatory singular integral operator T with polynomial phase P(x,y) on an atomic space $H_P^1$ related to the phase P. Moreover, we show that the cancellation condition to be imposed on T holds under more general conditions. To that purpose, we obtain a van der Corput type lemma with integrability at infinity.

As a natural extension of $L^p$ Sobolev spaces, we consider Hardy-Sobolev spaces and establish an atomic decomposition theorem, analogous to the atomic decomposition characterization of Hardy spaces. As an application, we deduce several embedding results for Hardy-Sobolev spaces.

The spectral synthesis theorem for Sobolev spaces of Hedberg and Wolff [7] has been applied in combination with duality, to problems of Lq approximation by analytic and harmonic functions. In fact, such applications were one of the main motivations to consider spectral synthesis problems in the Sobolev space setting. In this paper we go the opposite way in the context of the BMO-H1 duality: we prove a BMO approximation theorem by harmonic functions and then we apply the ideas in its proof to produce...

Let $G$ be the Lie group ${ℝ}^2⋉{ℝ}^{+}$ endowed with the Riemannian symmetric space structure. Let $X_0,X_1,X_2$ be a distinguished basis of left-invariant vector fields of the Lie algebra of $G$ and define the Laplacian $\Delta =-(X_0^2+X_1^2+X_2^2)$. In this paper we consider the first order Riesz transforms $R_i=X_i{\Delta }^{-1/2}$ and $S_i={\Delta }^{-1/2}{X}_i$, for $i=0,1,2$. We prove that the operators $R_i$, but not the $S_i$, are bounded from the Hardy space $H^1$ to $L^1$. We also show that the second-order Riesz transforms $T_{ij}=X_i{\Delta }^{-1}{X}_j$ are bounded from $H^1$ to $L^1$, while the transforms $S_{ij}={\Delta }^{-1}{X}_i{X}_j$ and $R_{ij}=X_i{X}_j{\Delta }^{-1}$, for $i,j=0,1,2$, are not.

The author investigates the boundedness of the higher order commutator of strongly singular convolution operator, $T_b^m$, on Herz spaces $K{̇}_q^{\alpha ,p}\left(ℝⁿ\right)$ and $K_q^{\alpha ,p}\left(ℝⁿ\right)$, and on a new class of Herz-type Hardy spaces $HK{̇}_{q,b,m}^{\alpha ,p,0}\left(ℝⁿ\right)$ and $H{K}_{q,b,m}^{\alpha ,p,0}\left(ℝⁿ\right)$, where 0 < p ≤ 1 < q < ∞, α = n(1-1/q) and b ∈ BMO(ℝⁿ).

We obtain the boundedness of Calderón-Zygmund singular integral operators $T$ of non-convolution type on Hardy spaces $H^p\left(𝒳\right)$ for $1/(1+ϵ)<p\le 1$, where $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss (1971), and $ϵ$ is the regularity exponent of the kernel of the singular integral operator $T$. Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast to what was...

The aim of this paper is to study singular integrals T generated by holomorphic kernels defined on a natural neighbourhood of the set $z{\zeta }^{-1}:z,\zeta \in ,z\ne \zeta$, where is a star-shaped Lipschitz curve, $=exp\left(iz\right):z=x+iA\left(x\right),A^{\text{'}}\in L^{\infty }[-\pi ,\pi ],A(-\pi )=A\left(\pi \right)$. Under suitable conditions on F and z, the operators are given by (1) $TF\left(z\right)=p.v.{\int }_{(}z{\eta }^{-1})F\left(\eta \right)(d\eta /\eta ).$ We identify a class of kernels of the stated type that give rise to bounded operators on $L^2(,|d\left|\right)$. We establish also transference results relating the boundedness of kernels on closed Lipschitz curves to corresponding results on periodic, unbounded curves.

Sufficient conditions for the boundedness of the Hausdorff operators in the Hardy spaces $H^p$, 0 < p < 1, on the real line are proved. Two related negative results are also given.