An RD-space $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type $𝒳$ having “dimension” $n$, there exists a $p_0\in (n/(n+1),1)$ such that for certain classes of distributions, the $L^p\left(𝒳\right)$ quasi-norms of their radial maximal functions and grand maximal functions are equivalent when $p\in (p_0,\infty ]$. This result yields a radial maximal function characterization for Hardy spaces on $𝒳$.

The aim of this article is to present “refined” Hardy-type inequalities. Those inequalities are generalisations of the usual Hardy inequalities, their additional feature being that they are invariant under oscillations: when applied to highly oscillatory functions, both sides of the refined inequality are of the same order of magnitude. The proof relies on paradifferential calculus and Besov spaces. It is also adapted to the case of the Heisenberg group.

In this paper, the authors introduce a kind of local Hardy spaces in Rn associated with the local Herz spaces. Then the authors investigate the regularity in these local Hardy spaces of some nonlinear quantities on superharmonic functions on R2. The main results of the authors extend the corresponding results of Evans and Müller in a recent paper.

For s>0, we consider bounded linear operators from $D\left({ℝ}^n\right)$ into $D^{\text{'}}\left({ℝ}^n\right)$ whose kernels K satisfy the conditions $|{\partial }_x^{\gamma }K(x,y)|\le C_{\gamma }{|x-y|}^{-n+s-\left|\gamma \right|}$ for x≠y, |γ|≤ [s]+1, $|{\nabla }_y{\partial }_x^{\gamma }K(x,y)|\le C_{\gamma }{|x-y|}^{-n+s-\left|\gamma \right|-1}$ for |γ|=[s], x≠y. We establish a new criterion for the boundedness of these operators from $L^2\left({ℝ}^n\right)$ into the homogeneous Sobolev space ${Ḣ}^s\left({ℝ}^n\right)$. This is an extension of the well-known T(1) Theorem due to David and Journé. Our arguments make use of the function T(1) and the BMO-Sobolev space. We give some applications to the Besov and Triebel-Lizorkin spaces as well as some other potential...

Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from $H_p\left(ℝ\right)$ to $L_p\left(ℝ\right)$ (1/(α+1) < p < ∞) and is of weak type (1,1), where $H_p\left(ℝ\right)$ is the classical Hardy space. As a consequence we deduce that the Riesz means of a function $⨍\in L_1\left(ℝ\right)$ converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on $H_p\left(ℝ\right)$ whenever 1/(α+1) < p < ∞. Thus, in case $⨍\in H_p\left(ℝ\right)$, the Riesz means converge...

In this paper we introduce atomic Hardy spaces on the product domain $S^{n-1}\times S^{m-1}$ and prove that rough singular integral operators with Hardy space function kernels are $L^p$ bounded on ${ℝ}^n\times {ℝ}^m$. This is an extension of some well known results.