Let $(X,d,\mu )$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator of order $m$ on $L^2\left(X\right)$. Assume that the semigroup ${\mathrm{e}}^{-tL}$ generated by $L$ satisfies the Davies-Gaffney estimate of order $m$ and $L$ satisfies the Plancherel type estimate. Let $H_L^p\left(X\right)$ be the Hardy space associated with $L.$ We show the boundedness of Stein’s square function ${𝒢}_{\delta }\left(L\right)$ arising from Bochner-Riesz means associated to $L$ from Hardy spaces $H_L^p\left(X\right)$ to $L^p\left(X\right)$, and also study the boundedness...

Let s ∈ ℝ, p ∈ (0,1] and q ∈ [p,∞). It is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from the Triebel-Lizorkin space ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ to a quasi-Banach space ℬ if and only if sup${\left|\right|T\left(a\right)\left|\right|}_{ℬ}$: a is an infinitely differentiable (p,q,s)-atom of ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ < ∞, where the (p,q,s)-atom of ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ is as defined by Han, Paluszyński and Weiss.

Some boundedness results are established for sublinear operators on the homogeneous Herz spaces. As applications, some new theorems about the boundedness on homogeneous Herz spaces for commutators of singular integral operators are obtained.

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.

In this paper, the boundedness properties for some Toeplitz type operators associated to the Riesz potential and general integral operators from Lebesgue spaces to Orlicz spaces are proved. The general integral operators include singular integral operator with general kernel, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

The main purpose of this paper is to investigate the behavior of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, non-doubling measures. It turns out that this condition is enough to build up a theory that contains the classical results based upon the Lebesgue measure on Euclidean space and their known extensions for doubling...

Let $ℒ=$ -div $\left(A\right(x\left)\nabla \right)$ be a second order elliptic operator with real, symmetric, bounded measurable coefficients on ${ℝ}^n$ or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed $p\&gt;2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform $\nabla {\left(ℒ\right)}^{-1/2}$ on the $L^p$ space. As an application, for $1\&lt;p\&lt;3+ϵ$, we establish the $L^p$ boundedness of Riesz transforms on Lipschitz domains for operators with $VMO$ coefficients. The range of $p$ is sharp. The closely related boundedness of ...

The aim of the present paper is to obtain an inequality of Brézis-Gallouët-Wainger type for Besov-Morrey spaces. We investigate these spaces in a self-contained manner. Also, we verify that our result is sharp.

A typical wavelet system constitutes an unconditional basis for various function spaces -Lebesgue, Besov, Triebel-Lizorkin, Hardy, BMO. One of the main reasons is the frequency localization of an element from such a basis. In this paper we study a wavelet-type system, called a brushlet system. In [3] it was noticed that brushlets constitute unconditional bases for classical function spaces such as the Triebel-Lizorkin and Besov spaces. In this paper we study brushlet expansions of functions in the...