The process of translation averaging is known to improve dyadic BMO to the space BMO of functions of bounded mean oscillation, in the sense that the translation average of a family of dyadic BMO functions is necessarily a BMO function. The present work investigates the effect of translation averaging in other dyadic settings. We show that translation averages of dyadic doubling measures need not be doubling measures, translation averages of dyadic Muckenhoupt weights need not be Muckenhoupt weights,...

If $E\left(f\right)=\{x:lim\; supf☆{\mu }_j\left(x\right)>lim\; inff☆{\mu }_j\left(x\right)\}$, we examine the type of convergence of $g_k$ to $f$ so that $\left|E\right(g_k\left)\right|\le M$, $k=1,2,\cdots$, implies $\left|E\right(f\left)\right|\le M$.

This paper develops some Littlewood-Paley theory for Hermite expansions. The main result is that certain analogues of Triebel-Lizorkin spaces are well-defined in the context of Hermite expansions.

In [HS] the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type were introduced. In this paper, the Triebel-Lizorkin spaces on spaces of homogeneous type are generalized to the case where $p_0<p\le 1\le q<\infty$, and a new atomic decomposition for these spaces is obtained. As a consequence, we give the Littlewood-Paley characterization of Hardy spaces on spaces of homogeneous type which were introduced by the maximal function characterization in [MS2].

Suppose that μ is a Radon measure on ${ℝ}^d$, which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀ > 0 such that for all x ∈ supp(μ) and r > 0, μ(B(x,r)) ≤ C₀rⁿ, where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spaces ${Ḟ}_{pq}^s\left(\mu \right)$ for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C₀, n and d. The method does not use the vector-valued maximal function inequality...

Doubling measures appear in relation to quasiconformal mappings of the unit disk of the complex plane onto itself. Each such map determines a homeomorphism of the unit circle on itself, and the problem arises, which mappings f can occur as boundary mappings?

In this article, we first improve the scalar maximal theorem for the Dunkl maximal operator by giving some precisions on the behavior of the constants of this theorem for a general reflection group. Next we complete the vector-valued theorem for the Dunkl-type Fefferman-Stein operator in the $\mathbb{Z}{₂}^d$ case by establishing a result of exponential integrability corresponding to the case p = +∞.

In this paper, we give a generalization of Fefferman-Stein inequality for the fractional one-sided maximal operator: $${\int }_{-\infty }^{+\infty }{M}_{\alpha }^{+}\left(f\right){\left(x\right)}^{p}w\left(x\right)dx\le A_p{\int }_{-\infty }^{+\infty }{\left|f\left(x\right)\right|}^p{M}_{\alpha p}^{-}\left(w\right)\left(x\right)dx,$$ where $0<\alpha <1$ and $1<p<1/\alpha$. We also obtain a substitute of dual theorem and weighted norm inequalities for the one-sided fractional integral $I_{\alpha }^{+}$.

New sufficient conditions on the weight functions u(.) and v(.) are given in order that the fractional maximal [resp. integral] operator Ms [resp. Is], 0 ≤ s &lt; n, [resp. 0 &lt; s &lt; n] sends the weighted Lebesgue space Lp(v(x)dx) into Lp(u(x)dx), 1 &lt; p &lt; ∞. As a consequence a characterization for this estimate is obtained whenever the weight functions are radial monotone.

Let φ: R → [0,∞) an integrable function such that φχ(-∞,0) = 0 and φ is decreasing in (0,∞). Let τhf(x) = f(x-h), with h ∈ R {0} and fR(x) = 1/R f(x/R), with R &gt; 0. In this paper we characterize the pair of weights (u, v) such that the operators Mτhφf(x) = supR&gt;0|f| * [τhφ]R(x) are of weak type (p, p) with respect to (u, v), 1 &lt; p &lt; ∞.

Let s* denote the maximal function associated with the rectangular partial sums $s_{mn}(x,y)$ of a given double function series with coefficients $c_{jk}$. The following generalized Hardy-Littlewood inequality is investigated: ${\left|\right|s*\left|\right|}_{p,\mu }\le C_{p,\alpha ,\beta }{{\Sigma }_{j=0}^{\infty }{\Sigma }_{k=0}^{\infty }{\left(j̅\right)}^{p-\alpha -2}{\left(k̅\right)}^{p-\beta -2}{\left|c_{jk}\right|}^p}^{1/p}$, where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on $c_{jk}$ and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property of $s_{mn}(x,y)$...