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; ∞.

The inequality (*) $({\sum }_{\left|n\right|=1}^{\infty }{\sum }_{\left|m\right|=1}^{\infty }{\left|nm\right|}^{p-2}{\left|f̂(n,m)\right|}^p{)}^{1/p}\le C_p\parallel ƒ{\parallel }_{H_p}$ (0 < p ≤ 2) is proved for two-parameter trigonometric-Fourier coefficients and for the two-dimensional classical Hardy space $H_p$ on the bidisc. The inequality (*) is extended to each p if the Fourier coefficients are monotone. For monotone coefficients and for every p, the supremum of the partial sums of the Fourier series is in $L_p$ whenever the left hand side of (*) is finite. From this it follows that under the same condition the two-dimensional trigonometric-Fourier series...

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 $s_{mn}(x,y)$...

Sufficient conditions for a two-weight norm inequality for potential type integral operators to hold are given in the case p > q > 0 and p > 1 in terms of the Hedberg-Wolff potential.

We give a characterization of the pairs of weights (v,w), with w in the class $A_{\infty }$ of Muckenhoupt, for which the fractional maximal function is a bounded operator from $L^p(X,vd\mu )$ to $L^q(X,wd\mu )$ when 1 < p ≤ q < ∞ and X is a space of homogeneous type.

Necessary and sufficient conditions are shown in order that the inequalities of the form $\varrho \left(M_{\mu }f>\lambda \right)\Phi \left(\lambda \right)\le C{ʃ}_X\Psi \left(C\right|f\left(x\right)\left|\right)\sigma \left(x\right)d\mu$, or $\varrho \left(M_{\mu }f>\lambda \right)\le C{ʃ}_X\Phi (C{\lambda }^{-1}|f\left(x\right)\left|\right)\sigma \left(x\right)d\mu$ hold with some positive C independent of λ > 0 and a μ-measurable function f, where (X,μ) is a space with a complete doubling measure μ, $M_{\mu }$ is the maximal operator with respect to μ, Φ, Ψ are arbitrary Young functions, and ϱ, σ are weights, not necessarily doubling.

Let $\mu$ be a nonnegative Borel measure on ${ℝ}^d$ satisfying that $\mu \left(Q\right)\le l{\left(Q\right)}^n$ for every cube $Q\subset {ℝ}^n$, where $l\left(Q\right)$ is the side length of the cube $Q$ and $0<n\le d$. We study the class of pairs of weights related to the boundedness of radial maximal operators of fractional type associated to a Young function $B$ in the context of non-homogeneous spaces related to the measure $\mu$. Our results include two-weighted norm and weak type inequalities and pointwise estimates. Particularly, we give an improvement of a two-weighted result for certain...