In [4] P. Jones solved the question posed by B. Muckenhoupt in [7] concerning the factorization of Ap weights. We recall that a non-negative measurable function w on Rn is in the class Ap, 1 < p < ∞ if and only if the Hardy-Littlewood maximal operator is bounded on Lp(Rn, w). In what follows, Lp(X, w) denotes the class of all measurable functions f defined on X for which ||fw1/p||Lp(X) < ∞, where X is a measure space and w is a non-negative measurable function on X.It has recently...

Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for ${\Phi }_2^{-1}(ʃ{\Phi }_2\left(w\left(x\right)\right|Tf\left(x\right)\left|\right)t\left(x\right)dx)\le {\Phi }_1^{-1}(ʃ{\Phi }_1\left(Cu\left(x\right)\right|f\left(x\right)\left|\right)v\left(x\right)dx)$ to hold when ${\Phi }_1$ and ${\Phi }_2$ are N-functions with ${\Phi }_2\circ {\Phi }_1^{-1}$ convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.

Applying discrete Calderón’s identity, we study weighted multi-parameter mixed Hardy space $H_{\mathrm{mix}}^p(\omega ,{ℝ}^{n_1}\times {ℝ}^{n_2})$. Different from classical multi-parameter Hardy space, this space has characteristics of local Hardy space and Hardy space in different directions, respectively. As applications, we discuss the boundedness on $H_{\mathrm{mix}}^p(\omega ,{ℝ}^{n_1}\times {ℝ}^{n_2})$ of operators in mixed Journé’s class.

We obtain weighted $L^p$ boundedness, with weights of the type $y^{\delta }$, δ > -1, for the maximal operator of the heat semigroup associated to the Laguerre functions, ${{ℒ}_k^{\alpha }}_k$, when the parameter α is greater than -1. It is proved that when -1 < α < 0, the maximal operator is of strong type (p,p) if p > 1 and 2(1+δ)/(2+α) < p < 2(1+δ)/(-α), and if α ≥ 0 it is of strong type for 1 < p ≤ ∞ and 2(1+δ)/(2+α) < p. The behavior at the end points of the intervals where there is strong type is studied...

We prove weighted norm inequalities for the averaging operator Af(x) = 1/x ∫0x f of monotone functions.

Let µ be a Borel measure on Rd which may be non doubling. The only condition that µ must satisfy is µ(B(x, r)) ≤ Crn for all x ∈ Rd, r &gt; 0 and for some fixed n with 0 &lt; n ≤ d. In this paper we introduce a maximal operator N, which coincides with the maximal Hardy-Littlewood operator if µ(B(x, r)) ≈ rn for x ∈ supp(µ), and we show that all n-dimensional Calderón-Zygmund operators are bounded on Lp(w dµ) if and only if N is bounded on Lp(w dµ), for a fixed p ∈ (1, ∞). Also, we prove...

The main purpose of this paper is to use some of the results and techniques in [9] to further investigate weighted norm inequalities for Hardy-Littlewood type maximal operators.

For some pairs of weight functions u, v which satisfy the well-known Muckenhoupt conditions, we derive the boundedness of the maximal fractional operator Ms (0 ≤ s &lt; n) from Lvp to Luq with q &lt; p.

A weighted theory describing Morrey boundedness of fractional integral operators and fractional maximal operators is developed. A new class of weights adapted to Morrey spaces is proposed and a passage to the multilinear cases is covered.

In this paper we establish weighted norm inequalities for singular integral operators with kernel satisfying a variant of the classical Hörmander's condition.