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...