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In [P] we characterize the pairs of weights for which the fractional integral operator $I_{\gamma }$ of order $\gamma$ from a weighted Lebesgue space into a suitable weighted $BMO$ and Lipschitz integral space is bounded. In this paper we consider other weighted Lipschitz integral spaces that contain those defined in [P], and we obtain results on pairs of weights related to the boundedness of $I_{\gamma }$ acting from weighted Lebesgue spaces into these spaces. Also, we study the properties of those classes of weights and compare...

In this work we give sufficient and necessary conditions for the boundedness of the fractional integral operator acting between weighted Orlicz spaces and suitable $BM{O}_{\phi }$ spaces, in the general setting of spaces of homogeneous type. This result generalizes those contained in [P1] and [P2] about the boundedness of the same operator acting between weighted $L^p$ and Lipschitz integral spaces on ${ℝ}^n$. We also give some properties of the classes of pairs of weights appearing in connection with this boundedness.

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...

In this work we prove some sharp weighted inequalities on spaces of homogeneous type for the higher order commutators of singular integrals introduced by R. Coifman, R. Rochberg and G. Weiss in Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976), 611–635. As a corollary, we obtain that these operators are bounded on $L^p\left(w\right)$ when $w$ belongs to the Muckenhoupt’s class $A_p$, $p>1$. In addition, as an important tool in order to get our main result, we prove a weighted Fefferman-Stein...

We prove two-weighted norm estimates for higher order commutator of singular integral and fractional type operators between weighted $L^p$ and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal parameters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces belong to a certain region out of which the classes of weights are satisfied by trivial weights. We also exhibit pairs of nontrivial...

In the context of variable exponent Lebesgue spaces equipped with a lower Ahlfors measure we obtain weighted norm inequalities over bounded domains for the centered fractional maximal function and the fractional integral operator.