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The main purpose of this work is to obtain a Harnack inequality and estimates for the Green function for the general class of degenerate elliptic operators described below.

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.

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.

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...

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.

We characterize the class of weights, invariant under dilations, for which a modified fractional integral operator $I_{\alpha }$ maps weak weighted Orlicz$-\phi$ spaces into appropriate weighted versions of the spaces $BM{O}_{\psi }$, where $\psi \left(t\right)=t^{\alpha /n}{\phi }^{-1}(1/t)$. This generalizes known results about boundedness of $I_{\alpha }$ from weak $L^p$ into Lipschitz spaces for $p>n/\alpha$ and from weak $L^{n/\alpha }$ into $BMO$. It turns out that the class of weights corresponding to $I_{\alpha }$ acting on weak$-L_{\phi }$ for $\phi$ of lower type equal or greater than $n/\alpha$, is the same as the one solving the problem for weak...