We prove that if the exponent function p(·) satisfies log-Hölder continuity conditions locally and at infinity, then the fractional maxi-We also prove a weak-type inequality corresponding to the weak (1, n/(n − α)) inequality for M α . We build upon earlier work on the Hardy-Littlewood maximal operator by Cruz-Uribe, Fiorenza and Neugebauer [3]. As a consequence of these results for M α , we show that the fractional integral operator I α satisfies the same norm inequalities. These in turn yield a generalization of the Sobolev embedding theorem to variable L p spaces.

We prove weighted strong and weak-type norm inequalities for the Hardy–Littlewood maximal operator on the variable Lebesgue space Lp(·). Our results generalize both the classical weighted norm inequalities on Lp and the more recent results on the boundedness \ud of the maximal operator on variable Lebesgue spaces

We consider weighted norm inequalities for the Riesz potentials I α , also referred to as fractional integral operators. First we prove mixed A p -A ∞ type estimates in the spirit of [13,15,17]. Then we prove strong and weak type inequalities in the case p < q using the so-called log bump conditions. These results complement the strong type inequalities of Pérez [30] and answer a conjecture from [3]. For both sets of results our main tool is a corona decomposition adapted to fractional averages.

Abstract. We prove two-weight norm inequalities in R n for the minimal operatorextending to higher dimensions results obtained by Cruz-Uribe, Neugebauer and Olesen [8] on the real line. As an application we extend to R n weighted norm inequalities for the geometric maximal operatorproved by Yin and Muckenhoupt [27]. We also give norm inequalities for the centered minimal operator, study powers of doubling weights and give sufficient conditions for the geometric maximal operator to be equal to the closely related limiting operator M * 0 f = lim r→0 M (|f | r )

We give a new proof of the sharp weighted $L^2$ inequality ||T||_{L^2(w)} \leq c [w]_{A_2} where $T$ is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift operators. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators.Comment: To appear in the Electronic Research Announcements in Mathematical Science