The Riesz potential as a multilinear operator into general BMOβ spaces

Abstract

Given α > 0 and a space of homogeneous type X, n-normal, with $ n \in {\mathbb{R}^{+} } $ , we consider an extension of the standard multilinear fractional integral on L p1 ×⋯× L pk for the range of 1/p i = 1/p 1⋯+1/p k −α/n ≤ 0. We show that the target space is an adequate space BMOβ defined through mean oscillations. For general spaces of homogeneous type this is a Banach space of classes of functions modulii constants and the range of β is [0, 1). However, if $ X = {\mathbb{R}^n}\left( {n \in \mathbb{N}} \right) $ , we can extend the result to β > 0 taking in account that BMOβ is a space of classes modulii polynomials of order lower than or equal to [β]. Bibliography: 15 titles.