Q spaces of several real variables

For alpha is an element of (-infinity, infinity), let Q(alpha)(R-n) be the space of all measurable functions with sup[l(I)](2 alpha -n) integral (I)integral (I)/f(x) - f(y)/(2)//x-y/(n+2 al pha) dxdy < <infinity>, where the supremum is taken over all cubes I with the edge length l(I) and the edges parallel to the coordinate axes in R-n. If alpha is an element of (-infinity, 0), then Q(alpha)(R-n) = BMO(R-n), and if alpha is an element of [1, infinity), then Q(alpha)(R-n) = (constants). In the present paper, w e discuss the case alpha is an element of [0,1). These spaces are new subsp aces of BMO(R-n) containing some special Besov spaces. We characterize func tions in Q(alpha)(R-n) by means of the Poisson extension, p-Carleson measur es, mean oscillation and wavelet coefficients, and give a dyadic counterpar t. Finally, we pose some open problems.