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\(Q\) spaces of several real variables. (English) Zbl 0984.46020

For \(\alpha\in (-\infty, +\infty)\), the space \(Q_{\alpha} ({\mathbf R}^n)\) is defined to be the space of all measurable functions \(f\) on \({\mathbf R}^n\) with \[ \sup [l(I)]^{2\alpha -n}\int_I \int_I |f(x)-f(y)|^2 / |x-y|^{n+2\alpha} dx dy<\infty, \] where the supremum is taken over all cubes \(I\) with edges parallel to coordinate axes, and \(l(I)\) is the length of \(I\). It is shown that \(Q_{\alpha} ({\mathbf R}^n)=\text{BMO}({\mathbf R}^n)\) for \(\alpha <0\) and \(Q_{\alpha} ({\mathbf R}^n)\) contains only the constant functions if \(\alpha\geq 1\). For \(\alpha\in[0, 1]\) various characteristics of \(Q_{\alpha} ({\mathbf R}^n)\) are given (in terms of Poisson integrals, \(p\)-Carleson measures, mean oscillation, and wavelet coefficients). Relationship between the spaces \(Q_{\alpha} ({\mathbf R}^n)\) and the scale of Besov spaces is discussed.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E15 Banach spaces of continuous, differentiable or analytic functions
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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