×

On the characterization of Hardy-Besov spaces on the dyadic group and its applications. (English) Zbl 0817.43004

Let \(P = \{0,1\}^ N\) denote the dyadic group. The author defines the Besov spaces \(B^ \alpha_{p,q}(P)\) on \(P\), with \(0 < p,q \leq \infty\) and \(\alpha \in \mathbb{R}\), in terms of the differences of certain Dirichlet kernels on \(P\). Let \(\Delta_ 0(x) = 1\) for all \(x \in P\) and let \(\Delta_ j(x) = D_{2^ j}(x) - D_{2^{j-1}}(x)\). A distribution \(f\) on \(P\) belongs to \(B^ \alpha_{p,q}(P)\) if \(\sum^ \infty_{j = 0} (2^{\alpha j} \| \Delta_ j * f\|_ p)^ q < \infty\). Characterizations of these Besov spaces are given in terms of oscillations, best approximations, \((\alpha,p)\)-atoms and \((\alpha,p)\)- molecules, as defined in the paper. The atomic characterization of the Besov spaces is used to prove on \(P\) a strong capacity inequality of the type of Maz’ya’s inequality [see V. G. Maz’ya and T. O. Shaposhnikova, Theory of multipliers in spaces of differentiable functions. Pitman, Boston (1985; Zbl 0645.46031)]. Furthermore, a weak- type \((1,p)\) inequality for the maximal Césaro means of order \(\beta\), \(\sup_ n | \sigma^ \beta_ n f(x)|\), with \(f \in B^ \alpha_{p,q}(P)\), is proved when \(\alpha > 0\), \(1/2 \leq p < 1\), \(0 < q \leq \infty\) and \(\beta = 1/p - 1\).

MSC:

43A75 Harmonic analysis on specific compact groups
26A16 Lipschitz (Hölder) classes
41A25 Rate of convergence, degree of approximation

Citations:

Zbl 0645.46031
PDFBibTeX XMLCite
Full Text: DOI EuDML