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On Besov spaces and absolute convergence of the Fourier transform on Heisenberg groups. (English) Zbl 1060.46512
Let \(B^s_{pq}(\mathbb H^n)\) denote the Besov space defined on a Heisenberg group \(\mathbb H^n\), \(s\in \mathbb R\), \(1<p<\infty \), \(1\leq q\leq \infty \). Let \(\pi _{\lambda }(x,y,t)\), \(\lambda \in \mathbb R\), \(x,y\in \mathbb R^n\) be an irreducible unitary representation of \(\mathbb H^n\) and, for \(\lambda \in \mathbb R\), \(\widehat f(\lambda )=\int _{\mathbb H^n}f(x,y,t)\pi _{\lambda }(-x,-y,-t)\,dx\,dy\,dt\) be the Fourier transform. In this paper, estimates of \(\| \,\| \widehat f(\lambda )\| _{HS}| \lambda | ^n| L_1(\mathbb R)\| \) and \(\| \,\| \widehat f(\lambda )\| _{HS}| \lambda | ^{n/2}| L_1(\mathbb R)\| \) by \(c\| f| B^{n+[n/2]+2}_{11}(\mathbb H^n)\| \) and by \(c\| f| B^{n+2}_{11}(\mathbb H^n)\| \), resp., with \(c\) independent of \(f\) are proved (where \(\| \,.\,\| \) is the corresponding Hilbert-Schmidt norm) via the atomic decomposition technique. Real interpolation then yields estimates of similar type for \(f\in B^{s_p}_{pp}(\mathbb H^n)\) and \(f\in B^{\sigma _{pq}}_{pq}(\mathbb H^n)\), where \(1<p<2\) and \(1\leq q\leq p\leq 2\), with explicit formulas for the smoothness parameters \(s_p\), \(\sigma _{pq}\).

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
22E25 Nilpotent and solvable Lie groups
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
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