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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 $$s_p$$, $$\sigma _{pq}$$.

##### MSC:
 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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