Cowling-Price and Hardy theorems on Chébli-Trimèche hypergroups. (English) Zbl 1122.43005

This paper establishes the following analogue of the Cowling-Price theorem for the generalized Fourier transform \({\mathcal F}\) on Chébli-Trimèche hypergroups \(({\mathbb R}_+,*(A))\) with Gaussian kernel \(E_t\) \((t>0)\): Let \(a,b>0\), \(1\leq p,q\leq +\infty\) such that at least one of \(p,q\) is finite, and let \(f\) be a measurable function on \({\mathbb R}_+\) for which the \(L^p\)-norm of \(E_a^{-1}f\) and the \(L^q\)-norm of \(e^{by^2}{\mathcal F}(f)\) are finite. If \(ab\geq 1/4\) then \(f=0\) a.e., while for \(ab<1/4\) there are many nonzero \(f\) satisfying the hypotheses. The case \(p=q=+\infty\) (the analogue of the Hardy theorem for \({\mathcal F}\)) is also investigated. It is shown that if \(ab>1/4\), then \(f=0\) a.e.; if \(ab=1/4\), then \(f(x)=C_0 E_{1/4a}(x)\) for some \(C_0\in {\mathbb R}\); while if \(ab<1/4\), then there are infinitely many nonzero \(f\) satisfying the hypotheses. To prove these results the author establishes an \(L^p\)-version of the Phragmén-Lindelöf theorem and makes decisive use of the generalized Riemann-Liouville and Weyl integral transforms on Chébli-Trimèche hypergroups.


43A62 Harmonic analysis on hypergroups
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
44A15 Special integral transforms (Legendre, Hilbert, etc.)