The two-sided quaternion Fourier transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization of Beurling's theorem, Hardy, Cowling-Price and Gelfand-Shilov theorems, is obtained for the two-sided quaternion Fourier transform.

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We consider the Chebychev semigroup dened on the interval À1; þ1 ½ by its Dirichlet form. We prove, via a method involving probabilistic techniques, a family of inequalities which interpolate between the Sobolev and Poincar e inequalities.

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In this paper, we prove an analog of Younis's result [Int J Math Math Sci 9(2): 301-312 1986, Theorem 5.2] on the image under the Fourier-Helgason transform of a set of functions satisfying the DiniLipschitz functions in L p (1 < p ≤ 2) for functions on noncompact rank 1 Riemannian symmetric spaces. Mathematics Subject Classification. 43A30.