In this paper we investigate the L p boundedness of the lacunary maximal function M lac H n associated to the spherical means A r f taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the L p improving property of the operator A r f and a continuity property of the difference A r f − τ y A r f , where τ y f (x) = f (xy −1 ) is the right translation operator.
We prove an uncertainty principle for certain eigenfunction expansions on L 2 (R + , w(r)dr) and use it to prove analogues of theorems of Chernoff and Ingham for Laplace-Beltrami operators on compact symmetric spaces, special Hermite operator on C n and Hermite operator on R n .
Let G be a noncompact semisimple Lie group with finite centre. Let X = G∕K be the associated Riemannian symmetric space and assume that X is of rank one. The generalized spectral projections associated to the Laplace-Beltrami operator are given by P f = f * Φ , where Φ are the elementary spherical functions on X. In this paper, we prove an Ingham type uncertainty principle for P f . Moreover, similar results are obtained in the case of generalized spectral projections associated to Dunkl Laplacian.
There is increasing interest in recent years in the structural chemistry and properties of layered metal oxides possessing the K2NiF4 or related structures. Many new oxides of this structure exhibiting novel properties are being reported from time to time in the literature. The crystal chemistry of the oxides of the general formula A2B04 with particular reference to the stability of the K2NiF4 structure and the relations between the different structures exhibited by this family of oxides is discussed. Non-stoichiometry in these oxides is another aspect of interest discussed in the article. While K2NiF4 itself is a well-known two-dimensional antiferromagnet, oxides of this structure with a variety of magnetic properties are examined in some detail. Besides the ternary A2BO4 oxides, the structure and magnetic properties of complex oxides, where the A or/and the B ions are partly substituted by other cations, is discussed. Some of the problems related to this family of oxides that are worth investigating are indicated. Much of the discussion in this article would have relevance in understanding the structure and properties of layered materials.
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