Abstract. We prove that if q is in (1, ∞), Y is a Banach space, and T is a linear operator defined on the space of finite linear combinations of (1, q)-atoms in R n with the property that sup{ T a Y : a is a (1, q)-atom} < ∞, then T admits a (unique) continuous extension to a bounded linear operator from H 1 (R n ) to Y . We show that the same is true if we replace (1, q)-atoms by continuous (1, ∞)-atoms. This is known to be false for (1, ∞)-atoms.
We study the dispersive properties of the wave equation associated with the shifted Laplace-Beltrami operator on real hyperbolic spaces and deduce new Strichartz estimates for a large family of admissible pairs. As an application, we obtain local well-posedness results for the nonlinear wave equation.
Let G be a noncompact connected Lie group, denote with ρ a right Haar measure and choose a family of linearly independent left-invariant vector fields X on G satisfying Hörmander's condition. Let χ be a positive character of G and consider the measure µχ whose density with respect to ρ is χ. In this paper, we introduce Sobolev spaces L p α (µχ) adapted to X and µχ (1 < p < ∞, α ≥ 0) and study embedding theorems and algebra properties of these spaces. As an application, we prove local well-posedness and regularity results of solutions of some nonlinear heat and Schrödinger equations on the group.
In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We realize the dual space Y k (M ) of the Hardy-type space X k (M ), introduced in a previous paper of the authors, as the class of all locally square integrable functions satisfying suitable BM O-like conditions, where the role of the constants is played by the space of global k-quasi-harmonic functions. Furthermore we prove that Y k (M ) is also the dual of the space X k fin (M ) of finite linear combination of X k -atoms. As a consequence, if Z is a Banach space and T is a Z-valued linear operator defined on X k fin (M ), then T extends to a bounded operator from X k (M ) to Z if and only if it is uniformly bounded on X k -atoms. To obtain these results we prove the global solvability of the generalized Poisson equation L k u = f with f ∈ L 2 loc (M ) and we study some properties of generalized Bergman spaces of harmonic functions on geodesic balls.2010 Mathematics Subject Classification. 30H10, 42B20, 42B35, 58C99.
In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below, positive injectivity radius and spectral gap b. We introduce a sequence X 1 (M ), X 2 (M ), . . . of new Hardy spaces on M , the sequence Y 1 (M ), Y 2 (M ), . . . of their dual spaces, and show that these spaces may be used to obtain endpoint estimates for purely imaginary powers of the Laplace-Beltrami operator and for more general spectral multipliers associated to the Laplace-Beltrami operator L on M . Under the additional condition that the volume of the geodesic balls of radius r is controlled by C r α e 2 √ br for some real number α and for all large r, we prove also an endpoint result for first order Riesz transforms ∇L −1/2 .In particular, these results apply to Riemannian symmetric spaces of the noncompact type.
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