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$L^p$-estimates for Riesz transforms on forms in the Poincar\'e space by Bruna, Joaquim is a Mathematics article available to read on EtoBox.

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Using hyperbolic form convolution with doubly isometry-invariant kernels, the explicit expression of the inverse of the de Rham laplacian ∆ acting on m-forms in the Poincaré space H n is found. Also, by means of some estimates for hyperbolic singular integrals, L p -estimates for the Riesz transforms ∇ i ∆ -1 , i ≤ 2, in a range of p depending on m, n are obtained. Finally, using these, it is shown that ∆ defines topological isomorphisms in a scale of Sobolev spaces H s m,p (H n ) in case m ≠ (n ± 1)/2, n/2. ## 1. STATEMENT OF RESULTS AND PRELIMINARIES 1.1. The main object of study in this paper is the Hodge-de Rham Laplacian ∆ acting on m-forms in the Poincaré hyperbolic space (H n , g). The aim is to prove that ∆ defines topological isomorphisms in a range H s m,p (H n ) of Sobolev spaces of forms defined as follows. For 0 ≤ m ≤ n, 1 ≤ p < ∞ and s ∈ N, the Sobolev space H s m,p (H n ) is the completion of the space D m (H n ) of smooth m-forms with compact support with respect to the norm Here ∇ (i) means the i-th covariant differential of η, and for a covariant tensor α α p = H n |α(x)| p dμ(x)

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Author

Bruna, Joaquim

Publisher

Indiana University Mathematics Journal; Indiana University Press (ISSN 0022-2518)

Published

2005

Language

EN

Field

Mathematics (Physical Sciences)