EUDML

Let $(M, d)$ be a metric space, equipped with a Borel measure $μ$ satisfying suitable compatibility conditions. An amalgam $A_{p}^{q} (M)$ is a space which looks locally like $L^{p} (M)$ but globally like $L^{q} (M)$ . We consider the case where the measure $μ (B (x, ρ)$ of the ball $B (x, ρ)$ with centre $x$ and radius $ρ$ behaves like a polynomial in $ρ$ , and consider the mapping properties between amalgams of kernel operators where the kernel $ker K (x, y)$ behaves like $d (x, y)^{- a}$ when $d (x, y) \leq 1$ and like $d (x, y)^{- b}$ when $d (x, y) \geq 1$ . As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems...

$L^{p} - L^{q}$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. III

Michael Cowling; Saverio Giulini; Stefano Meda — 2001

Annales de l’institut Fourier

Let $X$ be a symmetric space of the noncompact type, with Laplace–Beltrami operator $- ℒ$ , and let $[b, \infty)$ be the $L^{2} (X)$ -spectrum of $ℒ$ . For $τ$ in $ℂ$ such that $Re τ \geq 0$ , let $𝒫_{τ}$ be the operator on $L^{2} (X)$ defined formally as $\exp (- τ {(ℒ - b)}^{1 / 2})$ . In this paper, we obtain $L^{p} - L^{q}$ operator norm estimates for $𝒫_{τ}$ for all $τ$ , and show that these are optimal when $τ$ is small and when $| \arg τ |$ is bounded below $π / 2$ .

Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth

Michael Cowling; Saverio Giulini; Andrzej Hulanicki; Giancarlo Mauceri — 1994

Studia Mathematica

We prove that on Iwasawa AN groups coming from arbitrary semisimple Lie groups there is a Laplacian with a nonholomorphic functional calculus, not only for $L^{1} (A N),$ but also for $L^{p} (A N)$ , where 1 < p < ∞. This yields a spectral multiplier theorem analogous to the ones known for sublaplacians on stratified groups.

Contact and conformal maps on Iwasawa N groups

Michael Cowling; Filippo De Mari; Adam Korányi; Hans Martin Reimann — 2002

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The action of the conformal group $O (1, n + 1)$ on $R^{n} \cup \{\infty\}$ may be characterized in differential geometric terms, even locally: a theorem of Liouville states that a $C^{4}$ map between domains $U$ and $V$ in $R^{n}$ whose differential is a (variable) multiple of a (variable) isometry at each point of $U$ is the restriction to $U$ of a transformation $x \to g \cdot x$ , for some $g$ in $O (1, n + 1)$ . In this paper, we consider the problem of characterizing the action of a more general semisimple Lie group $G$ on the space $G / P$ , where $P$ is a parabolic subgroup. We solve...

A weak type (1,1) estimate for a maximal operator on a group of isometries of a homogeneous tree

Michael G. Cowling; Stefano Meda; Alberto G. Setti — 2010

Colloquium Mathematicae

We give a simple proof of a result of R. Rochberg and M. H. Taibleson that various maximal operators on a homogeneous tree, including the Hardy-Littlewood and spherical maximal operators, are of weak type (1,1). This result extends to corresponding maximal operators on a transitive group of isometries of the tree, and in particular for (nonabelian finitely generated) free groups.

Page 1

Download Results (CSV)

Advanced Search

Formula preview

Currently displaying 1 – 14 of 14

$L^{p} - L^{q}$ -estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. II.

An Application of Littlewood-Paley Theory in Harmonie Analysis.

Some Applications of Grothendieck's Theory of Topological Tensor Products in Harmonie Analysis.

On the Paley-Wiener theorem.

The Fourier-Stieltjes algebra of a semisimple group

Variants of Littlewood-Paley theory.

On Representations in Banach Spaces.

Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one.

Invariant operators on function spaces on homogeneous trees

Riesz potentials and amalgams

$L^{p} - L^{q}$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. III

Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth

Contact and conformal maps on Iwasawa N groups

A weak type (1,1) estimate for a maximal operator on a group of isometries of a homogeneous tree