arXiv subjects
Publications and source records attributed to Michael G. Cowling.
We show that irreducible strongly continuous representations of $\mathrm{SL}(2,\mathbb{R})$ on certain Banach spaces are admissible and that the admissibility of Banach space representations of SL(2,R) and the invariant subspace problem are intimately related.
We study the Brascamp--Lieb inequalities on locally compact nonabelian groups and the Brascamp--Lieb constants $\mathbf{BL}(G, \boldsymbolσ, \boldsymbol{p})$ associated to a Brascamp--Lieb datum: locally compact groups $G$ and $G_j$, a family of homomorphisms $σ_j: G \to G_j$ and Lebesgue indices $p_j$. We focus on homogeneous Lie groups and compact Lie groups. For homogeneous Lie groups $G$, we show that the constant $\mathbf{BL}(G, \boldsymbolσ, \boldsymbol{p})$ is equal to the constant $\mathbf{BL}(\mathfrak{g}, \boldsymbol{\mathrm{d}σ}, \boldsymbol{p})$, where $\mathfrak{g}$ is the Lie algebra of $G$ and $\mathrm{d}σ_j$ is the differential of $σ_j$. For Heisenberg-like groups $G$, we show that the only inequalities that can occur are multilinear Hölder inequalities. For compact Lie groups, we find necessary and sufficient conditions for finiteness of the constant $\mathbf{BL}(G, \boldsymbolσ, \boldsymbol{p})$ in terms of $\boldsymbolσ$ and $\boldsymbol{p}$ and find an explicit expression for the constant, similar to those found by Bennett and Jeong in the abelian case.
We establish a complete theory of the flag Hardy space on the Heisenberg group $\mathbb H^{n}$ with characterisations via atomic decompositions, area functions, square functions, maximal functions and singular integrals. We introduce several new techniques to overcome the difficulties caused by the noncommutative Heisenberg group multiplication, and the lack of a suitable Fourier transformation and Cauchy--Riemann type equations. Applications include the boundedness from the flag Hardy space to $L^1(\mathbb H^n)$ of various singular integral operators that arise in complex analysis, a sharp boundedness result on the flag Hardy space of the Marcinkiewicz-type multipliers introduced by Müller, Ricci and Stein, and the decomposition of flag BMO space via singular integrals.
Let $L$ be a positive self-adjoint operator on $L^2(X)$, where $X$ is a $σ$-finite metric measure space. When $α\in (0,1)$, the subordinated semigroup $\{\exp(-tL^α):t \in \mathbb{R}^+\}$ can be defined on $L^2(X)$ and extended to $L^p(X)$. We prove various results about the semigroup $\{\exp(-tL^α):t \in \mathbb{R}^+\}$, under different assumptions on $L$. These include the weak type $(1,1)$ boundedness of the maximal operator $f \mapsto \sup _{t\in \mathbb{R}^+}\exp(-tL^α)f$ and characterisations of Hardy spaces associated to the operator $L$ by the area integral and vertical square function.
We establish a structure theorem for the Brascamp--Lieb constant formulated in the general setting of locally compact abelian groups. This extends and unifies the finiteness characterisations previously known for euclidean spaces and for finitely generated groups and their duals. We place particular emphasis on Fourier invariance throughout, reflecting the fundamental Fourier invariance of Brascamp--Lieb multilinear forms in this context.
The aim of this note is to insert in the literature some easy but apparently not widely known facts about morphisms of locally compact groups, all of which are concerned with the openness of the morphism.
We establish hyperweak boundedness of area functions, square functions, maximal operators and Calderón--Zygmund operators on products of two stratified Lie groups.
A large part of the theory of Hardy spaces on products of Euclidean spaces has been extended to the setting of products of stratified Lie groups. This includes characterisation of Hardy spaces by square functions and by atomic decompositions, proof of the duality of Hardy spaces with BMO, and description of many interpolation spaces. Until now, however, two aspects of the classical theory have been conspicuously absent: the characterisation of Hardy spaces by singular integrals (of Christ--Geller type) or by (vertical or nontangential) maximal functions. In this paper we fill in these gaps by developing new techniques on products of stratified groups, using the ideas of Chen, Cowling, Lee, Li and Ottazzi on the Heisenberg group with flag structure.
Let $G$ be a connected semisimple Lie group with finite centre and $K$ be a maximal compact subgroup thereof. Given a function $u$ on $G$, we define $\mathcal{A} u$ to be the root mean square average over $K$, acting both on the left and the right, of $u$. We show that for all unitary representations $π$ of $G$, there exists a unique minimal positive-real-valued spherical function $ϕ_λ$ on $G$ such that $\mathcal{A} \langle π(\cdot) ξ, η\rangle \leq \Vert ξ\Vert_{\mathcal{H}_π} \Vert η\Vert_{\mathcal{H}_π} ϕ_λ$. This estimate has nice features of both asymptotic pointwise estimates and Lebesgue space estimates; indeed it is equivalent to pointwise estimates $\vert \langle π(\cdot) ξ, η\rangle \vert \leq C(ξ, η) \,ϕ_λ$ for $K$-finite or smooth vectors $ξ$ and $η$, and it exhibits different decay rates in different directions at infinity in $G$. Further, if we assume the latter inequality with arbitrary $C( ξ, η)$, we can prove the former inequality and then return to the latter inequality with explicit knowledge of $C( ξ, η)$. On the other hand, it holds everywhere in $G$, in contrast to asymptotic estimates which are not global. We also provide some applications.
We study homogeneous metric spaces, by which we mean connected, locally compact metric spaces whose isometry group acts transitively. After a review of some classical results, we use the Gleason-Iwasawa-Montgomery-Yamabe-Zippin structure theory to show that for all positive $ε$, each such space is $(1,ε)$-quasi-isometric to a connected metric Lie group. Next, we develop the structure theory of Lie groups to show that every homogeneous metric manifold is homeomorphically roughly isometric to a quotient space of a connected amenable Lie group, and roughly isometric to a simply connected solvable metric Lie group. Third, we investigate solvable metric Lie groups in more detail, and expound on and extend work of Gordon and Wilson and of Jablonski on these, showing, for instance, that connected, simply connected solvable Lie groups may be made isometric if and only if they have the same real-shadow. Finally, we extend a result of Kivioja and Le Donne to show that homogeneous metric spaces that admit a metric dilation are all metric Lie groups with an automorphic dilation.
Let $M^{(k)}$, $k=1,2,\ldots, n$, be the boundary of an unbounded polynomial domain $Ω^{(k)}$ of finite type in $\mathbb C ^2$, and let $\Box_b^{(k)}$ be the Kohn Laplacian on $M^{(k)}$. In this paper, we study multivariable spectral multipliers $m(\Box_b^{(1)},\ldots, \Box_b^{(n)})$ acting on the Shilov boundary $\widetilde{M}=M^{(1)} \times\cdots\times M^{(n)}$ of the product domain $Ω^{(1)}\times\cdots\times Ω^{(n)}$. We show that if a function $F(λ_1, \ldots ,λ_n)$ satisfies a Marcinkiewicz-type differential condition, then the spectral multiplier operator $m(\Box_b^{(1)}, \ldots, \Box_b^{(n)})$ is a product Calderón--Zygmund operator of Journé type.
We prove a nonlinear variant of the general Brascamp-Lieb inequality. Instances of this inequality are quite prevalent in analysis, and we illustrate this with substantial applications in harmonic analysis and partial differential equations. Our proof consists of running an efficient, or "tight", induction on scales argument, which uses the existence of gaussian near-extremisers to the underlying linear Brascamp-Lieb inequality (Lieb's theorem) in a fundamental way. A key ingredient is an effective version of Lieb's theorem, which we establish via a careful analysis of near-minimisers of weighted sums of exponential functions.
We consider a class of stratified groups with a CR structure and a compatible control distance. For these Lie groups we show that the space of conformal maps coincide with the space of CR and anti-CR diffeomorphisms. Furthermore, we prove that on products of such groups, all CR and anti-CR maps are product maps, up to a permutation isomorphism, and affine in each component.
Estimates for matrix coefficients of unitary representations of semisimple Lie groups have been studied for a long time, starting with the seminal work by Bargmann, by Ehrenpreis and Mautner, and by Kunze and Stein. Two types of estimates have been established: on the one hand, $L^p$ estimates, which are a dual formulation of the Kunze--Stein phenomenon, and which hold for all matrix coefficients, and on the other pointwise estimates related to asymptotic expansions at infinity, which are more precise but only hold for a restricted class of matrix coefficients. In this paper we prove a new type of estimate for the irreducibile unitary representations of $\mathrm{SL}(2,\mathbb{R})$ and for the so-called metaplectic representation, which we believe has the best features of, and implies, both forms of estimate described above. As an application outside representation theory, we prove a new $L^2$ estimate of dispersive type for the free Schrödinger equation in $\mathbb{R}^n$.
A sharp $L^p$ spectral multiplier theorem of Mihlin--Hörmander type is proved for a distinguished sub-Laplacian on quaternionic spheres. This is the first such result on compact sub-Riemannian manifolds where the horizontal space has corank greater than one. The proof hinges on the analysis of the quaternionic spherical harmonic decomposition, of which we present an elementary derivation.
We prove several results about the best constants in the Hausdorff-Young inequality for noncommutative groups. In particular, we establish a sharp local central version for compact Lie groups, and extend known results for the Heisenberg group. In addition, we prove a universal lower bound to the best constant for general Lie groups.
We estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of SL(2, R) as completely bounded multipliers of the Fourier algebra. Our results suggest that the known inequality relating the uniformly bounded norm of a representation and the completely bounded norm of its coefficients may not be optimal.
We show that a graded Lie algebra admits a Levi decomposition that is compatible with the grading.