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We show that irreducible strongly continuous representations of $\mathrm{SL}(2,\mathbb{R})$ on Hilbert spaces are admissible, modulo the recently proposed solution of the invariant subspace problem on Hilbert spaces., Comment: 13 pages, slightly revised to correct typos and to generalise the scope

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., Comment: 5 pages

We establish hyperweak boundedness of area functions, square functions, maximal operators and Calder\'on--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., Comment: Minor corrections and improvements made

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 $\pi$ of $G$, there exists a unique minimal positive-real-valued spherical function $\phi_{\lambda}$ on $G$ such that $\mathcal{A} \langle \pi(\cdot) \xi, \eta \rangle \leq \Vert \xi \Vert_{\mathcal{H}_\pi} \Vert \eta \Vert_{\mathcal{H}_\pi} \phi_{\lambda}$. This estimate has nice features of both asymptotic pointwise estimates and Lebesgue space estimates; indeed it is equivalent to pointwise estimates $\vert \langle \pi(\cdot) \xi, \eta \rangle \vert \leq C(\xi, \eta) \,\phi_{\lambda}$ for $K$-finite or smooth vectors $\xi$ and $\eta$, and it exhibits different decay rates in different directions at infinity in $G$. Further, if we assume the latter inequality with arbitrary $C( \xi, \eta )$, we can prove the former inequality and then return to the latter inequality with explicit knowledge of $C( \xi, \eta )$. On the other hand, it holds everywhere in $G$, in contrast to asymptotic estimates which are not global. We also provide some applications., Comment: 20 pages, revised version to improve readability, one new result added

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 M\"uller, Ricci and Stein, and the decomposition of flag BMO space via singular integrals., Comment: Revised version; minor errors corrected

Let $M^{(k)}$, $k=1,2,\ldots, n$, be the boundary of an unbounded polynomial domain $\Omega^{(k)}$ 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 $\Omega^{(1)}\times\cdots\times \Omega^{(n)}$. We show that if a function $F(\lambda_1, \ldots ,\lambda_n)$ satisfies a Marcinkiewicz-type differential condition, then the spectral multiplier operator $m(\Box_b^{(1)}, \ldots, \Box_b^{(n)})$ is a product Calder\'on--Zygmund operator of Journ\'e type.

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 Schr\"odinger equation in $\mathbb{R}^n$., Comment: 20 pages

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., Comment: 29 pages. This article subsumes the results of arXiv:1801.05214. To appear in the Duke Mathematical Journal

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., Comment: 29 pages

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.

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 compute the "norm" of irreducible uniformly bounded representations of SL2R. We show that the Kunze-Stein version of the uniformly bounded representations has minimal norm in the similarity class of uniformly bounded representations.

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 $\epsilon$, each such space is $(1,\epsilon)$-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., Comment: 60 pages; major rewrite

We show that a graded Lie algebra admits a Levi decomposition that is compatible with the grading.

A sharp $L^p$ spectral multiplier theorem of Mihlin--H\"ormander 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.

Let $\mathfrak{g}$ be a real semisimple Lie algebra with Iwasawa decomposition $\mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$. We show that, except for some explicit exceptional cases, every derivation of the nilpotent subalgebra $\mathfrak{n}$ that preserves its restricted root space decomposition is of the form $\text{ad}( W)$, where $W \in \mathfrak{m}\oplus ., Comment: 29 pages

Recent progress in multilinear harmonic analysis naturally raises questions about the local behaviour of the best constant (or bound) in the general Brascamp--Lieb inequality as a function of the underlying linear transformations. In this paper we prove that this constant is continuous, but is not in general differentiable., Comment: 8 pages. Author accepted version. To appear in Bull. London Math. Soc

The unit sphere $\mathbb{S}$ in $\mathbb{C}^n$ is equipped with the tangential Cauchy-Riemann complex and the associated Laplacian $\Box_b$. We prove a H\"ormander spectral multiplier theorem for $\Box_b$ with critical index $n-1/2$, that is, half the topological dimension of $\mathbb{S}$. Our proof is mainly based on representation theory and on a detailed analysis of the spaces of differential forms on $\mathbb{S}$., Comment: 28 pages

We show that globally defined quasiconformal mappings of rigid Carnot groups are affine, but that globally defined contact mappings of rigid Carnot groups need not be quasiconformal, and a fortiori not affine.

If f is a conformal mapping defined on a connected open subset of a Carnot group G, then either f is the composition of a translation, a dilation and an isometry, or G is the nilpotent Iwasawa component of a real rank 1 simple Lie group S, and f arises from the action of S on G, viewed as an open subset of S/P, where P is a parabolic subgroup of G and NP is open and dense in S.

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group $G$. In particular, we show that the operators $T_\alpha: f \mapsto |.|^{-\alpha} L^{-\alpha/2} f$, where $|.|$ is a homogeneous norm, $0 < \alpha < Q/p$, and $L$ is the sub-Laplacian, are bounded on the Lebesgue space $L^p(G)$. As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg-Pauli-Weyl inequality, relating the $L^p$ norm of a function $f$ to the $L^q$ norm of $|.|^\beta f$ and the $L^r$ norm of $L^{\delta/2} f$., Comment: 19 pages

We show that order-invariant injective maps on the noncompactly causal symmetric space $SO_0 (1,n)/SO_0 (1,n-1)$ belong to $O(1,n)^+$.

We prove a number of results on the geometry associated to the solutions of evolution equations given by first-order differential operators on manifolds. In particular, we consider distance functions associated to a first-order operator, and discuss the associated geometry, which is sometimes surprisingly different to riemannian geometry., Comment: 45 pages

In this paper, we generalise Hardy's uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the diagonal if the spectrum is also localised., Comment: 10 pages