We present a characterization of hyperbolicity for strongly continuous semigroups on Banach spaces in terms of Fourier multiplier properties of the resolvent of the generator. Hyperbolicity with respect to classical solutions is also considered. Our approach unifies and simplifies the M. Kaashoek-- S. Verduyn Lunel theory and multiplier-type results previously obtained by S. Clark, M. Hieber, S. Montgomery-Smith, F. Räbiger, T. Randolph, and L. Weis.
We present a spectral mapping theorem for continuous semigroups of operators on any Banach space $E$. The condition for the hyperbolicity of a semigroup on $E$ is given in terms of the generator of an evolutionary semigroup acting in the space of $E$-valued functions. The evolutionary semigroup generated by the propagator of a nonautonomous differential equation in $E$ is also studied. A ``discrete'' technique for the investigating of the evolutionary semigroup is developed and applied to describe the hyperbolicity (exponential dichotomy) of the nonautonomuos equation. File Length: 68K
An operator $M$ acting on the space of real analytic functions is called a multiplier if every monomial is its eigenvector. In this paper we state some results concerning the problem of generating strongly continuous semigroups by multipliers. In particular we show when the Euler differential operator of finite order is a generator and when it is not.
Let L be a non-negative, self-adjoint operator on L^2(Ω), where (Ω, d μ) is a space of homogeneous type. Assume that the semigroup {T_t}_{t>0} generated by -L satisfies Gaussian bounds, or more generally Davies-Gaffney estimates. We say that f belongs to the Hardy space H^1_L if the square function S_h f(x)=(\iint_{Γ(x)} |t^2 L e^{-t^2 L} f(y)|^2 \frac{dμ(y)}{μ(B_d(x,t))} \frac{dt}{t})^{1/2} belongs to L^1(Ω, dμ), where Γ(x)={(y,t) \in Ω\times (0,\infty): d(x,y)<t}. We prove spectral multiplier theorems for L on H^1_L.
We show that the analyticity of semigroups $(T_t)_{t \geq 0}$ of (not necessarily positive) selfadjoint contractive Fourier multipliers on $\mathrm{L}^p$-spaces of any abelian locally compact group is preserved by the tensorisation of the identity operator $\mathrm{Id}_X$ of a Banach space $X$ for a large class of K-convex Banach spaces, answering partially a conjecture of Pisier. The result is even new for semigroups of Fourier multipliers acting on $\mathrm{L}^p(\mathbb{R}^n)$. The proof relies on the use of noncommutative Banach spaces and we give a more general result for semigroups of Fou…
We study polynomial and exponential stability for $C_{0}$-semigroups using the recently developed theory of operator-valued $(L^{p},L^{q})$ Fourier multipliers. We characterize polynomial decay of orbits of a $C_{0}$-semigroup in terms of the $(L^{p},L^{q})$ Fourier multiplier properties of its resolvent. Using this characterization we derive new polynomial decay rates which depend on the geometry of the underlying space. We do not assume that the semigroup is uniformly bounded, our results depend only on spectral properties of the generator. As a corollary of our work on polynomial stability …
We show that the analyticity of semigroups $(T_t)_{t \geq 0}$ of selfadjoint contractive Fourier multipliers on $L^p$-spaces of compact abelian groups is preserved by the tensorisation of the identity operator of a Banach space for a large class of K-convex Banach spaces, answering partially a conjecture of Pisier. We also give versions of this result for some semigroups of Schur multipliers and Fourier multipliers on noncommutative $L^p$-spaces. Finally, we give a precise description of semigroups of Schur multipliers to which the result of this paper can be applied.
In order to investigate the relationship between weak amenability and the Haagerup property for groups, we introduce the weak Haagerup property, and we prove that having this approximation property is equivalent to the existence of a semigroup of Herz-Schur multipliers generated by a proper function. It is then shown that a (not necessarily proper) generator of a semigroup of Herz-Schur multipliers splits into a positive definite kernel and a conditionally negative definite kernel. We also show that the generator has a particularly pleasant form if and only if the group is amenable. In the sec…
The present paper, is devoted to investigation of operator--valued Fourier multiplier theorems from $B_{q_{1},r}^{s}$ to $B_{q_{2},r}^{s}$, optimal embedding of Besov spaces, the separability and positivity of differential operators. Here, we show that these differential operators generate analytic semigroup.
We study BMO spaces associated with semigroup of operators and apply the results to boundedness of Fourier multipliers. We prove a universal interpolation theorem for BMO spaces and prove the boundedness of a class of Fourier multipliers on noncommutative Lp spaces for all 1 < p < \infty, with optimal constants in p.
In this paper we obtain some noncommutative multiplier theorems and maximal inequalities on semigroups. As applications, we obtain the corresponding individual ergodic theorems. Our main results extend some classical results of Stein and Cowling on one hand, and simplify the main arguments of Junge-Le Merdy-Xu's related work [15].
In this work we investigate semigroups of operators acting on noncommutative $L^p$-spaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and $H^\infty$ functional calculus. We discuss several examples of noncommutative diffusion semigroups. This includes Schur multipliers, $q$-Ornstein-Uhlenbeck semigroups, and the noncommutative Poisson semigroup on free groups.
In this work, we solve the problem explicitly stated at the end of a paper of Junge, Mei and Parcet [JEMS2018, Problem C.5] for a large class of groups including all amenable groups and free groups. More precisely, we prove that the Hodge-Dirac operator of the canonical "hidden" noncommutative geometry associated with a Markov semigroup $(T_t)_{t \geq 0}$ of Fourier multipliers is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus on a bisector which implies a positive answer to the quoted problem. Our result can be seen as a strengthening of the dimension free estimates …
Let $X$ be a Riemannian symmetric space of noncompact type and $T$ be a linear translation-invariant operator which is bounded on $L^p(X)$. We shall show that if $T$ is not a constant multiple of identity then there exist complex constants $z$ such that $zT$ is chaotic on $L^p(X)$ when $p$ is in the sharp range $2<p<\infty$. This vastly generalizes the result that dynamics of the (perturbed) heat semigroup is chaotic on $X$ proved in [15, 17].
We define Hölder classes $Λ_α$ associated with a Markovian semigroup and prove that, when the semigroup satisfies the $Γ^2 \geq 0$ condition, the Riesz transforms are bounded between the Hölder classes. As a consequence, this bound holds in manifolds with nonnegative Ricci curvature. We also show, without the need for extra assumptions on the semigroup, that a version of the Morrey inequalities is equivalent to the ultracontractivity property. This result extends the semigroup approach to the Sobolev inequalities laid by Varopoulos. After that, we study certain families of operators between th…
Let $X$ be a space of homogeneous type and let $L$ be an injective, non-negative, self-adjoint operator on $L^2(X)$ such that the semigroup generated by $-L$ fulfills Davies-Gaffney estimates of arbitrary order. We prove that the operator $F(L)$, initially defined on $H^1_L(X)\cap L^2(X)$, acts as a bounded linear operator on the Hardy space $H^1_L(X)$ associated with $L$ whenever $F$ is a bounded, sufficiently smooth function. Based on this result, together with interpolation, we establish Hörmander type spectral multiplier theorems on Lebesgue spaces for non-negative, self-adjoint operators …
We prove the strong continuity of spectral multiplier operators associated with dilations of certain functions on the general Hardy space $H^1_L$ introduced by Hofmann, Lu, Mitrea, Mitrea, Yan. Our results include the heat and Poisson semigroups as well as the group of imaginary powers.
In this paper we study the behavior of some harmonic analysis operators associated with the discrete Laplacian $Δ_d$ in discrete Hardy spaces $\mathcal H^p(\mathbb Z)$. We prove that the maximal operator and the Littlewood-Paley $g$ function defined by the semigroup generated by $Δ_d$ are bounded from $\mathcal H^p(\mathbb Z)$ into $\ell^p(\mathbb Z)$, $0<p\leq 1$. Also, we establish that every $Δ_d$-spectral multiplier of Laplace transform type is a bounded operator from $\mathcal H^p(\mathbb Z)$ into itself, for every $0<p\leq 1$.
We investigate Fourier multipliers on the compact dual of arbitrary discrete groups. Our main result is a Hörmander-Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood-Paley type inequalities in group von Neumann algebras, prove $L_p$ estimates for noncommutative Riesz transforms and characterize $L_\infty \to \mathrm{BMO}$ boundedness for radial Fourier multipliers. The key novelties of our approach are to exploit group cocycles and cross products in Fourier multiplier theory in conjunction with BMO spaces associated to semigrou…
Let $L$ be a non-negative self adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ have Gaussian upper bounds but possess no regularity in variables $x$ and $y$. In this article, we study weighted $L^p$-norm inequalities for spectral multipliers of $L$. We show sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate $L^2$ estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers…
The operator-valued multiplier theorems in weighted abstract Besov spaces are studied. These results permit us to show embedding theorems in weighted Besov-Lions type spaces. The most regular class of interpolation space is found such that the mixed differential operator is bounded from Besov-Lions space to this and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By using these results the separability properties of degenerate differential operators are studied. Especially, we prove that the associated differential operators are positive and also are generators of analytic se…
We investigate Hörmander spectral multiplier theorems as they hold on $X = L^p(Ω),\: 1 < p < \infty,$ for many self-adjoint elliptic differential operators $A$ including the standard Laplacian on $\R^d.$ A strengthened matricial extension is considered, which coincides with a completely bounded map between operator spaces in the case that $X$ is a Hilbert space. We show that the validity of the matricial Hörmander theorem can be characterized in terms of square function estimates for imaginary powers $A^{it}$, for resolvents $R(λ,A),$ and for the analytic semigroup $\exp(-zA).$ We deduce Hörma…
Let A be a commutative Banach algebra such that uA = {0} for u $\in$ A \ {0} which possesses dense principal ideals. The purpose of the paper is to give a general framework to define F (--$λ$1$Δ$T 1 ,. .. , --$λ$ k $Δ$T k) where F belongs to a natural class of holomorphic functions defined on suitable open subsets of C k containing the "Arveson spectrum" of (--$λ$1$Δ$T 1 ,. .. , --$λ$ k $Δ$T k), where $Δ$T 1 ,. .. , $Δ$T k are the infinitesimal generators of commuting one-parameter semigroups of multipliers on A belonging to one of the following classes (1) The class of strongly continous semi…
We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new Hörmander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties which shape our approach are three. First, we control a broad class of Fourier multipliers by certain maximal operators in noncommutative $L_p$ spaces. This general principle ---exploited in Euclidean harmonic analysis during the last 40 years--- is of independent interest and might admit further applications. Second, we replace the formerly used cocycle dimensio…
We obtain dimension free estimates for noncommutative Riesz transforms associated to conditionally negative length functions on group von Neumann algebras. This includes Poisson semigroups, beyond Bakry's results in the commutative setting. Our proof is inspired by Pisier's method and a new Khintchine inequality for crossed products. New estimates include Riesz transforms associated to fractional laplacians in $\mathbb{R}^n$ (where Meyer's conjecture fails) or to the word length of free groups. Lust-Piquard's work for discrete laplacians on LCA groups is also generalized in several ways. In th…
We prove spectral multiplier theorems for Hörmander classes $\mathcal{H}^α\_p$ for 0-sectorial operators A on Banach spaces assuming a bounded $H^\infty(Σ\_σ)$ calculus for some $σ\in (0,π)$ and norm and certain R-bounds on one of the following families of operators: the semigroup $e^{--zA}$ on $\mathbb{C}\_+$, the wave operators $e^{isA}$ for $s \in \mathbb{R}$, the resolvent $(λ-- A)^{-1}$ on $\mathbb{C} \backslash \mathbb{R}$, the imaginary powers $A^{it}$ for $t \in \mathbb{R}$ or the Bochner-Riesz means $(1-A/u)^α\_+$ for $u > 0.$ In contrast to the existing literature we neither assume t…
We start by identifying a class of pseudo-differential operators, generated by the set of continuous negative definite functions, that are in the weak similarity (WS) orbit of the self-adjoint log-Bessel operator on the Euclidean space. These WS relations turn out to be useful to first characterize a core for each operator in this class, which enables us to show that they generate a class, denoted by $\mathscr{P}$, of non-self-adjoint $\mathcal{C}_0$-contraction positive semigroups. Up to a homeomorphism, $\mathscr{P}$ includes, as fundamental objects in probability theory, the family of self-…
Let L be a generator of a semigroup satisfying the Gaussian upper bounds. In this paper, we study further a new BMO_L space associated with L which was introduced recently by Duong and Yan. We discuss applications of the new BMO_L spaces in the theory of singular integration such as BMO_L estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space BMO_L might coincide with or might be essentially different from the classical BMO space.
We consider here one-parameter semigroups ${\bf T}=(T(t))_{t>0}$ of bounded operators on a Banach space $X$ which are weakly continuous in the sense of Arveson. For such a semigroup ${\bf T}$ denote by ${\mathcal M}_{ω_{\bf T}}$ the convolution algebra consisting in those measures $μ$ on $(0,+\infty)$ such that $\int_0^{+\infty}\Vert T(t)\Vert d\vert μ\vert (t)<+\infty.$ The Pettis integral $\int_0^{+\infty}T(t)dμ(t)$ defines for $μ\in {\mathcal M}_{ω_{\bf T}}$ a bounded operator $ϕ_{\bf T}(μ)$ on $X.$ Identifying the space $L^1_{ω_{\bf T}}$ of (classes of) measurable functions $f$ satisfying …
The aim of the article is to show a H{ö}rmander spectral multiplier theorem for an operator $A$ whose kernel of the semigroup $\exp(-zA)$ satisfies certain Poisson estimates for complex times $z.$ Here $\exp(-zA)$ acts on $L^p(Ω),\,1 < p < \infty,$ where $Ω$ is a space of homogeneous type with the additional condition that the measure of annuli is controlled. In most of the known H{ö}rmander type theorems in the literature, Gaussian bounds and self-adjointness for the semigroup are needed, whereas here the new feature is that the assumptions are the to some extend weaker Poisson bounds, and $\…
Consider the Schrödinger operator $\mathcal{L}=-Δ+V$ in $\mathbb{R}^n, n\ge 3,$ where $V$ is a nonnegative potential satisfying a reverse Hölder condition of the type \begin{equation*} \left( \frac{1}{|B|}\int_B V(y)^qdy\right)^{1/q}\le \frac{C}{|B|}\int_B V(y)dy, \, \text{ for some }q>n/2. \end{equation*} We define $Λ^α_{\mathcal{L}},\, 0<α<2,$ the class of measurable functions such that $$ \|ρ(\cdot)^{-α}f(\cdot)\|_\infty<\infty \quad \, \, \text{and}\:\: \quad \sup_{|z|>0}\frac{\|f(\cdot+z)+f(\cdot-z)-2f(\cdot)\|_\infty}{|z|^α}<\infty, $$ where $ρ$ is the critical radius function associated…
Local Hardy Spaces Associated with Ball Quasi-Banach Function Spaces and Non-negative Self-adjoint Operators on Spaces of Homogeneous Type and Their Applications
