A Hörmander-type spectral multiplier theorem for operators without heat kernel

Sönke Blunck

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Displaying similar documents to “A Hörmander-type spectral multiplier theorem for operators without heat kernel”

Weighted norm estimates and $L_{p}$ -spectral independence of linear operators

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Colloquium Mathematicae

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We investigate the $L_{p}$ -spectrum of linear operators defined consistently on $L_{p} (Ω)$ for p₀ ≤ p ≤ p₁, where (Ω,μ) is an arbitrary σ-finite measure space and 1 ≤ p₀ < p₁ ≤ ∞. We prove p-independence of the $L_{p}$ -spectrum assuming weighted norm estimates. The assumptions are formulated in terms of a measurable semi-metric d on (Ω,μ); the balls with respect to this semi-metric are required to satisfy a subexponential volume growth condition. We show how previous results on $L_{p}$ -spectral independence...

The unit group of some fields of the form $ℚ (\sqrt{2}, \sqrt{p}, \sqrt{q}, \sqrt{- l})$

Moha Ben Taleb El Hamam (2024)

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Let $p$ and $q$ be two different prime integers such that $p \equiv q \equiv 3 (mod 8)$ with $(p / q) = 1$ , and $l$ a positive odd square-free integer relatively prime to $p$ and $q$ . In this paper we investigate the unit groups of number fields $𝕃 = ℚ (\sqrt{2}, \sqrt{p}, \sqrt{q}, \sqrt{- l})$ .

Semi-elliptic operators generated by vector fields

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Let M be a $C^{\infty}$ -smooth n-dimensional manifold and ν₁, ..., νₙ be $C^{\infty}$ -smooth vector fields on M which span the tangent space $T_{x} M$ at each point x ∈ M. The vector fields ν₁, ..., νₙ may have nonzero commutators. We construct a calculus of pseudodifferential operators (ψDOs) which act on sections of vector bundles over M and have symbols belonging to anisotropic analogues of the Hörmander classes $S_{ϱ, δ}^{r}$ , and apply it to semi-elliptic operators generated by ν₁,...,νₙ. The results obtained include the...

Bicyclotomic polynomials and impossible intersections

David Masser, Umberto Zannier (2013)

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In a recent paper we proved that there are at most finitely many complex numbers $t \neq 0, 1$ such that the points $(2, \sqrt{2 (2 - t)})$ and $(3, \sqrt{6 (3 - t)})$ are both torsion on the Legendre elliptic curve defined by $y^{2} = x (x - 1) (x - t)$ . In a sequel we gave a generalization to any two points with coordinates algebraic over the field $Q (t)$ and even over $C (t)$ . Here we reconsider the special case $(u, \sqrt{u (u - 1) (u - t)})$ and $(v, \sqrt{v (v - 1) (v - t)})$ with complex numbers $u$ and $v$ .

Regularity of domains of parameterized families of closed linear operators

Teresa Winiarska, Tadeusz Winiarski (2003)

Annales Polonici Mathematici

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The purpose of this paper is to provide a method of reduction of some problems concerning families $A_{t} = {(A (t))}_{t \in}$ of linear operators with domains ${(_{t})}_{t \in}$ to a problem in which all the operators have the same domain . To do it we propose to construct a family ${(Ψ_{t})}_{t \in}$ of automorphisms of a given Banach space X having two properties: (i) the mapping $t \mapsto Ψ_{t}$ is sufficiently regular and (ii) $Ψ_{t} () =_{t}$ for t ∈ . Three effective constructions are presented: for elliptic operators of second order with the Robin boundary condition...

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Oleg Lihvoinen (2023)

Commentationes Mathematicae Universitatis Carolinae

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The strong closure of feasible states of families of operators is studied. The results are obtained for self-adjoint operators in reflexive Banach spaces and for more concrete case - families of elliptic systems encountered in the optimal layout of $r$ materials. The results show when it is possible to parametrize the strong closure by the same type of operators. The results for systems of elliptic operators for the case when number of unknown functions $m$ is less than the dimension $n$ of...

Isomorphic properties in spaces of compact operators

Ioana Ghenciu (2023)

Commentationes Mathematicae Universitatis Carolinae

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We introduce the definition of $p$ -limited completely continuous operators, $1 \leq p < \infty$ . The question of whether a space of operators has the property that every $p$ -limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$ -limited completely continuous evaluation operators.

Second order elliptic operators with complex bounded measurable coefficients in $L^{p}$ , Sobolev and Hardy spaces

Steve Hofmann, Svitlana Mayboroda, Alan McIntosh (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$ , such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^{p}$ , Sobolev, and some new Hardy spaces naturally associated to $L$ . First, we show...

The Embeddability of c₀ in Spaces of Operators

Ioana Ghenciu, Paul Lewis (2008)

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Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w *} (X *, Y)$ , as well as the complementation of the space $K_{w *} (X *, Y)$ of w*-w compact operators in the space $L_{w *} (X *, Y)$ of w*-w operators from X* to Y.

On a proof of the boundedness and nuclearity of pseudodifferential operators in $R^{n}$

Jan A. Rempała (1990)

Annales Polonici Mathematici

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Bilinear operators associated with Schrödinger operators

Chin-Cheng Lin, Ying-Chieh Lin, Heping Liu, Yu Liu (2011)

Studia Mathematica

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Let L = -Δ + V be a Schrödinger operator in $ℝ^{d}$ and $H ¹_{L} (ℝ^{d})$ be the Hardy type space associated to L. We investigate the bilinear operators T⁺ and T¯ defined by $T^{\pm} (f, g) (x) = (T ₁ f) (x) (T ₂ g) (x) \pm (T ₂ f) (x) (T ₁ g) (x)$ , where T₁ and T₂ are Calderón-Zygmund operators related to L. Under some general conditions, we prove that either T⁺ or T¯ is bounded from $L^{p} (ℝ^{d}) \times L^{q} (ℝ^{d})$ to $H ¹_{L} (ℝ^{d})$ for 1 < p,q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails. ...

On analytic rapidly decreasing functions of a real variable

Gianfranco Cimmino (1983)

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

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Condizione necessaria e sufficiente affinché una funzione rapidamente decrescente di variabile reale sia uniformemente analitica è che per i suoi coefficienti $\gamma_{0},\gamma_{1},\dots$ di Fourier-Hermite riesca $\gamma_{m}=0(e^{\sqrt{mt}})$ per $t>0$ abbastanza piccolo.

Interpolation by elementary operators

Bojan Magajna (1993)

Studia Mathematica

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Given two n-tuples $a = (a_{1}, . . ., a_{n})$ and $b = (b_{1}, . . ., b_{n})$ of bounded linear operators on a Hilbert space the question of when there exists an elementary operator E such that $E a_{j} = b_{j}$ for all j =1,...,n, is studied. The analogous question for left multiplications (instead of elementary operators) is answered in any C*-algebra A, as a consequence of the characterization of closed left A-submodules in $A^{n}$ .

On hyponormal operators in Krein spaces

Kevin Esmeral, Osmin Ferrer, Jorge Jalk, Boris Lora Castro (2019)

Archivum Mathematicum

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In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators $T$ for which there exists a fundamental decomposition $𝕂 = 𝕂^{+} \oplus 𝕂^{-}$ of the Krein space $𝕂$ with $𝕂^{+}$ and $𝕂^{-}$ invariant under $T$ .

Sharp spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates

Peng Chen (2013)

Colloquium Mathematicae

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We consider an abstract non-negative self-adjoint operator L acting on L²(X) which satisfies Davies-Gaffney estimates. Let $H_{L}^{p} (X)$ (p > 0) be the Hardy spaces associated to the operator L. We assume that the doubling condition holds for the metric measure space X. We show that a sharp Hörmander-type spectral multiplier theorem on $H_{L}^{p} (X)$ follows from restriction-type estimates and Davies-Gaffney estimates. We also establish a sharp result for the boundedness of Bochner-Riesz means on $H_{L}^{p} (X)$ . ...

Partially elliptic differential equations having distributions as their right members

H. Marcinkowska

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ContentsIntroduction.............................................................................................................................31. Definitions, notations and some auxiliary lemmas...................................................42. The definition of the spaces $H_{p, q; Y} (Ω, ℬ)$ ..........................................................73. Some properties of the spaces $H_{p, q; Y} (Ω, ℬ)$ ...................................................104. Some examples of the spaces $H_{p, q; Y} (Ω, ℬ)$ ....................................................155....

On a binary relation between normal operators

Takateru Okayasu, Jan Stochel, Yasunori Ueda (2011)

Studia Mathematica

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The main goal of this paper is to clarify the antisymmetric nature of a binary relation ≪ which is defined for normal operators A and B by: A ≪ B if there exists an operator T such that $E_{A} (Δ) \leq T * E_{B} (Δ) T$ for all Borel subset Δ of the complex plane ℂ, where $E_{A}$ and $E_{B}$ are spectral measures of A and B, respectively (the operators A and B are allowed to act in different complex Hilbert spaces). It is proved that if A ≪ B and B ≪ A, then A and B are unitarily equivalent, which shows that the relation ≪ is...

An $M_{q} ()$ -functional calculus for power-bounded operators on certain UMD spaces

Earl Berkson, T. A. Gillespie (2005)

Studia Mathematica

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For 1 ≤ q < ∞, let $_{q} ()$ denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes $_{q} ()$ as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction...