Spectral theory of SG pseudo-differential operators on $L^{p}(ℝⁿ)$

Aparajita Dasgupta; M. W. Wong

Displaying similar documents to “Spectral theory of SG pseudo-differential operators on $L^{p} (ℝ ⁿ)$ ”

Continuous pseudo-hairy spaces and continuous pseudo-fans

Janusz R. Prajs (2002)

Fundamenta Mathematicae

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A compact metric space X̃ is said to be a continuous pseudo-hairy space over a compact space X ⊂ X̃ provided there exists an open, monotone retraction $r : X ̃ \binom{o n t o}{⟶} X$ such that all fibers $r^{- 1} (x)$ are pseudo-arcs and any continuum in X̃ joining two different fibers of r intersects X. A continuum $Y_{X}$ is called a continuous pseudo-fan of a compactum X if there are a point $c \in Y_{X}$ and a family ℱ of pseudo-arcs such that $⋃ ℱ = Y_{X}$ , any subcontinuum of $Y_{X}$ intersecting two different elements of ℱ contains c, and ℱ is homeomorphic...

Pseudo-boundaries and pseudo-interiors for topological convexities

M. Van de Vel

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CONTENTS0. Introduction......... .............................................................51. Topological convexity structures.......................................62. Half-spaces and related results......................................123. Pseudo-boundaries........................................................254. The Krein-Milman theorem.............................................335. Pseudo-interiority...........................................................406. The existence...

Pseudodifferential operators on non-quasianalytic classes of Beurling type

C. Fernández, A. Galbis, D. Jornet (2005)

Studia Mathematica

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We introduce pseudodifferential operators (of infinite order) in the framework of non-quasianalytic classes of Beurling type. We prove that such an operator with (distributional) kernel in a given Beurling class $_{(ω)}^{'}$ is pseudo-local and can be locally decomposed, modulo a smoothing operator, as the composition of a pseudodifferential operator of finite order and an ultradifferential operator with constant coefficients in the sense of Komatsu, both operators with kernel in the same class...

The continuity of pseudo-differential operators on weighted local Hardy spaces

Ming-Yi Lee, Chin-Cheng Lin, Ying-Chieh Lin (2010)

Studia Mathematica

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We first show that a linear operator which is bounded on $L ²_{w}$ with w ∈ A₁ can be extended to a bounded operator on the weighted local Hardy space $h ¹_{w}$ if and only if this operator is uniformly bounded on all $h ¹_{w}$ -atoms. As an application, we show that every pseudo-differential operator of order zero has a bounded extension to $h ¹_{w}$ .

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...

Pseudo-spectrum for a class of semi-classical operators

Karel Pravda-Starov (2008)

Bulletin de la Société Mathématique de France

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We study in this paper a notion of pseudo-spectrum in the semi-classical setting called injectivity pseudo-spectrum. The injectivity pseudo-spectrum is a subset of points in the complex plane where there exist some quasi-modes with a precise rate of decay. For that reason, these values can be considered as some ‘almost eigenvalues’ in the semi-classical limit. We are interested here in studying the absence of injectivity pseudo-spectrum, which is characterized by a global a priori estimate....

Propagation of singularities for operators with multiple involutive characteristics

Johannes Sjöstrand (1976)

Annales de l'institut Fourier

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Let $P$ be a classical pseudodifferential operator of order $m$ on a paracompact $C^{\infty}$ manifold $X$ . Let $p_{m}$ be the principal symbol and assume that $Σ = p_{m}^{- 1} (0)$ is an involutive $C^{\infty}$ sub-manifold of $T^{*} X ∖ 0$ , satisfying a certain transversality condition. We assume that $p_{m}$ vanishes exactly to order $M$ on $Σ$ and that the derivatives of order $M$ satisfy a certain condition, inspired from the Calderòn uniqueness theorem (usually empty when $M = 2$ ). Suppose that a Levi condition is valid for the lower order symbols. If $u \in 𝒟^{'} (X)$ , $P u \in C^{\infty} (X)$ , then...

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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Pseudo-homotopies of the pseudo-arc

Alejandro Illanes (2012)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a continuum. Two maps $g, h : X \to X$ are said to be pseudo-homotopic provided that there exist a continuum $C$ , points $s, t \in C$ and a continuous function $H : X \times C \to X$ such that for each $x \in X$ , $H (x, s) = g (x)$ and $H (x, t) = h (x)$ . In this paper we prove that if $P$ is the pseudo-arc, $g$ is one-to-one and $h$ is pseudo-homotopic to $g$ , then $g = h$ . This theorem generalizes previous results by W. Lewis and M. Sobolewski.

Multiple summing operators on $l_{p}$ spaces

Dumitru Popa (2014)

Studia Mathematica

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We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of $l_{p}$ spaces. This characterization is used to show that multiple s-summing operators on a product of $l_{p}$ spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators $T : l_{4 / 3} \times l_{4 / 3} \to l ₂$ such that none of the associated linear operators...

Absolutely continuous linear operators on Köthe-Bochner spaces

(2011)

Banach Center Publications

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Let E be a Banach function space over a finite and atomless measure space (Ω,Σ,μ) and let $(X, | | \cdot | |_{X})$ and $(Y, | | \cdot | |_{Y})$ be real Banach spaces. A linear operator T acting from the Köthe-Bochner space E(X) to Y is said to be absolutely continuous if $| | T (1_{A ₙ} f) {| |}_{Y} \to 0$ whenever μ(Aₙ) → 0, (Aₙ) ⊂ Σ. In this paper we examine absolutely continuous operators from E(X) to Y. Moreover, we establish relationships between different classes of linear operators from E(X) to Y.

Besov spaces and 2-summing operators

M. A. Fugarolas (2004)

Colloquium Mathematicae

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Let Π₂ be the operator ideal of all absolutely 2-summing operators and let $I_{m}$ be the identity map of the m-dimensional linear space. We first establish upper estimates for some mixing norms of $I_{m}$ . Employing these estimates, we study the embedding operators between Besov function spaces as mixing operators. The result obtained is applied to give sufficient conditions under which certain kinds of integral operators, acting on a Besov function space, belong to Π₂; in this context, we also...

Multivalued pseudo-contractive mappings defined on unbounded sets in Banach spaces

Claudio H. Morales (1992)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a real Banach space. A multivalued operator $T$ from $K$ into $2^{X}$ is said to be pseudo-contractive if for every $x, y$ in $K$ , $u \in T (x)$ , $v \in T (y)$ and all $r > 0$ , $∥ x - y ∥ \leq ∥ (1 + r) (x - y) - r (u - v) ∥$ . Denote by $G (z, w)$ the set ${u \in K : ∥ u - w ∥ \leq ∥ u - z ∥}$ . Suppose every bounded closed and convex subset of $X$ has the fixed point property with respect to nonexpansive selfmappings. Now if $T$ is a Lipschitzian and pseudo-contractive mapping from $K$ into the family of closed and bounded subsets of $K$ so that the set $G (z, w)$ is bounded for some $z \in K$ and some $w \in T (z)$ , then $T$ has a fixed point in $K$ . ...

Semi-elliptic operators generated by vector fields

E. Shargorodsky

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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...

Modulation space estimates for multilinear pseudodifferential operators

Árpád Bényi, Kasso A. Okoudjou (2006)

Studia Mathematica

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We prove that for symbols in the modulation spaces $ℳ^{p, q}$ , p ≥ q, the associated multilinear pseudodifferential operators are bounded on products of appropriate modulation spaces. In particular, the symbols we study here are defined without any reference to smoothness, but rather in terms of their time-frequency behavior.

On rough maximal operators and Marcinkiewicz integrals along submanifolds

H. M. Al-Qassem, Y. Pan (2009)

Studia Mathematica

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We investigate the $L^{p}$ boundedness for a class of parametric Marcinkiewicz integral operators associated to submanifolds and a class of related maximal operators under the $L {(l o g L)}^{α} (^{n - 1})$ condition on the kernel functions. Our results improve and extend some known results.

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

Peer C. Kunstmann, Hendrik Vogt (2007)

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...

Displaying similar documents to “Spectral theory of SG pseudo-differential operators on L p ( ℝ ⁿ ) ”

Displaying similar documents to “Spectral theory of SG pseudo-differential operators on $L^{p} (ℝ ⁿ)$ ”