Theory of Bessel potentials. III : potentials on regular manifolds

Robert Adams; Nachman Aronszajn; M. S. Hanna

Displaying similar documents to “Theory of Bessel potentials. III : potentials on regular manifolds”

Rate of convergence in singular perturbations

Wilfred M. Greenlee (1968)

Annales de l'institut Fourier

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Soit $D \subset R^{n}$ un domaine et $ϵ$ un paramètre réel positif. Considérons les deux problèmes aux limites sur $D$ , $(ϵ 𝒰 + ℬ w_{ϵ} = f$ et $ℬ u = f$ , où $𝒰$ et $ℬ$ sont des opérateurs différentiels elliptiques et où le degré de $𝒰$ est supérieur au degré de $ℬ$ . En utilisant l’interpolation quadratique entre espaces de Hilbert, on étudie les problèmes suivants : 1) Déterminer les normes pour lesquelles $w_{ϵ}$ converge vers $u$ ; 2) Estimer la rapidité de convergence de $w_{ϵ}$ vers $u$ , pour ces normes. ...

Theory of Bessel potentials. II

Robert Adams, Nachman Aronszajn, K. T. Smith (1967)

Annales de l'institut Fourier

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Dans cette partie de la théorie des potentiels besseliens on considère les restrictions de potentiels de la classe $P^{a} (R^{n})$ aux domaines ouverts $D \subset R^{n}$ . On cherche à caractériser de manière intrinsèque la classe $P^{a} (D)$ ainsi obtenue. On attaque ce problème en définissant de manière directe (§ 2) une classe ${\overset{ˇ}{P}}^{a} (D) \subset P^{a} (D)$ qui, pour des domaines assez réguliers, est égale à $P^{a} (D)$ . L’égalité $P^{a} (D) = P^{a} (D)$ est équivalente à l’existence d’un opérateur-extension $E : {\overset{ˇ}{P}}^{a} (D) \to P^{a} (R^{n})$ , linéaire et continu, tel que $E u$ soit une extension...

Metrics in the set of partial isometries with finite rank

Esteban Andruchow, Gustavo Corach (2005)

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

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Let $I_{(\infty)}$ be the set of partial isometries with finite rank of an infinite dimensional Hilbert space $H$ . We show that $I_{(\infty)}$ is a smooth submanifold of the Hilbert space $B_{2} (H)$ of Hilbert-Schmidt operators of $H$ and that each connected component is the set $I_{N}$ , which consists of all partial isometries of rank $N < \infty$ . Furthermore, $I_{(\infty)}$ is a homogeneous space of $U_{(\infty)} \times U_{(\infty)}$ , where $U_{(\infty)}$ is the classical Banach-Lie group of unitary operators of $H$ , which are Hilbert-Schmidt perturbations of the identity. We introduce two Riemannian...

Conjugate norms in ℂⁿ and related geometrical problems

Baran Mirosław

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AbstractWe consider ℂⁿ as a normed space equipped with a complex norm F and we investigate some geometrical problems related with the notion of a conjugate norm F*. A crucial role in our considerations is played by the classical Shmul'yan theorem on exposed points in dual spaces. Many applications of this theorem are given for different problems including characterization of linear (biholomorphic) equivalence for a class of balls in ℂⁿ, calculation of the group of linear automorphisms...

A commutant lifting theorem on analytic polyhedra

Calin Ambrozie, Jörg Eschmeier (2005)

Banach Center Publications

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In this note a commutant lifting theorem for vector-valued functional Hilbert spaces over generalized analytic polyhedra in ℂⁿ is proved. Let T be the compression of the multiplication tuple $M_{z}$ to a *-invariant closed subspace of the underlying functional Hilbert space. Our main result characterizes those operators in the commutant of T which possess a lifting to a multiplier with Schur class symbol. As an application we obtain interpolation results of Nevanlinna-Pick and Carathéodory-Fejér...

A counterexample to the Γ-interpolation conjecture

Adama S. Kamara (2015)

Annales Polonici Mathematici

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Agler, Lykova and Young introduced a sequence $C_{ν}$ , where ν ≥ 0, of necessary conditions for the solvability of the finite interpolation problem for analytic functions from the open unit disc into the symmetrized bidisc Γ. They conjectured that condition $C_{n - 2}$ is necessary and sufficient for the solvability of an n-point interpolation problem. The aim of this article is to give a counterexample to that conjecture.

Geometry of manifolds which admit conservation laws

David E. Blair, Alexander P. Stone (1971)

Annales de l'institut Fourier

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Let $M$ be an $(n + 1)$ -dimensional Riemannian manifold admitting a covariant constant endomorphism $h$ of the localized module of 1-forms with distinct non-zero eigenvalues. After it is shown that $M$ is locally flat, a manifold $N$ immersed in $M$ is studied. The manifold $N$ has an induced structure with $n$ of the same eigenvalues if and only if the normal to $N$ is a fixed direction of $h$ . Finally conditions under which $N$ is invariant under $h$ , $N$ is totally geodesic and the induced structure has vanishing...

A functional calculus description of real interpolation spaces for sectorial operators

Markus Haase (2005)

Studia Mathematica

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For a holomorphic function ψ defined on a sector we give a condition implying the identity ${(X, (A^{α}))}_{θ, p} = x \in X | t^{- θ R e α} ψ (t A) \in L ⁎^{p} ((0, \infty); X)$ where A is a sectorial operator on a Banach space X. This yields all common descriptions of the real interpolation spaces for sectorial operators and allows easy proofs of the moment inequalities and reiteration results for fractional powers.

Measure of weak noncompactness under complex interpolation

Andrzej Kryczka, Stanisław Prus (2001)

Studia Mathematica

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Logarithmic convexity of a measure of weak noncompactness for bounded linear operators under Calderón’s complex interpolation is proved. This is a quantitative version for weakly noncompact operators of the following: if T: A₀ → B₀ or T: A₁ → B₁ is weakly compact, then so is $T : A_{[θ]} \to B_{[θ]}$ for all 0 < θ < 1, where $A_{[θ]}$ and $B_{[θ]}$ are interpolation spaces with respect to the pairs (A₀,A₁) and (B₀,B₁). Some formulae for this measure and relations to other quantities measuring weak noncompactness are...

$H^{\infty}$ functional calculus in real interpolation spaces, II

Giovanni Dore (2001)

Studia Mathematica

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Let A be a linear closed one-to-one operator in a complex Banach space X, having dense domain and dense range. If A is of type ω (i.e.the spectrum of A is contained in a sector of angle 2ω, symmetric about the real positive axis, and ${| | λ (λ I - A)}^{- 1} | |$ is bounded outside every larger sector), then A has a bounded $H^{\infty}$ functional calculus in the real interpolation spaces between X and the intersection of the domain and the range of the operator itself.

On complex interpolation and spectral continuity

Karen Saxe (1998)

Studia Mathematica

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Let ${[X_{0}, X_{1}]}_{t}$ , 0 ≤ t ≤ 1, be Banach spaces obtained via complex interpolation. With suitable hypotheses, linear operators T that act boundedly on both $X_{0}$ and $X_{1}$ will act boundedly on each ${[X_{0}, X_{1}]}_{t}$ . Let $T_{t}$ denote such an operator when considered on ${[X_{0}, X_{1}]}_{t}$ , and $σ (T_{t})$ denote its spectrum. We are motivated by the question of whether or not the map $t \to σ (T_{t})$ is continuous on (0,1); this question remains open. In this paper, we study continuity of two related maps: $t \to {(σ (T_{t}))}^{\land}$ (polynomially convex hull) and $t \to \partial_{e} (σ (T_{t}))$ (boundary of the polynomially...