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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

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 that the...

Section spaces of real analytic vector bundles and a theorem of Grothendieck and Poly

Dietmar Vogt (2010)

Banach Center Publications

The structure of the section space of a real analytic vector bundle on a real analytic manifold X is studied. This is used to improve a result of Grothendieck and Poly on the zero spaces of elliptic operators and to extend a result of Domański and the author on the non-existence of bases to the present case.

Seeking a network characterization of Corson compacta

Ziqin Feng (2021)

Commentationes Mathematicae Universitatis Carolinae

We say that a collection 𝒜 of subsets of X has property ( C C ) if there is a set D and point-countable collections 𝒞 of closed subsets of X such that for any A ∈ 𝒜 there is a finite subcollection ℱ of 𝒞 such that A = D ∖ ⋃ ℱ . Then we prove that any compact space is Corson if and only if it has a point- σ - ( C C ) base. A characterization of Corson compacta in terms of (strong) point network is also given. This provides an answer to an open question in “A Biased View of Topology as a Tool in Functional Analysis” (2014) by...

Segal algebras, approximate identities and norm irregularity in C₀(X,A)

Jussi Mattas (2013)

Studia Mathematica

We study three closely related concepts in the context of the Banach algebra C₀(X,A). We show that, to a certain extent, Segal extensions, norm irregularity and the existence of approximate identities in C₀(X,A) can be deduced from the corresponding features of A and vice versa. Extensive use is made of the multiplier norm and the tensor product representation of C₀(X,A).

Selecting basic sequences in φ-stable Banach spaces

Tadeusz Figiel, Ryszard Frankiewicz, Ryszard A. Komorowski, Czesław Ryll-Nardzewski (2003)

Studia Mathematica

In this paper we make use of a new concept of φ-stability for Banach spaces, where φ is a function. If a Banach space X and the function φ satisfy some natural conditions, then X is saturated with subspaces that are φ-stable (cf. Lemma 2.1 and Corollary 7.8). In a φ-stable Banach space one can easily construct basic sequences which have a property P(φ) defined in terms of φ (cf. Theorem 4.5). This leads us, for appropriate functions φ, to new results on the existence of unconditional...

Self-similar solutions and Besov spaces for semi-linear Schrödinger and wave equations

Fabrice Planchon (1999)

Journées équations aux dérivées partielles

We prove that the initial value problem for the semi-linear Schrödinger and wave equations is well-posed in the Besov space B ˙ 2 n 2 - 2 p , ∞ ( 𝐑 n ) , when the nonlinearity is of type u p , for p ∈ 𝐍 . This allows us to obtain self-similar solutions, as well as to recover previously known results for the solutions under weaker smallness assumptions on the data.

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