|
| |
|
|
Analysis in Banach spaces
Pernecká, Eva ; Hájek, Petr (advisor) ; Johanis, Michal (referee) ; Godefroy, Gilles (referee)
The thesis consists of two papers and one preprint. The two papers are de- voted to the approximation properties of Lipschitz-free spaces. In the first pa- per we prove that the Lipschitz-free space over a doubling metric space has the bounded approximation property. In particular, the Lipschitz-free space over a closed subset of Rn has the bounded approximation property. We also show that the Lipschitz-free spaces over ℓ1 and over ℓn 1 admit a monotone finite-dimensional Schauder decomposition. In the second paper we improve this work and obtain even a Schauder basis in the Lipschitz-free spaces over ℓ1 and ℓn 1 . The topic of the preprint is rigidity of ℓ∞ and ℓn ∞ with respect to uniformly differentiable map- pings. Our main result is a non-linear analogy of the classical result on rigidity of ℓ∞ with respect to non-weakly compact linear operators by Rosenthal, and it generalises the theorem on non-complementability of c0 in ℓ∞ due to Phillips. 1
|
|
|
Analysis in Banach spaces
Pernecká, Eva ; Hájek, Petr (advisor) ; Johanis, Michal (referee) ; Godefroy, Gilles (referee)
The thesis consists of two papers and one preprint. The two papers are de- voted to the approximation properties of Lipschitz-free spaces. In the first pa- per we prove that the Lipschitz-free space over a doubling metric space has the bounded approximation property. In particular, the Lipschitz-free space over a closed subset of Rn has the bounded approximation property. We also show that the Lipschitz-free spaces over ℓ1 and over ℓn 1 admit a monotone finite-dimensional Schauder decomposition. In the second paper we improve this work and obtain even a Schauder basis in the Lipschitz-free spaces over ℓ1 and ℓn 1 . The topic of the preprint is rigidity of ℓ∞ and ℓn ∞ with respect to uniformly differentiable map- pings. Our main result is a non-linear analogy of the classical result on rigidity of ℓ∞ with respect to non-weakly compact linear operators by Rosenthal, and it generalises the theorem on non-complementability of c0 in ℓ∞ due to Phillips. 1
|
|
| |
|
|
Compactness of operators on function spaces
Pernecká, Eva ; Gurka, Petr (referee) ; Pick, Luboš (advisor)
Hardy-type operators involving suprema have turned out to be a useful tool in the theory of interpolation, for deriving Sobolev-type inequalities, for estimates of the non-increasing rearrangements of fractional maximal functions or for the description of norms appearing in optimal Sobolev embeddings. This thesis deals with the compactness of these operators on weighted Banach function spaces. We de ne a category of pairs of weighted Banach function spaces and formulate and prove a criterion for the compactness of a Hardy-type operator involving supremum which acts between a couple of spaces belonging to this category. Further, we show that the category contains speci c pairs of weighted Lebesgue spaces determined by a relation between the exponents. Besides, we bring an extension of the criterion to all weighted Lebesgue spaces, in proof of which we use characterization of the compactness of operators having the range in the cone of non-negative non-increasing functions, introduced as a separate result.
|
guest ::
