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Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions

100%

Pełczyński A.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction................................................................................................................................................. 5 Preliminaries.............................................................................................................................................. 9 § 1. Regular operators and their products............................................................................................ 11 § 2. Exaves. Extension and averaging operators................................................................................. 15 § 3. Linear multiplicative exaves and retractions. Localization principle......................................... 21 § 4. Integral representations and compositions of linear exaves.................................................... 22 § 5. Milutin spaces..................................................................................................................................... 27 § 6. Dugundji spaces................................................................................................................................ 34 § 7. Exaves and topological groups....................................................................................................... 37 § 8. Application to linear topological classification of spaces of continuous functions............... 40 § 9. Linear averaging operators and projections onto spaces of continuous functions.............. 47 Notes and Remarks.................................................................................................................................. 59 Appendix: Category-theoretical approach............................................................................................. 75 Bibliography................................................................................................................................................ 80

2

Uwagi o miarach wektorowych

93%

Pełczyński A.

Commentationes Mathematicae

|

1959

|

tom 3

|

nr 1

EN

The article contains no abstract

3

Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm

60%

Pełczyński A., Wojciechowski M.

Studia Mathematica

|

1993

|

tom 107

|

nr 1

61-100

EN

Let E be a Banach space. Let $L¹_{(1)}(ℝ^d,E)$ be the Sobolev space of E-valued functions on $ℝ^d$ with the norm $ʃ_{ℝ^d} ∥f∥_E dx + ʃ_{ℝ^d} ∥∇f∥_E dx = ∥f∥₁ + ∥∇f∥₁$. It is proved that if $f ∈ L¹_{(1)}(ℝ^d,E)$ then there exists a sequence $(g_m) ⊂ L_{(1)}¹(ℝ^d,E)$ such that $f = ∑_m g_m$; $∑_m (∥g_m∥₁ + ∥∇g_m ∥₁) < ∞$; and $∥g_m∥_∞^{1/d} ∥g_m∥₁^{(d-1)/d} ≤ b∥∇g_m∥₁$ for m = 1, 2,..., where b is an absolute constant independent of f and E. The result is applied to prove various refinements of the Sobolev type embedding $L_{(1)}¹(ℝ^d,E) ↪ L²(ℝ^d,E)$. In particular, the embedding into Besov spaces $L¹_{(1)} (ℝ^d,E) ↪ B_{p,1}^{θ(p,d)}(ℝ^d,E)$ is proved, where $θ(p,d) = d(p^{-1} + d^{-1} -1)$ for 1 < p ≤ d/(d-1), d=1,2,... The latter embedding in the scalar case is due to Bourgain and Kolyada.

4

On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in L²

60%

Ovsepian R. I., Pełczyński A.

Studia Mathematica

|

1975-1976

|

tom 54

|

nr 2

149-159

5

Estimated extension theorem, homogeneous collections and skeletons and their applications to topological classification of linear metric spaces and convex sets

60%

Bessaga C., Pełczyński A.

Fundamenta Mathematicae

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1970

|

tom 69

|

nr 2

153-190

6

Addendum to the paper "On isomorphisms of anisotropic Sobolev spaces with "classical Banach spaces" and a Sobolev type embedding theorem"

60%

Pełczyński A., Senator K.

Studia Mathematica

|

1986

|

tom 84

|

nr 2

217-218

7

Banach spaces with finite dimensional expansions of identity and universal bases of finite dimensional subspaces

59%

Pełczyński A., Wojtaszczyk P.

Studia Mathematica

|

1971

|

tom 40

|

nr 1

91-108

8

On the non-existence of linear isomorphisms between Banach spaces of analytic functions of one and several complex variables

59%

Mitiagin B. S., Pełczyński A.

Studia Mathematica

|

1976

|

tom 56

|

nr 2

175-186

9

Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basic

59%

Pełczyński A.

Studia Mathematica

|

1971

|

tom 40

|

nr 3

239-243

10

p-integral operators commuting with group representations and examples of quasi-p-integral operators which are not p-integral

59%

Pełczyński A.

Studia Mathematica

|

1969

|

tom 33

|

nr 1

63-70

11

Approximative dimension of linear topological spaces and some of its applications

48%

Bessaga C., Pełczyński A., Rolewicz S.

Studia Mathematica

|

1963

|

tom Special Series

|

nr 1

27-29

12

Spaces of continuous functions (IV). (On isomorphical classification of spaces of continuous functions).

48%

Bessaga C., Pełczyński A.

Studia Mathematica

|

1960

|

tom 19

|

nr 1

53-62

13

On isomorphisms of anisotropic Sobolev spaces with "classical Banach spaces" and a Sobolev type embedding theorem

48%

Pełczyński A., Senator K.

Studia Mathematica

|

1986

|

tom 84

|

nr 2

169-215

14

On simultaneous extension of continuous functions (A generalization of theorems of Rudin-Carleson and Bishop)

47%

Pełczyński A.

Studia Mathematica

|

1964

|

tom 24

|

nr 3

285-304

15

All separable Banach space admit for every ε>0 fundamental total and bounded by 1 + ε biorthogonal sequences

47%

Pełczyński A.

Studia Mathematica

|

1976

|

tom 55

|

nr 3

295-304

16

Computing norms and critical exponents of some operators in $L^{p}$-spaces

42%

Figiel T., Iwaniec T., Pełczyński A.

Studia Mathematica

|

1984

|

tom 79

|

nr 3

227-274

17

On Banach spaces X for which $Π_{2}(ℒ_{∞},X)=B(ℒ_{∞},X)$

42%

Dubinsky E., Pełczyński A., Rosenthal H. P.

Studia Mathematica

|

1972

|

tom 44

|

nr 6

617-648

18

A criterion for compositions of (p,q)-absolutely summing operators to be compact

42%

Maurey B., Pełczyński A.

Studia Mathematica

|

1975-1976

|

tom 54

|

nr 3

291-300

19

Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$

42%

Kadec M. I., Pełczyński A.

Studia Mathematica

|

1961-1962

|

tom 21

|

nr 2

161-176

20

Absolutely summing operators in $ℒ_{p}$-spaces and their applications

42%

Lindenstrauss J., Pełczyński A.

Studia Mathematica

|

1966-1968

|

tom 29

|

nr 3

275-326

Ograniczanie wyników

Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions

Uwagi o miarach wektorowych

Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm

On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in L²

Estimated extension theorem, homogeneous collections and skeletons and their applications to topological classification of linear metric spaces and convex sets

Addendum to the paper "On isomorphisms of anisotropic Sobolev spaces with "classical Banach spaces" and a Sobolev type embedding theorem"

Banach spaces with finite dimensional expansions of identity and universal bases of finite dimensional subspaces

On the non-existence of linear isomorphisms between Banach spaces of analytic functions of one and several complex variables

Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basic

p-integral operators commuting with group representations and examples of quasi-p-integral operators which are not p-integral

Approximative dimension of linear topological spaces and some of its applications

Spaces of continuous functions (IV). (On isomorphical classification of spaces of continuous functions).

On isomorphisms of anisotropic Sobolev spaces with "classical Banach spaces" and a Sobolev type embedding theorem

On simultaneous extension of continuous functions (A generalization of theorems of Rudin-Carleson and Bishop)

All separable Banach space admit for every ε>0 fundamental total and bounded by 1 + ε biorthogonal sequences

Computing norms and critical exponents of some operators in $L^{p}$-spaces

On Banach spaces X for which $Π_{2}(ℒ_{∞},X)=B(ℒ_{∞},X)$

A criterion for compositions of (p,q)-absolutely summing operators to be compact

Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$

Absolutely summing operators in $ℒ_{p}$-spaces and their applications