An alternative approach to the uniqueness of unconditional basis of for

We give an alternative and much simpler proof of the uniqueness of unconditional basis (up to equivalence and permutation) in the quasi-Banach spaces ℓ_p(c₀) for 0<p<1 and its complemented subspaces with unconditional basis. The new approach uses the fact that the Banach envelope of these spaces is not sufficiently Euclidean with the lattice structure induced by its unconditional basis.     

Uniqueness of unconditional bases in nonlocally convex c<Subscript>0</Subscript>-products

We show that the c ₀-product (X ⊕ X ⊕ ... ⊕ X ⊕ ...)₀ of a natural quasi-Banach space X with strongly absolute unconditional basis has a unique unconditional basis up to permutation. Our results apply to a wide range of cases, including most of the c ₀-products of the nonlocally convex classical quasi-Banach spaces.     

Uniqueness of unconditional bases in quasi-banach spaces with applications to Hardy spaces

We prove some general results on the uniqueness of unconditional bases in quasi-Banach spaces. We show in particular that certain Lorentz spaces have unique unconditional bases answering a question of Nawrocki and Ortynski. We then give applications of these results to Hardy spaces by showing the spacesH _p (T ⁿ ) are mutually non-isomorphic for differing values ofn when 0<p<1. The research of the first two authors was partially supported by NSF-grant DMS 8901636.     

On unconditional polynomial bases inL p and bergman spaces

In this paper we consider unconditional bases inL _p(T), 1<p<∞,p ≠ 2, consisting of trigonometric polynomials. We give a lower bound for the degree of polynomials in such a basis (Theorem 3.4) and show that this estimate is best possible. This is applied to the Littlewood-Paley-type decompositions. We show that such a decomposition has to contain exponential gaps. We also consider unconditional polynomial bases inH _p as bases in Bergman-type spaces and show that they provide explicit isomorphisms between Bergman-type spaces and natural sequences spaces.     

关于Banach空间的无条件基

给出具有唯一无条件基的无穷维Banach空间,并给出其无条件基的若干性质.     

Uniqueness of symmetric basis in quasi-Banach spaces

We show that if X is a nonlocally convex natural quasi-Banach space with symmetric basis whose Banach envelope is isomorphic to ?₁, then all symmetric bases of X are equivalent. The scope of this result is quite ample since the Banach envelopes of natural quasi-Banach spaces with basis always exhibit an ?₁-like behavior, in the sense that they contain copies of 's uniformly complemented.     

Uniqueness of unconditional bases in Banach spaces

We prove a general results on complemented unconditional basic sequences in Banach lattices and apply it to give some new examples of spaces with unique unconditional basis. We show that Tsirelson space and certain Nakano spaces have unique unconditional bases. We also construct an example of a space with a unique unconditional basis with a complemented subspace failing to have a unique unconditional basis. Both authors were supported by NSF Grant DMS-9201357.     

Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions

Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolevtype spaces on abstract metric measure spaces defined via (weak) upper gradients. Our main focus lies on metric spaces with a doubling measure that support a Poincaré inequality. Absolute continuity of the function lattice quasi-norm is shown to be crucial for approximability by (locally) Lipschitz functions. The proof of the density result uses, among other facts, the fact that a suitable maximal operator is locally weakly bounded. In particular, various sufficient conditions for such boundedness on quasi-Banach function lattices (and rearrangement-invariant spaces, in particular) are established and applied.     

On shrinking and boundedly complete Schauder frames of Banach spaces

This paper studies Schauder frames in Banach spaces, a concept which is a natural generalization of frames in Hilbert spaces and Schauder bases in Banach spaces. The associated minimal and maximal spaces are introduced, as are shrinking and boundedly complete Schauder frames. Our main results extend the classical duality theorems on bases to the situation of Schauder frames. In particular, we will generalize James' results on shrinking and boundedly complete bases to frames. Secondly we will extend his characterization of the reflexivity of spaces with unconditional bases to spaces with unconditional frames.     

Harmonic analysis in (UMD)-spaces: Applications to the theory of bases

In the paper, a general method for the construction of bases and unconditional finite-dimensional basis decompositions for spaces with the property of unconditional martingale differences is proposed. The construction makes use of a certain strongly continuous representation of Cantor's group in these spaces. The results are applied to vector function spaces and symmetric spaces of measurable operators associated with factors of type II.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 890–905, December, 1995.     

Nonlinear Approximation with Local Fourier Bases

It is shown that local Fourier bases are unconditional bases for the modulation spaces on R, including the Bessel potential spaces and the Segal algebra S ₀ . As a consequence, the abstract function spaces, that are defined by the approximation properties with respect to a local Fourier basis, are precisely the modulation space s. April 22, 1998. Date accepted: May 18, 1999.     

NEW WAVELET BASES FOR NON－HOMOGENEOUS SYMBOLIC SPACE OpS1,1^m AND RELATED KERNEL－DISTRIBUTION SPACES

This paper constructs several classes of new wavelet bases, which are unconditional bases for related operator spaces. Using these bases, the author analyzes non-homogeneous symbolic space OpSm1,1 and two related kernel-distribution spaces, and characterizes them in two wavelet coefficients spaces. Besides, some properties for singular integral operators are studied.     

NEW WAVELET BASES FOR NON-HOMOGENEOUS SYMBOLIC SPACE OpSm1,1 AND RELATED KERNEL-DISTRIBUTION SPACES

This paper constructs several classes of new wavelet bases, which are unconditional bases for related operator spaces. Using these bases, the author analyzes non-homogeneous symbolic space OpSm1,1 and two related kernel-distribution spaces, and characterizes them in two wavelet coefficients spaces. Besides, some properties for singular integral operators are studied.     

Wilson Bases and Modulation Spaces

We show that the recently discovered WILSON bases of exponential decay are unconditional bases for all modulation spaces on R, including the classical BESSEL potential spaces, the Segal algebra S_o, and the SCHWARTZ space. As a consequence we obtain new bases for spaces of entire functions. On the other hand, the WILSON bases are no unconditional bases for the ordinary L^p-spaces for p ≠ 2.     

NEW WAVELET BASES FOR NON-HOMOGENEOUS SYMBOLIC SPACE $OpS_{1,1}^m$ AND RELATED KERNEL-DISTRIBUTION SPACES

This paper constructs several classes of new wavelet bases, which are unconditional bases for related operator spaces. Using these bases, the author analyzes non-homogeneous symbolic space $OpS_{1,1}^m$ and two related kernel-distribution spaces, and characterizes them in two wavelet coefficients spaces. Besides, some properties for singular integral operators are studied.     

Remarks on the Existence of Unconditional Bases for Weighted Countably-Hilbert Spaces and Their Complemented Subspaces

We give versions of a criterion for existence of unconditional bases for countably-Hilbert spaces. As applications we obtain theorems on existence of unconditional bases for certain classes of countably-Hilbert function spaces and for their complemented subspaces under additional constraints on the space and the corresponding projections to the complemented subspaces. These classes include generalizations of power series spaces of finite type and Kothe spaces determined by Dragilev-type functions.     

On the equivalence of brushlet and wavelet bases

We prove that the Meyer wavelet basis and a class of brushlet systems associated with exponential type partitions of the frequency axis form a family of equivalent (unconditional) bases for the Besov and Triebel-Lizorkin function spaces. This equivalence is then used to obtain new results on nonlinear approximation with brushlets in Triebel-Lizorkin spaces.     

Uniqueness of unconditional bases in quasi-banach spaces with applications to hardy spaces, II

We prove that a wide class of quasi-Banach spaces has a unique up to a permutation unconditional basis. This applies in particular to Hardy spacesH _p forp<1. We also investigate the structure of complemented subspaces ofH _p (D). The proofs use in essential way matching theory. This research was supported in part by KBN grant N. 2P301004.06 and by the Overseas Visiting Scholarship of St. John’s College, Cambridge.     

Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization

Our first aim in this paper is to prove the boundedness of some sublinear operators on Herz spaces with variable exponent. As an application, we give characterizations and unconditional bases of the spaces in terms of wavelets.     

A note on cones associated to Schauder bases

We study the existence of infima of subsets in Banach spaces ordered by normal cones associated to shrinking Schauder bases. Under these conditions we prove the existence of infima for a class of subsets verifying a weakly compactness property. Moreover we prove that a normal cone associated to a Schauder basis in a reflexive Banach space is strongly minihedral extending the known result for unconditional Schauder bases. Several examples are also discussed. Miguel Sama: The work of this author is partially supported by Ministerio de Educación y Ciencia (Spain), project MTM2006-02629 and Ingenio Mathematica (i-MATH) CSD2006-00032 (ConsoliderIngenio 2010).