On the Fréchet space L p −

This paper contains a study of the structure of the Fréchet space L _p ⁻, 1< p ≤∞, defined as the intersection of L _q [0,1] for q<p, and endowed with the projective topology. The main topics covered are: normable, Schwartz and nuclear subspaces of L _p ⁻; construction of uncomplemented copies of ?₂ inside L _p ⁻ for p<2; construction of Montel non-Schwartz subspaces; the space L _p ⁻ is primary. Received: 30 October 1996 / Revised version: 1 February 1998     

A new class of unbalanced haar wavelets that form an unconditional basis for Lp on general measure spaces

Given a complete separable σ-finite measure space (X,Σ, μ) and nested partitions of X, we construct unbalanced Haar-like wavelets on X that form an unconditional basis for L_p (X,Σ, μ) where1<p<∞. Our construction and proofs build upon ideas of Burkholder and Mitrea. We show that if(X,Σ, μ) is not purely atomic, then the unconditional basis constant of our basis is (max(p, q) −1). We derive a fast algorithm to compute the coefficients.     

Compactness in Vector-valued Banach Function Spaces

We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces L_X^p, where X is a Banach space and 1≤ p<∞, and extend the result to vector-valued Banach function spaces E_X, where E is a Banach function space with order continuous norm. The author is supported by the ‘VIDI subsidie’ 639.032.201 in the ‘Vernieuwingsimpuls’ programme of the Netherlands Organization for Scientific Research (NWO) and by the Research Training Network HPRN-CT-2002-00281.     

Triangular Dirichlet Kernels and Growth of L p Lebesgue Constants

Let P be a polygon in ℤ² and consider the mapping of an L¹(\mathbbT²)L^{1}(\mathbb{T}^{2}) function into the partial sum of its Fourier series determined by the dilate of P by the integer N. If the image space is endowed with the L ^p norm, 1<p<∞, then the operator norm will be given by the L ^p norm of ∑_(m,n)∈NP e ^{2π i(mx+ny)}. The size of this operator norm is shown to be O(N ^2(1−1/p)) when the polygon is a triangle. The estimate is independent of the shape of the triangle. For a k sided polygon the corresponding estimate is O(kN ^2(1−1/p)).     

On Marcinkiewicz-Zygmund Type Inequalities for Irregular Knots in Lp-Spaces, 0 < p ≦ +∞

An equivalence of a discrete norm and a continuous norm of a trigonometric polynomial is proved for the case of irregular knots in L_p-spaces, where 0 < p ≦ +∞.     

A Generalization of the Mean Size Formula of Wavelet Packets in Lp

The purpose of this paper is to investigate the mean size formula of wavelet packets in Lp for 0 〈 p ≤ ∞. We generalize a mean size formula of wavelet packets given in terms of the p-norm joint spectral radius and we also give some asymptotic formulas for the Lp-norm or quasi-norm on the subdivision trees. All results will be given in the general setting,     

Besov spaces andK-functionals inL p, 0<p<1*

We investigate Besov spaces and their connection with trigonometric polynomial approximation inL _p[−π,π], algebraic polynomial approximation inL _p[−1,1], algebraic polynomial approximation inL _p(S), and entire function of exponential type approximation inL _p(R), and characterizeK-functionals for certain pairs of function spaces including (L _p[−π,π],B _s ^a(L _p[−π,π])), (L _p(R),_s ^a(L_p(R))), , and , where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r. This project is supported by the National Science Foundation of China.     

齐型空间上带非光滑核的奇异积分算子的双权不等式

该文在齐型空间( X, d,μ)上建立带非光滑核的奇异积分算子的双权、弱型不等式, 即对于1< p≤ q <∞, 此算子是L^p( X, v)到 L^q,∞( X, u)有界的, 只要权函数对(u, v)满足在权 u 上增加一个"Orlicz-bump" '的 A_p 型条件.     

Cardinal Interpolation with Gaussian Kernels

In this paper, interpolation by scaled multi-integer translates of Gaussian kernels is studied. The main result establishes L _p Sobolev error estimates and shows that the error is controlled by the L _p multiplier norm of a Fourier multiplier closely related to the cardinal interpolant, and comparable to the Hilbert transform. Consequently, its multiplier norm is bounded independent of the grid spacing when 1<p<∞, and involves a logarithmic term when p=1 or ∞.     

Embedding of C q and R q into noncommutative L p -spaces, 1≤p<q≤2

We prove that a quotient of a subspace of C_p⊕_pR_p (1≤p<2) embeds completely isomorphically into a noncommutative L_p -space, where C_p and R_p are respectively the p-column and p-row Hilbertian operator spaces. We also represent C_q and R_q (p<q≤2) as quotients of subspaces of C_p⊕_pR_p. Consequently, C_q and R_q embed completely isomorphically into a noncommutative L_p (M). We further show that the underlying von Neumann algebra M cannot be semifinite.     

Subspaces and Orthogonal Decompositions Generated by Bounded Orthogonal Systems

We investigate properties of subspaces of L₂ spanned by subsets of a finite orthonormal system bounded in the L_∞ norm. We first prove that there exists an arbitrarily large subset of this orthonormal system on which the L₁ and the L₂ norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two orthogonal subspaces of L₂ⁿ, complementary to each other and each of dimension roughly n/2, spanned by ± 1 vectors (i.e. Kashin’s splitting) and in logarithmic distance to the Euclidean space. The same method applies for p > 2, and, in connection with the Λ_p problem (solved by Bourgain), we study large subsets of this orthonormal system on which the L₂ and the L_p norms are close (again, up to a logarithmic factor). Partially supported by an Australian Research Council Discovery grant. This author holds the Canada Research Chair in Geometric Analysis.     

Korn type inequalities in Orlicz spaces

We exhibit balance conditions between a Young function A and a Young function B   for a Korn type inequality to hold between the L^B$L^B$ norm of the gradient of vector-valued functions and the L^A$L^A$ norm of its symmetric part. In particular, we extend a standard form of the Korn inequality in L^p$L^p$, with 1<p<∞$1<p<\infty$, and an Orlicz version involving a Young function A   satisfying both the _Δ2${\mathrm{\Delta }}_2$ and the _∇2${\mathrm{\nabla }}_2$ condition.     

Primarite deL p (l r , 1<p,r<∞

In this article we show that if 1<p, r<∞, the spaceL _p (l _r ) is primary. If we let (h _k ) be the Haar system inL _p and (e _i ) the usual base ofl _r , we give sufficient conditions on a subsequence of (h _k ⊗e _i ) _{k ≧1,i≧1} for it to generate a space isomorphic toL _p (l _r ). We deduce the primarity ofL _p (l _r ).      

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

Noncommutative Burkholder/Rosenthal inequalities II: Applications

We show norm estimates for the sum of independent random variables in noncommutative L _p -spaces for 1 < p < ∞, following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. As applications, we derive an equivalence for the p-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative L _p for 2 < p < ∞. The first author is partially supported by the National Science Foundation DMS-0301116. The second author is partially supported by the Agence Nationale de Recherche 06-BLAN-0015.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-Cohomology and Normal Solvability

We consider some problems concerning the L _p,q -cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L_p^k (M) into L^k+1_q (M). In the second part, we prove that the first L _p,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q. Received: 17 January 2006 Supported by INTAS (Grant 03–51–3251) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh 311.2003.1, NSh 8526.2006.1).     

Dunford-Pettis like properties on tensor products

In this paper we study equivalent formulations of the DP^? P_p (1 < p < ∞). We show that X has the DP^? P_p if and only if every weakly-p-Cauchy sequence in X is a limited subset of X. We give su?cient conditions on Banach spaces X and Y so that the projective tensor product X ?_π Y, the dual (X ?_? Y)^? of their injective tensor product, and the bidual (X ?_π Y)^?? of their projective tensor product, do not have the DP P_p, 1 < p < ∞. We also show that in some cases, the projective and the injective tensor products of two spaces do not have the DP^? P_p, 1 < p < ∞.     

Some properties of the injective tensor product of L[0,1] and a Banach space

Let X be a (real or complex) Banach space and 1<p,p′<∞ such that 1/p+1/p′=1. Then , the injective tensor product of L^p[0,1] and X, has the Radon-Nikodym property (resp. the analytic Radon-Nikodym property, the near Radon-Nikodym property, contains no copy of c₀, is weakly sequentially complete) if and only if X has the same property and each continuous linear operator from L^p′[0,1] to X is compact.     

Weak* Hypercyclicity and Supercyclicity of Shifts on ℓ∞

We study hypercyclicity and supercyclicity of weighted shifts on ℓ^∞, with respect to the weak * topology. We show that there exist bilateral shifts that are weak * hypercyclic but fail to be weak * sequentially hypercyclic. In the unilateral case, a shift T is weak * hypercyclic if and only if it is weak * sequentially hypercyclic, and this is equivalent to T being either norm, weak, or weak-sequentially hypercyclic on c₀ or ℓ^p (1 ≤ p < ∞). We also show that the set of weak * hypercyclic vectors of any unilateral or bilateral shift on ℓ^∞ is norm nowhere dense. Finally, we show that ℓ^∞ supports an isometry that is weak * sequentially supercyclic.     

The coarse Lipschitz geometry of \ell_p\oplus\ell_q

We show that for 1 < p < ∞ with p ≠ 2 the space L _p (0,1) is not uniformly homeomorphic to . We also show that if 1 < p < 2 < q < ∞ the space has unique uniform structure, answering a question of Johnson, Lindenstrauss and Schechtman (Geom. Funct. Anal. 6:430–470, 1996). The first author was supported by NSF grant DMS-0555670 and the second author was supported by NSF grant DMS-0701097.