An extension of a result of ribe

It is proved that for every 1≦p<∞, 1≦q<∞ and for every sequence {p _n}, 1≦p _n<∞,p _n→p, the spaceX=(Σ⊕l _{p n}) _q (resp.U=(Σ⊕L _{p n}(0, 1)) _q ) is uniformly homeomorphic toX⊕l _p (resp.U⊕L _p(0, 1)). This extends Ribe’s result from the casep=1 to generalp<∞ and thus provides examples of uniformly convex, uniformly homeomorphic Banach spaces which are not Lipschitz equivalent.     

Some non-uniformly homeomorphic spaces

in this paper we prove that for 0<p, q≤1 the real F-spacesL _q[0, 1] and ℓ _p are not uniformly homeomorphic. The particular casep=q=1 is due to Enflo and our work is motivated by his. Partially supported by NSF grant #DMS-9104040. The author would like to dedicate this paper to Nancy and Bernie Weston on the occasion of their 40th wedding anniversary.     

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.     

The H?lder-Poincaré duality for L q,p -cohomology

We prove the following version of Poincaré duality for reduced L _q,p -cohomology: For any 1 < q, p < ∞, the L _q,p -cohomology of a Riemannian manifold is in duality with the interior L _p',q'-cohomology for 1/p + 1/p′ = 1/q + 1/q′ = 1.     

Recovery of band limited functions via cardinal splines

The main result of this paper asserts that if a function f is in the class B_π,p, 1 <p < ∞; that is, those p-integrable functions whose Fourier transforms are supported in the interval [ - π, π], then f and its derivatives f^(j) j = 1, 2, …, can be recovered from its sampling sequence{f(k)} via the cardinal interpolating spline of degree m in the metric ofL _q(ℝ)), 1 <p=q < ∞, or 11 <p=q < ⩽ ∞.     

On an inclusion between operator ideals

Let 1 ⩽ q < p < ∞ and 1/r:= 1/p max(q/2, 1). We prove that L _r,p ^(c), the ideal of operators of Gel’fand type l _r,p , is contained in the ideal Π _p,q of (p, q)-absolutely summing operators. For q > 2 this generalizes a result of G. Bennett given for operators on a Hilbert space.     

Universal non-compact operators between super-reflexive Banach spaces and the existence of a complemented copy of Hilbert space

Suppose that 1<p≦2, 2≦q<∞. The formal identity operatorI:l _p→l _qfactorizes through any given non-compact operator from ap-smooth Banach space into aq-convex Banach space. It follows that ifX is a 2-convex space andY is an infinite dimensional subspace ofX which is isomorphic to a Hilbert space, thenY contains an isomorphic copy ofl ₂ which is complemented inX.     

Uniform convexity of unitary ideals

ItE is a symmetric Banach sequence which isq-concave with the constant equal to 1 (where 2≦q<∞), thenS _E isq-PL-convex. IfE isq-concave andp-convex with the constants equal to 1 (where 1<p≦2≦q<∞), thenS _E is uniformly convex with modulus of convexity of power typeq and uniformly smooth with modulus of smoothness of power typep.     

On Gaussian Marginals of Uniformly Convex Bodies

Recently, Bo’az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag’s quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies with power type 2, and power type p>2 with some additional type condition. In particular, our results apply to all unit-balls of subspaces of quotients of L _p for 1<p<∞. The same is true when L _p is replaced by S _p ^m , the l _p -Schatten class space. We also extend our results to arbitrary uniformly convex bodies with power type p, for 2≤p<4. These results are obtained by putting the bodies in (surprisingly) non-isotropic positions and by a new concentration of volume observation for uniformly convex bodies. Supported in part by BSF and ISF.     

Greedy Bases for Besov Spaces

We prove that the Banach space (?_n=1^￥l_pⁿ)_lq(\bigoplus_{n=1}^{\infty}\ell_{p}^{n})_{\ell_{q}}, which is isomorphic to certain Besov spaces, has a greedy basis whenever 1≤p≤∞ and 1<q<∞. Furthermore, the Banach spaces (?_n=1^￥l_pⁿ)_l1(\bigoplus_{n=1}^{\infty}\ell _{p}^{n})_{\ell_{1}}, with 1<p≤∞, and (?_n=1^￥l_pⁿ)_c0(\bigoplus_{n=1}^{\infty}\ell_{p}^{n})_{c_{0}}, with 1≤p<∞, do not have a greedy basis. We prove as well that the space (?_n=1^￥l_pⁿ)_lq(\bigoplus_{n=1}^{\infty}\ell _{p}^{n})_{\ell_{q}} has a 1-greedy basis if and only if 1≤p=q≤∞.     

Rearrangement invariant subspaces of Lorentz function spaces

The main result is that for 2≦q≦p<∞ the only subspaces of the Lorentz function spaceL _pq [0, 1] which are isomorphic to r.i. function spaces on [0, 1] are, up to equivalent renormings,L _pq [0, 1] andL ₂[0, 1].     

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     

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 = ∞.     

On the Diameter of Wenger Graphs

Let q be a prime power, the field of q elements, and n≥1 a positive integer. The Wenger graph W _n (q) is defined as follows: the vertex set of W _n (q) is the union of two copies P and L of (n+1)-dimensional vector spaces over , with two vertices (p ₁,p ₂,…,p _n+1)∈P and [l ₁,l ₂,…,l _n+1]∈L being adjacent if and only if l _i +p _i =p ₁ l _i−1 for 2≤i≤n+1. Graphs W _n (q) have several interesting properties. In particular, it is known that when connected, their diameter is at most 2n+2. In this note we prove that the diameter of connected Wenger graphs is 2n+2 under the assumption that 1≤n≤q−1.     

Banded perturbations of the unit vector basis in some sequence spaces

Perturbations of the unit vector basis of the formX _n =Σ_|j−n|≦m a _nj e _j wherem is a fixed positive integer are investigated. It is shown that if |a _nj |≦1 and if {x _n } possesses a biorthogonal sequence uniformly bounded inl _p for some 1<=p<∞, then {x _n } is a seminormalized basic sequence in some reflexive Orlicz spacel _N, then {x_n} is equivalent to {e _n} inl _N.     

On perfectly homogeneous bases in Banach spaces

It is proved that a Banach space is isomorphic toc _o or tol _p if and only if it has a normalized basis {χ_i i _{} i=1} ^∞ which is equivalent to every normalized block-basis with respect to {χ_i i _{} i=1} ^∞ . This is part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Prof. A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank Dr. Lindenstrauss for his helpful guidance and for the interest he showed in the paper, and the referee for his valuable remakrs.     

Subspaces of<Emphasis Type="Italic">l</Emphasis><Superscript>∞</Superscript> (Γ) without quasicomplements

J. Lindenstrauss proves in [L] thatc ₀(Γ) is not quasicomplemented inl ^∞(Γ) while H. P. Rosenthal in [R] proves that subspaces, whose dual balls are weak* sequentially compact and weak* separable, are quasicomplemented inl ^∞(Γ). In this note it is proved that weak* separability of the dual is the precise condition determining whether a subspace, without isomorphic copies ofl ₁ and whose dual balls are weak* sequentially compact, is quasicomplemented or not inl ^∞(Γ). Especially spaces isomorphic tol _p(Γ), for 1<p<∞, have no quasicomplements inl ^∞(Γ) if Γ is uncountable.     

On the projection and macphail constants ofl n p spaces

We prove that the projection and Macphail constants ofl _n ^p (1≦p≦2) are asymptotically equivalent ton ^1/2 andn ^−1/2 respectively. We also obtain some relations linking certain parameters of general finite dimensional real Banach spaces. This note is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. J. Lindenstrauss, to whom the author wishes to express his thanks and appreciation.     

Compactness properties of certain integral operators related to fractional integration

Suppose 1≤p,q≤∞ and α > (1/p−1/q)₊. Then we investigate compactness properties of the integral operator when regarded as operator from L_p[0,1] into L_q[0,1]. We prove that its Kolmogorov numbers tend to zero faster than exp(−c_αn^1/2). This extends former results of Laptev in the case p=q=2 and of the authors for p=2 and q=∞. As application we investigate compactness properties of related integral operators as, for example, of the difference between the fractional integration operators of Riemann–Liouville and Weyl type. It is shown that both types of fractional integration operators possess the same degree of compactness. In some cases this allows to determine the strong asymptotic behavior of the Kolmogorov numbers of Riemann–Liouville operators. In memoria of Eduard (University of the West Indies) who passed away in October 2004.     

A bilinear estimate for biharmonic functions in Lipschitz domains

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain Ω is equivalent to the solvability of the Dirichlet problem for the biharmonic equation in Ω. As a result, we prove that for any given bounded Lipschitz domain Ω in _boxclose^d{\mathbb{R}^{d}} and 1 < q < ∞, the solvability of the L ^q Dirichlet problem for Δ ² u = 0 in Ω with boundary data in WA ^1,q (∂Ω) is equivalent to that of the L ^p regularity problem for Δ ² u = 0 in Ω with boundary data in WA ^2,p (∂Ω), where \frac1p + \frac1q=1{\frac{1}{p} + \frac{1}{q}=1}. This duality relation, together with known results on the Dirichlet problem, allows us to solve the L ^p regularity problem for d ≥ 4 and p in certain ranges.