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.     

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.     

On Hille-Tamarkin operators and Schatten classes

We determine the smallest Schatten class containing all integral operators with kernels inL _p(L_{p', q})^symm, where 2 <p∞ and 1≦q≦∞. In particular, we give a negative answer to a problem posed by Arazy, Fisher, Janson and Peetre in [1]. Supported in part by DGICYT (SAB-90-0033).     

Uniform embeddings of Banach spaces

It is proved that for 1<-p≤2,L _p(0,1) andl _p are uniformly equivalent to bounded subsets of themselves. It is also shown that for 1<=p<=2, 1≦q<∞,L _p is uniformly equivalent to a subset ofl _q. This is a part of the author’s Ph. D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss. The author wishes to thank Professor Lindenstrauss for his guidance.     

Subspaces without the approximation property

It is proved that the Banach spacel _p with 1≦p<2 contains a subspace without AP (the case 2<p≦∞ follows from the Enflo’s construction and also from the present one). The result generalizes to the following one: if the supremum of types ofX is strictly less than 2 or if the infimum of cotypes ofX is strictly more than 2 thenX contains a subspace without AP.     

Nonreflexive spaces of type 2

The nonreflexive and uniformly nonoctahedral spacesX _pgr are known to be of typep if 1≦p<2 and ρ is sufficiently large. It is shown thatX _ρ is of type 2 if ρ>2.     

Interior points of convex hulls

If a setX ⊂E ⁿ has non-emptyk-dimensional interior, or if some point isk-dimensional surrounded, then the classic theorem of E. Steinitz may be extended. For example ifX ⊂E ⁿ has int _k X ≠ 0, (0 ≦k≦n) and ifp ɛ int conX, thenp ɛ int conY for someY ⊂X with cardY≦2n−k+1.     

Bounded pseudo-differential operators

This lecture gives an inside look into the proof of the continuity of pseudo-differential operators of orderm and typep, δ₁, δ₂ for 0≦p≦δ₁=1, 0≦p≦δ₂<1, andm/n≦p≦(δ₁+δ₂)/2. Applications are mentioned.     

Mixing of Cartesian squares of positive operators

LetT be a power bounded positive operator inL ₁(X, Σ, m)of a probability space, given by a transition measureP (x, A). The Cartesian squareS is the operator onL ₁ (X × X, Σ × Σ, m × m) induced by the transition measure Q((x, y), A × B)=P(x, A)P(y, B).T iscompletely mixing if ∝u e dm=0 impliesT ⁿ u→0 weakly (where 0≦e ∈L _∞ withT ^* e=e).Theorem. IfT has no fixed points, thenT is completely mixing if and only ifS is completely mixing. Part of this research was done at The Hebrew University of Jerusalem.     

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].     

Large solutions to the p-Laplacian for large p

In this work we consider the behaviour for large values of p of the unique positive weak solution u _p to Δ _p u = u ^q in Ω, u = +∞ on , where q > p − 1. We take q = q(p) and analyze the limit of u _p as p → ∞. We find that when q(p)/p → Q the behaviour strongly depends on Q. If 1 < Q < ∞ then solutions converge uniformly in compacts to a viscosity solution of with u = +∞ on . If Q = 1 then solutions go to ∞ in the whole Ω and when Q = ∞ solutions converge to 1 uniformly in compact subsets of Ω, hence the boundary blow-up is lost in the limit.     

On estimating the rate of best trigonometric approximation by a modulus of smoothness

Best trigonometric approximation in L _p , 1≦p≦∞, is characterized by a modulus of smoothness, which is equivalent to zero if the function is a trigonometric polynomial of a given degree. The characterization is similar to the one given by the classical modulus of smoothness. The modulus possesses properties similar to those of the classical one.     

Isolation and Component Structure in Spaces of Composition Operators

We establish a condition that guarantees isolation in the space of composition operators acting between H^p(B_N) and H^q(B_N), for 0 < p ≤ ∞, 0 < q < ∞, and N ≥ 1. This result will allow us, in certain cases where 0 < q < p ≤ ∞, completely to characterize the component structure of this space of operators.     

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 < ⩽ ∞.     

Functions with slowly-growing area and harmonic majorants

Letf be a function holomorphic inU={|z|<1}, and letA(R,f) be the area off(U)∩{|w|<R}, not counting multiplicities. IfA(R,f)=O(R ^γ) asR→∞ for a γ, 0≦γ<2, then the subharmonic function exp |f| ^p has a harmonic majorant inU for eachp, 0<p<2−γ. If 0≦γ<1 further, thene ^f is of Hardy classH ^p for eachp, 0<p<∞.     

Norm optimization problem for linear operators in classical Banach spaces

The main result of the paper shows that, for 1 < p < ∞ and 1 ≤ q < ∞, a linear operator T: ℓ _p → ℓ _q attains its norm if, and only if, there exists a not weakly null maximizing sequence for T (counterexamples can be easily constructed when p = 1). For 1 < p ≠ q < ∞, as a consequence of the previous result we show that any not weakly null maximizing sequence for a norm attaining operator T: ℓ _p → ℓ _q has a norm-convergent subsequence (and this result is sharp in the sense that it is not valid if p = q). We also investigate lineability of the sets of norm-attaining and non-norm attaining operators.     

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.     

Espaces de Banach stables

We define the notion of “stable Banach space” by a simple condition on the norm. We prove that ifE is a stable Banach space, then every subspace ofL ^p(E) (1≦p<∞) is stable. Our main result asserts that every infinite dimensional stable Banach space containsl ^p, for somep, 1≦p<∞. This is a generalization of a theorem due to D. Aldous: every infinite dimensional subspace ofL ¹ containsl ^p, for somep in the interval [1, 2].     

Embedding unconditional stable banach spaces into symmetric stable banach spaces

In this paper we prove the following result which solves a question raised by A. Pelczynski: “Every stable Banach space with an unconditional basis is isomorphic to a complemented subspace of some stable Banach space with a symmetric basis.” Moreover, we show that all the interpolation spacesl ^p ,l ^q _θ,X,1 1≦p, q<∞ andX stable, are stable.     

Some new thin sets of integers in harmonic analysis

We randomly construct various subsets A of the integers which have both smallness and largeness properties. They are small since they are very close, in various senses, to Sidon sets: the continuous functions with spectrum in Λ have uniformly convergent series, and their Fourier coefficients are in ℓ_p for all p > 1; moreover, all the Lebesgue spaces L _Λ ^q are equal forq < +∞. On the other hand, they are large in the sense that they are dense in the Bohr group and that the space of the bounded functions with spectrum in Λ is nonseparable. So these sets are very different from the thin sets of integers previously known.