L p -dual Quermassintegral sums

In this paper, we first introduce a concept of L _p -dual Quermassintegral sum function of convex bodies and establish the polar projection Minkowski inequality and the polar projection Aleksandrov-Fenchel inequality for L _p -dual Quermassintegral sums. Moreover, by using Lutwak’s width-integral of index i, we establish the L _p -Brunn-Minkowski inequality for the polar mixed projection bodies. As applications, we prove some interrelated results. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10271071), Zhejiang Provincial Natural Science Foundation of China (Grant No. Y605065) and Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

On volume distribution in 2-convex bodies

We consider convex sets whose modulus of convexity is uniformly quadratic. First, we observe several interesting relations between different positions of such “2-convex” bodies; in particular, the isotropic position is a finite volume-ratio position for these bodies. Second, we prove that high dimensional 2-convex bodies posses one-dimensional marginals that are approximately Gaussian. Third, we improve the known bounds on the isotropic constant of quotients of subspaces of L _p and S _p ^m , the Schatten Class space, for 1 < p ≤ 2. supported by the Clay Mathematics Institute and by NSF grant #DMS-0456590. supported in part by BSF and ISF.     

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.     

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.     

Dominated estimates of convex combinations of commuting isometries

The principal result of this paper is that the convex combination of two positive, invertible, commuting isometries ofL _p(X,F, μ) 1<p<+∞, one of which is periodic, admits a dominated estimate with constantp/p−1. In establishing this, the following analogue of Linderholm’s theorem is obtained: Let σ and ε be two commuting non-singular point transformations of a Lebesgue Space with τ periodic. Then given ε>O, there exists a periodic non-singular point transformation σ′ such that σ′ commutes with τ and μ(x:σ′x≠σx}<ε. Byan approximation argument, the principal result is applied to the convex combination of two isometries ofL _p (0, 1) induced by point transformations of the form τx=x ^k,k>0 to show that such convex combinations admit a dominated estimate with constantp/p−1. Research supported in part by NSF Grant No. GP-7475. A portion of the contents of this paper is based on the author’s doctoral dissertation written under the direction of Professor R. V. Chacon of the University of Minnesota.     

On estimates of biharmonic functions on Lipschitz and convex domains

Using Maz ’ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in ℝⁿ. Forn ≥ 8, combinedwitharesultin[18], these estimates lead to the solvability of the L^p Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of p. In the case of convex domains, the estimates allow us to show that the L^p Dirichlet problem is uniquely solvable for any 2 − ε < p < ∞ and n ≥ 4.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt Foundation.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.     

The maximal (C, α, β) operator of two-parameter walsh-fourier series

The two-parameter dyadic martingale Hardy spacesH _p are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from H_p to L_p (1/(α+1), 1/(β+1)<p<∞) and is of weak type (H ₁ ^# , L₁), where the Hardy space H ₁ ^# is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f∈H ₁ ^# converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on H_p whenever 1/(α+1), 1/(β+1)<p<∞. Thus in case f∈H_p, the (C, α, β) means converge to f in H_p norm. The same results are proved for the conjugate (C, α, β) means, too.     

Convex polynomial and spline approximation inL p,O<p<∞

We prove that a convex functionf ∈ L _p[−1, 1], 0<p<∞, can be approximated by convex polynomials with an error not exceeding Cω ₃ ^ϕ (f,1/n)_p where ω ₃ ^ϕ (f,·) is the Ditzian-Totik modulus of smoothness of order three off. We are thus filling the gap between previously known estimates involving ω ₃ ^ϕ (f,1/n)_p, and the impossibility of having such estimates involving ω₄. We also give similar estimates for the approximation off by convexC ⁰ andC ¹ piecewise quadratics as well as convexC ² piecewise cubic polynomials. Communicated by Dietrich Braess     

On the Cauchy problem for elliptic equations in a disk

Letφ, ψ be smooth functions on the boundary of the unit diskB ₁. A second order uniformly elliptic operatorL and a functionu with second order derivatives inL ^p (1<p<2) are constructed with the following properties:u solvesLu=0 inB ₁ and satisfies the Cauchy dataφ, ψ on∂B ₁.     

Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent

We prove the boundedness of the maximal operator Mr in the spaces L^p（·）（Г,p） with variable exponent p（t） and power weight p on an arbitrary Carleson curve under the assumption that p（t） satisfies the log-condition on Г. We prove also weighted Sobolev type L^p（·）（Г, p） → L^q（·）（Г, p）-theorem for potential operators on Carleson curves.     

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.     

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     

Kolmogorov Entropy for Classes of Convex Functions

Kolmogorov ε-entropy of a compact set in a metric space measures its metric massivity and thus replaces its dimension which is usually infinite. The notion quantifies the compactness property of sets in metric spaces, and it is widely applied in pure and applied mathematics. The ε-entropy of a compact set is the most economic quantity of information that permits a recovery of elements of this set with accuracy ε. In the present article we study the problem of asymptotic behavior of the ε-entropy for uniformly bounded classes of convex functions in L _p -metric proposed by A.I.   Shnirelman. The asymptotic of the Kolmogorov ε-entropy for the compact metric space of convex and uniformly bounded functions equipped with L _p -metric is ε ^−1/2, ε→0₊.      

On the Riesz transform associated with the ultraspherical polynomials

We define and investigate the Riesz transform associated with the differential operatorL _λ f(θ)=−f"(θ)−2λ cot’θ. We prove that it can be defined as a principal value and that it is bounded onL ^P ([0, π],dm _λ (θ)),dm _λ(θ)=sin^2λ θdθ, for every 1<p<∞ and of weak type (1,1). The same boundedness properties hold for the maximal operator of the truncated operators. The speed of convergence of the truncated operators is measured in terms of the boundedness inL ^P (dm _λ ), 1<p<∞, and weak type (1,1) of the oscillation and ρ-variation associated to them. Also, a multiplier theorem is proved to get the boundedness of the conjugate function studied by Muckenhoupt and Stein for 1<p<∞ as a corollary of the results for the Riesz transform. Moreover, we find a condition on the weightv which is necessary and sufficient for the existence of a weightu such that the Riesz transform is bounded fromL ^P (v dm _λ ) intoL ^P (u dm _λ ). The authors were partially supported by RTN Harmonic Analysis and Related Problems contract HPRN-CT-2001-00273-HARP. The first and fourth authors were supported in part by KBN grant 1-P93A 018 26. The second and third authors were partially supported by BFM grant 2002-04013-C02-02.     

von Neumann��s mean ergodic theorem on complete random inner product modules

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on L _ℱ ^p (ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L ^p (ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and L _ℱ ^p (ℰ(E,H) the complete random normed module generated by L ^p (ℰ, H).     

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.     

An L0（F,R）-valued Function＇s Intermediate Value Theorem and Its Applications to Random Uniform Convexity

Let (Ω , F , P ) be a probability space and L0 ( F, R ) the algebra of equivalence classes of real- valued random variables on (Ω , F , P ). When L0 ( F, R ) is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0 ( F, R ) to L0 ( F, R ). As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module ( S,|| · ||) is random uniformly convex iff Lp ( S ) is uniformly convex for each fixed positive number p such that 1 p + ∞ .     

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.