A grothendieck factorization theorem on 2-convex schatten spaces

We prove that for every bounded linear operatorT:C ^2p →H(1≤p<∞,H is a Hilbert space,C ^{2 p} p is the Schatten space) there exists a continuous linear formf onC ^p such thatf≥0, ‖f‖(C _C ^p)*=1 and $$\forall x \in C^{2p} , \left\| {T(x)} \right\| \leqslant 2\sqrt 2 \left\| T \right\|< f\frac{{x * x + xx*}}{2} > 1/2$$ . Forp=∞ this non-commutative analogue of Grothendieck’s theorem was first proved by G. Pisier. In the above statement the Schatten spaceC ^2p can be replaced byE _E ² whereE ⁽²⁾ is the 2-convexification of the symmetric sequence spaceE, andf is a continuous linear form onC _E. The statement can also be extended toL _E{(su2)}(M, τ) whereM is a Von Neumann algebra,τ a trace onM, E a symmetric function space.     

Characterization of measures by potentials

Let μ be a measure in a Banach spaceE, f be an even function onR. We consider the potentialg(a)=f _E f(‖x?a‖)dμ(x). The question is as follows: For whichf does the potentialg determine μ uniquely? In this article we give answers in the cases whereE=l _∞ ⁿ and wheref(t)=|t| ^p andE is a finite dimensional Banach space with symmetric analytic norm. Calculating the Fourier transform of the functionf(‖x‖ _∞) we give a new proof of the J. Misiewicz's result that the functionf(‖x‖ _∞) is positive definite only iff is a constant function.     

Distributional estimates for the bilinear Hilbert transform

We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power −2/3 (modulo some logarithmic factors). These results yield all known L^p bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on L^p1 × L^p2 when either 1/p₁ + 1/p₂ = 3/2 or one of p₁, p₂ is equal to 1. As a consequence of this work we also obtain that the square root of the bilinear Hilbert transform of two characteristic functions is exponentially integrable over any compact set.     

Многомерные вариаци и множеств и их контин генции

LetE be a compact set inR ⁿ (n≧2), and denote byV ₀(E) the number of the components ofE. Letp=1,2, ...,n?1;k=0,1, ...,p, and $$V_k (E;n,p) = \int\limits_{\Omega _k^n } {V_0 (E \cap \tau )^{{{(n - p)} \mathord{\left/ {\vphantom {{(n - p)} {(n - k)}}} \right. \kern-\nulldelimiterspace} {(n - k)}}} d\mu _\tau ,}$$ whereΩ _k ⁿ is the set of all (n-k)-dimensional hyperplanesτ?R ⁿ anddμ _τ is the Haar measure on the spaceΩ _k ⁿ ; furthermore, let $$V_n (E;n,n - 1) = mes_n E.$$ . Theorem. Let E?Rⁿ, p=1, 2 ..., n?1, V_p+1(E;n,p)=0, and V_k(E; n, p)<∞ for k= =0,1, ..., p. Then the contingency of the set E at a point x∈E coincides with a certain p-dimensional hyperplane for almost all points x∈E in the sense of Hausdorff p-measure.     

On rational approximations of functions of complex variable integrable over plane domains

пУстьE — ИжМЕРИМОЕ пО лЕБЕгУ ОгРАНИЧЕННОЕ МНОжЕстВО пОлОжИтЕльНОИ плОЩА ДИ mes₂ E кОМплЕксНОИ плОск ОстИ с. кАк ОБыЧНО, пРИp≧1 ОБОжНАЧИМ ЧЕРЕжL ^p (E) БА НАхОВО пРОстРАНстВО ИжМЕРИ Мых пО лЕБЕгУ НАE кОМплЕксНОжНАЧНых Ф УНкцИИf с сУММИРУЕМО Иp—стЕпЕНьУ Их МОДУль И ОБыЧНОИ НОРМОИ \(\left\| \cdot \right\|_p = \left\| \cdot \right\|_{L_p (E)}\) . ЧЕР ЕжL ^p R _n (f,E) ОБОжНАЧИМ НАИМЕН ьшЕЕ УклОНЕНИЕf?L ^p (E) От РАц ИОНАльНых ФУНкцИИ ст ЕпЕНИ ≦n кОМплЕксНОгО пЕРЕМЕ ННОгОz пО НОРМЕ ∥ · ∥. пОлОжИМf(z)=0 Дльz?¯CE,E _δ —δ-ОкРЕстНОсть МНО жЕстВАE (δ>0), И $$\omega _p (\delta ,f) = \mathop {\sup {\mathbf{ }}}\limits_{\left| h \right|< \delta } \{ \int\limits_{E_\sigma } {\int {{\mathbf{ }}|f(z + h) - f(z)|^p } d\sigma } \} ^{1/p} .$$ тЕОРЕМА.пУсть 1≦p<2,f?L _p (E),n≧4.тОгДА $$\begin{array}{*{20}c} {L^p R_n (f,E) \leqq 12\omega _p \left( {\frac{{\delta + \ln n}}{{\sqrt n }},f} \right){\mathbf{ }}npu{\mathbf{ }}p = 1,} \\ {L^p R_n (f,E) \leqq \frac{{24}}{{(p - 1)(2 - p)}}\omega _p (n^{(p - 2)/2p} ,f){\mathbf{ }}npu{\mathbf{ }}1< p< 2,} \\ {L^1 R_n (\bar z,[0,1] \times [0,1]) \geqq \frac{1}{{32\sqrt n }}.} \\ \end{array} $$ .     

Dilations,Product Systems,and Weak Dilations

We generalize Bhat's construction of product systems of Hilbert spaces from E₀-semigroups on B(H) for some Hilbert space H to the construction of product systems of Hilbert modules from E₀-semigroups on B^a(E) for some Hilbert module E. As a byproduct we find the representation theory for B^a(E) if E has a unit vector. We establish a necessary and sufficient criterion for the conditional expectation generated by the unit vector to define a weak dilation of a CP-semigroup in the sense of [1]. Finally, we also show that white noises on general von Neumann algebras in the sense of [2] can be extended to white noises on a Hilbert module.     

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c₀ and ?_p for 1?p<∞. We add a new member to this family by showing that there are exactly four closed ideals in for the Banach space E?(⊕?₂ⁿ)_c0, that is, E is the c₀-direct sum of the finite-dimensional Hilbert spaces ?₂¹,?₂²,…,?₂ⁿ,… .     

Spherical Maximal Operators on Radial Functions

Let A_tf(x) denote the mean of f over a sphere of radius t and center x. We prove sharp estimates for the maximal function M_E f(X) = sup_t∈_E |Atf(x)| where E is a fixed set in IR⁺ and f is a radial function ∈ L^p(IR^d). Let P_d = d/(d?1) (the critical exponent for Stein's maximal function). For the cases (i) p < p_d, d ? 2, and (ii) p = p_d, d ? 3, and for p ? q ? ∞ we prove necessary and sufficient conditions on E for M_E to map radial functions in L^p to the Lorentz space L^P,q.     

DFT for cyclic code overF p +uF p + ··· +u k-1 F p

The transform domain characterization of cyclic codes over finite fields using Discrete Fourier Transform(DFT) over an appropriate extension field is well known. In this paper, we extend this transform domain characterization for cyclic codes overF _p +uF _p + ··· +u ^k-1 F _p . We give a way to characterize cyclic codes overF _p +uF _p + ··· +u ^k-1 F _p by Mattson-Solomon polynomials and multiple defining sets.     

Extension of isometries on the unit sphere of L p spaces

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of Lp (μ) (1 p ∞, p≠2) and a Banach space E can be extended to a linear isometry from Lp(μ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of Lp(μ), then E is linearly isometric to Lp(μ). We also prove that every surjective 1-Lipschitz or anti-1-Lipschitz map between the unit spheres of Lp (μ1, H1) and Lp(μ2,H2) must be an isometry and can be extended to a linear isometry from Lp (μ1,H1) to Lp (μ2,H2), where H1 and H2 are Hilbert spaces.     

On the bounded part of a Hilbert module over a locally C *-algebra

We will show that the bounded part of the locally C^*-algebra of all adjointable operators on the Hilbert A-module E is isomorphic to the C^*-algebra L _b(A)(b(E)) of all adjointable operators on the Hilbert b(A)-module b(E). This revised version was published online in June 2006 with corrections to the Cover Date.     

Variations on a theme of Boole and Stein-Weiss

We give an alternative proof of a theorem of Stein and Weiss: The distribution function of the Hilbert transform of a characteristic function of a set E only depends on the Lebesgue measure |E| of such a set. We exploit a rational change of variable of the type used by George Boole in his paper “On the comparison of transcendents, with certain applications to the theory of definite integrals” together with the observation that if two functions f and g have the same Lp norm in a range of exponents p₁<p<p₂ then their distribution functions coincide.     

Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially metacyclic

We study the subgroup structure of some two-generator p-groups and apply the obtained results to metacyclic p-groups. For metacyclic p-groups G, p > 2, we do the following: (a) compute the number of nonabelian subgroups with given derived subgroup, show that (ii) minimal nonabelian subgroups have equal order, (c) maximal abelian subgroups have equal order, (d) every maximal abelian subgroup is contained in a minimal nonabelian subgroup and all maximal subgroups of any minimal nonabelian subgroup are maximal abelian in G. We prove the same results for metacyclic 2-groups (e) with abelian subgroup of index p, (f) without epimorphic image ? D₈. The metacyclic p-groups containing (g) a minimal nonabelian subgroup of order p ⁴, (h) a maximal abelian subgroup of order p ³ are classified. We also classify the metacyclic p-groups, p > 2, all of whose minimal nonabelian subgroups have equal exponent. It appears that, with few exceptions, a metacyclic p-group has a chief series all of whose members are characteristic.     

Positive definite symmetric functions on linear spaces

If Φ is a positive definite function on a real linear space E of infinite dimension and Φ enjoys certain symmetry conditions we are able to show that Φ is expressible as a certain Laplace-Stieltjes transform. Conversely, if Φ is given by such a transform we can often show that Φ is positive definite on E. In particular, our results apply to the L^p spaces, 0 < p < ∞, as well as to other Orlicz spaces. We also are able to show that the only positive definite continuous sup-norm symmetric functions on C(T), the space of bounded real continuous functions on T, are constants whenever C(T) contains a sequence of functions with sup-norm one and disjoint support. Finally, we apply these ideas to obtain a result on radial exponentially convex functions on a Hilbert space.     

On a second-order matrix differential operator

We consider the second-order matrix differential operator $$N = \left( {\begin{array}{*{20}c} { - \frac{d}{{dx}}\left( {p_0 \frac{d}{{dx}}} \right) + p_1 } \\ r \\ \end{array} \begin{array}{*{20}c} r \\ { - \frac{d}{{dx}}\left( {q_0 \frac{d}{{dx}}} \right) + q_1 } \\ \end{array} } \right)$$ determined by the expression Nφ, [0 ?x < ∞), where \(\phi = \left( {\begin{array}{*{20}c} U \\ V \\ \end{array} } \right)\) . It has been proved that if p₀, q₀, p₁, q₁,r satisfy certain conditions, then N is in the limit point case at ∞. It has been also shown that certain differential operators in the Hilbert space L² of vectors, generated by the operator N, are symmetric and self-adjoint.     

Interpolation and frames in certain Banach spaces of entire functions

The important class of generalized bases known as frames was first introduced by Duffin and Schaeffer in their study of nonharmonic Fourier series in L ² (?π, π) [4]. Here we consider more generally the classical Banach spacesE ^p(1 ≤ p ≤ ∞) consisting of all entire functions of exponential type at most π that belong to L^p (?∞, ∞) on the real axis. By virtue of the Paley-Wiener theorem, the Fourier transform establishes an isometric isomorphism between L ² (?π, π) andE ² . When p is finite, a sequence {λ _n} of complex numbers will be called aframe forE ^p provided the inequalities $$A\left\| f \right\|^p \leqslant \sum {\left| {f\left( {\lambda _\pi } \right)} \right|^p } \leqslant B\left\| f \right\|^p $$ hold for some positive constants A and B and all functions f inE ^p. We say that {λ _n} is aninterpolating sequence forE ^p if the set of all scalar sequences {f (λ _n)}, with f εE ^p, coincides with ?^p. If in addition {λ _n} is a set of uniqueness forE ^p, that is, if the relations f(λ _n)=0(?∞<n<∞), with f εE ^p, imply that f ≡0, then we call {λ _n} acomplete interpolating sequence. Plancherel and Pólya [7] showed that the integers form a complete interpolating sequence forE ^p whenever1<p<∞. In Section 2 we show that every complete interpolating sequence forE ^p(1<p<∞) remains stable under a very general set of displacements of its elements. In Section 3 we use this result to prove a far-reaching generalization of another classical interpolation theorem due to Ingham [6].     

On the p 2-Rank of Tame Kernels of Quadratic Fields

For any odd prime p, we prove some results connecting the p²-rank of the tame kernel of a quadratic field F with the p²-rank Cl(𝒪_E1 ), where E₁ is the maximal real subfield of F(ζ_p² ).     

Characterization of the Weak-Type Boundedness of the Hilbert Transform on Weighted Lorentz Spaces

We characterize the weak-type boundedness of the Hilbert transform H on weighted Lorentz spaces $\varLambda^{p}_{u}(w)$ , with p>0, in terms of some geometric conditions on the weights u and w and the weak-type boundedness of the Hardy–Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of H on weighted Lebesgue spaces L ^p (u) and Muckenhoupt weights A _p , and the theory on classical Lorentz spaces Λ ^p (w) and Ariño-Muckenhoupt weights B _p .     

A characterization of the Wishart exponential families by an invariance property

E is the space of real symmetric (d, d) matrices, andS and \(\bar S\) are the subsets ofE of positive definite and semipositive-definite matrices. Let there be ap in $$\Lambda = \left\{ {\frac{1}{2},1,\frac{3}{2}, \ldots \frac{{d - 1}}{2}} \right\} \cup \left] {\frac{{d - 1}}{2}, + \infty } \right[$$ The Wishart natural exponential family with parameterp is a set of probability distributions on \(\bar S\) defined by $$F_p = \{ \exp [ - \tfrac{1}{2}Tr(\Gamma x)](det\Gamma )^p \mu _p (dx);\Gamma \in S\} $$ where μ_p is a suitable measure on \(\bar S\) . LetGL(?^d) be the subset ofE of invertible matrices. Fora inGL(?^d), define the automorphismg _a ofE byg _a(x)=^t axa, where ^t a is the transpose ofa. The aim of this paper is to show that a natural exponential familyF onE is invariant byg _a for alla inGL(?^d) if and only if there existsp in Λ such that eitherF=F _p, orF is the image ofF _p byx??x. (Theorem).     

Markov processes associated with L p -resolvents and applications to stochastic differential equations on Hilbert space

We give general conditions on a generator of a C₀-semigroup (resp. of a C₀-resolvent) on L^p(E,μ), p ≥ 1, where E is an arbitrary (Lusin) topological space and μ a σ-finite measure on its Borel σ-algebra, so that it generates a sufficiently regular Markov process on E. We present a general method how these conditions can be checked in many situations. Applications to solve stochastic differential equations on Hilbert space in the sense of a martingale problem are given. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday