On the existence of principal values for the cauchy integral on weighted lebesgue spaces for non-doubling measures

Let T be a Calderón-Zygmund operator in a “non-homogeneous” space ( , d, μ), where, in particular, the measure μ may be non-doubling. Much of the classical theory of singular integrals has been recently extended to this context by F. Nazarov, S. Treil, and A. Volberg and, independently by X. Tolsa. In the present work we study some weighted inequalities for T_*, which is the supremum of the truncated operators associated with T. Specifically, for1<p<∞, we obtain sufficient conditions for the weight in one side, which guarantee that another weight exists in the other side, so that the corresponding L^p weighted inequality holds for T_*. The main tool to deal with this problem is the theory of vector-valued inequalities for T_* and some related operators. We discuss it first by showing how these operators are connected to the general theory of vector-valued Calderón-Zygmund operators in non-homogeneous spaces, developed in our previous paper [6]. For the Cauchy integral operator C, which is the main example, we apply the two-weight inequalities for C_* to characterize the existence of principal values for functions in weighted L^p.     

Remarques sur les systemes dynamiques donnes avec plusieurs facteurs

Two Bernoulli shifts are given, (X, T) and (X′, T′), with independent generatorsR=P ∨Q andR′=P′ ∨Q′ respectively. (R andR′ are finite). One can chooseR such that if (X′, T′) can be made a factor of (X, T) in such a way that (P′) _T′ and (Q′) _T′ are full entropy factors of (P) _T and (Q) _T respectively thend (P ∨Q)=d(P′ ∨Q′). In addition it is proved that if (X, T) is a Bernoulli shift and ifS is a measure preserving transformation ofX that has the same factor algebras asT thenS=T orS=T ⁻¹. A tool for this proof, which may be of independent interest is a relative version for very weak Bernoullicity.

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Equipe de Recherche n^o 1 “Processus stochastique et applications” dépendant de la Section n^o 1 “Mathématiques, Informatique” associée au C.N.R.S.     

Unique solvability of the Cauchy problem for a system of Maxwell equations with initial data on a timelike surface

In this paper, a uniqueness theorem on the solution of the Cauchy problem for a system of Maxwell equations is proved in the case where the coefficients ε and μ are analytic functions of coordinates and the initial data are given on an “immovable” surface Σ=Γ×[0≤t≤2T], where Γ is an analytic surface in R³. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 30–37. Translated by N.S. Zabavnikova.     

On regularity att=0 for semilinear ADEs and its numerical applications

Consider the following initial value problem for the semilinearabstract differential equation (ADE):.u=Lu+f(t,u), t ε (O,T]; u(O)=u _o ε D(L), whereL: (D(L)⊂-X)→X is the infitesimal generatory of aC ^o-semigroup on the Banach spaceX. Extending a well-known result, we show that, iff: [O,T]×X→X is Lipschitz continuous and the space is reflexive, then the solution iscontinuously differentiable on [O,T], and satisfies the equation also att=0. Applications are given to the convergence analysis of implicit and semi-implicit Euler-type discretization schemes. Parabolic and hyperbolic examples are discussed. This work has been suported by the MURST Numerical Analysis funds (40%-funds), and by the “Istituto Nazionale di Alta Matematica F. Severi”. The author wishes to thank Professor R. Spigler for some useful discussions.     

On order preserving contractions

Let (Ω,Σ,μ) be a measure space and letP be an operator onL ₂(Ω,Σ,μ) with ‖P‖≦1,Pf≧0 a.e. wheneverf≧0. If the subspaceK is defined byK={x| ||P ⁿ x||=||P ^*n x||=||x||,n=1,2,...} thenK=L ₂(Ω,Σ₁,μ), where Σ₁ ⊂ Σ and onK the operatorP is “essentially” a measure preserving transformation. Thus the eigenvalues ofP of modulus one, form a group under multiplication. This last result was proved by Rota for finiteμ here finiteness is not assumed) and is a generalization of a theorem of Frobenius and Perron on positive matrices. The research reported in this document has been sponsored in part by Air Force Office of Scientific Research, OAR through the European Office, Aerospace Research, United States Air Force.     

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.     

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” u^f (·, T) is studied, where u^f (x,t) is the solution of the initial boundary-value problem u_tt−Δu=0 in Ω×(0,T), u|_t<0=0, u|_∂Ω×(0,T)=f, and the (singular) control f runs over the class L²((0,T); H^−m (∂Ω)) (m>0). The following result is established. Let Ω^T={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝⁿ (diam Ω<∞) filled with waves by a final instant of time t=T, let T_*=inf{T : Ω^T=Ω} be the time of filling the whole domain Ω. We introduce the notation D_m=Dom((−Δ)^m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H²(Ω)∩H ₀ ¹ (Ω);D_−m=(D_m)′;D_−m(Ω^T)={y∈D_−m:supp y ⋐ Ω^T. If T<T., then the reachable set R _m ^T ={u^t(·, T): f ∈ L₂((0,T), H^−m (∂Ω))} (∀m>0), which is dense in D_−m(Ω^T), does not contain the class C ₀ ^∞ (Ω^T). Examples of a ∈ C ₀ ^∞ , a ∈ R _m ^T , are presented. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21. Translated by T. N. Surkova.     

On the filter of computably enumerable supersets of an r-maximal set

We study the filter ℒ^*(A) of computably enumerable supersets (modulo finite sets) of an r-maximal set A and show that, for some such set A, the property of being cofinite in ℒ^*(A) is still Σ⁰ ₃-complete. This implies that for this A, there is no uniformly computably enumerable “tower” of sets exhausting exactly the coinfinite sets in ℒ^*(A). Received: 6 November 1999 / Revised version: 10 March 2000 /?Published online: 18 May 2001     

Decreasing Subsequences in Permutations and Wilf Equivalence for Involutions

In a recent paper, Backelin, West and Xin describe a map φ^* that recursively replaces all occurrences of the pattern k... 21 in a permutation σ by occurrences of the pattern (k−1)... 21 k. The resulting permutation φ^*(σ) contains no decreasing subsequence of length k. We prove that, rather unexpectedly, the map φ^* commutes with taking the inverse of a permutation. In the BWX paper, the definition of φ^* is actually extended to full rook placements on a Ferrers board (the permutations correspond to square boards), and the construction of the map φ^* is the key step in proving the following result. Let T be a set of patterns starting with the prefix 12... k. Let T′ be the set of patterns obtained by replacing this prefix by k... 21 in every pattern of T. Then for all n, the number of permutations of the symmetric group _n that avoid T equals the number of permutations of _n that avoid T′. Our commutation result, generalized to Ferrers boards, implies that the number of involutions of _n that avoid T is equal to the number of involutions of _n avoiding T′, as recently conjectured by Jaggard. Both authors were partially supported by the European Commission's IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe”     

On the Holmgren-John uniqueness theorem for the wave equation with piecewise analytic coefficients

In this paper a uniqueness theorem is proved for the wave equation in the domain Q^2T=Ω×(0,2T), where Ω is a piecewise analytic Riemannian manifold (Riemannian polyhedron). Initial data are assumed to be given on a part Γ₀ × (0, 2T) of the space-time boundary of the cylinder Q^2T, Γ₀. The uniqueness of a weak solution is proved “in the large,” in a domain formed by the corresponding characteristics of the wave equation. Bibliography:24 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 113–136. Translated by T. N. Surkova.     

Non Linear Perturbations of Kerr Spacetime in External Regions and the Peeling Decay

We prove, outside the influence region of a ball of radius R ₀ centred in the origin of the initial data hypersurface, Σ₀, the existence of global solutions near to Kerr spacetime, provided that the initial data are sufficiently near to those of Kerr. This external region is the “far” part of the outer region of the perturbed Kerr spacetime. Moreover, if we assume that the corrections to the Kerr metric decay sufficiently fast, o(r ⁻³), we prove that the various null components of the Riemann tensor decay in agreement with the “Peeling theorem”.     

Homomorphisms between Stone-Cech Homomorphisms into Stone-Cech Remainders of Countable Groups

We study properties of continuous homomorphisms from β S into T^* and from S^* into T^*, where S denotes a countably infinite semigroup and T denotes a countably infinite group. We show that they have striking algebraic properties if they do not arise as continuous extensions of homomorphisms from S to T.     

Bernoulli convolutions and an intermediate value theorem for entropies of<Emphasis Type="Italic">K</Emphasis>-partitions

We establish a strong regularity property for the distributions of the random sums Σ±λ ⁿ , known as “infinite Bernoulli convolutions”: For a.e. λ ∃ (1/2, 1) and any fixed ℓ, the conditional distribution of (w _n+1...,w _n+ℓ) given the sum Σ _n=0 ^∞ w _n λ ⁿ , tends to the uniform distribution on {±1}^ℓ asn → ∞. More precise results, where ℓ grows linearly inn, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system of entropyh hasK-partitions of any prescribed conditional entropy in [0,h]. This answers a question of Rokhlin and Sinai from the 1960’s, for the case of Bernoulli systems. The authors were partially supported by NSF grants DMS-9729992 (E. L.), DMS-9803597 (Y. P.) and DMS-0070538 (W. S.).     

On contractions with compact defects

In 2005, the following question was posed by Duggal, Djordjević, and Kubrusly: Assume that T is a contraction of the class C ₁₀ such that I − T ^* T is compact and the spectrum of T is the unit disk. Can the isometric asymptote of T be a reductive unitary operator? In this paper, we give a positive answer to this question. We construct two kinds of examples. One of them are the operators of multiplication by independent variable in the closure of analytic polynomials in L ²(ν),where ν is an appropriate positive finite Borel measure on the closed unit disk. The second kind of examples is based on a theorem by Chevreau, Exner, and Pearcy. We obtain a contraction T satisfying all the needed conditions and such that I − T ^* T belongs to the Schatten–von Neumann classes \mathfrakS_p {\mathfrak{S}_p} for all p > 1. We give an example of a contraction T such that I − T ^* T belongs to \mathfrakS_p {\mathfrak{S}_p} for all p > 1, T is quasisimilar to a unitary operator and has “more” invariant subspaces than this unitary operator. Also, following Bercovici and Kérchy, we show that if a subset of the unit circle is the spectrum of a contraction quasisimilar to a given absolutely continuous unitary operator, then this contraction T can be chosen so that I − T*T is compact. Bibliography: 29 titles.     

On triangular subalgebras of groupoidC*-algebras

Let ℬ be an AFC*-algebra with Stratila-Voiculescu masaD and letU be a maximal triangular subalgebra of ℬ with diagonalD. Peters, Poon and Wagner showed thatU need not be aC*-subdiagonal subalgebra of ℬ in the sense of Kawamura and Tomiyama. We investigate and explain this phenomena here from the perspective of groupoidC*-algebras by representing257-7 as the “incidence algebra” associated with a topological partial order. A number of examples are given showing what can keep a maximal triangular algebra from beingC*-subdiagonal. Supported by a grant from the National Science Foundation.     

Fractional Poisson equations and ergodic theorems for fractional coboundaries

For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT _α=Σ _j=1 ^∞ a _j T ^j , where {a _j } are the coefficients in the power series expansion (1-t)^α=1-Σ _j=1 ^∞ a _j t ^j in the open unit disk, which satisfya _j >0 anda _j >0 and Σ _j=1 ^∞ a _j =1. The operator calculus justifies the notation(I−T) ^α :=I−T _α (e.g., (I−T _1/2)²=I−T). A vectory∈X is called an, α-fractional coboundary for T if there is anx∈X such that(I−T) ^α x=y, i.e.,y is a coboundary forT _α . The fractional Poisson equation forT is the Poisson equation forT _α . We show that if(I−T)X is not closed, then(I−T) ^α X strictly contains(I−T)X (but has the same closure). ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ _k=1 ^∞ T ^k y/k ^1-α converges in norm, and conclude that lim _n ‖(1/n ^1-α)Σ _k=1 ⁿ T ^k y‖=0 for suchy. For a Dunford-Schwartz operatorT onL ₁ of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T) ^α L ₁ for some 0<α<1, then the one-sided Hilbert transform Σ _k=1 ^∞ T ^k f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T) ^α L _p with α>1−1/p=1/q, then Σ _k=1 ^∞ T ^k f/k ^1/p converges a.e., and thus (1/n ^1/p ) Σ _k=1 ⁿ T ^k f converges a.e. to zero. Whenf∈(I−T) ^1/q L _p (the case α=1/q), we prove that (1/n ^1/p (logn)^1/q )Σ _k=1 ⁿ T ^k f converges a.e. to zero.     

Simple sheaves along dihedral Lagrangians

We prove the unicity of a complex of sheavesF whose microsupport is carried by a “dihedral” Lagrangian Λ ofT ^* X (X=a real manifold) and which is simple with a prescribed shift at a regular point of Λ. Our method consists in reducing Λ, by a real contact transformation, to the conormal bundle to aC ¹-hypersuface, and then in using [K-S 1, Prop. 6.2.1] in the variant of [D'A-Z 1]. This is similar to [Z 2] but more general, since complex contact transformations and calculations of shifts are not required. We then consider the case of a complex manifoldX, and obtain some vanishing theorems for the complex of “microfunctions along Λ” similar to those of [A-G], [A-H], [K-S 1] (cf. also [D'A-Z 3 5], [Z 2]).     

Double extensions of dynamical systems and constructing mixing filtrations

Let T: X→X be an automorphism (a measurable invertible measure-preserving transformation) of a probability space (X, F, μ) and let two μ-symmetric Markov generators A_u and A_s acting on the space L₂=L₂ (X, F, μ) be “eigenfunctions” of the automorphism T with eigenvaluesθ _u > 1 andθ _s < 1, respectively. We construct an extension of the automorphism T having increasing and decreasing filtrations by means of a transformation on the path space of these processes. Under additional conditions, we give an estimate of the maximal correlation coefficient between the δ-fields chosen from these filtrations. Hyperbolic toral automorphisms are considered as an example. Applications to limit theorems are given. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 61–72. Translated by M. I. Gordin.     

Spaces with “small” annihilators

In this note one investigates the properties of subspaces G of C(S), such that G¹ is “not a very large part” of the space C(S)^*. The fundamental result is: if G¹ is reflexive, then every operator from G^* into ℓ₂ is absolutely summable. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 192–195, 1976.