On a theorem of Marcinkiewicz and Zygmund

Let {ϕn(x)} be an orthonormal system on the closed interval [0,1], and let ∥ϕ _n ∥_∞ ≤M _n . In 1937 Marcinkiewicz and Zygmund obtained an estimate of the norm inL _q [0,1] of the sum of the series ∑ _n=1 ^∞ c _n ϕ _n (x) under the condition that {M _n } is monotone increasing. In this paper it is shown that this condition cannot be discarded. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 386–390, March, 1998. The author wishes to thank V. I. Kolyada for setting the problem and for his permanent attention to this work.     

On perturbations of differentiable semigroups

LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC ₀-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:

Beurling's theorem and invariant subspaces for the shift on Hardy spaces

Let G be a bounded open subset in the complex plane and let H~2(G) denote the Hardy space on G. We call a bounded simply connected domain W perfectly connected if the boundary value function of the inverse of the Riemann map from W onto the unit disk D is almost 1-1 with respect to the Lebesgue measure on D and if the Riemann map belongs to the weak-star closure of the polynomials in H~∞(W). Our main theorem states: in order that for each M∈Lat (M_z), there exist u∈H~∞(G) such that M=∨{uH~2(G)}, it is necessary and sufficient that the following hold: (1) each component of G is a perfectly connected domain; (2) the harmonic measures of the components of G are mutually singular; (3) the set of polynomials is weak-star dense in H~∞(G). Moreover, if G satisfies these conditions, then every M∈Lat (M_z) is of the form uH~2(G), where u∈H~∞(G) and the restriction of u to each of the components of G is either an inner function or zero.     

Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold

Let M ⁿ be an n-dimensional compact C ^∞-differentiable manifold, n ≥ 2, and let S be a C ¹-differential system on M ⁿ . The system induces a one-parameter C ¹ transformation group φ _t (−∞ < t < ∞) over M ⁿ and, thus, naturally induces a one-parameter transformation group of the tangent bundle of M ⁿ . The aim of this paper, in essence, is to study certain ergodic properties of this latter transformation group. Among various results established in the paper, we mention here only the following, which might describe quite well the nature of our study. (A) Let M be the set of regular points in M ⁿ of the differential system S. With respect to a given C ^∞ Riemannian metric of M ⁿ , we consider the bundle of all (n−2) spheres Q _x ⁿ⁻², x∈M, where Q _x ⁿ⁻² for each x consists of all unit tangent vectors of M ⁿ orthogonal to the trajectory through x. Then, the differential system S gives rise naturally to a one-parameter transformation group ψ _t ^# (−∞<t<∞) of . For an l-frame α = (u ₁, u ₂,⋯, u _l ) of M ⁿ at a point x in M, 1 ≥ l ≥ n−1, each u _i being in , we shall denote the volume of the parallelotope in the tangent space of M ⁿ at x with edges u ₁, u ₂,⋯, u _l by υ(α), and let . This is a continuous real function of t. Let

Lp-decay rates to nonlinear diffusion waves for p-system with nonlinear damping

In this paper, we study the L ^p (2 ⩽ p ⩽ +∞) convergence rates of the solutions to the Cauchy problem of the so-called p-system with nonlinear damping. Precisely, we show that the corresponding Cauchy problem admits a unique global solution (v(x,t), u(x,t)) and such a solution tends time-asymptotically to the corresponding nonlinear diffusion wave (ῡ(x,t), ū(x,t)) governed by the classical Darcys’s law provided that the corresponding prescribed initial error function (w ₀(x), z ₀(x)) lies in (H ³ × H ²) (ℝ) and |v ₊ − v ₋| + ∥w ₀∥₃ + ∥z ₀∥₂ is sufficiently small. Furthermore, the L ^p (2 ⩽ p ⩽ +∞) convergence rates of the solutions are also obtained.     

L P boundedness of area function associated with operators on product spaces

Let L be the infinitesimal generator of an analytic semigroup on L2 (Rn) with suitable upper bounds on its heat kernels. Assume that L has a bounded holomorphic functional calculus on L2(Rn). In this paper,we define the Littlewood- Paley g function associated with L on Rn × Rn, denoted by GL(f)(x1, x2), and decomposition, we prove that ‖SL(f)‖p ≈‖GL(f)‖p ≈‖f‖p for 1 ＜ p ＜∞.     

On Small Deviations of Series of Weighted Random Variables

Let ξ,ξ ₁,ξ ₂,… be positive i.i.d. random variables, S=∑ _j=1^∞ a(j)ξ _j , where the coefficients a(j)≥0 are such that P(S<∞)=1. We obtain an explicit form of the asymptotics of −ln P(S<x) as x→0 for the following three cases:

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

Multiparameter inverse problem of approximation by functions with given supports

Let L _p (S), 0 < p < +∞, be a Lebesgue space of measurable functions on S with ordinary quasinorm ∥·∥ _p . For a system of sets {B _t |t ∈ [0, +∞) ⁿ } and a given function ψ: [0, +∞) ⁿ ↦ [ 0, +∞), we establish necessary and sufficient conditions for the existence of a function f ∈ L _p (S) such that inf {∥f − g∥ _p ^p g ∈ L _p (S), g = 0 almost everywhere on S\B _t } = ψ (t), t ∈ [0, +∞) ⁿ . As a consequence, we obtain a generalization and improvement of the Dzhrbashyan theorem on the inverse problem of approximation by functions of the exponential type in L ₂. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1116–1127, August, 2006.     

The ultrametric corona problem

Let K be a complete ultrametric algebraically closed field and let A be the K-Banach algebra of bounded analytic functions in the disk D: |x| < 1. Let Mult(A, ∥ · ∥) be the set of continuous multiplicative semi-norms of A, let Mult _m (A, ∥ · ∥) be the subset of the ϕ ∈ Mult(A, ∥ · ∥) whose kernel is a maximal ideal and let Mult _a (A, ∥ · ∥) be the subset of the ϕ ∈ Mult _m (A, ∥ · ∥) whose kernel is of the form (x − a)A, a ∈ D ( if ϕ ∈ Mult _m (A, ∥ · ∥) \ Mult _a (A, ∥ · ∥), the kernel of ϕ is then of infinite codimension). We examine whether Mult _a (A, ∥ · ∥) is dense inside Mult _m (A, ∥ · ∥) with respect to the topology of simple convergence. This a first step to the conjecture of density of Mult _a (A, ∥ · ∥) in the whole set Mult(A, ∥ · ∥): this is the corresponding problem to the well-known complex corona problem. We notice that if ϕ ∈ Mult _m (A, ∥ · ∥) is defined by an ultrafilter on D, then ϕ lies in the closure of Mult _a (A, ∥ · ∥). Particularly, we show that this is case when a maximal ideal is the kernel of a unique ϕ ∈ Multm(A, ∥ · ∥). Particularly, when K is strongly valued all maximal ideals enjoy this property. And we can prove this is also true when K is spherically complete, thanks to the ultrametric holomorphic functional calculus. More generally, we show that if ψ ∈ Mult(A, ∥ · ∥) does not define the Gauss norm on polynomials (∥ · ∥), then it is defined by a circular filter, like on rational functions and analytic elements. As a consequence, if ψ ∈ Multm(A, ∥ · ∥) \ Multa(A, ∥ · ∥) or if φ does not lie in the closure of Mult _a (A, ∥ · ∥), then its restriction to polynomials is the Gauss norm. The first situation does happen. The second is unlikely. The text was submitted by the authors in English.     

Hardy spaces via distribution spaces

Let ℐ(ℝⁿ) be the Schwartz class on ℝⁿ and ℐ_∞(ℝⁿ) be the collection of functions ϕ ∊ ℐ(ℝⁿ) with additional property that

Possible limit laws for entrance times of an ergodic aperiodic dynamical system

LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ ₀ ^∞ G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ _U (x)=inf{k⩾1:T ^k xεU}, and defineG _U (t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U _n ) _n≥1 of neighbourhoods ofx such that {x}=∩ _n U _n , and for anyG ∈G, there exists a subsequence (n _k ) _k≥1 withG _{U n k} ↑U weakly. We also construct a uniquely ergodic Toeplitz flowO(x ^∞,S, μ), the orbit closure of a Toeplitz sequencex ^∞, such that the above conclusion still holds, with moreover the requirement that eachU _n be a cylinder set. In memory of Anzelm Iwanik     

Transient phenomena for random walks in the absence of the expected value of jumps

Let ξ, ξ₁, ξ₂, ... be independent identically distributed random variables, and S _n :=Σ _j=1 ⁿ ,ξ _j , $ \bar S $ \bar S := sup _n≥0 S _n . If Eξ = −a < 0 then we call transient those phenomena that happen to the distribution $ \bar S $ \bar S as a → 0 and $ \bar S $ \bar S tends to infinity in probability. We consider the case when Eξ fails to exist and study transient phenomena as a → 0 for the following two random walk models:

An Engel condition with generalized derivations on multilinear polynomials

Let R be a prime ring with extended centroid C, g a nonzero generalized derivation of R, f (x ₁,..., x _n) a multilinear polynomial over C, I a nonzero right ideal of R. If [g(f(r ₁,..., r _n)), f(r ₁,..., r _n)] = 0, for all r ₁, ..., r _n ∈ I, then either g(x) = ax, with (a − γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds:

Stochastic Equations with Time-Dependent Drift Driven by Levy Processes

The stochastic equation dX _t =dS _t +a(t,X _t )dt, t≥0, is considered where S is a one-dimensional Levy process with the characteristic exponent ψ(ξ),ξ∈ℝ. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X ₀=x ₀∈ℝ when (ℛe  ψ(ξ))⁻¹=o(|ξ|⁻¹) as |ξ|→∞. These conditions coincide with those found by Tanaka, Tsuchiya and Watanabe (J. Math. Kyoto Univ. 14(1), 73–92, 1974) in the case of a(t,x)=a(x). Our approach is based on Krylov’s estimates for Levy processes with time-dependent drift. Some variants of those estimates are derived in this note.     

Ramanujan and extensions and contractions of continued fractions

If a continued fraction K _n=1^∞ a _n /b _n is known to converge but its limit is not easy to determine, it may be easier to use an extension of K _n=1^∞ a _n /b _n to find the limit. By an extension of K _n=1^∞ a _n /b _n we mean a continued fraction K _n=1^∞ c _n /d _n whose odd or even part is K _n=1^∞ a _n /b _n . One can then possibly find the limit in one of three ways:

ON THE DIFFUSION PHENOMENON OF QUASILINEAR HYPERBOLICWAVES

Introduction1.1.ConsiderthefollowingquasilinearhyperbolicCauchyproblemwithlineardamping{:;;!OTt=-:i<:,;>>L06,(11)wherexER",t20,anda(.)isasmoothfunctionsatisfyinga(y)~1 O(lyl")aslyl-0,orEN.(1.2)Thepurposeofthispaperistoshowthat,atleastwhenn53,theasymptoticprofileofthesolutionu(x,t)of(l.1)isgivenbythesolutionv(x,t)ofthecorrespondingparabolicproblem{:;.t>ivj:相似文献   

On the growth of mass for a viscous Hamilton-Jacobi equation

We investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equationu _t = Δu + |Δu| ^p ,t>0,x ∈ ℝ ^N , wherep≥1 andu(0,.)=u ₀≥0,u ₀≢0,u ₀∈L ¹. DenotingI _∞=lim _t→∞‖u(t)‖₁≤∞, we show that the asymptotic behavior of the mass can be classified along three cases as follows:

Subexponential ultracontractivity and L p ? L q functional calculus

 We prove that if a symmetric submarkovian semigroup (T _t ) _t>0 satisfies an estimate of the form

Clustering of linearly interacting diffusions and universality of their long-time limit distribution

Let K⊂ℝ ^d (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K ^Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX _i (t) = ∑ _j a(j−i) (X _j (t) −X _i (t))dt + σ (X _i (t))dB _i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a _S (i) = a(i) + a(−i) is recurrent, then each component X _i (∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a _S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a _S on Abelian groups Λ. Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000