Normalized Lebesgue Measure Research Articles

A prediction problem in $L^2 (w)$

For a nonnegative integrable weight function w on the unit circle T, we provide an expression for p = 2, in terms of the series coefficients of the outer function of w, for the weighted L P distance inf f ∫ T |1 - f[Pwdμ, where μ is the normalized Lebesgue measure and f ranges over trigonometric polynomials with frequencies in [{..., -3, -2, -1}\{-n}]Ufm}, m > 0, n ≥ 2. The problem is open for p # 2.

The Complete Continuity Property in Banach Spaces

Let X be a complex Banach space. We show that the following are equivalent: (i) X has the complete continuity property, (ii) for every (or equivalently for some) 1 < p < ∞, for f ∈ h(D, X) and rn ↑ 1, the sequence frn is p-Pettis-Cauchy, where frn is defined by frn(t) = f(rne ) for t ∈ [0, 2π], (iii) for every (or equivalently for some) 1 < p < ∞, for every μ ∈ V (X), the bounded linear operator T : L(0, 2π) → X defined by Tφ = ∫ 2π 0 φdμ is compact, where 1/q+1/p = 1, (iv) for every (or equivalently for some) 1 < p <∞, each μ ∈ V (X) has a relatively compact range. Through this note (X, ‖ · ‖) denotes a complex Banach space, D denotes the open unit disc in the complex plane. λ will be the normalized Lebesgue measure on [0, 2π]. For a Banach space Y , we denote by BY the closed unit ball of Y . Given 1 < p <∞ we denote by hp(D, X) the space of all X-valued harmonic functions f on D such that ‖f‖p = sup 0<r<1 ( ∫ 2π 0 ‖f(re)‖dλ(t)) <∞. h∞(D, X) will be the space of all X-valued bounded harmonic functions on D equipped with the norm ‖f‖∞ = supz∈D ‖f(z)‖. For f ∈ hp(D, X) and n ∈ Z, we define the Fourier coefficient f(n) by

COMPOSITION OPERATORS ON THE PRIVALOV SPACES OF THE UNIT BALL OF ℂn

Let B and S be the unit ball and the unit sphere in <TEX>$\mathbb{C}^n$</TEX>, respectively. Let <TEX>${\sigma}$</TEX> be the normalized Lebesgue measure on S. Define the Privalov spaces <TEX>$N^P(B)\;(1\;<\;p\;<\;{\infty})$</TEX> by <TEX>$$N^P(B)\;=\;\{\;f\;{\in}\;H(B) : \sup_{0<r<1}{\Bigint}_S \{log(1+|f(\gamma\zeta)|)\}^{p}d\sigma(\zeta)\;<\;\infty\;\}$$</TEX> where H(B) is the space of all holomorphic functions in B. Let <TEX>${\varphi}$</TEX> be a holomorphic self-map of B. Let <TEX>${\mu}$</TEX> denote the pull-back measure <TEX>${\sigma}o({\varphi}^{\ast})^{-1}$</TEX>. In this paper, we prove that the composition operator <TEX>$C_{\varphi}$</TEX> is metrically bounded on <TEX>$N^P$</TEX>(B) if and only if <TEX>${\mu}(S(\zeta,\delta)){\le}C{\delta}^n$</TEX> for some constant C and <TEX>$C_{\varphi}$</TEX> is metrically compact on <TEX>$N^P(B)$</TEX> if and only if <TEX>${\mu}(S(\zeta,\delta))=o({\delta}^n)$</TEX> as <TEX>${\delta}\;{\downarrow}\;0$</TEX> uniformly in <TEX>${\zeta}\;\in\;S</TEX>. Our results are an analogous results for Mac Cluer's Carleson-measure criterion for the boundedness or compactness of <TEX>$C_{\varphi}$</TEX> on the Hardy spaces <TEX>$H^P(B)$</TEX>.

Scarred Eigenstates for Quantum Cat Maps of Minimal Periods

In this paper we construct a sequence of eigenfunctions of the ``quantum Arnold's cat map'' that, in the semiclassical limit, shows a strong scarring phenomenon on the periodic orbits of the dynamics. More precisely, those states have a semiclassical limit measure that is the sum of 1/2 the normalized Lebesgue measure on the torus plus 1/2 the normalized Dirac measure concentrated on any a priori given periodic orbit of the dynamics. It is known (the Schnirelman theorem) that ``most'' sequences of eigenfunctions equidistribute on the torus. The sequences we construct therefore provide an example of an exception to this general rule. Our method of construction and proof exploits the existence of special values of ¯h for which the quantum period of the map is relatively ``short'', and a sharp control on the evolution of coherent states up to this time scale. We also provide a pointwise description of these states in phase space, which uncovers their ``hyperbolic'' structure in the vicinity of the fixed points and yields more precise localization estimates.

A characterization of weighted Bergman-Orlicz spaces on the unit ball in Cn

AbstractLet B denote the unit ball in Cn, and ν the normalized Lebesgue measure on B. For α &gt; −1, define Here cα is a positive constant such that να(B) = 1. Let H(B) denote the space of all holomorphic functions in B. For a twice differentiable, nondecreasing, nonnegative strongly convex function ϕ on the real line R, define the Bergman-Orlicz space Aϕ(να) by In this paper we prove that a function f ∈ H(B) is in Aϕ(να) if and only if where is the radial derivative of f.

Measures induced by analytic functions and a problem of Walter Rudin

The measure μ φ \mu _\varphi induced by a bounded analytic function φ \varphi on the unit disk U U may be defined by μ φ ( E ) = m ( φ − 1 ( E ) ) \mu _\varphi (E)=m(\varphi ^{-1}(E)) , where m m is normalized Lebesgue measure on ∂ U \partial U . We discuss the problem of characterizing such measures, and produce some necessary conditions which we conjecture are sufficient. Our main results are a construction showing that our conjectured sufficient conditions are sufficient for a measure to be weakly approximable by induced measures, and a construction of a function φ \varphi , not a constant multiple of an inner function, whose induced measure is rotationally symmetric. This function is not inner, but satisfies ∫ φ ( e i θ ) m φ ( e i θ ) n ¯ d θ 2 π = 0 \int \varphi \left (e^{i\theta }\right )^m\overline {\varphi \left (e^{i\theta }\right )^n} \frac {d\theta }{2\pi }=0 if m ≠ n m\ne n , thus answering a question posed by Walter Rudin.

A CHARACTERIZATION OF WEIGHTED BERGMAN-PRIVALOV SPACES ON THE UNIT BALL OF Cn

Let B denote the unit ball in , and ν the normalized Lebesgue measure on B. For > -1, define (z) ＝ dν(z), z B. Here is a positive constant such that (B) ＝ 1. Let H(B) denote the space of all holomorphic functions in B. For , define the Bergman-Privalov space by = : $\int_B{log(1+\midf\mid)}^pdv_\alpha\; is in if and only if in the case 1＜p＜, or in the case p ＝ 1, where f is the gradient of f with respect to the Bergman metric on B. This is an analogous result to the characterization of the Hardy spaces by M. Stoll [18] and that of the Bergman spaces by C. Ouyang-W. Yang-R. Zhao [13].

Isometries of weighted Bergman-Privalov spaces on the unit ball of Cn

Let B denote the unit ball in Cn, and v the normalized Lebesgue measure on B. For α>-1, define dvα(z)=Γ(n+α+1)/{Γ(n+1)Γ(α+1)}(1-|z|2)αdv(z), z∈B. Let H(B) denote the space of holomorphic functions in B. For p≥1, define ANpνα=f∈HB:f≡∫Blog1+fpdνα1p<∞. ANpνα is an F-space with respect to the metric ρfg≡f-g. In this paper we prove that every linear isometry T of ANpνα into itself is of the form Tf=cf∘ψ for all f∈ANpνα , where c is a complex number with c=1 and ψ is a holomorphic self-map of B which is measure-preserving with respect to the measure να.

Differential properties for a class of Sobolev orthogonal polynomials

We study the orthogonal polynomials with respect to a Sobolev inner product of the following type: 〈f,g〉 s= ∫ 0 2π f( e iθ ) g( e iθ ) |B h( e iθ )| 2 dθ 2π + 1 λ ∫ 0 2π f′( e iθ ) g′( e iθ ) dθ 2π , z= e iθ , where B h ( z) is a complex polynomial of degree h, d θ/2 π is the normalized Lebesgue measure and λ is a positive real number. The asymptotic behavior in the complex plane, as well as the differential equations satisfied by the orthogonal polynomials are obtained. As an application, two differential problems are solved, one of them is like a Dirichlet boundary value problem.

Differential properties for Sobolev orthogonality on the unit circle

The aim of this paper is to study differential properties of the sequence of monic orthogonal polynomials with respect to the following Sobolev inner product: 〈f,g〉 s = ∫ 0 2 π f( e iθ ) g( e iθ ) dμ(θ)+ 1 λ ∫ 0 2 π f′( e iθ ) g′( e iθ ) dθ 2 π , where μ is a finite positive Borel measure on [0,2π] verifying the following conditions: the Carathéodory function associated with μ has an analytic extension outside the unit disk and the induced norm is equivalent to the Lebesgue norm in the space L 2. Here d θ/2π is the normalized Lebesgue measure and λ is a positive real number. The nonhomogeneous second-order differential equations satisfied by the sequence of monic Sobolev orthogonal polynomials are obtained. Moreover, as an application, a sample of Dirichlet boundary value problem is solved.

Boundedness of the Riesz projection on spaces with weights

Let ∂D be the unit circle in the complex plane, define the function χ on ∂D by χ(eiθ) = eiθ, and set P = {p : p = ∑Nk=−N ckχ}. Let σ be normalized Lebesgue measure on ∂D. The Riesz projection P+ is defined on P by the formula P+( ∑N k=−N ckχ k) = ∑N k=0 ckχ k. In [4], Paul Koosis proved: Theorem 1 (Koosis). Given a non-negative function w ∈ L1, there exists a non-negative, non-trivial function v ∈ L1 such that ∫

Markov Chains with Positive Transitions Are Not Determined by Any p -Marginals

We prove that if \(\) is a finite valued stationary Markov Chain with strictly positive probability transitions, then for any natural number p, there exists a continuum of finite valued non Markovian processes which have the p-marginal distributions of X and with positive entropy, whereas for an irrational rotation \(\) and essentially bounded real measurable function f with no zero Fourier coefficient on the unit circle with normalized Lebesgue measure, the process \(\) is uniquely determined by its three-dimensional distributions in the class of ergodic processes. We give also a family of Gaussian non-Markovian dynamical systems for which the symbolic dynamic associated to the time zero partition has the two-dimensional distributions of a reversible mixing Markov Chain.

Bernstein-szegö-lebesgue sobolev orthogonal polynomials on the unit circle

In this paper we consider the following contineous Sobolev inner product on the unit circle where λ>0,dμ(φ) is a Bernstein— Szegö measure and dφ/2π is the normalized Lebesgue measure on [0,2π] We study the corresponding sequence of orthogonal polynomials Algebraic properties,asymptotic behaviour and asymptotic distribution of zeros for such polynomials are obtained.

A note on weighted Bergman spaces and the Cesaro Operator

Let B denote the unit ball in ℂn, and dV(z) normalized Lebesgue measure on B. For α &gt; -1, define dVα(z) = (1 - \z\2)αdV(z). Let (B) denote the space of holomorhic functions on B, and for 0 &lt; p &lt; ∞, let p(dVα) denote Lp(dVα) ∩ (B). In this note we characterize p(dVα) as those functions in (B) whose images under the action of a certain set of differential operators lie in Lp(dVα). This is valid for 1 &lt; p &lt; oo. We also show that the Cesàro operator is bounded on p(dVα) for 0 &lt; p &lt; oo. Analogous results are given for the polydisc.

Fuchsian Groups, Quasiconformal Groups, and Conical Limit Sets

We construct examples showing that the normalized Lebesgue measure of the conical limit set of a uniformly quasiconformal group acting discontinuously on the disc may take any value between zero and one. This is in contrast to the cases of Fuchsian groups acting on the disc, conformal groups acting discontinuously on the ball in dimension three or higher, uniformly quasiconformal groups acting discontinuously on the ball in dimension three or higher, and discrete groups of biholomorphic mappings acting on the ball in several complex dimensions. In these cases the normalized Lebesgue measure is either zero or one.

A family of Sobolev orthogonal polynomials on the unit circle

The aim of this paper is to study the polynomials orthogonal with respect to the following Sobolev inner product: 〈P,Q〉 s 1 = ∫ 0 2π P( e iθ ) Q( e iθ ) dμ(θ)+ 1 λ ∫ 0 2π P′( e iθ ) Q′( e iθ ) dν(θ), z= e iθ , λ>0, where ν is the normalized Lebesgue measure and μ is a rational modification of ν. In this situation we analyse the algebraic results and the asymptotic behaviour of such orthogonal polynomials. Moreover some properties about the distribution of their zeros are given.

Conservative Exact Rational Maps of the Sphere

In this paper we investigate conditions for conservativity and exactness ofd-to-1 endomorphisms. We prove that all rational maps of degree greater than 1 are either conservative and exact with respect to normalized Lebesgue measure on the sphere, or have an attracting property inside the Julia set. We provide examples of rational maps which can be shown to be conservative and exact by using the previous result which was not formerly known to have this property.

Lebesgue Sobolev orthogonality on the unit circle

This paper is devoted to the study of asymptotic properties of the orthogonal polynomials with respect to a Sobolev inner product 〈f(z),g(z)〉 s = ∫ 0 2π f(e iθ)) g(e iθ) dμ(θ)+∑ k=1 p λ k∫ 0 2π f k(e iθ) g(e iθ) dθ 2π , z=e iθ, with d μ( θ) a finite positive Borel measure on [0, 2π] with an infinite set as support verifying the Szegő condition, λ 1 > 0, λ k ⩾ 0 ( k = 2,…, p) and dθ 2π the normalized Lebesgue measure on [0, 2π]. Our aim is to extend some previous results that we have obtained in [2, 3] when the measure μ belongs to the Bernstein-Szegő class and p = 1.

Rigged non-tangential maximal function associated with Toeplitz operators and Hankel operators

Let T denote the unit circle and dm the normalized Lebesgue measure on T . For 1 ≤ p ≤ ∞, L stands for L(T, dm). As usual, H is the Hardy subspace of L. Let P : L → H be the orthogonal projection. For f ∈ L, the Toeplitz operator Tf and the Hankel operator Hf are defined by the formulas Tfφ = Pfφ and Hfφ = (1 − P )fφ, φ ∈ H, whenever these expressions make sense. Thus the domains of Tf and Hf contain at least H∞. Let D be the open disc {z ∈ C : |z| < 1}. We denote the Poisson kernel associated with z ∈ D by Pz. That is, Pz(τ) = (1−|z|2)/|1− τ z|2. We write f(z) = ∫ T fPzdm, z ∈ D, for f ∈ L. In particular, for any measurable set E ⊂ T , χE(z) = ∫ E Pzdm is the value of the harmonic extension of χE at z. The well-known probabilistic interpretation of χE(z) may help put the results stated below in perspective: χE(z) is the probability that a Brownian walker starting at the point z will exit D through E. In this note we will address a question raised by Sarason in [5, 6], namely when is the product TgTf a bounded operator on H? This problem, which is non-trivial only if at least one symbol function is unbounded, arose from Sarason’s study of exposed points in the unit ball of H. He showed that, for outer functions f, g ∈ H, a necessary condition for TgTf to be bounded is that sup |z|<1 ∫

Pointwise Ergodic Theorems for Radial Averages on the Heisenberg Group

LetH=Hn=Cn×R denote the Heisenberg group, and letσrdenote the normalized Lebesgue measure on the sphere {(z, 0):|z|=r}. Let (X, B, m) be a standard Borel probability space on whichHacts measurably and ergodically by measure preserving transformations, and letπ(σr) denote the operator canonically associated withσronLp(X). We prove maximal and pointwise ergodic theorems inLp, for radial averagesσron the Heisenberg groupHn,n>1. The results are best possible for actions of the reduced Heisenberg group. The method of proof is to use the spectral theory of the Banach algebra of radial measures on the group and decay estimates for its characters to establish maximal inequalities using spectral methods, in particular Littlewood–Paley–Stein square-functions and analytic interpolation.