On the index of invariant subspaces in Hilbert spaces of vector-valued analytic functions

Let H be a Hilbert space of analytic functions on the unit disc D with ‖Mz‖?1, where Mz denotes the operator of multiplication by the identity function on D. Under certain conditions on H it has been shown by Aleman, Richter and Sundberg that all invariant subspaces have index 1 if and only if for all f∈H, f?0 [A. Aleman, S. Richter, C. Sundberg, Analytic contractions and non-tangential limits, Trans. Amer. Math. Soc. 359 (7) (2007) 3369-3407]. We show that the natural counterpart to this statement in Hilbert spaces of Cn-valued analytic functions is false and prove a correct generalization of the theorem. In doing so we obtain new information on the boundary behavior of functions in such spaces, thereby improving the main result of [M. Carlsson, Boundary behavior in Hilbert spaces of vector-valued analytic functions, J. Funct. Anal. 247 (1) (2007)].     

Vector-valued Hardy spaces in non-smooth domains

We characterize the Radon-Nikodým property of a Banach space X in terms of the existence of non-tangential limits of X-valued harmonic functions u defined in a domain D⊂Rn, n>2, with Lipschitz boundary and belonging to maximal Hardy spaces. This extends the same result previously known for the unit disk of C. We also prove an atomic decomposition of the Borel X-valued measures in ∂D that arise as boundary limits of X-valued harmonic functions whose non-tangential maximal function is integrable with respect to harmonic measure of ∂D.     

Wandering subspaces and the Beurling type Theorem I

An elementary proof of the Aleman, Richter and Sundberg theorem concerning the invariant subspaces of the Bergman space is given.     

Beurling type theorem on the Bergman space via the Hardy space of the bidisk

In this paper,by lifting the Bergman shift as the compression of an isometry on a subspace of the Hardy space of the bidisk,we give a proof of the Beurling type theorem on the Bergman space of Aleman,Richter and Sundberg(1996) via the Hardy space of the bidisk.     

Wandering Subspace Theorems

We consider the approximation relation (0.1) below as well as some stronger statements phrased in terms of summability of the series (0.4). The principal new result is an estimate of Fourier multiplier type for this series. The results obtained also include strengthened forms of previous results by S. Richter [7], A. Aleman, S. Richter and C. Sundberg [1], and S. M. Shimorin [8].     

Spatial energy decay estimate in dynamical problems for a micropolar viscoelastic solid

We consider a linear micropolar viscoelastic solid occupying a domainB in dynamical conditions. First, on assuming thatB is of the kindB={∈R:x’ =(x ₁,x ₂)∈D(x ₃);x ₃∈R⁺⁺}, and that the body is subjected to boundary data different from zero only onD(0), we estimate for any fixedt>0, in terms of the initial and boundary data, the «energy» of the portions of the solid at distance greater thanz fromD(0)(g _t(z)) and its norm inL ¹(0,t) (G_t(z)). Moreover we show that, if there exists somez ₀≥0, such that past histories vanish onD(z) withz≥z ₀, then for any fixedt>0 the points (x’’, z) withz?z ₀≥Vt are at rest, while forz?z ₀≤Vt, G_t(z) decays withz?z ₀, the decay rate being described by the factor $1 - \frac{{z - z_0 }}{{Vt}}$ .V is a computable positive constant depending on the relaxation functions, the mass density and the microinertial tensor. Finally these last results are extended to more general domains under the hypothesis that the initial and boundary data have a bounded support. In our analysis we make use of a Maximal Free Energy which allows us to impose very mild restrictions on the relaxation functions.     

Generalized 3G theorem and application to relativistic stable process on non-smooth open sets

Let G(x,y) and GD(x,y) be the Green functions of rotationally invariant symmetric α-stable process in Rd and in an open set D, respectively, where 0<α<2. The inequality GD(x,y)GD(y,z)/GD(x,z)?c(G(x,y)+G(y,z)) is a very useful tool in studying (local) Schrödinger operators. When the above inequality is true with c=c(D)∈(0,∞), then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded κ-fat open set, which includes a bounded John domain. The 3G we consider is of the form GD(x,y)GD(z,w)/GD(x,w), where y may be different from z. When y=z, we recover the usual 3G. The 3G form GD(x,y)GD(z,w)/GD(x,w) appears in non-local Schrödinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on κ-fat open sets. As an application, we discuss relativistic α-stable processes (relativistic Hamiltonian when α=1) in κ-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic α-stable processes in κ-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in κ-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Lévy processes.     

On the radius of α-convexity of certain classes of starlike functions

LetM (α) denote the class of α-convex functions, α real, that is the class of analytic functions? (z) =z + Σ _n=2/^∞ a _n z ⁿ in the unit discD = {z: |z | < 1} which satisfies inD the condition ?′ (z) ?(z)/z ≠ 0 and $$\operatorname{Re} \left\{ {(1 - a) \frac{{z f'(z)}}{{f (z)}} + a \left( {1 + \frac{{z f''(z)}}{{f' (z)}}} \right)} \right\} > 0. Let W (a) $$ denote the class of meromorphic α-convex functions. α real, that is the class of analytic functions ? (z) =z ^?1 + Σ _n=0/^∞ b _n z ⁿ inD* = {z: 0 < |z | < 1} which satisfies inD* the conditionsz?′(z)/?(z) ≠ 0 and $$\operatorname{Re} \left\{ {(1 - a) \frac{{z\phi ' (z)}}{{\phi (z)}} + a \left( {1 + \frac{{z\phi ''(z)}}{{\phi ' (z)}}} \right)} \right\}< 0. $$ In this paper we obtain the relation betweenM (a) and W(α). The radius of α-convexity for certain classes of starlike functions is also obtained.     

Weyl-Titchmarsh functions of vector-valued Sturm-Liouville operators on the unit interval

The matrix-valued Weyl-Titchmarsh functions M(λ) of vector-valued Sturm-Liouville operators on the unit interval with the Dirichlet boundary conditions are considered. The collection of the eigenvalues (i.e., poles of M(λ)) and the residues of M(λ) is called the spectral data of the operator. The complete characterization of spectral data (or, equivalently, N×N Weyl-Titchmarsh functions) corresponding to N×N self-adjoint square-integrable matrix-valued potentials is given, if all N eigenvalues of the averaged potential are distinct.     

The support points of several classes of analytic functions with fixed coefficients

Let B denote the set of functions ?(z) that are analytic in the unit disk D and satisfy |?(z)|?1(|z|<1). Let P denote the set of functions p(z) that are analytic in D and satisfy p(0)=1 and Rep(z)>0(|z|<1). Let T denote the set of functions f(z) that are analytic in D, normalized by f(0)=0 and f^′(0)=1 and satisfy that f(z) is real if and only if z is real (|z|<1). In this article we investigate the support points of the subclasses of B, P and T of functions with fixed coefficients.     

A Laurent-type expansion for solutions of Stokes' equation in sectorial regions

We study solutions of Stokes' equation in a regionD (in the complex plane) being the intersection of a sector with vertex at the origin and a ring about the origin, subject to no-slip boundary conditions on the radial boundaries ofD. Using a result of Kratz and Peyerimhoff, we represent solutionsv(z) by means of two analytic functionsv ₁(z) andv ₂(z), and for these we obtain expansions into infinite series, quite analogous to Laurent series, but in complex powers ofz, the exponents depending upon the angular opening ofD. Forv(z), this leads to an expansion quite analogous to the one stated without proof by Moffat in 1964 in a more special situation.     

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.     

Strong Hypercontractivity and Relative Subharmonicity

A Hermitian metric, g, on a complex manifold, M, together with a smooth probability measure, μ, on M determine minimal and maximal Dirichlet forms, Q_D and Q_max, given by Q(f)=∫_M g(grad f(z), grad f(z)) dμ(z). Q_D is the form closure of Q on C^∞_c(M) and Q_max is the form closure of Q on C¹_b(M). The corresponding operators, A_D and A_max, generate semigroups having standard hypercontractivity properties in the scale of L^p spaces, p>1, when the corresponding form, Q, satisfies a logarithmic Sobolev inequality. It was shown by the author (1999, Acta Math.182, 159-206) that the semigroup e^−tA_D has even stronger hypercontractivity properties when restricted to certain holomorphic subspaces of L^p. These results are extended here to A_max. When (M, g) is not complete it is necessary that the elliptic differential operator A_max degenerate on the boundary of M. A second proof of these strong hypercontractive inequalities for both A_D and A_max is given, which depends on an extension of the submean value property of subharmonic functions. The Riemann surface for z^1/n and the weighted Bergman spaces in the unit disc are given as examples.     

On the Carleson duality

As a tool for solving the Neumann problem for divergence-form equations, Kenig and Pipher introduced the space ${\mathcal{X}}$ of functions on the half-space, such that the non-tangential maximal function of their L ₂ Whitney averages belongs to L ₂ on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of ${\mathcal{X}}$ , and characterize the pointwise multipliers from ${\mathcal{X}}$ to L ₂ on the half-space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to L _p generalizations of the space ${\mathcal{X}}$ . Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.     

Non-linear extremal problems for analytic two-dimensional Riemann-Stieltjes integrals

We examine certain non-linear extremal problems for two-dimensional Riemann-Stieltjes integrals \(\varphi (z) \equiv \int {_D \int {g(z,\zeta )d\mu (\zeta ),} \zeta \in D \equiv [\zeta |\left| \zeta \right|} \leqslant 1]\) ,z∈Δ≡[z‖|z|<1] whereg(z, ζ) is a continuous function in (z, ζ)∈[Δ×D] and an analytic function forz∈Δ and μ(ζ) is a unit mass measure onD. In particular, if the mass is distributed on the segment [a, b], we obtain the well-known Ruscheweyh results for the one-dimensional Riemann-Stieltjes integrals \(\varphi (z) \equiv \int_a^b {g(z,t)d\mu (t),z \in \Delta } \) . In particular, ifg(z,σ)≡z/(1—zσ), we determine the maximal domain of univalence and the radii of starlikeness and convexity of order α, ?∞<α<1, of the corresponding functions ?(z). A particular study is made of the functions of classesS ₁(D) andS ₂(D) which is similar to the study of the functions of the corresponding classesS ₁(C) andS ₂(C) of Schwarz analytic functions. In addition to obtaining maximal domains of univalence, we also determine the unique extremal functions for each of the functional studied.     

An explicit extension formula of bounded holomorphic functions from analytic varieties to strictly convex domains

We consider a strictly convex domain D ⁿ and m holomorphic functions, φ₁,…, φ_m, in a domain . We set V = {z ε Ω: φ₁(z) = ··· = φ_m(z) = 0}, M = V ∩ D and ∂M = V ∩ ∂D. Under the assumptions that the variety V has no singular point on ∂M and that V meets ∂D transversally we construct an explicit kernel K(ζ, z) defined for ζ ε ∂M and z ε D so that the integral operator Ef(z) = ∝ _{ζ ε ∂M} f(ζ) K(ζ, z) (z ε D), defined for f ε H^∞(M) (using the boundary values f(ζ) for a.e. ζ ε ∂M), is an extension operator, i.e., Ef(z) = f(z) for z ε M and furthermore E is a bounded operator from H^∞ to H^∞(D).     

Characterization of solutions to the generalized Cauchy-Riemann system

For a generalized biaxially symmetric potential U on a semi-disk D⁺, a harmonic conjugate V is defined by the generalized Cauchy-Riemann system. There is an associated boundary value theory for the Dirichlet problem. The converse to the Dirichlet problem is considered by determining the boundary functions to which U and V converge. The unique limits are hyperfunctions on the ?D⁺. In fact, the space of hyperfunctions is isomorphic to the spaces of generalized biaxially symmetric potentials and their harmonic conjugates. A representation theorem is given for U and V terms of convolutions of certain Poisson kernels with continuous functions that satisfy a growth condition on the ?D⁺.     

Comportement asymptotique des fonctions harmoniques en courbure négative

LetM be a complete simply connected Riemannian manifold whose sectional curvatures are bounded between two negative constants. It is shown that, for a given harmonic function onM, non-tangential properties of convergence, boundedness and finiteness of energy are equivalent for almost every point of the geometric boundary. This is a “geometric” analogue of Calderón-Stein theorem in the euclidean half-space. The proof is using Brownian motion, like J. Brossard's one for the euclidean case.     

An improvement of Montel's criterion

Let F be a family of holomorphic functions in a domain D, and let a(z), b(z) be two holomorphic functions in D such that a(z)?b(z), and a(z)?a^′(z) or b(z)?b^′(z). In this paper, we prove that: if, for each f∈F, f(z)−a(z) and f(z)−b(z) have no common zeros, f^′(z)=a(z) whenever f(z)=a(z), and f^′(z)=b(z) whenever f(z)=b(z) in D, then F is normal in D. This result improves and generalizes the classical Montel's normality criterion, and the related results of Pang, Fang and the first author. Some examples are given to show the sharpness of our result.     

On mappings related to the gradient of the conformal radius

We establish a criterion for the gradient ?R(D, z) of the conformal radius of a convex domain D to be conformal: the boundary ?D must be a circle. We obtain estimates for the coefficients K(r) for the K(r)-quasiconformal mappings ?R(D, z), D(r) ? D, 0 < r < 1, and supplement the results of Avkhadiev and Wirths concerning the structure of the boundary under diffeomorphic mappings of the domain D.