Smirnov Spaces Research Articles

Weighted composition operators on Hardy–Smirnov spaces

Abstract Operators of type f → ψf ◦ φ acting on function spaces are called weighted composition operators. If the weight function ψ is the constant function 1, then they are called composition operators. We consider weighted composition operators acting on Hardy–Smirnov spaces and prove that their unitarily invariant properties are reducible to the study of weighted composition operators on the classical Hardy space over a disc. We give examples of such results, for instance proving that Forelli’s theorem saying that the isometries of non–Hilbert Hardy spaces over the unit disc need to be special weighted composition operators extends to all non–Hilbert Hardy–Smirnov spaces. A thorough study of boundedness of weighted composition operators is performed.

Liouville dynamical percolation

We construct and analyze a continuum dynamical percolation process which evolves in a random environment given by a \(\gamma \)-Liouville measure. The homogeneous counterpart of this process describes the scaling limit of discrete dynamical percolation on the rescaled triangular lattice. Our focus here is to study the same limiting dynamics, but where the speed of microscopic updates is highly inhomogeneous in space and is driven by the \(\gamma \)-Liouville measure associated with a two-dimensional log-correlated field h. Roughly speaking, this continuum percolation process evolves very rapidly where the field h is high and barely moves where the field h is low. Our main results can be summarized as follows. First, we build this inhomogeneous dynamical percolation, which we call \(\gamma \)-Liouville dynamical percolation (LDP), by taking the scaling limit of the associated process on the triangular lattice. We work with three different regimes each requiring different tools: \(\gamma \in [0,2-\sqrt{5/2})\), \(\gamma \in [2-\sqrt{5/2}, \sqrt{3/2})\), and \(\gamma \in (\sqrt{3/2},2)\). When \(\gamma <\sqrt{3/2}\), we prove that \(\gamma \)-LDP is mixing in the Schramm–Smirnov space as \(t\rightarrow \infty \), quenched in the log-correlated field h. On the contrary, when \(\gamma >\sqrt{3/2}\) the process is frozen in time. The ergodicity result is a crucial piece of the Cardy embedding project of the second and fourth coauthors, where LDP for \(\gamma =\sqrt{1/6}\) is used to study the scaling limit of a variant of dynamical percolation on uniform triangulations. When \(\gamma <\sqrt{3/4}\), we obtain quantitative bounds on the mixing of quad crossing events.

Double bases from generalized Faber polynomials with complex-valued coefficients in weighted Lebesgue spaces

In the paper it is considered the generalized Faber polynomials defined inside and outside a regular curve on the complex plane. The weighted Smirnov spaces corresponding to bounded and unbounded regions are defined. It is proved that the generalized Faber polynomials forms a basis in weighted Smirnov spaces, if the weight function satisfies the Muckenhoupt condition on the regular curve. The double system of generalized Faber polynomials with complex-valued coefficients is also considered and the basis properties of such a system in weighted Lebesgue spaces over regular curves are studied.

Reachable states and holomorphic function spaces for the 1-D heat equation

The description of the reachable states of the heat equation is one of the central questions in control theory. The aim of this work is to present new results for the 1-D heat equation with boundary control on the segment [0,π]. In this situation it is known that the reachable states are holomorphic in a square D the diagonal of which is given by [0,π]. The most precise results obtained recently say that the reachable space is contained between two well known spaces of analytic functions: the Smirnov space E2(D) and the Bergman space A2(D). We show that the space of reachable states is exactly the sum of two Bergman spaces on sectors the intersection of which is D. In order to get a more precise information on this sum of Bergman spaces, we also prove that it contains a certain weighted Bergman space on D.

From the reachable space of the heat equation to Hilbert spaces of holomorphic functions

This work considers systems described by the heat equation on the interval [0, π] with L^2 boundary controls and it studies the reachable space at some instant τ > 0. The main results assert that this space is generally sandwiched between two Hilbert spaces of holomorphic functions defined on a square in the complex plane and which has [0, π] as one of the diagonals. More precisely, in the case Dirichlet boundary controls acting at both ends we prove that the reachable space contains the Smirnov space and it is contained in the Bergman space associated to the above mentioned square. The methodology, quite different of the one employed in previous literature, is a direct one. We first represent the input-to-state map as an integral operator whose kernel is a sum of Gaussians and then we study the range of this operator by combining the theory of Riesz bases for Smirnov spaces in polygons and the theory developed by Aikawa, Hayashi and Saitoh on the range of integral transforms, in particular those associated with the heat kernel.

Closure of an exponential system in some Hardy–Smirnov spaces

Let $$\Omega $$ be an open, simply connected, bounded subset of the complex plane $$\mathbb {C}$$ with a rectifiable boundary $${\partial {\Omega }}$$ . We investigate the relation between the Hardy–Smirnov space $$H^2(\Omega )$$ and the closed span of the exponential system $$\{e^{nz}\}_{n=1}^{\infty }$$ with respect to the Hardy–Smirnov norm $$\Vert \cdot \Vert _\Omega $$ . Depending on the “height” of $$\Omega $$ , the two spaces may coincide or not. In the latter case we characterize the closure of the system by proving that any element f extends analytically in some half-plane as a Dirichlet series $$f(z)=\sum _{n=1}^{\infty } c_n e^{nz}$$ . Finally we consider the converse problem: assuming that such a Dirichlet series is in $$H^2(\Omega )$$ , we provide some sufficient conditions for f to be in the closed linear span of the system with respect to the Hardy–Smirnov norm $$\Vert \cdot \Vert _\Omega $$ .

Approximation in weighted rearrangement invariant Smirnov spaces

In the present work, we investigate the approximation problems in weighted rearrangement invariant Smirnov spaces. We prove a direct theorem for polynomial approximation of functions in certain subclasses of weighted rearrangement invariant Smirnov spaces

Trace ideal criteria for embeddings and composition operators on model spaces

Let Kϑ be a model space generated by an inner function ϑ. We study the Schatten class membership of composition operators Cφ:Kϑ→H2(D) with a holomorphic function φ:D→D, and, more generally, of embeddings Iμ:Kθ→L2(μ) with a positive measure μ in D¯. In the case of one-component inner functions ϑ we show that the problem can be reduced to the study of natural extensions of I and Cφ to the Hardy–Smirnov space E2(D) in some domain D⊃D. In particular, we obtain a characterization of Schatten membership of Cφ in terms of Nevanlinna counting function. By example this characterization does not hold true for general ϑ.

Generalized growth and approximation of entire function solution of Helmholtz equation in Banach spaces

In this paper, we study the generalized growth and polynomial approximation of entire function solution of Helmholtz equation in \(R^2\) in Smirnov spaces [\(\varepsilon _p (S)\) and \(\varepsilon ^{^\prime }_p(S), 1\le p\le \infty \)] where S is finitely simply connected domain in the complex plane with the boundary that belongs to the Al’per class (Izv AN SSSR Ser Matem 19(3):423–444, 1955). Some bounds on generalized order and generalized type of entire solution of Helmholtz equation have been obtained in terms of the coefficients and approximation errors using function theoretic methods. Our results extend and improve the results of Kumar (J Appl Anal 18:179–196, 2012).

Self-commutators of Toeplitz operators and isoperimetric inequalities

For a hyponormal operator, C. R. Putnam's inequality gives an upper bound on the norm of its self-commutator. In the special case of a Toeplitz operator with analytic symbol in the Smirnov space of a domain, there is also a geometric lower bound shown by D. Khavinson (1985) that when combined with Putnam's inequality implies the classical isoperimetric inequality. For a nontrivial domain, we compare these estimates to exact results. Then we consider such operators acting on the Bergman space of a domain, and we obtain lower bounds that also reflect the geometry of the domain. When combined with Putnam's inequality they give rise to the Faber-Krahn inequality for the fundamental frequency of a domain and the Saint-Venant inequality for the torsional rigidity (but with non-sharp constants). We conjecture an improved version of Putnam's inequality within this restricted setting.

Approximation to the function z α by rational fractions in a domain with zero external angle

Rational approximations to the function zα, α ∈ ℝ \ ℤ, were studied by Newman, Gonchar, Bulanov, Vyacheslavov, Andersson, Stahl, and others. The present paper deals with the order of best rational approximations to this function in a domain with zero external angle and vertex at the point z = 0. In particular, the obtained results show that the conditions imposed on the boundary of the domain in the Jackson-type inequality proved by the author in 2001 for the best rational approximations in Smirnov spaces cannot be weakened significantly.

Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth

Abstract This work deals with basic approximation problems such as direct, inverse and simultaneous theorems of trigonometric approximation of functions of weighted Lebesgue spaces with a variable exponent on weights satisfying a variable Muckenhoupt A p(·) type condition. Several applications of these results help us transfer the approximation results for weighted variable Smirnov spaces of functions defined on sufficiently smooth finite domains of complex plane ℂ.

On converse theorems of trigonometric approximation in weighted variable exponent Lebesgue spaces

[1] R. Akgu¨n, Approximating polynomials for functions of weighted Smirnov-Orlicz spaces, J.Funct. Spaces Appl. (to appear).[2] —, Sharp Jackson and converse theorems of trigonometric approximation in weightedLebesgue spaces, Proc. A. Razmadze Math. Inst. 152 (2010), 1–18.[3] —, Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces withnonstandard growth, Georgian Math. J. (to appear).

Summation of p-Faber series by the Abel–poisson method in the integral metric

We establish conditions on the boundary $$ \Gamma $$ of a bounded simply connected domain $$ \Omega \subset \mathbb{C} $$ under which the p-Faber series of an arbitrary function from the Smirnov space $$ {E_p}\left( \Omega \right),1 \leqslant p < \infty $$ , can be summed by the Abel–Poisson method on the boundary of the domain up to the limit values of the function itself in the metric of the space $$ {L_p}\left( \Gamma \right) $$ .

On dimensional properties of the generalized Smirnov's spaces

We show that the transfinite inductive dimensions modulo P P -trind and P -trInd introduced in M.G. Charalambous (1997) [2] differ by simple spaces, where P is the absolutely additive Borel class A ( α ) or the absolutely multiplicative Borel class M ( α ) , 0 ⩽ α < ω 1 .

MULTIPLIER THEOREMS IN WEIGHTED SMIRNOV SPACES

The analogues of Marcinkiewicz multiplier theorem and Littlewood-Paley theorem are proved for p-Faber series in weighted Smirnov spaces defined on bounded and unbounded components of a rectifiable Jordan curve.

Compact Hankel Forms on Planar Domains

A Hankel form on a Hilbert function space is a bounded, symmetric, bilinear form [., .] satisfying [fx, y] = [x, fy] for a class of multipliers f. We prove analogs of Weyl–Horn and Ky Fan inequalities for compact Hankel forms, and apply them to estimate the related eigenvalues, both for Hardy–Smirnov and Bergman spaces norms associated to multiply connected planar domains. In the case of the unit disk, we investigate the asymptotic of some measures constructed by eigenfunctions of Hankel operators with certain Markov functions as symbols.

Approximation by polynomials and rational functions in weighted rearrangement invariant spaces

Let Γ be a Dini-smooth curve in the complex plane, and let G : = Int Γ . We prove some direct and inverse theorems of approximation theory by algebraic polynomials and rational functions in the weighted rearrangement invariant Smirnov spaces E X ( G , ω ) defined on G.

Improved Inverse Theorems in Weighted Lebesgue and Smirnov Spaces

The improvement of the inverse estimation of approximation theory by trigonometric polynomials in the weighted Lebesgue spaces was obtained and its application in the weighted Smirnov spaces was considered.

Weighted Composition Operators on H 2 and Applications

Operators on function spaces acting by composition to the right with a fixed selfmap φ of some set are called composition operators of symbol φ. A weighted composition operator is an operator equal to a composition operator followed by a multiplication operator. We summarize the basic properties of bounded and compact weighted composition operators on the Hilbert Hardy space on the open unit disk and use them to study composition operators on Hardy–Smirnov spaces.