Poletsky–Stessin Hardy Spaces on Complex Ellipsoids in $$\mathbb {C}^{n}$$ C n

We provide a precise description of the lattice of invariant subspaces of composition operators acting on the classical Hardy space, whose inducing symbol is a parabolic non-automorphism.This is achieved with an explicit isomorphism between the Hardy space and the Sobolev Banach algebra W 1,2 [0, ∞) that induces a bijection between the lattice of the composition operator and the closed ideals of W 1,2 [0, ∞).In particular, each invariant subspace of parabolic non-automorphism composition operator always consists of the closed span of a set of eigenfunctions.As a consequence, such composition operators have no non-trivial reducing subspaces.For the sake of completeness, we also include a characterization of the closed ideals of the Banach algebra W 1,2 [0, ∞).Although such a characterization is known, the proof we provide here is somehow different.

Isoperimetric type inequalities for harmonic functions

In this paper, the authors study the boundedness of the operator [μΩ,b], the commutator generated by a functionb∈Lip β (ℝn)(0<β≤1) and the Marcinkiewicz integrals \gm\gW, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω∈Lipα(S n−1)(0<≤1).

In this paper, we will study the Hardy and BMO spaces associated to the generalized Hardy operator L_{alpha }= (-Delta )^{alpha /2}+a|x|^{-alpha }. Similarly to the classical Hardy and BMO spaces, we will prove that our new function spaces will enjoy some important results such as molecular decomposition and duality. As applications, we show the boundedness of the spectral multiplier of Laplace transform type and the Sobolev norm inequalities involving the generalized Hardy operator.

Abstract. Let $\varphi : \mathbb R^n \times [0,\infty) \rightarrow [0,\infty)$ satisfy that $\varphi(x,\cdot)$ is an Orlicz function for any given $x \in \mathbb R^n$, and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty$ weight uniformly in $t \in (0,\infty)$. The weak Musielak-Orlicz Hardy space $WH^\varphi(\mathbb R^n)$ is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak-Orlicz space $WL^\varphi(\mathbb R^n)$. In this paper, we discuss the boundedness of the Calderon-Zygmund operator with variable kernel from $WH^\varphi(\mathbb R^n)$ to $WL^\varphi(\mathbb R^n)$. These results are new even for the classical weighted weak Hardy space and probably new for the classical weak Hardy space.

Let X be a ball quasi-Banach function space satisfying some mild assumptions and H X (R n ) the Hardy space associated with X. In this article, the authors introduce both the Hardy space H X (R n+1 + ) of harmonic functions and the Hardy space H X (R n+1 + ) of harmonic vectors, associated with X, and then establish the isomorphisms among , where H X,2 (R n+1 + ) and H X,2 (R n+1 + ) are, respectively, certain subspaces of H X (R n+1 + ) and H X (R n+1 + ). Using these isomorphisms, the authors establish the first order Riesz transform characterization of H X (R n ). The higher order Riesz transform characterization of H X (R n ) is also obtained. The results obtained in this article have a wide range of generality and can be applied to the classical Hardy space, the weighted Hardy space, the Herz-Hardy space, the Lorentz-Hardy space, the variable Hardy space, the mixed-norm Hardy space, the local generalized Herz-Hardy space, and the mixed-norm Herz-Hardy space.

We give a complete description of the possible ranges of real Smirnov functions (quotients of two bounded analytic functions on the open unit disk where the denominator is outer and such that the radial boundary values are real almost everywhere on the unit circle). Our techniques use the theory of unbounded symmetric Toeplitz operators, some general theory of unbounded symmetric operators, classical Hardy spaces, and an application of the uniformization theorem. In addition, we completely characterize the possible valences for these real Smirnov functions when the valence is finite. To do so we construct Riemann surfaces we call disk trees by welding together copies of the unit disk and its complement in the Riemann sphere. We also make use of certain trees we call valence trees that mirror the structure of disk trees.

Let Hp(m), 0 < p ^ oo, be the Hardy spaces on a quotient K of the Bohr group. In this paper we completely determine the isometries of Hp{m), p Φ 2, onto itself. Our result is a generalization of a recent work of Muhly who determined the isometries of Hp(m) onto itself under the assumption that the dual group of K is countable, and it may be regarded as a partial answer to a question posed by Muhly. 1* Introduction* Many results have been obtained concerning isometries of Hardy spaces in the theory of uniform algebras. The most fundamental result in this direction is due to de Leeuw, Rudin, and Wermer [2], which states that an automorphism of the classical Hardy space H°°{T) is induced via composition with the unit circle T of a fractional linear transformation of the unit disc onto itself. Their work was carried on independent of Nagasawa [13], who also described the isometries of ίί oo(Tr) onto itself. On the other hand, Arens [1] completely determined the automorphisms of the uniform algebra of analytic functions on a compact abelian group K whose dual group Γ is archimedean ordered (cf. [11]). This result was extended by Muhly [11] to the uniform algebra of analytic functions induced by a flow which has no periodic orbits. Moreover Muhly [12] has recently given, among other things, the following interesting generalization of this result of Arens to the case of isometries of Hardy spaces Hp(m), p Φ 2, on K: Under the assumption that Γ is countable, every isometry of Hp(m), p Φ 2, is induced via composition with an affine map of K such that the adjoint of the additive factor of this map preserves the order of Γ. The purpose of this paper is to remove the assumption on Γ. This result provide a partial positive answer to the following question posed by Muhly in [12; §5]: Is it possible to describe the isometries of ergodic Hardy spaces

Average radial integrability spaces of analytic functions

The paper deals with the invertibility of Toeplitz plus Hankel operators T(a)+H(b) acting on classical Hardy spaces on the unit circle T. It is supposed that the generating functions a and b satisfy the condition a(t)a(1/t) = b(t)b(1/t), t ∈ T. Special attention is paid to the case of piecewise continuous generating functions. In some cases the dimensions of null spaces of the operator T(a) + H(b) and its adjoint are described. Fredholm properties of Toeplitz plus Hankel operatorsT(a)+H(b) with piecewise continuous generating functions a and b have been studied for many years. These operators are considered in various Banach and Hilbert spaces, and the results ob- tained show that the structure of the algebras generated by such operators is much more complicated than the structure of the algebras generated by one-dimensional singular integral operators with piecewise continuous coefficients defined on closed smooth curves. Moreover, calculating the index of Toeplitz plus Hankel operators, one encounters even more difficult problems, and the difficulti es grow if one attempts to study invertibility or one-sided invertibility of such operators. It is worth noting that, in general, Fredholm operators T(a) +H(b) are not one-sided invertible. Nev- ertheless, a few works where one-sided invertibility was discussed, have appeared in literature recently. They are mainly concerned with two special cases of Toeplitz plus Hankel operators—viz. with the operators having the form M(a) := T(a) +H(a) or the form f

We construct a non-exposed extreme function f of the unit ball of H1, the classical Hardy space on the unit disc of the plane, which has the property: f(z)/(1−q(z))2 ∉ H1 for any nonconstant inner function q(z). This function constitutes a counterexample to a conjecture in D. Sarason [7].

Via the boundedness of intrinsic g-functions from the Hardy spaces with variable exponent, ℋp(·)(ℝn), into Lebesgue spaces with variable exponent, Lp(·)(ℝn), and establishing some estimates on a discrete Littlewood-Paley g-function and a Peetre-type maximal function, we obtain several equivalent characterizations of ℋp(·)(ℝn) in terms of wavelets, which extend the wavelet characterizations of the classical Hardy spaces. The main ingredients are that, we overcome the difficulties of the quasi-norms of ℋp(·)(ℝn) by elaborately using an observation and the Fefferman-Stein vector-valued maximal inequality on Lp(·)(ℝn), and also overcome the difficulty of the failure of q = 2 in the atomic decomposition of ℋp(·)(ℝn) by a known idea.

Sampling theorem for bandlimited Hardy space functions generated by Regge problem

Functions in Hardy spaces on multiply-connected domains in the plane are given an explicit characterization in terms of a boundary condition inspired by the two-dimensional Ising model. The key underlying property is the positivity of a certain operator constructed inductively on the number of components of the boundary.

In this chapter we study several problems related to invariant subspaces of Bergman spaces. First, we show by explicit examples that there exist invariant subspaces of index n for all 0 ≤ n ≤ +∞. Then we prove a theorem that can be considered an analogue to the classical Beurling’s theorem on invariant subspaces of the Hardy space. It states that in the spaces A2 α, with — 1 < α ≤ 0, each invariant subspace I is generated by I ⊖z I. In the classical Hardy space case, I ⊖zI is one-dimensional, and spanned by a classical inner function (unless I = {0}). In A2α, the dimension may be bigger, but all elements of I ⊖ zl of unit norm are A2α-inner functions