Hardy Space Research Articles

Extremal Distance and Conformal Mappings in Hardy and Bergman Spaces

We prove necessary and sufficient integral conditions involving extremal distance for a conformal mapping of the unit disk to belong to the Hardy or weighted Bergman spaces. We also give characterizations for the Hardy number and the Bergman number of a simply connected domain in terms of extremal distance.

Alternative proofs of Mandrekar’s theorem

We present two alternative proofs of Mandrekar’s theorem, which states that an invariant subspace of the Hardy space on the bidisc is of Beurling type precisely when the shifts satisfy a doubly commuting condition [Proc. Amer. Math. Soc. 103 (1988), pp. 145–148]. The first proof uses properties of Toeplitz operators to derive a formula for the reproducing kernel of certain shift invariant subspaces, which can then be used to characterize them. The second proof relies on the reproducing property in order to show that the reproducing kernel at the origin must generate the entire shift invariant subspace.

Functional calculus for a bounded C0-semigroup on Hilbert space

We introduce a new Banach algebra A(C+) of bounded analytic functions on C+={z∈C:Re(z)>0} which is an analytic version of the Figa-Talamanca-Herz algebras on R. Then we prove that the negative generator A of any bounded C0-semigroup on Hilbert space H admits a bounded (natural) functional calculus ρA:A(C+)→B(H). We prove that this is an improvement of the bounded functional calculus B0(C+)→B(H) recently devised by Batty-Gomilko-Tomilov on a certain Besov algebra B0(C+) of analytic functions on C+, by showing that B0(C+)⊂A(C+) and B0(C+)≠A(C+). In the Banach space setting, we give similar results for negative generators of γ-bounded C0-semigroups. The study of A(C+) involves dealing with Fourier multipliers on the Hardy space H1(R)⊂L1(R) of analytic functions.

Contractive inequalities between Dirichlet and Hardy spaces

We prove a conjecture of Brevig, Ortega-Cerdà, Seip and Zhao about contractive inequalities between Dirichlet and Hardy spaces and discuss its consequent connection with the Riesz projection.

Vertical and horizontal square functions on a class of non-doubling manifolds

We consider a class of non-doubling manifolds M that are the connected sum of a finite number of N-dimensional manifolds of the form Rni×Mi. Following on from the work of Hassell and the second author [20], a particular decomposition of the resolvent operators (Δ+k2)−M, for M∈N⁎, will be used to demonstrate that the vertical square function operatorSf(x):=(∫0∞|t∇(I+t2Δ)−Mf(x)|2dtt)12 is bounded on Lp(M) for 1<p<nmin=mini⁡ni and weak-type (1,1). In addition, it will be proved that the reverse inequality ‖f‖p≲‖Sf‖p holds for p∈(nmin′,nmin) and that S is unbounded for p≥nmin provided 2M<nmin.Similarly, for M>1, this method of proof will also be used to ascertain that the horizontal square function operatorsf(x):=(∫0∞|t2Δ(I+t2Δ)−Mf(x)|2dtt)12 is bounded on Lp(M) for all 1<p<∞ and weak-type (1,1). Hence, for p≥nmin, the vertical and horizontal square function operators are not equivalent and their corresponding Hardy spaces Hp do not coincide.

Atomic decompositions for noncommutative martingales

We prove an atomic type decomposition for the noncommutative martingale Hardy space hp for all 0<p<2 by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of hp for all 0<p<1, and provide a constructive proof of the atomic decomposition for p=1 which resolves a main problem on the subject left open for the last twelve years. We also study (p,∞)c-atoms, and show that every (p,2)c-atom can be decomposed into a sum of (p,∞)c-atoms; consequently, for every 0<p≤1, the (p,q)c-atoms lead to the same atomic space for all 2≤q≤∞. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space hp (0<p<1) as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to prove some sharp martingale inequalities.

On the Structure of Coisometric Extensions

If T is a bounded linear operator on a Hilbert space H and V is a given linear isometry on a Hilbert space K, we present necessary and sufficient conditions on T in order to ensure the existence of a linear isometry π:H→K such that πT=V*π (i.e., (π,V*) extends T). We parametrize the set of all solutions π of this equation. We show, for example, that for a given unitary operator U on a Hilbert space E and for the multiplication operator by the independent variable Mz on the Hardy space HD2(D), there exists an isometric operator π:H→E⊕HD2(D) such that (π,(U⊕Mz)*) extends T if and only if T is a contraction, the defect index δT≤dimD and, for some Y:AT→E, (Y,U*) extends the isometric operator AT1/2h↦AT1/2Th on the space AT=ATH¯, where AT is the asymptotic limit associated with T. We also prove that if T is isometric and V is unitary, there exists an isometric operator π:H→K such that (π,V) extends T if and only if (a) the spectral measures of the unitary part of T (in its Wold decomposition) and the restriction of V to one of its reducing subspaces K0 possess identical multiplicity functions and (b) dim(kerT*)=dim(K1⊖VK1) for a certain subspace K1 of K that contains K0 and is invariant under V. The precise form of π, in each situation, and characterizations of the minimality conditions are also included. Several examples are given for illustrative purposes.

Components of linear fractional composition operators on spaces of holomorphic functions

In this paper we consider the topological structure problem for the space C(X) of all composition operators on a space X of holomorphic functions over the unit disc, where X is one of the following function spaces: the Hardy space H2, Bergman spaces Aαp, the space H∞, weighted Banach spaces Hv with sup-norm, the classical Bloch space B, and weighted Bloch spaces Bv. In particular, we give a necessary and sufficient condition for two linear fractional composition operators to be in the same component of C(X). A characterization of isolated such composition operators in C(X) is also established.

Dual spaces and inequalities of new weak martingale Hardy spaces

Dual spaces and inequalities of new weak martingale Hardy spaces

Maximal operators of Walsh–Nörlund means on the dyadic Hardy spaces

We establish necessary and sufficient conditions for the maximal operator of Walsh–Nörlund means with non-increasing weights to be bounded from the dyadic Hardy space \(H_{p}(\mathbb{I})\) to the space \( L_{p}(\mathbb{I})\).

BOUNDEDNESS OF PSEUDO-DIFFERENTIAL OPERATORS ON WEIGHTED HARDY SPACES AND VARIABLE EXPONENTS HARDY LOCAL MORREY SPACES

In this paper, we use the atomic decomposition to establish the boundedness of pseudo-differential operators belonging to Hörmander class on weighted Hardy spaces $H^p(\omega)$ and on variable exponents Hardy local Morrey spaces $HLM_{p(\cdot)}^{u(\cdot)}$.

Two coefficient conjectures for nonvanishing Hardy functions, II

Recently the author proved that the Hummel-Scheinberg-Zalcman conjecture of 1977 on coefficients of nonvanishing $H^p$ functions is true for all $p = 2m, \ m \in \mathbb N$, i.e., for the Hilbertian Hardy spaces $H^{2m}$. As a consequence, this also implies the proof of the Krzyz conjecture for bounded nonvanishing functions, which originated this direction. In the present paper, we solve the problem for all spaces $H^p$ with $p \ge 2$.

Zeros Under Unitary Weighted Composition Operators in the Hardy and Bergman Spaces

Let mathbb {D} denote the unit disk of the complex plane and let mathcal {A}^2(mathbb {D}) be the Bergman space that consists of those analytic functions on mathbb {D} that are of integrable square modulus with respect to the normalized area measure. Let varphi : mathbb {D} rightarrow mathbb {D} be an automorphism of the disk and consider C_varphi f=f circ varphi the operator defined from mathcal {A}^2(mathbb {D}) onto itself. Consider the unitary operator U_varphi f = varphi ^prime f circ varphi . Then if f in mathcal {A}^2(mathbb {D}) is even and U_varphi f is odd, then f is the zero function. The same is true if f in mathcal {A}^2(mathbb {D}) is odd and U_varphi f is even. Similar results can be proved for the Hardy space of the unit disk, that is, the space of analytic functions on mathbb {D}, whose Taylor coefficients are of summable square modulus. The result remains true for the Dirichlet space, that is, the space of analytic functions on mathbb {D}, whose derivatives are in mathcal {A}^2(mathbb {D}).

Nonlinear open mapping principles, with applications to the Jacobian equation and other scale-invariant PDEs

For a nonlinear operator T satisfying certain structural assumptions, our main theorem states that the following claims are equivalent: i) T is surjective, ii) T is open at zero, and iii) T has a bounded right inverse. The theorem applies to numerous scale-invariant PDEs in regularity regimes where the equations are stable under weak⁎ convergence. Two particular examples we explore are the Jacobian equation and the equations of incompressible fluid flow.For the Jacobian, it is a long standing open problem to decide whether it is onto between the critical Sobolev space and the Hardy space. Towards a negative answer, we show that, if the Jacobian is onto, then it suffices to rule out the existence of surprisingly well-behaved solutions.For the incompressible Euler equations, we show that, for any p<∞, the set of initial data for which there are dissipative weak solutions in LtpLx2 is meagre in the space of solenoidal L2 fields. Similar results hold for other equations of incompressible fluid dynamics.

An extension of the $\mathrm{VMO}$-$H^1$ duality

In 1977, Coifman and Weiss gave a proof of the $\mathrm{VMO}$-$H^1$ duality. We consider generalized Campanato spaces and atomic Hardy spaces with variable growth condition and give an extension of the duality to these spaces. We also apply this duality to the Riesz transforms.

Commuting operators over Pontryagin spaces with applications to system theory

In this paper we extend vessel theory, or equivalently, the theory of overdetermined 2D systems to the Pontryagin space setting. We focus on realization theorems of the various characteristic functions associated to such vessels. In particular, we develop an indefinite version of de Branges-Rovnyak theory over real compact Riemann surfaces. To do so, we use the theory of contractions in Pontryagin spaces and the theory of analytic kernels with a finite number of negative squares. Finally, we utilize the indefinite de Branges-Rovnyak theory on compact Riemann surfaces in order to prove a Beurling type theorem on indefinite Hardy spaces on finite bordered Riemann surfaces.

On Korn’s First Inequality in a Hardy-Sobolev Space

AbstractKorn’s first inequality states that there exists a constant such that the ${\mathcal {L}}^{2}$ L 2 -norm of the infinitesimal displacement gradient is bounded above by this constant times the ${\mathcal {L}}^{2}$ L 2 -norm of the infinitesimal strain, i.e., the symmetric part of the gradient, for all infinitesimal displacements that are equal to zero on the boundary of a body ℬ. This inequality is known to hold when the ${\mathcal {L}}^{2}$ L 2 -norm is replaced by the ${\mathcal {L}}^{p}$ L p -norm for any $p\in (1,\infty )$ p ∈ ( 1 , ∞ ) . However, if $p=1$ p = 1 or $p=\infty $ p = ∞ the resulting inequality is false. It was previously shown that if one replaces the ${\mathcal {L}}^{\infty}$ L ∞ -norm by the $\operatorname{BMO}$ BMO -seminorm (Bounded Mean Oscillation) then one maintains Korn’s inequality. (Recall that ${\mathcal {L}}^{\infty}({\mathcal {B}})\subset \operatorname{BMO}({\mathcal {B}}) \subset {\mathcal {L}}^{p}({\mathcal {B}})\subset {\mathcal {L}}^{1}({ \mathcal {B}})$ L ∞ ( B ) ⊂ BMO ( B ) ⊂ L p ( B ) ⊂ L 1 ( B ) , $1&lt; p&lt;\infty $ 1 &lt; p &lt; ∞ .) In this manuscript it is shown that Korn’s inequality is also maintained if one replaces the ${\mathcal {L}}^{1}$ L 1 -norm by the norm in the Hardy space ${\mathcal {H}}^{1}$ H 1 , the predual of $\operatorname{BMO}$ BMO . One caveat: the results herein are only applicable to the pure-displacement problem with the displacement equal to zero on the entire boundary of ℬ.

The Coburn Lemma and the Hartman–Wintner–Simonenko Theorem for Toeplitz Operators on Abstract Hardy Spaces

Let X be a Banach function space on the unit circle {mathbb {T}}, let X' be its associate space, and let H[X] and H[X'] be the abstract Hardy spaces built upon X and X', respectively. Suppose that the Riesz projection P is bounded on X and ain L^infty {setminus }{0}. We show that P is bounded on X'. So, we can consider the Toeplitz operators T(a)f=P(af) and T({overline{a}})g=P({overline{a}}g) on H[X] and H[X'], respectively. In our previous paper, we have shown that if X is not separable, then one cannot rephrase Coburn’s lemma as in the case of classical Hardy spaces H^p, 1<p<infty , and guarantee that T(a) has a trivial kernel or a dense range on H[X]. The first main result of the present paper is the following extension of Coburn’s lemma: the kernel of T(a) or the kernel of T({overline{a}}) is trivial. The second main result is a generalisation of the Hartman–Wintner–Simonenko theorem saying that if T(a) is normally solvable on the space H[X], then 1/ain L^infty .

Average radial integrability spaces, tent spaces and integration operators

We deal with a Carleson measure type problem for the tent spaces ATpq(α) in the unit disc of the complex plane. They consist of the analytic functions of the tent spaces Tpq(α) introduced by Coifman, Meyer and Stein. Well known spaces like the Bergman spaces arise as a special case of this family. Let s,t,p,q∈(0,∞) and α>0. We find necessary and sufficient conditions on a positive Borel measure μ of the unit disc in order to exist a positive constant C such that∫T(∫Γ(ξ)|f(z)|tdμ(z))s/t|dξ|≤C‖f‖Tpq(α)s,f∈ATpq(α), where Γ(ξ)=ΓM(ξ)={z∈D:|1−ξ¯z|<M(1−|z|2)}, M>1/2 and ξ is a boundary point of the unit disc. This problem was originally posed by D. Luecking. We apply our results to the study of the action of the integration operator Tg, also known as Pommerenke operator, between the average integrability spaces RM(p,q), for p,q∈[1,∞). These spaces have appeared recently in the work of the first author with M.D. Contreras and L. Rodríguez-Piazza. We also consider the action from an RM(p,q) to a Hardy space Hs, where p,q,s∈[1,∞).

ON SOME NEW UNIFORM ESTIMATES AND MAXIMAL THEOREMS FOR H^p SPACES

We obtain some new uniform estimates and maximal theorems in classical Hardy spaces in the unit disk related with the Bergman projection, thus extending some previously well-known inequalities on Hardy spaces.