Hardy Space H2 Research Articles

Relative weak compactness in infinite-dimensional Fefferman-Meyer duality

Let E be a Banach space such that E′ has the Radon-Nikodým property. The aim of this work is to connect relative weak compactness in the E-valued martingale Hardy space H1(μ,E) to a convex compactness criterion in a weaker topology, such as the topology of uniform convergence on compacts in measure. These results represent a dynamic version of the deep result of Diestel, Ruess, and Schachermayer on relative weak compactness in L1(μ,E). In the reflexive case, we obtain a Kadec-Pełczyński dichotomy for H1(μ,E)-bounded sequences, which decomposes a subsequence into a relatively weakly compact part, a pointwise weakly convexly convergent part, and a null part converging to zero uniformly on compacts in measure. As a corollary, we investigate a parameterized version of the vector-valued Komlós theorem without the assumption of H1(μ,E)-boundedness.

Sharp analytic version of Fefferman's inequality

Let T be a unit circle. Assume further that f is an element of the Hardy space H1(T) and g belongs to the analytic BMO space on T. The paper contains the identification of the optimal universal constant C in the estimate|12π∫Tf(ζ)‾g(ζ)dζ|≤C‖f‖H1(T)‖g‖BMO(T). Actually, the inequality is studied in the stronger form, involving the Littlewood-Paley function on the left and the sharp maximal function of g on the right. The proof rests on the construction of an appropriate plurisuperharmonic function on a parabolic domain and the application of probabilistic techniques.

Linear equations with infinitely many derivatives and explicit solutions to zeta nonlocal equations

We summarize our theory on existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives of the formf(∂t)ϕ=J(t),t≥0, where f is an analytic function such as the (analytic continuation of the) Riemann zeta function. We explain how to analyse initial value problems for these equations, and we prove rigorously that the functionϕ(t)=∑n≥1μ(n)nhJ(t−ln⁡(n)), in which μ is the Möbius function and J satisfies some technical conditions to be specified in Section 4, is the solution to the zeta nonlocal equationζ(∂t+h)ϕ=J(t),t≥0, in which ζ is the Riemann zeta function and h>1. We also present explicit examples of solutions to initial value problems for this equation. Our constructions can be interpreted as highlighting the importance of the cosmological daemon functions considered by Aref'eva and Volovich (2011) [1]. Our main technical tool is the Laplace transform as a unitary operator between the Lebesgue space L2 and the Hardy space H2.

Information Design and Sharing in Supply Chains

We study the interplay between inventory replenishment policies and information sharing in the context of a two-tier supply chain with a single supplier and a single retailer serving an independent and identically distributed Gaussian market demand. We investigate how the retailer’s inventory policy impacts the supply chain’s cumulative expected long-term average inventory costs [Formula: see text] in two extreme information-sharing cases: (a) full information sharing and (b) no information sharing. To find the retailer’s inventory policy that minimizes [Formula: see text], we formulate an infinite-dimensional optimization problem whose decision variables are the MA([Formula: see text]) coefficients that characterize a stationary ordering policy. Under full information sharing, the optimization problem admits a simple solution and the optimal policy is given by an MA(1) process. On the other hand, to solve the optimization problem under no information sharing, we reformulate the optimization from its time domain formulation to an equivalent z-transform formulation in which the decision variables correspond to elements of the Hardy space H2. This alternative representation allows us to use a number of results from H2 theory to compute the optimal value of [Formula: see text] and characterize a sequence of ϵ-optimal inventory policies under some mild technical conditions. By comparing the optimal solution under full information sharing and no information sharing, we derive a number of important practical takeaways. For instance, we show that there is value in information sharing if and only if the retailer’s optimal policy under full information sharing is not invertible with respect to the sequence of demand shocks. Furthermore, we derive a fundamental mathematical identity that reveals the value of information sharing by exploiting the canonical Smirnov–Beurling inner–outer factorization of the retailer’s orders when viewed as an element of H2. We also show that the value of information sharing can grow unboundedly when the cumulative supply chain costs are dominated by the supplier’s inventory costs. Funding: R. Caldentey acknowledges the University of Chicago Booth School of Business for financial support. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.008 .

Norm of a generalized Hilbert linear and bilinear form on the Hardy spaces

In this paper, the precise norm of a generalized Hilbert linear form HLλ on the Hardy space H1 and that of the corresponding bilinear form HBλ on the Hardy space pair (Hp,Hq) are given for all λ∈{x∈R:x≠0,−1,−2,⋯}. Furthermore, the classical Hardy's inequality on the absolute convergence of HL1 is generalized to all HLλ with λ>0.

Measure induced Hankel and Toeplitz type operators on weighted Dirichlet spaces

For a complex Borel measure μ \mu on the open unit disk, and for a weighted Dirichlet space H s \mathcal {H}_s with 0 > s > 1 0>s>1 , we characterize the boundedness of the measure induced Hankel type operator H μ , s : H s → H s ¯ H_{\mu ,s}: \mathcal {H}_s \to \overline {\mathcal {H}_s} , extending the results of Xiao [Bull. Austral. Math. Soc. 62 (2000), pp. 135–140] for the classical Hardy space H 2 = H 1 H^2=\mathcal {H}_1 , and of Arcozzi, Rochberg, Sawyer, and Wick [J. Lond. Math. Soc. (2) 83 (2011), pp. 1–18] for the classical Dirichlet space D = H 0 \mathcal {D}= \mathcal {H}_0 . Our approach relies on some recent results about weak products of complete Nevanlinna-Pick reproducing kernel Hilbert spaces. We also include some related results on Hankel measures, Carleson measures, and Toeplitz type operators on weighted Dirichlet spaces H s \mathcal {H}_s , 0 > s > 1 0>s>1 .

Paired Kernels and Their Applications

This paper considers paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^2$$\\end{document}. The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly S∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S^*$$\\end{document}-invariant subspace of H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^2$$\\end{document}, is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.

Products and commutators of martingales in H1 and BMO

Let f:=(fn)n∈Z+ and g:=(gn)n∈Z+ be two martingales related to the probability space (Ω,F,P) equipped with the filtration (Fn)n∈Z+. Assume that f is in the martingale Hardy space H1 and g is in its dual space, namely the martingale BMO. Then the semi-martingale f⋅g:=(fngn)n∈Z+ may be written as the sumf⋅g=G(f,g)+L(f,g). Here L(f,g):=(L(f,g)n)n∈Z+ with L(f,g)n:=∑k=0n(fk−fk−1)(gk−gk−1) for any n∈Z+, where f−1:=0=:g−1. The authors prove that L(f,g) is a process with bounded variation and limit in L1, while G(f,g) belongs to the martingale Hardy-Orlicz space Hlog associated with the Orlicz functionΦ(t):=tlog⁡(e+t),∀t∈[0,∞). The above bilinear decomposition L1+Hlog is sharp in the sense that, for particular martingales, the space L1+Hlog cannot be replaced by a smaller space having a larger dual. As an application, the authors characterize the largest subspace of H1, denoted by H1b with b∈BMO, such that the commutators [T,b] with classical sublinear operators T are bounded from H1b to L1. This endpoint boundedness of commutators allows the authors to give more applications. On the one hand, in the martingale setting, the authors obtain the endpoint estimates of commutators for both martingale transforms and martingale fractional integrals. On the other hand, in harmonic analysis, the authors establish the endpoint estimates of commutators both for the dyadic Hilbert transform beyond doubling measures and for the maximal operator of Cesàro means of Walsh–Fourier series.

Complete Continuity of Composition-Differentiation Operators on the Hardy Space H 1

We study composition-differentiation operators on the Hardy space H 1 on the unit disk. We prove that if φ is an analytic self-map of the unit disk such that the composition-differentiation operator induced by φ is bounded on the Hardy space H 1 , then it is completely continuous. This result is stronger than the similar result for composition operators which says that the composition operator induced by φ is completely continuous if and only if φ e i θ < 1 almost everywhere on the unit circle.

Composition operators and generalized primes

We study composition operators on the Hardy space H 2 \mathcal {H}^2 of Dirichlet series with square summable coefficients. Our main result is a necessary condition, in terms of a Nevanlinna-type counting function, for a certain class of composition operators to be compact on H 2 \mathcal {H}^2 . To do that we extend our notions to a Hardy space H Λ 2 \mathcal {H}_{\Lambda }^2 of generalized Dirichlet series, induced in a natural way by a sequence of Beurling’s primes.

H ∞ interpolation constrained by Beurling–Sobolev norms

Abstract We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc, constrained by Beurling–Sobolev norms. We find sharp asymptotics of the corresponding interpolation quantities, thereby improving the known estimates. On our way we obtain a S. M. Nikolskii type inequality for rational functions whose poles lie outside of the unit disc. It shows that the embedding of the Hardy space H 2 into the Wiener algebra of absolutely convergent Fourier/Taylor series is invertible on the subset of rational functions of a given degree, whose poles remain at a given distance from the unit circle.

Cesàro-type operators associated with Borel measures on the unit disc acting on some Hilbert spaces of analytic functions

Given a complex Borel measure μ on the unit disc D={z∈C:|z|<1}, we consider the Cesàro-type operator Cμ defined on the space Hol(D) of all analytic functions in D as follows:If f∈Hol(D), f(z)=∑n=0∞anzn (z∈D), then Cμ(f)(z)=∑n=0∞μn(∑k=0nak)zn, (z∈D), where, for n≥0, μn denotes the n-th moment of the measure μ, that is, μn=∫Dwndμ(w).We study the action of the operators Cμ on some Hilbert spaces of analytic function in D, namely, the Hardy space H2 and the weighted Bergman spaces Aα2 (α>−1). Among other results, we prove that, if we set Fμ(z)=∑n=0∞μnzn (z∈D), then Cμ is bounded on H2 or on Aα2 if and only if Fμ belongs to the mean Lipschitz space Λ1/22. We prove also that Cμ is a Hilbert-Schmidt operator on H2 if and only if Fμ belongs to the Dirichlet space D, and that Cμ is a Hilbert-Schmidt operator on Aα2 if and only if Fμ belongs to the Dirichlet-type space D−1−α2.

Brown-Halmos type theorems of Toeplitz+Hankel operators

In this paper, we discuss Brown-Halmos type theorems of Toeplitz +Hankel operators Tf+Hg acting on the Hardy space H2(T) of the unit circle T, especially operators Mf≜Tf+Hf. Basor and Ehrhardt (1999 [3] and 2004 [4]) noted that if f is co-analytic or g is even, then the product of Mf and Mg equals Mfg, we show that the converse is true and obtain a necessary and sufficient condition for the product of two arbitrary Toeplitz+Hankel operators is a Toeplitz+Hankel operator. Moreover, we prove that Mf and Mg commute if and only if either both of f and g be co-analytic, or both of f and g be even, or one be a linear function of the other.

Maximal norm Hankel operators

A Hankel operator Hφ on the Hardy space H2 of the unit circle with analytic symbol φ has minimal norm if ‖Hφ‖=‖φ‖2 and maximal norm if ‖Hφ‖=‖φ‖∞. The Hankel operator Hφ has both minimal and maximal norm if and only if |φ| is constant almost everywhere on the unit circle or, equivalently, if and only if φ is a constant multiple of an inner function. We show that if Hφ is norm-attaining and has maximal norm, then Hφ has minimal norm. If |φ| is continuous but not constant, then Hφ has maximal norm if and only if the set at which |φ|=‖φ‖∞ has nonempty intersection with the spectrum of the inner factor of φ. We obtain further results illustrating that the case of maximal norm is in general related to “irregular” behavior of log⁡|φ| or the argument of φ near a “maximum point” of |φ|. The role of certain positive functions coined apical Helson–Szegő weights is discussed in the former context.

Disjoint frequent hypercyclicity of composition operators

We give a sufficient condition for two operators to be disjointly frequently hypercyclic. We apply this criterion to composition operators acting on H(D) or on the Hardy space H2(D). We simplify a result on disjoint frequent hypercyclicity of pseudo shifts of a recent paper of Martin et al. and we exhibit two disjointly frequently hypercyclic weighted shifts.

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.

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.

Coefficient multipliers in the Hardy space associated with Jacobi expansions

In this paper a multiplier theorem in the Hardy space H 1 ( T ) H^1(\mathbb {T}) associated with Jacobi expansions of exponential type is proved, that is, a bilateral sequence { λ n } n = − ∞ ∞ \left \{\lambda _n\right \}_{n=-\infty }^{\infty } is a multiplier from H 1 ( T ) H^1(\mathbb {T}) into the sequence space ℓ 1 ( Z ) \ell ^1(\mathbb {Z}) associated with Jacobi expansions of exponential type, if \[ sup N ∑ k = 1 ∞ ( ∑ k N &gt; | j | ≤ ( k + 1 ) N | λ j | ) 2 &gt; ∞ . \sup _N\sum _{k=1}^{\infty }\left (\sum _{kN&gt;|j|\le (k+1)N}|\lambda _j|\right )^2&gt;\infty . \] This is a generalization of a multiplier theorem on usual Fourier expansions in the Hardy space H 1 ( T ) H^1(\mathbb {T}) , and for λ n = ( | n | + 1 ) − 1 \lambda _n=(|n|+1)^{-1} , a Hardy type inequality for Jacobi expansions is immediate which has ever been proved by Kanjin and Sato [Math. Inequal. Appl. 7 (2004), pp. 551–555].

Posinormal composition operators on H2

A bounded linear operator A on a Hilbert space is posinormal if there exists a positive operator P such that AA⁎=A⁎PA. Posinormality of A is equivalent to the inclusion of the range of A in the range of its adjoint A⁎. Every hyponormal operator is posinormal, as is every invertible operator. We characterize both the posinormal and coposinormal composition operators Cφ on the Hardy space H2 of the open unit disk D when φ is a linear-fractional selfmap of D. Our work reveals that there are composition operators that are both posinormal and coposinormal yet have powers that fail to be posinormal.

Sequences of [formula omitted]-term deviations in Hilbert space

Let D be a dictionary in a Hilbert space H, that is, a set of unit elements whose linear combinations are dense in H. We consider the least m-term deviation σm(x) of an element x∈H: this is the distance from x to the set of all m-term linear combinations of elements of D. We prove a dichotomy result: for any dictionary D, either the sequence {σm(x)}m=0∞ decreases exponentially for every x∈H, or the rate of convergence σm(x)→0 can be arbitrarily slow. We seek universal dictionaries realizing all strictly decreasing null sequences as sequences of m-term deviations. All commonly used dictionaries turn out not to be universal. In particular, the least rational deviations in Hardy space H2 do not form certain strictly monotone null sequences. There are no universal dictionaries in finite dimensional Hilbert spaces. We construct a universal dictionary in every infinite dimensional Hilbert space.