Space H1 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.

Treatment of 3D diffusion problems with discontinuous coefficients and Dirac curvilinear sources

Three-dimensional diffusion problems with discontinuous coefficients and unidimensional Dirac sources arise in a number of fields. The statement we pursue is a singular-regular expansion where the singularity, capturing the stiff behavior of the potential, is expressed by a convolution formula using the Green kernel of the Laplace operator. The correction term, aimed at restoring the boundary conditions, fulfills a variational Poisson equation set in the Sobolev space H1, which can be approximated using finite element methods. The mathematical justification of the proposed expansion is the main focus, particularly when the variable diffusion coefficients are continuous, or have jumps. A computational study concludes the paper with some numerical examples. The potential is approximated by a combined method: (singularity, by integral formulas, correction, by linear finite elements). The convergence is discussed to highlight the practical benefits brought by different expansions, for continuous and discontinuous coefficients.

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 .

The class of grim reapers in [formula omitted]

We study translators of the mean curvature flow in the product space H2×R. In H2×R there are three types of translations: vertical translations due to the factor R and parabolic and hyperbolic translations from H2. A grim reaper in H2×R is a translator invariant by a one-parameter group of translations. The variety of translators and translations in H2×R makes the family of grim reapers particularly rich. In this paper we give a full classification of the grim reapers of H2×R with a description of their geometric properties. In some cases, we obtain explicit parametrizations of the surfaces.

Orbital stability of smooth solitons for the modified Camassa-Holm equation

The modified Camassa-Holm equation with cubic nonlinearity is completely integrable and is considered a model for the unidirectional propagation of shallow-water waves. The localized smooth-wave solution exists uniquely, up to translation, within a certain range of the linear dispersive parameter. By constructing conserved H1 and L1 quantities in terms of the momentum variable m, this study demonstrates that the smooth soliton, when regarded as a solution of the initial-value problem for the modified Camassa-Holm equation, is orbitally stable to perturbations in the Sobolev space H3. Furthermore, the global well-posedness of the solution is established for certain initial data in Hs with s≥3.

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.

More on doubled Hilbert space in double-scaled SYK

We continue our study of the doubled Hilbert space formalism of double-scaled SYK model initiated in Okuyama (2024) [7]. We show that the 1-particle Hilbert space introduced by Lin and Stanford (2023) [6] is related to the doubled Hilbert space H0⊗H0 of the 0-particle Hilbert space H0 by some linear isomorphism C. It turns out that the entangler and disentangler appeared in our previous study are given by the dual (or adjoint) of C−1 and C.

From Schrödinger equation to tempered distribution space on Minkowski chronotope and Schwartz Linear Algebra

In this paper, we propose a rational and almost obliged path leading us from the most natural assumption regarding the SchrÅNodinger wave functions (wave functions classically belonging to the domain of the classic Schrödinger equation) to the space of tempered distributions defined upon the Minkowski space time. It’s extremely natural to consider the space E of complex valued smooth functions as the natural domain of the Schrödinger equation, as it was the obvious implicit assumption of any partial differential equation with constant coefficients at the time of Schrödinger himself. The orthodox Born interpretation of the normalized wave functions and the intervention of von Neumann (a Hilbert‘s pupil) have deviated the straightforward understanding of that domain towards the unnecessary and unnatural Hilbert space L2. The Hilbert space of square integrable functions is clearly incompatible with any partial differential equation, which would require at least Sobolev Spaces to live in; unfortunately, the inner product of the Sobolev space H2 is not good for quantum mechanics and the de Broglie solutions of the Schrödinger equation do not belong to L2. We show in this article that moving inside the very good space E - and finally inside the huge tempered distribution space S′ - represents the first framework in which any desirable property of the key characters and actors of basic Quantum Mechanics is satisfied.

Asymptotic stability of peakons for the two-component Novikov equation

We study the asymptotic stability of peaked solitons under H1 × H1-perturbations of the two-component Novikov equation involving interaction between two components. This system, as a two-component generalization of the Novikov equation, is a completely integrable system which has Lax pair and bi-Hamiltonian structure. Interestingly, it admits the two-component peaked solitons with different phases, which are the weak solutions in the sense of distribution and lie in the energy space H1 × H1. It is shown that the peakons are asymptotically stable in the energy space H1 × H1 with non-negative momentum density by establishing a rigidity theorem for H1 × H1-almost localized solutions. Our proof generalizes the arguments for studying the Camassa-Holm and Novikov equations. There are three new ingredients in our proof. One is a new characteristic describing interaction of the two-components; the second is new additional conserved densities for establishing the main inequalities; while the third one is a new Lyapunov functional used to overcome the difficulty caused by the loss of momentum.

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.

Asymptotics for the semi-dissipative 2D Boussinesq system

In this article, we consider the convergence from the semi-dissipative 2D Boussinesq equations to the 2D Navier–Stokes equations in some sense. When the temperature variable θ tends to 0, we prove that the component uθ of the solution of the semi-dissipative 2D Boussinesq equations is converging to the solution u of the 2D Navier–Stokes equations in the space H and uθ is asymptotically converging to u in the space H2(Ω) with the initial data (u0, θ0) ∈ H × L2(Ω). Moreover, we obtain the continuity of the local attractor Ar of the semi-dissipative 2D Boussinesq equations with respect to A×{0} in the space H2(Ω) × L2(Ω) at r = 0, here A×{0} is a natural embedding of the global attractor A of the 2D Navier–Stokes equations into the space H × L2(Ω).

Global energy conservation solution for the [formula omitted] family of Camassa–Holm type equation

In this paper, we investigate the N−abc family of Camassa–Holm type equation with (N+1)-order nonlinearities. This quasi-linear equation is nonlocal with higher order nonlinearities, compared to the Camassa–Holm equation (N=1) and Novikov equation (N=2). Using both the lower order and the higher order energy conservation laws, as well as the characteristic method, we establish the global existence and uniqueness of the Hölder continuous energy weak solution to the N−abc family of Camassa–Holm type equation in the energy space H1(R)×W1,2N(R). Moreover, we show that a very natural and interesting problem is to study how the regularity of solution changes with respect to N. More precisely, we establish Hölder continuous energy weak solutions with the exponent 1−12N. This result precisely shows how the regularity of solution changes with respect to the power of nonlinear wave speed N, and it reveals an intrinsic relation between Camassa–Holm equation (linear wave speed u1), Novikov equation (quadratic wave speed u2) and the N−abc family of Camassa–Holm type equation (wave speed uN).

Regular Tessellations of Maximally Symmetric Hyperbolic Manifolds

We first briefly summarize several well-known properties of regular tessellations of the three two-dimensional maximally symmetric manifolds, E2, S2, and H2, by bounded regular tiles. For instance, there exist infinitely many regular tessellations of the hyperbolic plane H2 by curved hyperbolic equilateral triangles whose vertex angles are 2π/d for d=7,8,9,… On the other hand, we prove that there is no curved hyperbolic regular tetrahedron which tessellates the three-dimensional hyperbolic space H3. We also show that a regular tessellation of H3 can only consist of the hyperbolic cubes, hyperbolic regular icosahedra, or two types of hyperbolic regular dodecahedra. There exist only two regular hyperbolic space-fillers of H4. If n>4, then there exists no regular tessellation of Hn.

A note on non-supercyclic vectors of Hilbert space operators

In this note it is shown that there is a bounded linear operator T on the Hardy Hilbert space H2 and a vector f in H2 such that the closure of the set {αTnf:α∈ℂ,n≥0} is not H2, but for every subsequence (nk)k=1∞ the closed linear span of {Tnkf:k≥1} is the whole space H2. Furthermore, the closure of {Tng:n≥0} is H2 for some g∈H2.

The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula

For a Hermitian Lie group G of tube type we find the contribution of the holomorphic discrete series to the Plancherel decomposition of the Whittaker space L2(G/N,ψ), where N is the unipotent radical of the Siegel parabolic subgroup and ψ is a certain non-degenerate unitary character on N. The holomorphic discrete series embeddings are constructed in terms of generalized Whittaker vectors for which we find explicit formulas in the bounded domain realization, the tube domain realization and the L2-model of the holomorphic discrete series. Although L2(G/N,ψ) does not have finite multiplicities in general, the holomorphic discrete series contribution does.Moreover, we obtain an explicit formula for the formal dimensions of the holomorphic discrete series embeddings, and we interpret the holomorphic discrete series contribution to L2(G/N,ψ) as boundary values of holomorphic functions on a domain Ξ in a complexification GC of G forming a Hardy type space H2(Ξ,ψ).

Positive solutions to the planar logarithmic Choquard equation with exponential nonlinearity

In this paper we study the following nonlinear Choquard equation −Δu+u=ln1|x|∗F(u)f(u),inR2,where f∈C1(R,R) and F is the primitive of the nonlinearity f vanishing at zero. We use an asymptotic approximation approach to establish the existence of positive solutions to the above problem in the standard Sobolev space H1(R2). We give a new proof and at the same time extend part of the results established in (Cassani-Tarsi, Calc.Var.PDE, 2021) [11].

Estimates of Bergman kernels and Bergman metric on compact Picard surfaces

Let Γ⊂SU((2,1),C) be a torsion-free cocompact subgroup. Let B2 denote the 2-dimensional complex ball endowed with the hyperbolic metric μhyp, and let XΓ:=Γ﹨B2 denote the quotient space, which is a compact complex manifold of dimension 2. Let Λ:=ΩXΓ2 denote the line bundle on XΓ, whose sections are holomorphic (2,0)-forms. For any k≥1, the hyperbolic metric induces a point-wise metric on H0(XΓ,Λ⊗k), which we denote by |⋅|hyp. For any k≥1, let BΛk denote the Bergman kernel of the complex vector space H0(XΓ,Λ⊗k). For any k≥3, and z,w∈XΓ, the first main result of the article is the following off-diagonal estimate of the Bergman kernel BΛk|BΛk(z,w)|hyp=OXΓ(k2cosh3k−8⁡(dhyp(z,w)/2)), where dhyp(z,w) denotes the geodesic distance between the points z and w on XΓ, and the implied constant depends only on the Picard surface XΓ.For any k≥1, let μberk(z):=−i2π∂z∂z‾log⁡|BΛk(z,z)|hyp denote the Bergman metric associated the line bundle Λ⊗k, and let μberk,vol denote the associated volume form. For k≫1 sufficiently large, and ϵ>0, the second main result of the article is the following estimatesupz∈XΓ⁡|μberk,vol(z)μhypvol(z)|=OXΓ,ϵ(k4+ϵ), where μhypvol denotes the volume form associated to the hyperbolic metric μhyp, and the implied constant depends on the Picard surface XΓ, and on the choice of ϵ>0.

Expected integration approximation under general equal measure partition

In this paper, we first use an L2−discrepancy bound to give the expected uniform integration approximation for functions in the Sobolev space H1(K) equipped with a reproducing kernel. The concept of stratified sampling under general equal measure partition is introduced into the research. For different sampling modes, we obtain a better convergence order O(N−1−1d) for the stratified sampling set than for the Monte Carlo sampling method and the Latin hypercube sampling method. Second, we give several expected uniform integration approximation bounds for functions equipped with boundary conditions in the general Sobolev space Fd,q∗, where 1p+1q=1. Probabilistic Lp−discrepancy bound under general equal measure partition, including the case of Hilbert space-filling curve-based sampling are employed. All of these give better general results than simple random sampling, and in particular, Hilbert space-filling curve-based sampling gives better results than simple random sampling for the appropriate sample size.