We investigate Hardy spaces H1L(X) corresponding to self-adjoint operators L. Our main aim is to obtain a description of H1L(X) in terms of atomic decompositions similar to such characterization of the classical Hardy spaces H1(Rd). Under suitable assumptions, such a description was obtained by Yan and the authors in [Trans. Amer. Math. Soc. 375 (2022), 6417–6451], where the atoms associated with an L-harmonic function are considered. Here we continue this study and modify the previous definition of atoms. The modified approach allows us to investigate settings when the generating operator is related to a system of linearly independent harmonic functions. In this context, the cancellation condition for atoms is adjusted to fit this system. In an explicit example, we consider a symmetric manifold with ends Rd#Rd. For this manifold the space of bounded harmonic functions is two-dimensional. Any element from the Hardy space H1L(X) has to be orthogonal to all of the harmonic functions in the system.
Answering questions of A. Avilés, F. Cabello Sánchez, J. Castillo, M. González and Y. Moreno we show that the following statements are independent of the usual {\sf ZFC} axioms with arbitrarily large continuum: for every (some) ω<κ<2ω, (1) any bounded linear operator T:c0(κ)→ℓ∞/c0 extends to any superspace of c0(κ), (2) any isomorphism between any two copies of c0(κ) inside ℓ∞/c0 extends to an automorphism of ℓ∞/c0. This contrasts with Boolean, Banach-algebraic or isometric levels, where the objects known as Hausdorff gap and Luzin gap witness the failure in {\sf ZFC} of the corresponding properties for the corresponding structures already at the first uncountable cardinal κ=ω1. In particular, consistently, any two pairwise disjoint families in ℘(N)/Fin of the same cardinality ω<κ<2ω can be mapped onto each other by a linear automorphism of ℓ∞/c0 regardless of their different combinatorial, algebraic or topological positions in ℘(N)/Fin. Our positive consistency results use a restricted version of Martin’s axiom for a partial order that adds an infinite block diagonal matrix of an operator on ℓ∞ which induces an operator on ℓ∞/c0. The construction of its finite blocks relies on a lemma of Bourgain and Tzafriri on finite-dimensional Banach spaces. Our negative consistency results rely on an analysis of almost disjoint families of N, the embeddings of c0(κ) into ℓ∞/c0 they induce and their extensions to ℓc∞(κ).
Let triangle d defined be the discrete Laplacian on Zd, d >= 1, defined by triangle df(x) = Xd j =1-[f(x + ej) + f(x-ej)-2f(x)], x is an element of Zd, where {ej : j = 1, ... , d} is the standard basis for Rd. The main aim of this paper is to prove the & ell;p(Zd) estimates for the solutions to the Schr & ouml;dinger equation i partial derivative partial derivative tu +triangle du= 0, u(x, 0) = f(x). Our approach is inspired by harmonic analysis methods such as the estimates of joint spectral multipliers and the theory of Hardy spaces associated with the discrete Laplacian.
This paper explores the interplay between boundary conditions and invariant subspaces for one-dimensional Laplacians, extending these concepts to Walsh’s spider process on a star-like graph. We establish a precise correspondence between the transmission condition characterizing this process and a specific subspace within a larger function space. This correspondence is facilitated by relating the cosine family associated with the spider process to the basic cosine family of unrestricted Brownian motion. Furthermore, we introduce a complementary subspace, leading to a novel decomposition of the function space that generalizes known results for simpler boundary conditions. This decomposition reveals a fundamental relationship between two distinct transmission conditions, highlighting their complementary nature. Our findings provide new insights into the structure of Walsh’s spider process and offer a framework for further analysis, including the study of its limiting behavior as the stickiness parameter varies.
Khabibullin established the best estimate in the Paley problem for a plurisubharmonic (psh) function u of lower order, 0 <=rho <= 1. For rho > 1, obtaining a sharp estimate has remained an open question. In this work, we solve this problem. We also provide estimates for the types of characteristic functions T (r, u) and M(r, u). Finally, we compare our results with those of Dahlberg for subharmonic functions and show that the latter are not optimal for psh functions of finite lower order rho > 1.
