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.
We study random dynamical systems of certain continuous functions on the unit interval. We use bounded variation to provide sufficient conditions for unique ergodicity of these systems. Several classes of examples are provided.
Let T be a strongly Kreiss bounded linear operator on Lp. We obtain a bound on the rate of growth of the norms of the powers of T. The bound is optimal with respect to the polynomial scale. The proof makes use of Fourier multipliers, in particular of the Littlewood-Paley inequalities on arbitrary intervals as initiated by Rubio de Francia and developed by Kislyakov and Parilov.
Kalton and Mitrea (1998) characterized complex interpolation spaces of quasi-Banach function spaces as Calder & oacute;n products if both interpolants are separable. We show that one separability assumption may be omitted and establish a Wolff-reiteration result with one non-separable endpoint space.
Let f is an element of L1(N), where N = L infinity(Gm) circle times & strns; M, Gm is a bounded Vilenkin group and M is a semifinite von Neumann algebra. We prove the noncommutative weak type maximal inequality parallel to(Dn(f))n >= 1 parallel to Lambda 1,infinity(N,& ell;infinity) <= C parallel to f parallel to L1(N), where Dn(f) represents the Vilenkin derivative of the integral function If. The main strategy in the proof is to exploit the recent advances on the noncommutative Calder & oacute;n-Zygmund decomposition established by Cadilhac, Conde-Alonso and Parcet.
Let A be a weakly sequentially complete Banach algebra containing a bounded approximate identity that is an ideal in its second dual A(& lowast;& lowast;). We refer to such an algebra as a WeSeBai algebra. In the present paper, we examine the Arens regularity properties of closed ideals of algebras in this class and observe that, although WeSeBai algebras themselves are always strongly Arens irregular, a variety of Arens regularity properties can be found among their closed ideals. After characterizing Arens regular ideals and strongly Arens irregular ideals, we focus on the main examples of WeSeBai algebras: the convolution group algebras L-1(G), G compact, and the Fourier algebras A(Gamma), Gamma discrete and amenable. We find examples of Arens regular ideals in L-1(G) and A(Gamma), reflexive and nonreflexive, and examples of strongly Arens irregular ideals that are not in the WeSeBai class. For this, we introduce a new class of Riesz sets that are not Lambda(p), for any p > 1, and show how to obtain them in a broad family of noncommutative groups. As a consequence of our approach, we prove that every infinite Abelian group contains a Rosenthal set that is not Lambda(p), for any p > 0.
We derive rates of convergence for the mixing of operators under infinitely divisible measures in the framework of linear dynamics on Banach spaces. Our approach is based on the characterization of mixing in terms of codifference functionals and control measures, and extends previous results obtained in the Gaussian setting via the use of covariance operators. Explicit mixing rates are obtained for weighted shifts under compound Poisson, α-stable, and tempered α-stable measures.
We show that generic continuous linear cocycles over shifts and other zero-dimensional systems admit no quasiconformal orbits, thus providing a partial answer to a question of Nassiri, Rajabzadeh, and Reshadat. The proof relies on a new result about towers for homeomorphisms of zero-dimensional spaces, which may be of independent interest.
We initiate the theory of real noncommutative (nc) convex sets, the real case of the recent and profound complex theory developed by Davidson and Kennedy (2025). The present paper focuses on the real case of the topics from the first several sections of their memoir. Later results will be discussed in future papers. We develop here some of the infrastructure of real nc convexity, giving many foundational structural results for real operator systems and their associated nc convex sets, and elucidate how the complexification interacts with the basic convexity theory constructions. Several new features appear in the real case, including the novel notion of the complexification of a nc convex set.
A version of the recent functional inequality between the Hessians of the square root and the logarithm of positive functions is proven in spaces with non-zero curvature.
This paper is a continuation of our work on the functional-analytic core of the classical Furstenberg-Zimmer theory. We introduce and study (in the framework of lattice-ordered spaces) the notions of total order-boundedness and uniform total order-boundedness. Either one generalizes the concept of ordinary precompactness known from metric space theory. These new notions are then used to define and characterize "compact extensions" of general measure-preserving systems (with no restrictions on the underlying probability spaces nor on the acting groups). In particular, it is (re)proved that compact extensions and extensions with discrete spectrum are one and the same thing. Finally, we show that under natural hypotheses a subset of a Kaplansky-Banach module is totally order bounded if and only if it is cyclically compact (in the sense of Kusraev).
A ring is called clean if every element is the sum of an invertible element and an idempotent. This paper investigates the cleanness of AW*-algebras. We prove that all finite AW*-algebras are clean, affirmatively solving a question posed by Vas. We also prove that all countably decomposable infinite AW*-factors are clean. A *-ring is called almost *-clean if every element can be expressed as the sum of a non-zero-divisor and a projection. We show that an AW*-algebra is almost *-clean if and only if it is finite.
For two countable ordinals alpha and beta, a basis of a Banach space X is said to be (alpha, beta)-quasi-greedy if it is center dot quasi-greedy, center dot S alpha-unconditional but not S alpha+1-unconditional, and center dot S beta-democratic but not S beta +1-democratic. If alpha or beta is replaced with infinity, then the basis is required to be unconditonal or democratic, respectively. Previous work constructed a (0,0)-quasi-greedy basis, an (alpha, infinity)-quasi-greedy basis, and an (infinity, alpha)-quasi-greedy basis. In this paper, we construct (alpha, beta)-quasi-greedy bases for beta <= alpha + 1 (except the already solved case alpha = beta = 0).
Let G be a locally elliptic group, (Phi, Psi) a complementary pair of Young functions, and omega : G -+ [1, oo) a weight function on G such that the weighted Orlicz space L Phi (G, omega) is a Banach *-algebra when equipped with the convolution product and involution f & lowast;(x) := f(x-1) (f E L Phi (G, omega)). Such a weight always exists on G and we call studies the spectral theory and primitive ideal structure of L Phi(G, omega). In particular, we focus on studying the Hermitian, Wiener and *-regularity properties on this algebra, along with some related questions on spectral synthesis. It is shown that L Phi(G, omega) is always quasi-Hermitian, weakly-Wiener and *-regular. Thus, if L Phi(G, omega) is Hermitian, then it is Hermitianness of L1(G) implies Hermitianness of L Phi (G, omega) if omega is subadditive. We give numerous examples of locally elliptic groups G for which L1(G) is Hermitian and subadditive L Phi-weights on these groups. In the weighted L1 case, even stronger Hermitianness results are formulated.
We revisit the results of Kim, and of Katsoulis and Ramsey concerning hyperrigidity for non-degenerate C & lowast;-correspondences. We show that the tensor algebra is hyperrigid, if and only if Katsura's ideal acts non-degenerately, if and only if Katsura's ideal acts non-degenerately under any representation. This gives a positive answer to the question of Katsoulis and Ramsey, showing that their necessary condition and their sufficient condition for hyperrigidity of the tensor algebra are equivalent. Non-degeneracy of the left action of Katsura's ideal was also shown by Kim to be equivalent to hyperrigidity for the selfadjoint operator space associated with the C & lowast;-correspondence, and our approach provides a simplified proof of this result as well. In the process we study unitisations of selfadjoint operator spaces in the sense of Werner, and revisit Arveson's criterion connecting maximality with the unique extension property and hyperrigidity, in conjunction with the work of Salomon on generating sets.
We investigate the general properties and structure of C∗-extreme points within the C∗-convex set UCP(A,B(H)) of all unital completely positive (UCP) maps from a unital real C∗-algebra A to the algebra B(H) of all bounded real linear maps on a real Hilbert space H. We analyze the differences in the structure of C∗-extreme points between the real and complex C∗-algebra cases. In particular, we show that the necessary and sufficient conditions for a UCP map between matrix algebras to be a C∗-extreme point are identical in both the real and complex matrix algebra cases. We also observe significant differences in the structure of C∗-extreme points when A is a commutative real C∗-algebra compared to when A is a commutative complex C∗-algebra. We provide a complete classification of the C∗-extreme points of UCP(A,B(H)), where A is a unital commutative real C∗-algebra and H is a finite-dimensional real Hilbert space. As an application, we classify all C∗-extreme points in the C∗-convex set of all contractive skew-symmetric matrices in Mn(R).
The paper is devoted to the properties of a complex matrix “twisted,” otherwise called “spectral,” cocycle, associated with substitution dynamical systems. Following a recent finding of Rajabzadeh and Safaee [arXiv:2501.16824] of an invariant section for the twisted cocycle, we indicate that this implies presence of a zero Lyapunov exponent. This has consequences for the spectral properties of substitution dynamical systems; in particular, this extends the scope and simplifies the proof of singular spectrum for a large class of substitutions on two symbols. We also obtain some results on positivity of the top exponent. In the appendix we compute the Lebesgue almost everywhere local dimension of spectral measures of some “simple” test functions, for almost every irrational rotation. This sheds some light on the earlier work of Bufetov and the author, relating the local dimension of spectral measures to pointwise Lyapunov exponents of the twisted cocycle. It should be noted that the paper has some (mutually acknowledged) overlap with [arXiv:2501.16824, Appendix].
Let (X, T, mu, d) be a metric measure-preserving system for which 3-fold correlations decay exponentially for Lipschitz continuous observables. Suppose that (Mk) is a sequence satisfying some weak decay conditions and suppose there exist open balls Bk(x) around x such that mu (Bk(x)) = Mk. Under a short return time assumption, we prove a strong Borel-Cantelli lemma, including an error term, for recurrence, i.e., for mu -a.e. x is an element of X, Xn k =1 1Bk(x)(Tkx) = Phi(n) + O(Phi(n)1/2(log Phi(n))3/2+epsilon), where Phi(n) = Enk =1 mu (Bk(x)). Applications to systems include some non-linear piecewise expanding interval maps and hyperbolic automorphisms of T2.
