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This note is devoted to developing Morrey regularity theory for the following system of Rivière \begin{equation*} -\Delta u=\Omega \cdot \nabla u+f \qquad \text {in }B^{2}, \end{equation*} under the assumption that f belongs to some Morrey space. Our results extend the L^p regularity theory of Sharp and Topping [Trans. Amer. Math. Soc. 365 (2013), pp. 2317–2339], and also generalize a Hölder continuity result of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24] on harmonic mappings. Potential applications of our results are also possible in second order conformally invariant geometrical problems as that of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24]. [ABSTRACT FROM AUTHOR]

The goal of this paper is to establish new Maximum Principles for parabolic equations in the framework of mixed local/nonlocal operators. In particular, these results apply to the case of mixed local/nonlocal Neumann boundary conditions, as introduced by Dipierro, Proietti Lippi, and Valdinoci [Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023), pp. 1093–1166]. Moreover, they play an important role in the analysis of population dynamics involving the so-called Allee effect, which is performed by Dipierro, Proietti Lippi, and Valdinoci [J. Math. Biol. 89 (2024), Paper No. 19]. This is particularly relevant when studying biological populations, since the Allee effect detects a critical density below which the population is severely endangered and at risk of extinction. [ABSTRACT FROM AUTHOR]

The chord log-Minkowski problem asks for necessary and sufficient conditions for a finite Borel measure on the unit sphere so that it is the cone-chord measure of a convex body. The chord log-Minkowski problem has been extensively studied by Guo, Xi, and Zhao [Math. Ann. (2023), DOI 10.1007/s00208-023-02721-8]; Lutwak, Xi, Yang, and Zhang [Commun. Pure Appl. Math. (2023), DOI 10.1002/cpa.22190]; Qin [Adv. Math. 427 (2023), Paper No. 109132]. In this paper, we solve the chord log-Minkowski problem when q\in (0,1), without symmetry assumptions. [ABSTRACT FROM AUTHOR]

The question of when a vertex algebra is a quantization of the arc space of its associated scheme has recently received a lot of attention in both the mathematics and physics literature. This property was first studied by Tomoyuki Arakawa and Anne Moreau (see their paper in the references), and was given the name \lq\lq classical freeness" by Jethro van Ekeren and Reimundo Heluani [Comm. Math. Phys. 386 (2021), no. 1, pp. 495-550] in their work on chiral homology. Later, it was extended to vertex superalgebras by Hao Li [Eur. J. Math. 7 (2021), pp. 1689–1728]. In this note, we prove the classical freeness of the simple affine vertex superalgebra L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{m|2r}) for all positive integers m,n,r satisfying -\frac {m}{2} + r +n+1 > 0. In particular, it holds for the rational vertex superalgebras L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{1|2r}) for all positive integers r,n. [ABSTRACT FROM AUTHOR]

The objective of this paper is to characterize the structure of the set \Theta for a continuous ergodic upper probability \mathbb {V}=\sup _{P\in \Theta }P \Theta contains a finite number of ergodic probabilities; Any invariant probability in \Theta is a convex combination of those ergodic ones in \Theta; Any probability in \Theta coincides with an invariant one in \Theta on the invariant \sigma-algebra. The last property has already been obtained in Cerreia-Vioglio, Maccheroni, and Marinacci [Proc. Amer. Math. Soc. 144 (2016), pp. 3381–3396], which first studied the ergodicity of such capacities. As an application of the characterization, we prove an ergodicity result, which improves the result of Cerreia-Vioglio, Maccheroni, and Marinacci [Proc. Amer. Math. Soc. 144 (2016), pp. 3381–3396] in the sense that the limit of the time means of \xi is bounded by the upper expectation \sup _{P\in \Theta }E_P[\xi ], instead of the Choquet integral. Generally, the former is strictly smaller. [ABSTRACT FROM AUTHOR]

We continue the program first initiated by Laptev, Schimmer, and Takhtajan [Geom. Funct. Anal. 26 (2016), pp. 288–305] and develop a modification of the technique introduced in that paper to study the spectral asymptotics, namely the Riesz means and eigenvalue counting functions, of functional difference operators H_0 = \mathcal {F}^{-1} M_{\cosh (\xi)} \mathcal {F} with potentials of the form W(x) = \left \lvert {x} \right \rvert ^pe^{\left \lvert {x} \right \rvert ^\beta } for either \beta = 0 and p > 0 or \beta \in (0, 2] and p \geq 0. We provide a new method for studying general potentials which includes the potentials studied by Laptev, Schimmer, and Takhtajan [Geom. Funct. Anal. 26 (2016), pp. 288–305] and [J. Math. Phys. 60 (2019), p. 103505]. The proof involves dilating the variance of the gaussian defining the coherent state transform in a controlled manner preserving the expected asymptotics. [ABSTRACT FROM AUTHOR]

In this paper, we study long time behavior of solution to the quantum Zakharov system in two dimensions. We construct a small global solution to the quantum Zakharov system which scatters to a given free solution by using space-time resonance method developed by Gustafson-Nakanishi-Tsai [Commun. Contemp. Math. 11 (2009), pp. 657–707] and Germain-Masmoudi-Shatah [Int. Math. Res. Not. IMRN 3 (2009), 414–432; J. Math. Pures Appl. (9) 97 (2012), pp. 505–543] etc. [ABSTRACT FROM AUTHOR]

Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [ Frieze matrices and infinite frieze patterns with coefficients , Preprint, arXiv: 2207.04120 , 2022] can all be obtained as consequences of our result. [ABSTRACT FROM AUTHOR]

In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D and any integers i,j (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

A study of the class \Delta consisting of topological \Delta-spaces was originated by Jerzy Ka̧kol and Arkady Leiderman [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99; Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 267–280]. The main purpose of this paper is to introduce and investigate new classes \Delta _2 \subset \Delta _1 properly containing \Delta. We observe that for every first-countable X the following equivalences hold: X\in \Delta _1 iff X\in \Delta _2 iff each countable subset of X is G_{\delta }. Thus, new proposed concepts provide a natural extension of the family of all \lambda-sets beyond the separable metrizable spaces. We prove that (1) A pseudocompact space X belongs to the class \Delta _1 iff countable subsets of X are scattered. (2) Every regular scattered space belongs to the class \Delta _2. We investigate whether the classes \Delta _1 and \Delta _2 are invariant under the basic topological operations. Similarly to \Delta, both classes \Delta _1 and \Delta _2 are invariant under the operation of taking countable unions of closed subspaces. In contrast to \Delta, they are not preserved by closed continuous images. Let Y be l-dominated by X, i.e. C_p(X) admits a continuous linear map onto C_p(Y). We show that Y \in \Delta _1 whenever X \in \Delta _1. Moreover, we establish that if Y is l-dominated by a compact scattered space X, then Y is a pseudocompact space such that its Stone–Čech compactification \beta Y is scattered. [ABSTRACT FROM AUTHOR]

In this paper we show that the geodesic flow of a Finsler metric is Anosov if and only if there exists a C^2 open neighborhood of Finsler metrics all of whose closed geodesics are hyperbolic. For surfaces this result holds also for Riemannian metrics. This follows from a recent result of Contreras and Mazzucchelli [Duke Math. J. 173 (2024), pp. 347–390]. Furthermore, geodesic flows of Riemannian or Finsler metrics on surfaces are C^2 stably ergodic if and only if they are Anosov. [ABSTRACT FROM AUTHOR]

In the present paper, we study the shifted hypergeometric function f(z)=z_{2}F_{1}(a,b;c;z) for real parameters with 0

This paper studies the higher differentials of the classical Adams spectral sequence at odd primes. In particular, we follow the "cofiber of \tau philosophy" of Gheorghe, Isaksen, Wang, and Xu [Acta Math. 226 (2021), pp. 319–407; Publ. Math. Inst. Hautes Études Sci. 137 (2023), pp. 107–243] to show that higher Adams differentials agree with their corresponding higher algebraic Novikov differentials in a certain range. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the relationship between sampling measure \mu, Berezin transform \tilde {\mu } and r-averaging transform \widehat {\mu }_r on Bergman spaces. Compared with some results of Luecking [Amer. J. Math. 107 (1985), pp. 85–111], our results provide an equivalent description of sampling measures, which reveals the reason why 1/\hat {\mu }_r\in L^\infty or 1/\tilde {\mu }\in L^\infty does not make sure that \mu is a sampling measure on Bergman spaces. [ABSTRACT FROM AUTHOR]

In this paper, using the method of blow-up analysis, we establish a trace Trudinger-Moser inequality involving L^p-norm and obtain the corresponding extremal on a compact Riemann surface with a smooth boundary. The result generalizes those of Li-Liu [Math. Z. 250 (2005), pp. 363–686] and Zhang [Commun. Pure Appl. Anal. 20 (2021), pp. 1721–1735]. [ABSTRACT FROM AUTHOR]

Let \mu (n) be the Möbius function and e(\alpha)=e^{2\pi i\alpha }. In this paper, we study upper bounds of the classical sum \[ S(x,\alpha)≔\sum _{1\leq n\leq x}\mu (n)e(\alpha n). \] We can improve some classical results of Baker and Harman [J. London Math. Soc. (2) 43 (1991), pp. 193–198]. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

This paper is devoted to the isochronicity of two kinds of planar piecewise polynomial potential systems separated by x-axis and y-axis, respectively. By means of the expansion of the period function near the infinity, we obtain in this paper that for these two cases, the piecewise potential system has an isochronous center at the origin if and only if the subsystems are linear. It is however not true in the analytic scenario; one can find in the paper by F. Mañosas and P. J. Torres [Proc. Amer. Math. Soc. 133 (2005), pp. 3027–3035] that the isochronicity of the center of the piecewise analytic potential system cannot imply the linearity of the two subsystems. [ABSTRACT FROM AUTHOR]

In 2000, Héthelyi and Külshammer [Bull. London Math. Soc. 32 (2000), pp. 668–672] proposed that if G is a finite group, p is a prime dividing the group order, and k(G) is the number of conjugacy classes of G, then k(G)\geq 2\sqrt {p-1}, and they proved this conjecture for solvable G and showed that it is sharp for those primes p for which \sqrt {p-1} is an integer. This initiated a flurry of activity, leading to many generalizations and variations of the result; in particular, today the conjecture is known to be true for all finite groups. In this note, we put forward a natural new and stronger conjecture, which is sharp for all primes p, and we prove it for solvable groups, and when p is large, also for arbitrary groups. [ABSTRACT FROM AUTHOR]

Quite recently, we have obtained two main theorems dealing with absolute weighted arithmetic mean summability factors of infinite series and trigonometric Fourier series [C. R. Math. Acad. Sci. Paris 359 (2021), pp. 323–328]. In this paper, we have generalized these theorems for a general summability method. We have also obtained some new and known results for certain absolute summability methods. [ABSTRACT FROM AUTHOR]

In 1934, G. H. Hardy and J. E. Littlewood calculated [Proc. London Math. Soc. (2) 36 (1934), pp. 516–531] the optimal Cesàro exponent for Hardy spaces. In this paper we calculate it for mixed norm spaces, hence including the Bergman spaces in particular. The main technical challenge lies in the analysis of the example needed for the critical case. [ABSTRACT FROM AUTHOR]

Hovey and Palmieri [ The structure of the Bousfield lattice , Amer. Math. Soc., Providence, RI, 1999] proposed the retract conjecture on the Bousfield lattice of the stable homotopy category. The author, Shimomura and Tatehara [Publ. Res. Inst. Math. Sci. 50 (2014), pp. 497–513] defined the notion of monoidal posets as a generalization of the Bousfield lattice. In this paper, we prove that an analogue of the retract conjecture holds on a sublattice of monoidal posets. [ABSTRACT FROM AUTHOR]

Kurokawa and Wakayama [Proc. Amer. Math. Soc. 132 (2004), pp. 935–943] defined a q-analogue of the Euler constant and proved the irrationality of certain numbers involving q-Euler constant. In this paper, we improve their results and prove the linear independence result involving q-analogue of the Euler constant. Further, we derive the closed-form of a q-analogue of the k-th Stieltjes constant \gamma _k(q). These constants are the coefficients in the Laurent series expansion of a q-analogue of the Riemann zeta function around s=1. Using a result of Nesterenko [C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), pp. 909–914], we also settle down a question of Erdős regarding the arithmetic nature of the infinite series \sum _{n\geq 1}{\sigma _1(n)}/{t^n} for any integer t > 1. Finally, we study the transcendence nature of some infinite series involving \gamma _1(2). [ABSTRACT FROM AUTHOR]

A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. In this paper we show that every quasiconformal tree bi-Lipschitz embeds in some Euclidean space, with the ambient dimension and the bi-Lipschitz constant depending only on the doubling and bounded turning constants of the tree. This answers Question 1.6 of David and Vellis [Illinois J. Math. 66 (2022), pp. 189–244]. [ABSTRACT FROM AUTHOR]

Geodesic trapping is an obstruction to dispersive estimates for solutions to the Schrödinger equation. Surprisingly little is known about solutions to the Schrödinger equation on manifolds with degenerate trapping, since the conditions for degenerate trapping are not stable under perturbations. In this paper we extend some of the results of Christianson and Metcalfe [Indiana Univ. Math. J. 63 (2014), pp. 969–992] on inflection-transmission type trapping on warped product manifolds to the case of multi -warped products. The main result is that the trapping on one cross section does not interact with the trapping on other cross sections provided the manifold has only one infinite end and only inflection-transmission type trapping. [ABSTRACT FROM AUTHOR]

A recent paper of Shekhar compares the ranks of elliptic curves E1 and E2 for which there is an isomorphism E1[p] ∼ E2[p]asGal(.../Q)- modules, where p is a prime of good ordinary reduction for both curves. In this paper we prove an analogous result in the case where p is a prime of good supersingular reduction. [ABSTRACT FROM AUTHOR]

We provide an elementary proof of a result by V. P. Fonf and C. Zanco on point-finite coverings of separable Hilbert spaces. Indeed, by using a variation of the famous argument introduced by J. Lindenstrauss and R. R. Phelps [Israel J. Math. 6 (1968), pp. 39–48] to prove that the unit ball of a reflexive infinite-dimensional Banach space has uncountably many extreme points, we prove the following result. Let X be an infinite-dimensional Hilbert space satisfying dens(X)< 2ℵ0, then X does not admit point-finite coverings by open or closed balls, each of positive radius. In the second part of the paper, we follow the argument introduced by V. P. Fonf, M. Levin, and C. Zanco in [J. Geom. Anal. 24 (2014), pp. 1891–1897] to prove that the previous result holds also in infinite-dimensional Banach spaces that are both uniformly rotund and uniformly smooth. [ABSTRACT FROM AUTHOR]

A well known conjecture due to Hassett asserts that a cubic fourfold X whose transcendental cohomology TX cannot be realized as the transcendental cohomology of a K3 surface is irrational. Since the geometry of cubic fourfolds is intricately related to the existence of algebraic 2-cycles on them, it is natural to ask for the most algebraic cubic fourfolds X to which this conjecture is still applicable. In this paper, we show that for an appropriate "algebraicity index" κX ∈ Q+, there exists a unique class of cubics maximizing κX, not having an associated K3 surface; namely, the cubic fourfolds with an Eckardt point (previously investigated in by Laza, Pearlstein, and Zhang [Adv. Math. 340 (2018), pp. 684-722]). Arguably, they are the most algebraic conjecturally irrational cubic fourfolds, and thus a good testing ground for Hassett's irrationality conjecture for cubic fourfolds. [ABSTRACT FROM AUTHOR]

We prove that a parabolic subgroup P contained in another parabolic subgroup P' of an Artin group A is a parabolic subgroup of P'. This answers a question of Godelle which is not obvious despite appearances. In order to achieve our result we construct a set-retraction A \to P of the inclusion map from a parabolic subgroup P into A. This retraction was implicitly constructed in a previous paper by Charney and the second author [Bull. Lond. Math. Soc. 46 (2014), pp. 1248–1255]. [ABSTRACT FROM AUTHOR]

Levin, Przytycki and Shen [Invent. Math. 205 (2016), pp. 363–382] proved for a polynomial map f_c(z)=z^d+c, d\geq 2 and c \in \mathbb C, with Julia set J(f) of positive measure that for a.e. z \in J(f) the Lyapunov exponent \chi _s(z)=0. The aim of this paper is to show that the extension to non-entire transcendental meromorphic functions is not possible. [ABSTRACT FROM AUTHOR]

In this paper a multiplier theorem in the Hardy space H^1(\mathbb {T}) associated with Jacobi expansions of exponential type is proved, that is, a bilateral sequence \left \{\lambda _n\right \}_{n=-\infty }^{\infty } is a multiplier from H^1(\mathbb {T}) into the sequence space \ell ^1(\mathbb {Z}) associated with Jacobi expansions of exponential type, if \[ \sup _N\sum _{k=1}^{\infty }\left (\sum _{kN<|j|\le (k+1)N}|\lambda _j|\right)^2<\infty.\] This is a generalization of a multiplier theorem on usual Fourier expansions in the Hardy space H^1(\mathbb {T}), and for \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]. [ABSTRACT FROM AUTHOR]

This paper provides a refinement of the study of asymptotic behaviour for a class of nonlocal in time telegraph equations with positively singular kernels. Based on fundamental properties of relaxation functions and recent representation of the fundamental solution in [Nonlinear Anal. 193 (2020), 111411], we establish the asymptotic expansions of the variance of the stochastic process for both long-time and short-time, which sharply improves the main result in [Proc. Amer. Math. Soc. 149 (2021), 2067–2080] by removing their technical conditions on the regularly varying behaviours and reformulating the asymptotic expansion in a more natural form. By analysing a new noncommutative operation on a subclass of completely positive functions, we provide a new way to construct finitely many ultraslow subdiffusion processes that are rapidly slower than a given ultraslow kernel. Consequently, we show that for a given completely monotonic ultraslow kernel, there is an induced kernel whose corresponding mean square displacement is logarithmic. [ABSTRACT FROM AUTHOR]

We show that the celebrated operad of pre-Lie algebras is very rigid: it has no "non-obvious" degrees of freedom from either of the three points of view: deformations of maps to and from the "three graces of operad theory", homotopy automorphisms, and operadic twisting. Examining the latter, it is possible to answer two questions of Markl from 2005 [Czechoslovak Math. J. 57 (2007), pp. 253–268; J. Lie Theory 17 (2007), pp. 241–261], including a Lie-theoretic version of the Deligne conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we correct some errors that appeared in ''The Łojasiewicz exponent of a continuous subanalytic function at an isolated zero'', Proc. Amer. Math. Soc. 139 (2011), no. 1, 1-9. [ABSTRACT FROM AUTHOR]

For any n\in \mathbb {N}=\{0,1,2,\ldots \} and b,c\in \mathbb {Z}, the generalized central trinomial coefficient T_n(b,c) denotes the coefficient of x^n in the expansion of (x^2+bx+c)^n. Let p be an odd prime. In this paper, we determine the summations \sum _{k=0}^{p-1}T_k(b,c)^2/m^k modulo p^2 for integers m with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), pp. 1375–1400]. [ABSTRACT FROM AUTHOR]

Let M be the collection of the multipliers of the Drury-Arveson space H2n, n ≥ 2. In a recent paper [Adv. Math. 335 (2018), pp. 372-404], Aleman et al. showed that for ƒ ∈ H2n, the condition sup|z|<1 Re(ƒ,K_zƒ) < ∞ is sufficient for the membership ƒ ∈ M. We show that this condition is not necessary for ƒ ∈ M. Moreover, we show that the condition sup|z|<1 Re(ƒ,Kzƒ) < ∞ only captures a nowhere dense subset of M. [ABSTRACT FROM AUTHOR]

We study the fundamental solutions of parabolic magnetic Schrödinger operators and uniform parabolic operators with nonnegative potentials in the reverse Hölder class. The main aim of the paper is to give pointwise estimates of the heat kernel of the operators above, which improve and generalize the main results by Kurata [J. London Math. Soc. (2) 62 (2000), pp. 885–903]. [ABSTRACT FROM AUTHOR]

In this note we fix an error in the proof of Theorem 2.1 in our paper ''On composition of maximal operator and Bochner-Riesz operator at the critical index,'' which appeared in Proc. Amer. Math. Soc. 148 (2020), no. 4, 1545-1554. [ABSTRACT FROM AUTHOR]

In this paper we study the regularity of solutions of a transmission problem arising in studying a fiber-reinforced composite media. It is described through a divergence type equation div(A∇u) = h, in an open set D, where h is a bounded function and A is a uniformly elliptic matrix, bounded and with piecewise Hölder continuous coefficients. The subdomains Di where A is of class Cα have disjoint closure and are of class C1,α. Exploiting an idea contained in a paper by Li and Vogelius, we obtain the optimal regularity result for solutions, proving that they are of class C1,α(D i). [ABSTRACT FROM AUTHOR]

The paper deals with the interplay between boundedness, order and ring structures in function lattices on the line and related metric spaces. It is shown that the lattice of all Lipschitz functions on a normed space E is isomorphic to its sublattice of bounded functions if and only if E has dimension one. The lattice of Lipschitz functions on E carries a "hidden" f-ring structure with a unit, and the same happens to the (larger) lattice of all uniformly continuous functions for a wide variety of metric spaces. An example of a metric space whose lattice of uniformly continuous functions supports no unital f-ring structure is provided. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the boundedness of the Hardy-Littlewood maximal operator on the Orlicz space Lp(·)(log L)q(·)(X) of two variable exponents over unbounded quasi-metric measure spaces, as an extension of [Math Scand. 116 (2015), pp. 5–22]. The result is new even for the variable exponent Lebesgue space Lp(·)(X) in that the underlying spaces need not be bounded and that the underlying measure need not be doubling. [ABSTRACT FROM AUTHOR]

By a result of Schur [J. Reine Angew. Math. 140 (1911), pp. 1–28], the entrywise product M ∘ N of two positive semidefinite matrices M,N is again positive. Vybíral [Adv. Math. 368 (2020), p. 9] improved on this by showing the uniform lower bound M ∘ M ≥ En/n for all n × n real or complex correlation matrices M, where En is the all-ones matrix. This was applied to settle a conjecture of Novak [J. Complexity 15 (1999), pp. 299–316] and to positive definite functions on groups. Vybíral (in his original preprint) asked if one can obtain similar uniform lower bounds for higher entrywise powers of M, or for M ∘ N when N ≠ M, M. A natural third question is to ask for a tighter lower bound that does not vanish as n → ∞, i.e., over infinite-dimensional Hilbert spaces. In this note, we affirmatively answer all three questions by extending and refining Vybíral's result to lower-bound M ∘ N, for arbitrary complex positive semidefinite matrices M,N. Specifically: we provide tight lower bounds, improving on Vybíral's bounds. Second, our proof is 'conceptual' (and self-contained), providing a natural interpretation of these improved bounds via tracial Cauchy–Schwarz inequalities. Third, we extend our tight lower bounds to Hilbert–Schmidt operators. As an application, we settle Open Problem 1 of Hinrichs–Krieg–Novak–Vybíral [J. Complexity 65 (2021), Paper No. 101544, 20 pp.], which yields improvements in the error bounds in certain tensor product (integration) problems. [ABSTRACT FROM AUTHOR]

This erratum concerns Lemma 3.1 in the original paper [Proc. Amer. Math. Soc. 147 (2019), pp. 1657-1669]. The statement of that lemma needs to be slightly changed since the statement of Claim 3.5 must be changed. The proof of the lemma is mostly still kept as the original one with some lines modified. This change does not effect the proof of the main results. [ABSTRACT FROM AUTHOR]