Harmonic functions of general graph Laplacians

We study harmonic functions on general weighted graphs which allow for a compatible intrinsic metric. We prove an \(L^{p}\) Liouville type theorem which is a quantitative integral \(L^{p}\) estimate of harmonic functions analogous to Karp’s theorem for Riemannian manifolds. As corollaries we obtain Yau’s \(L^{p}\) -Liouville type theorem on graphs, identify the domain of the generator of the semigroup on \(L^{p}\) and get a criterion for recurrence. As a side product, we show an analogue of Yau’s \(L^{p}\) Caccioppoli inequality. Furthermore, we derive various Liouville type results for harmonic functions on graphs and harmonic maps from graphs into Hadamard spaces.     

A converse of Baer’s theorem

Schur’s classical theorem states that for a group $G$ , if $G/Z(G)$ is finite, then $G'$ is finite. Baer extended this theorem for the factor group $G/Z_n(G)$ , in which $Z_n(G)$ is the $n$ -th term of the upper central series of $G$ . Hekster proved a converse of Baer’s theorem as follows: If $G$ is a finitely generated group such that $\gamma _{n+1}(G)$ is finite, then $G/Z_n(G)$ is finite where $\gamma _{n+1}(G)$ denotes the $(n+1)$ st term of the lower central series of $G$ . In this paper, we generalize this result by obtaining the same conclusion under the weaker hypothesis that $G/Z_n(G)$ is finitely generated. Furthermore, we show that the index of the subgroup $Z_n(G)$ is bounded by a precisely determined function of the order of $\gamma _{n+1}(G)$ . Moreover, we prove that the mentioned theorem of Hekster is also valid under a weaker condition that $Z_{2n}(G)/Z_{n}(G)$ is finitely generated. Although in this case the bound for the order of $\gamma _{n+1}(G)$ is not achieved.     

The Lattice of Varieties of Completely Regular Semigroups

Polák’s theorem on the structure of the (lattice of) varieties of completely regular semigroups provides an isomorphic copy of the interval $[{\cal S,CR}]$ of varieties which contain semilattices in terms of certain functions. We give a variant of this theorem for the lattice ${\cal L(CR)}$ of all varieties of completely regular semigroups in terms of pairs with componentwise inclusion. The first entry of these pairs is a band variety and the second consists of a ?₀-tuple of members of ${\cal K}_0$ . Here ${\cal K}_0$ is the set of varieties which satisfy ${\cal V}_K={\cal V}$ where ${\cal V}_K$ is the least element of the K-class containing ${\cal V}$ . We have based the proof of our theorem on Polák’s theorem for the sake of expediency and comparison. It utilizes a set of varieties which we term canonical. Several corollaries treat various special cases.     

On locally analytic Beilinson–Bernstein localization and the canonical dimension

Let $\mathbf{G}$ be a connected split reductive group over a $p$ -adic field. In the first part of the paper we prove, under certain assumptions on $\mathbf{G}$ and the prime $p$ , a localization theorem of Beilinson–Bernstein type for admissible locally analytic representations of principal congruence subgroups in the rational points of $\mathbf{G}$ . In doing so we take up and extend some recent methods and results of Ardakov–Wadsley on completed universal enveloping algebras (Ardakov and Wadsley, Ann. Math., 2013) to a locally analytic setting. As an application we prove, in the second part of the paper, a locally analytic version of Smith’s theorem on the canonical dimension.     

Factorization of Lipschitz functions and absolute convergence of Vilenkin-Fourier series

LetG be a Vilenkin group (i.e., an infinite, compact, metrizable, zero-dimensional Abelian group). Our main result is a factorization theorem for functions in the Lipschitz spaces \(\mathfrak{L}\mathfrak{i}\mathfrak{p}\) (α,p; G). As colloraries of this theorem, we obtain (i) an extension of a factorization theorem ofY. Uno; (ii) a convolution formula which says that \(\mathfrak{L}\mathfrak{i}\mathfrak{p} (\alpha , r; G) = \mathfrak{L}\mathfrak{i}\mathfrak{p} (\beta , l; G)*\mathfrak{L}\mathfrak{i}\mathfrak{p} (\alpha - \beta , r; G)\) for 0<β<α<∞ and 1≤r≤∞; and (iii) an analogue, valid for allG, of a classical theorem ofHardy andLittlewood. We also present several results on absolute convergence of Fourier series defined onG, extending a theorem ofC. W. Onneweer and four results ofN. Ja. Vilenkin andA. I. Rubinshtein. The fourVilenkin-Rubinshtein results are analogues of classical theorems due, respectively, toO. Szász, S. B. Bernshtein, A. Zygmund, andG. G. Lorentz.     

Scalarization of \epsilon -Super Efficient Solutions of Set-Valued Optimization Problems in Real Ordered Linear Spaces

In this paper, we investigate the scalarization of \(\epsilon \) -super efficient solutions of set-valued optimization problems in real ordered linear spaces. First, in real ordered linear spaces, under the assumption of generalized cone subconvexlikeness of set-valued maps, a dual decomposition theorem is established in the sense of \(\epsilon \) -super efficiency. Second, as an application of the dual decomposition theorem, a linear scalarization theorem is given. Finally, without any convexity assumption, a nonlinear scalarization theorem characterized by the seminorm is obtained.     

Weighted Dunkl transform inequalities and application on radial Besov spaces

In Dunkl theory on $\mathbb R ^d$ which generalizes classical Fourier analysis, we prove first weighted inequalities for certain Hardy-type averaging operators. In particular, we deduce for specific choices of the weights the $d$ -dimensional Hardy inequalities whose constants are sharp and independent of $d$ . Second, we use the weight characterization of the Hardy operator to prove weighted Dunkl transform inequalities. As consequence, we obtain Pitt’s inequality which gives an integrability theorem for this transform on radial Besov spaces.     

Simplicity of normal subgroups and conjugacy class sizes

Given a finite group \(G\) which possesses a non-abelian simple normal subgroup \(N\) having exactly four \(G\) -class sizes, we prove that \(N\) is isomorphic to PSL \((2, 2^a)\) with \(a\ge 2\) . Thus, we obtain an extension for normal subgroups of the classic N. Itô’s theorem which characterizes those finite simple groups with exactly four class sizes.     

Modular Decomposition of the Orlik-Terao Algebra

Let ${\mathcal{A}}$ be a collection of n linear hyperplanes in ${\mathbb{k}^\ell}$ , where ${\mathbb{k}}$ is an algebraically closed field. The Orlik-Terao algebra of ${\mathcal{A}}$ is the subalgebra ${{\rm R}(\mathcal{A})}$ of the rational functions generated by reciprocals of linear forms vanishing on hyperplanes of ${\mathcal{A}}$ . It determines an irreducible subvariety ${Y (\mathcal{A})}$ of ${\mathbb{P}^{n-1}}$ . We show that a flat X of ${\mathcal{A}}$ is modular if and only if ${{\rm R}(\mathcal{A})}$ is a split extension of the Orlik-Terao algebra of the subarrangement ${\mathcal{A}_X}$ . This provides another refinement of Stanley’s modular factorization theorem [34] and a new characterization of modularity, similar in spirit to the fibration theorem of [27]. We deduce that if ${\mathcal{A}}$ is supersolvable, then its Orlik-Terao algebra is Koszul. In certain cases, the algebra is also a complete intersection, and we characterize when this happens.     

A generalizaton of Hardy’s uncertainty principle on compact extensions of \mathbb R ^n

Let \(K\) be a compact subgroup of automorphisms of \(\mathbb R ^n\) . We prove in this paper a generalization of Hardy’s uncertainty principle on the semi-direct product \(K\ltimes \mathbb R ^n\) , and we solve the sharpness problem. As a consequence, a complete analogue of classical Hardy’s theorem is obtained. The representation theory and the Plancherel formula play an important role in the proofs.     

Moment Functions and Central Limit Theorem for Jacobi Hypergroups on [0,\infty [

In this paper, we derive sharp estimates and asymptotic results for moment functions on Jacobi type hypergroups. Moreover, we use these estimates to prove a central limit theorem (CLT) for random walks on Jacobi hypergroups with growing parameters $\alpha ,\beta \rightarrow \infty $ . As a special case, we obtain a CLT for random walks on the hyperbolic spaces ${H}_d(\mathbb F )$ with growing dimensions $d$ over the fields $\mathbb F =\mathbb R ,\ \mathbb C $ or the quaternions $\mathbb H $ .     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\) ). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\) – \(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\) -bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\) – \(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin carathéodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Hölder spaces, but our approach is different from them.     

On the representation of tight functionals as integrals

We consider a vector lattice $\mathcal L $ of bounded real continuous functions on a topological space $X$ that separates the points of $X$ and contains the constant functions. A notion of tightness for linear functionals is defined, and by an elementary argument we prove with the aid of the classical Riesz representation theorem that every tight continuous linear functional on $\mathcal L $ can be represented by integration with respect to a Radon measure. This result leads incidentally to an simple proof of Prokhorov’s existence theorem for the limit of a projective system of Radon measures.     

Harmonic mappings in Bergman spaces

In this paper, we investigate the properties of mappings in harmonic Bergman spaces. First, we discuss the coefficient estimate, the Schwarz-Pick Lemma and the Landau-Bloch theorem for mappings in harmonic Bergman spaces in the unit disk $\mathbb D $ of $\mathbb C $ . Our results are generalizations of the corresponding ones in Chen et al. (Proc Am Math Soc 128:3231–3240, 2000), Chen et al. (J Math Anal Appl 373:102–110, 2011), Chen et al. (Ann Acad Sci Fenn Math 36:567–576, 2011). Then, we study the Schwarz-Pick Lemma and the Landau-Bloch theorem for mappings in harmonic Bergman spaces in the unit ball $\mathbb B ^{n}$ of $\mathbb C ^{n}$ . The obtained results are generalizations of the corresponding ones in Chen and Gauthier (Proc Am Math Soc 139:583–595 2011). At last, we get a characterization for mappings in harmonic Bergman spaces on $\mathbb B ^{n}$ in terms of their complex gradients.     

The structure theorem for comodule algebras over Hopf algebroids

We obtain the structure theorem for $(H, \mathcal{H})$ -Hopf bimodules over Hopf algebroids, where H is the total algebra of the Hopf algebroid  $\mathcal{H}$ . Based on this theorem, we investigate the structure theorem for comodule algebras over Hopf algebroids.     

Rigidity Theorems for Spherical Hyperexpansions

The class of spherical hyperexpansions is a multi-variable analog of the class of hyperexpansive operators with spherical isometries and spherical 2-isometries being special subclasses. It is known that in dimension one, an invertible $2$ -hyperexpansion is unitary. This rigidity theorem allows one to prove a variant of the Berger–Shaw Theorem which states that a finitely multi-cyclic $2$ -hyperexpansion is essentially normal. In the present paper, we seek for multi-variable manifestations of this rigidity theorem. In particular, we provide several conditions on a spherical hyperexpansion which ensure it to be a spherical isometry. We further carry out the analysis of the rigidity theorems at the Calkin algebra level and obtain some conditions for essential normality of a spherical hyperexpansion. In the process, we construct several interesting examples of spherical hyperexpansions which are structurally different from the Drury-Arveson $m$ -shift.     

Pointwise convergence of vector-valued Fourier series

We prove a vector-valued version of Carleson’s theorem: let $Y=[X,H]_\theta $ be a complex interpolation space between an unconditionality of martingale differences (UMD) space $X$ and a Hilbert space $H$ . For $p\in (1,\infty )$ and $f\in L^p(\mathbb T ;Y)$ , the partial sums of the Fourier series of $f$ converge to $f$ pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form $Y=[X,H]_\theta $ . In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions.     

Menger’s theorem in {{\Pi^1_1\tt{-CA}_0}}

We prove Menger’s theorem for countable graphs in ${{\Pi^1_1\tt{-CA}_0}}$ . Our proof in fact proves a stronger statement, which we call extended Menger’s theorem, that is equivalent to ${{\Pi^1_1\tt{-CA}_0}}$ over ${{\tt{RCA}_0}}$ .     

Limit theorems for von Mises statistics of a measure preserving transformation

For a measure preserving transformation \(T\) of a probability space \((X,\mathcal{F },\mu )\) and some \(d \ge 1\) we investigate almost sure and distributional convergence of random variables of the form $$\begin{aligned} x \rightarrow \frac{1}{C_n} \sum _{0\le i_1,\ldots ,\,i_d where \(C_1, C_2,\ldots \) are normalizing constants and the kernel \(f\) belongs to an appropriate subspace in some \(L_p(X^d\!,\, \mathcal{F }^{\otimes d}\!,\,\mu ^d)\) . We establish a form of the individual ergodic theorem for such sequences. Using a filtration compatible with \(T\) and the martingale approximation, we prove a central limit theorem in the non-degenerate case; for a class of canonical (totally degenerate) kernels and \(d=2\) , we also show that the convergence holds in distribution towards a quadratic form \(\sum _{m=1}^{\infty } \lambda _m\eta ^2_m\) in independent standard Gaussian variables \(\eta _1, \eta _2, \ldots \) .