Search

This paper gives an explicit formula of the asymptotic expansion of the Kobayashi–Royden metric on the punctured sphere ℂ ⁢ ℙ 1 ∖ { 0 , 1 , ∞ } {\mathbb{CP}^{1}\setminus\{0,1,\infty\}} in terms of the exponential Bell polynomials. We prove a local quantitative version of the Little Picard’s Theorem as an application of the asymptotic expansion. Furthermore, the approach in the paper leads to the interesting consequence that the coefficients in the asymptotic expansion are rational numbers. Furthermore, the explicit formula of the metric and the conclusion regarding the coefficients apply to more general cases of ℂ ⁢ ℙ 1 ∖ { a 1 , … , a n } {\mathbb{CP}^{1}\setminus\{a_{1},\ldots,a_{n}\}} , n ≥ 3 {n\geq 3} , as well, and the metric on ℂ ⁢ ℙ 1 ∖ { 0 , 1 3 , - 1 6 ± 3 6 ⁢ i } {\mathbb{CP}^{1}\setminus\{0,\frac{1}{3},-\frac{1}{6}\pm\frac{\sqrt{3}}{6}i\}} will be given as a concrete example of our results.

This paper proves the boundedness of the generalized Riesz potentials I ρ , μ , τ ⁢ f {I_{\rho,\mu,\tau}f} of functions in the Morrey space L ( 1 , φ ; κ ) ⁢ ( G ) {L^{(1,\varphi;\kappa)}(G)} over a general measure space X, with G a bounded open set in X (or G is X ) {X)} , as an extension of earlier results. The modification parameter τ is introduced for the purpose of including the case where the underlying measure does not satisfy the doubling condition. What is new in the present paper is that ρ depends on x ∈ X {x\in X} . An example in the end of this article convincingly explains why the modification parameter τ must be introduced.

We define the concepts of commutator (Krylov) transitive and strongly commutator (Krylov) transitive Abelian p-groups. These two innovations are respectively non-trivial generalizations of the notions of commutator fully transitive and strongly commutator fully transitive p-groups from a paper of Chekhlov and Danchev (J. Group Theory, 2015). They are also commutator socle-regular in the sense of Danchev and Goldsmith (J. Group Theory, 2014). Various results from there and from a paper of Goldsmith and Strüngmann (Houston J. Math., 2007) are considerably extended to this new point of view. We also define and explore the concept of a commutator weakly transitive Abelian p-group, comparing its properties with those of the aforementioned two group classes. Some affirmations, sounding quite curiously, are detected in order to illustrate the pathology of the commutators in the endomorphism rings of p-primary Abelian groups.

This paper studies the bounded approximation property (BAP) in quasi-Banach spaces. In the first part of the paper, we show that the kernel of any surjective operator ℓ p → X {\ell_{p}\to X} has the BAP when X has it and 0 < p ≤ 1 {0 , which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec–Pełczyński–Wojtaszczyk complementably universal spaces for Banach spaces with the BAP.

As a continuation of a previous paper, we prove Perelman’s assertion, that is, for ancient solutions to the Ricci flow with bounded nonnegative curvature operator, uniformly bounded entropy is equivalent to κ-noncollapsing on all scales. We also establish an equality between the asymptotic entropy and the asymptotic reduced volume, which is a result similar to a paper by Xu (2017), where he assumes the Type I curvature bound.

This paper is the complementary work of [S. Cho, Group schemes and local densities of ramified hermitian lattices in residue characteristic 2: Part I, Algebra Number Theory 10 2016, 3, 451–532]. Ramified quadratic extensions E / F {E/F} , where F is a finite unramified field extension of ℚ 2 {\mathbb{Q}_{2}} , fall into two cases that we call Case 1 and Case 2. In our previous work, we obtained the local density formula for a ramified hermitian lattice in Case 1. In this paper, we obtain the local density formula for the remaining Case 2, by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with [W. T. Gan and J.-K. Yu, Group schemes and local densities, Duke Math. J. 105 2000, 3, 497–524] and our previous work, allows the computation of the mass formula for any hermitian lattice ( L , H ) {(L,H)} , when a base field is unramified over ℚ {\mathbb{Q}} at a prime ( 2 ) {(2)} .

In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) Condition), which means that in each tangent plane, we can find a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) Condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove that any Lie group whose Lie algebra is a rank $2$ non-Abelian compact Lie algebra admits a left invariant Finsler metric satisfying the (FP) condition. As by-products, we find the first example of non-compact coset space $S^3\times \mathbb{R}$ which admits homogeneous flag-wise positively curved Finsler metrics. Moreover, we find some non-negatively curved Finsler metrics on $S^2\times S^3$ and $S^6\times S^7$ which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on $S^3\times S^3$, shedding some light on the long standing general Hopf conjecture., 23 pages. The newest version has strengthened the main results in the paper, and provides more examples. We add a short survey on the most recent progress inspired by this paper in the introduction section

Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher-order Hochschild homology, also called Hochschild–Pirashvili homology. In this paper, we generalize all these results to high-dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of π 0 {\pi_{0}} .

This paper is a companion paper to [G4], where sharp estimates are proven for Fourier transforms of compactly supported functions built out of two-dimensional real-analytic functions. The theorems of [G4] are stated in a rather general form. In this paper, we expand on the results of [G4] and show that there is a class of "well-behaved" functions that contains a number of relevant examples for which such estimates can be explicitly described in terms of the Newton polygon of the function. We will further see that for a subclass of these functions, one can prove noticeably more precise estimates, again in an explicitly describable way., 13 pages

The classification of subfactors of small index revealed several new subfactors. The first subfactor above index 4, the Haagerup subfactor, is increasingly well understood and appears to lie in a (discrete) infinite family of subfactors where the ℤ \mathbb{Z} /3 ℤ \mathbb{Z} symmetry is replaced by other finite abelian groups. The goal of this paper is to give a similarly good description of the Asaeda–Haagerup subfactor which emerged from our study of its Brauer–Picard groupoid. More specifically, we construct a new subfactor 𝒮 {\mathcal{S}} which is a ℤ \mathbb{Z} /4 ℤ \mathbb{Z} × \times ℤ \mathbb{Z} /2 ℤ \mathbb{Z} analogue of the Haagerup subfactor and we show that the even parts of the Asaeda–Haagerup subfactor are higher Morita equivalent to an orbifold quotient of 𝒮 {\mathcal{S}} . This gives a new construction of the Asaeda–Haagerup subfactor which is much more symmetric and easier to work with than the original construction. As a consequence, we can settle many open questions about the Asaeda–Haagerup subfactor: calculating its Drinfeld center, classifying all extensions of the Asaeda–Haagerup fusion categories, finding the full higher Morita equivalence class of the Asaeda–Haagerup fusion categories, and finding intermediate subfactor lattices for subfactors coming from the Asaeda–Haagerup categories. The details of the applications will be given in subsequent papers.

We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s = 0 s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.

The first part of this paper is a generalization of the Feix-Kaledin theorem on the existence of a hyperkahler metric on a neighbourhood of the zero section of the cotangent bundle of a Kahler manifold. We show that the problem of constructing a hyperkahler structure on a neighbourhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix-Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kahler type has a hyperkahler structure on a neighbourhood of its identity section. More generally, we reduce the existence of a hyperkahler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin's unobstructedness theorem., 20 pages. To appear in Journal fur die reine und angewandte Mathematik (Crelle's Journal)

Every simply connected and connected solvable Lie group $G$ admits a simply transitive action on a nilpotent Lie group $H$ via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups $G$ can act simply transitive on which Lie groups $H$. So far the focus was mainly on the case where $G$ is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action $\rho: G \to \operatorname{Aff}(H)$ is simply transitive by looking only at the induced morphism $\varphi: \mathfrak{g} \to \operatorname{aff}(\mathfrak{h})$ between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group $G$ acts simply transitive on a given nilpotent Lie group $H$, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension $4$., Comment: 22 pages, 8 tables. Comments are welcome

Let 𝕍 {{\mathbb{V}}} be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety S. In this paper, we show that the union of the non-factor special subvarieties for ( S , 𝕍 ) {(S,{\mathbb{V}})} , which are of Shimura type with dominant period maps, is a finite union of special subvarieties of S. This generalizes previous results of Clozel and Ullmo (2005) and Ullmo (2007) on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part of a conjecture of Klingler on the André–Oort conjecture for variations of Hodge structures.

In this paper, we construct a class of new modules for the quantum group U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) which are free of rank 1 when restricted to C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] . The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) -module of rank 1 is isomorphic to one of the modules we constructed, and their isomorphism classes are obtained. We also investigate the tensor products of the C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free modules with finite-dimensional simple modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) , and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) .

The main result of this paper is to give that if b ∈ Lip ⁡ ( ℝ n ) {b\in\operatorname{Lip}(\mathbb{R}^{n})} , h j ∈ BMO ⁡ ( ℝ n ) {h_{j}\in\operatorname{BMO}(\mathbb{R}^{n})} , j = 1 , … , k {j=1,\ldots,k} , k ∈ ℤ + {k\in\mathbb{Z}^{+}} and w ∈ A p {w\in A_{p}} , 1 < p < ∞ {1 , then the multilinear Calderón commutators T Ω , b , h → {T_{\Omega,b,\vec{h}}} with variable kernels are bounded on L p ⁢ ( w ) {L^{p}(w)} . In addition, the authors extend the above result to the Morrey space.

In this paper, we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire’s sense) invariant measure has, for each q > 0 {q>0} , zero lower q-generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift system ( X , T ) {(X,T)} (where X = M ℤ {X=M^{\mathbb{Z}}} is endowed with a sub-exponential metric and the alphabet M is a compact and perfect metric space), for which we show that a typical invariant measure has, for each q > 1 {q>1} , infinite upper q-correlation dimension. Under the same conditions, we show that a typical invariant measure has, for each s ∈ ( 0 , 1 ) {s\in(0,1)} and each q > 1 {q>1} , zero lower s-generalized and infinite upper q-generalized dimensions.

The Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

In this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in {\mathbb{R}^{n}} , whose level set {\{u=0\}} is contained in a half-space. If {n\leq 3} , we prove that the solution must be one-dimensional. In dimension {n\geq 4} , we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.

In recent years, a number of papers have been devoted to the study of roots of period polynomials of modular forms. Here, we study cohomological analogues of the Eichler-Shimura period polynomials corresponding to higher $L$-derivatives. We state general conjectures about the locations of the roots of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative "period polynomials" in the case of cusp forms. We prove a special case of this conjecture in the case of Eisenstein series., 21 pages

In this paper we study smooth orientation-preserving free actions of the cyclic group $\mathbb Z/m$ on a class of $(n-1)$-connected $2n$-manifolds, $\sharp g (S^n \times S^n)\sharp \Sigma$, where $\Sigma$ is a homotopy $2n$-sphere. When $n=2$ we obtain a classification up to topological conjugation. When $n=3$ we obtain a classification up to smooth conjugation. When $n \ge 4$ we obtain a classification up to smooth conjugation when the prime factors of $m$ are larger than a constant $C(n)$., Comment: 18 pages

In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander-Reiten quiver of the subcategories is obtained only by the Ausalander-Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, whenever $\Lambda$ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the $2 \times 2$ upper triangular matrix algebra over $\Lambda$ and the stable Auslander algebra of $\Lambda$ can be estimated by the category of modules over the stable Cohen-Macaulay Auslander algebra of $\Lambda$., Comment: Accepted for publication in Forum Mathematicum

We consider iterated integrals of $\log\zeta(s)$ on certain vertical and horizontal lines. Here, the function $\zeta(s)$ is the Riemann zeta-function. It is a well known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of $\int_{0}^{t} \log \zeta(1/2 + it')dt'$ under the Riemann Hypothesis. Moreover, we show that, for any $m\geq 2$, the denseness of the values of an $m$-times iterated integral on the critical line is equivalent to the Riemann Hypothesis., Comment: 15 pages

Recently, the problem of bounding the sup norms of L 2 {L^{2}} -normalized cuspidal automorphic newforms ϕ on GL 2 {\mathrm{GL}_{2}} in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character χ of ϕ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general χ. If the level N is a square, our result reduces to ∥ ϕ ∥ ∞ ≪ N 1 4 + ϵ , \|\phi\|_{\infty}\ll N^{\frac{1}{4}+\epsilon}, at least under the Ramanujan Conjecture. In particular, when χ has conductor N, this improves upon the previous best known bound ∥ ϕ ∥ ∞ ≪ N 1 2 + ϵ {\|\phi\|_{\infty}\ll N^{\frac{1}{2}+\epsilon}} in this setup (due to [A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 2017, 1009–1045]) and matches a lower bound due to [N. Templier, Large values of modular forms, Camb. J. Math. 2 2014, 1, 91–116], thus our result is essentially optimal in this case.

This paper concerns the study of the Schwartz differential equation { h , τ } = s ⁢ E 4 ⁡ ( τ ) {\{h,\tau\}=s\operatorname{E}_{4}(\tau)} , where E 4 {\operatorname{E}_{4}} is the weight 4 Eisenstein series and s is a complex parameter. In particular, we determine all the values of s for which the solutions h are modular functions for a finite index subgroup of SL 2 ⁡ ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}})} . We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of SL 2 ⁡ ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}})} . This also leads to the solutions to the Fuchsian differential equation y ′′ + s ⁢ E 4 ⁡ y = 0 {y^{\prime\prime}+s\operatorname{E}_{4}y=0} .

In this paper, we establish an improved variable coefficient version of the square function inequality, by which the local smoothing estimate L α p → L p {L^{p}_{\alpha}\to L^{p}} for the Fourier integral operators satisfying cinematic curvature condition is further improved. In particular, we establish almost sharp results for 2 < p ⩽ 3 {2 and push forward the estimate for the critical point p = 4 {p=4} . As a consequence, the local smoothing estimate for the wave equation on the manifold is refined. We generalize the results in [S. Lee and A. Vargas, On the cone multiplier in ℝ 3 \mathbb{R}^{3} , J. Funct. Anal. 263 2012, 4, 925–940; J. Lee, A trilinear approach to square function and local smoothing estimates for the wave operator, preprint 2018, https://arxiv.org/abs/1607.08426v5] to its variable coefficient counterpart. The main ingredients in the argument includes multilinear oscillatory integral estimate [J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 2006, 2, 261–302] and decoupling inequality [D. Beltran, J. Hickman and C. D. Sogge, Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds, Anal. PDE 13 2020, 2, 403–433].

In this paper, we study the structure of the pointed-Gromov-Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger-Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any Ricci shrinker limit space is convex, inspired by Colding-Naber's original idea of parabolic smoothing of the distance functions., Comment: revised version, to appear in J. Reine Angew. Math

In this paper we discuss coefficient problems for functions in the class 𝒞 0 ⁢ ( k ) {{\mathcal{C}}_{0}(k)} . This family is a subset of 𝒞 {{\mathcal{C}}} , the class of close-to-convex functions, consisting of functions which are convex in the positive direction of the real axis. Our main aim is to find some bounds of the difference of successive coefficients depending on the fixed second coefficient. Under this assumption we also estimate | a n + 1 | - | a n | {|a_{n+1}|-|a_{n}|} and | a n | {|a_{n}|} . Moreover, it is proved that Re ⁡ { a n } ≥ 0 {\operatorname{Re}\{a_{n}\}\geq 0} for all f ∈ 𝒞 0 ⁢ ( k ) {f\in{\mathcal{C}}_{0}(k)} .

A generalised Postnikov tower for a space $X$ is a tower of principal fibrations with fibres generalised Eilenberg-MacLane spaces, whose inverse limit is weakly homotopy equivalent to $X$. In this paper we give a characterisation of a polyhedral product $Z_K(X,A)$ whose universal cover either admits a generalised Postnikov tower of finite length, or is a homotopy retract of a space admitting such a tower. We also include $p$-local and rational versions of the theorem. We end with a group theoretic application., 24 pages

We generalize the Alvis–Curtis duality to the abstract representations of reductive groups with Frobenius maps. Similar to the case of representations of finite reductive groups, we show that the Alvis–Curtis duality of infinite type, which we define in this paper, also interchanges the irreducible representations in the principal representation category.

A skew generalized power series ring R ⁢ [ [ S , ω , ≤ ] ] {R[[S,\omega,\leq]]} consists of all functions from a strictly ordered monoid ( S , ≤ ) {(S,\leq)} to a ring R whose support is artinian and narrow, with pointwise addition, and with multiplication given by convolution twisted by an action ω of the monoid S on the ring R. Special cases of this ring construction are skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Mal’cev–Neumann series rings, the “unskewed” versions of all of these, and generalized power series rings. In this paper, we characterize the skew generalized power series rings R ⁢ [ [ S , ω , ≤ ] ] {R[[S,\omega,\leq]]} that are left (right) Archimedean domains in the case where the order ≤ {\leq} is total, or ≤ {\leq} is semisubtotal and the monoid S is commutative torsion-free cancellative, or ≤ {\leq} is trivial and S is totally orderable. We also answer four open questions posed by Moussavi, Padashnik and Paykan regarding the rings in the title.

After the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂ ¯ {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are L p ⁢ ( ℝ n ) {L^{p}({\mathbb{R}^{n}})} bounded for 1 < p < ∞ {1 , but only bounded on local Hardy spaces h p ⁢ ( ℝ n ) {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for 0 < p ≤ 1 {0 . Though much work has been done on the L p ⁢ ( ℝ n 1 × ℝ n 2 ) {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 1 < p < ∞ {1 and Hardy H p ⁢ ( ℝ n 1 × ℝ n 2 ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 0 < p ≤ 1 {0 for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of 0 < p ≤ 1 {0 . The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces h p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for 0 < p ≤ 1 {0 recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

Let the self-affine measure μ M , D {\mu_{M,D}} be generated by an expanding real matrix M = diag ⁡ ( ρ 1 - 1 , ρ 2 - 1 ) {M=\operatorname{diag}(\rho_{1}^{-1},\rho_{2}^{-1})} and an integer digit set D = { ( 0 , 0 ) t , ( α 1 , α 2 ) t , ( β 1 , β 2 ) t } {D=\{(0,0)^{t},(\alpha_{1},\alpha_{2})^{t},(\beta_{1},\beta_{2})^{t}\}} with α 1 ⁢ β 2 - α 2 ⁢ β 1 ≠ 0 {\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\neq 0} . In this paper, the sufficient and necessary conditions for L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} to contain an infinite orthogonal set of exponential functions are given.

In this paper, we prove the conjectural endoscopic character identities for tempered representations of metaplectic group $Mp_{2n}$ based on the formalism of endoscopy theory by J. Adams, D. Renard and W.W. Li., This is part of my thesis. Submitted

In this paper the authors produce a projective indecomposable module for the Frobenius kernel of a simple algebraic group in characteristic p that is not the restriction of an indecomposable tilting module. This yields a counterexample to Donkin’s longstanding Tilting Module Conjecture. The authors also produce a Weyl module that does not admit a p-Weyl filtration. This answers an old question of Jantzen, and also provides a counterexample to the ( p , r ) {(p,r)} -Filtration Conjecture.

In this paper, we study dimension-free L p {L^{p}} estimates for UMD lattice-valued q-variations of Hardy–Littlewood averaging operators associated with the Euclidean balls.

Iwasawa algebras of compact p-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of p-adic Lie groups. We previously determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over ℤ p {\mathbb{Z}_{p}} which were uniform pro-p groups in the sense of Dixon, du Sautoy, Mann and Segal. In this paper, for prime p > n + 1 {p>n+1} , we determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-p Iwahori subgroup of GL n ⁢ ( ℤ p ) {\mathrm{GL}_{n}(\mathbb{Z}_{p})} which is not, in general, a uniform pro-p group.

The purpose of this paper is to study a new class of function spaces, called 𝒩 ⁢ ( p , q , s ) {\mathcal{N}(p,q,s)} -type spaces, in the unit ball 𝔹 {{\mathbb{B}}} of ℂ n {{\mathbb{C}}^{n}} . The Carleson measure on such spaces is investigated. Some embedding theorems among 𝒩 ⁢ ( p , q , s ) {\mathcal{N}(p,q,s)} -type spaces, weighted Bergman spaces and weighted Hardy spaces are established. As for applications, the Hadamard products and random power series on 𝒩 ⁢ ( p , q , s ) {\mathcal{N}(p,q,s)} -type spaces are also studied.

In our recent work [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380], the multi-parameter local Hardy space h p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} has been introduced by using the continuous inhomogeneous Littlewood–Paley–Stein square functions. In this paper, we will first establish the new discrete multi-parameter local Calderón’s identity. Based on this identity, we will define the local multi-parameter Hardy space h dis p ⁢ ( ℝ n 1 × ℝ n 2 ) {h_{\mathrm{dis}}^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} by using the discrete inhomogeneous Littlewood–Paley–Stein square functions. Then we prove that these two multi-parameter local Hardy spaces are actually the same. Moreover, the norms of the multi-parameter local Hardy spaces under the continuous and discrete Littlewood–Paley–Stein square functions are equivalent. This discrete version of the multi-parameter local Hardy space is also critical in establishing the duality theory of the multi-parameter local Hardy spaces.

Let μ M , D {\mu_{M,D}} be a self-affine measure generated by an expanding diagonal matrix M ∈ M 3 ⁢ ( ℝ ) {M\in M_{3}(\mathbb{R})} with entries ρ 1 , ρ 2 , ρ 3 {\rho_{1},\rho_{2},\rho_{3}} and the digit set D = { ( 0 , 0 , 0 ) t , ( 1 , 0 , 0 ) t , ( 0 , 1 , 0 ) t , ( 0 , 0 , 1 ) t } {D=\{(0,0,0)^{t},(1,0,0)^{t},(0,1,0)^{t},(0,0,1)^{t}\}} . In this paper, we prove that for any ρ 1 , ρ 2 , ρ 3 ∈ ( 1 , ∞ ) {\rho_{1},\rho_{2},\rho_{3}\in(1,\infty)} , if ρ 1 , ρ 2 , ρ 3 ∈ { ± x 1 r : x ∈ ℚ + , r ∈ ℤ + } {\rho_{1},\rho_{2},\rho_{3}\in\{\pm x^{\frac{1}{r}}:x\in\mathbb{Q}^{+},r\in% \mathbb{Z}^{+}\}} , then L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} contains an infinite orthogonal set of exponential functions if and only if there exist two numbers of ρ 1 , ρ 2 , ρ 3 {\rho_{1},\rho_{2},\rho_{3}} that are in the set { ± ( p q ) 1 r : p ∈ 2 ⁢ ℤ + , q ∈ 2 ⁢ ℤ + - 1 ⁢ and ⁢ r ∈ ℤ + } {\{\pm(\frac{p}{q})^{\frac{1}{r}}:p\in 2\mathbb{Z}^{+},q\in 2\mathbb{Z}^{+}-1% \text{ and }r\in\mathbb{Z}^{+}\}} . In particular, if ρ 1 , ρ 2 , ρ 3 ∈ { p q : p , q ∈ 2 ⁢ ℤ + 1 } {\rho_{1},\rho_{2},\rho_{3}\in\{\frac{p}{q}:p,q\in 2\mathbb{Z}+1\}} , then there exist at most 4 mutually orthogonal exponential functions in L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} , and the number 4 is the best possible.

This is the last in a series of articles where we are concerned with normal elements of noncommutative Iwasawa algebras over SL n ⁢ ( ℤ p ) {\mathrm{SL}_{n}(\mathbb{Z}_{p})} . Our goal in this portion is to give a positive answer to an open question in [D. Han and F. Wei, Normal elements of noncommutative Iwasawa algebras over SL 3 ⁢ ( ℤ p ) \mathrm{SL}_{3}(\mathbb{Z}_{p}) , Forum Math. 31 2019, 1, 111–147] and make up for an earlier mistake in [F. Wei and D. Bian, Normal elements of completed group algebras over SL n ⁢ ( ℤ p ) \mathrm{SL}_{n}(\mathbb{Z}_{p}) , Internat. J. Algebra Comput. 20 2010, 8, 1021–1039] simultaneously. Let n ( n ≥ 2 {n\geq 2} ) be a positive integer. Let p ( p > 2 {p>2} ) be a prime integer, ℤ p {\mathbb{Z}_{p}} the ring of p-adic integers and 𝔽 p {\mathbb{F}_{p}} the finite filed of p elements. Let G = Γ 1 ⁢ ( SL n ⁢ ( ℤ p ) ) {G=\Gamma_{1}(\mathrm{SL}_{n}(\mathbb{Z}_{p}))} be the first congruence subgroup of the special linear group SL n ⁢ ( ℤ p ) {\mathrm{SL}_{n}(\mathbb{Z}_{p})} and Ω G {\Omega_{G}} the mod-p Iwasawa algebra of G defined over 𝔽 p {\mathbb{F}_{p}} . By a purely computational approach, for each nonzero element W ∈ Ω G {W\in\Omega_{G}} , we prove that W is a normal element if and only if W contains constant terms. In this case, W is a unit. Also, the main result has been already proved under “nice prime” condition by Ardakov, Wei and Zhang [Non-existence of reflexive ideals in Iwasawa algebras of Chevalley type, J. Algebra 320 2008, 1, 259–275; Reflexive ideals in Iwasawa algebras, Adv. Math. 218 2008, 3, 865–901]. This paper currently provides a new proof without the “nice prime” condition. As a consequence of the above-mentioned main result, we observe that the center of Ω G {\Omega_{G}} is trivial.

In this paper, we study the Hardy type space H ℒ p ⁢ ( ℝ n ) {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} by means of local maximal functions associated with the heat semigroup e - t ⁢ ℒ {e^{-t\mathcal{L}}} generated by - ℒ {-\mathcal{L}} , where ℒ = - Δ + μ {\mathcal{L}=-\Delta+\mu} is the generalized Schrödinger operator in ℝ n {{\mathbb{R}^{n}}} ( n ≥ 3 {n\geq 3} ) and μ ≢ 0 {\mu\not\equiv 0} is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. Via the equivalence of the norms between various local maximal functions, we show that the norms ∥ f ∥ H ℒ p ⁢ ( ℝ n ) p {\lVert f\rVert_{H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})}^{p}} and ∥ f ∥ H m p , q ⁢ ( ℝ n ) p {\lVert f\rVert_{H_{m}^{p,q}({\mathbb{R}^{n}})}^{p}} are equivalent for n n + δ ′ < p ≤ 1 ≤ q ≤ ∞ {\frac{n}{n+\delta^{\prime}} ( p ≠ q {p\neq q} ) with some δ ′ > 0 {\delta^{\prime}>0} . As applications, we prove that Calderón–Zygmund operators related to the auxiliary function m ⁢ ( x , μ ) {m(x,\mu)} are bounded from H ℒ p ⁢ ( ℝ n ) {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} into L p ⁢ ( ℝ n ) {L^{p}({\mathbb{R}^{n}})} for n n + γ 1 < p ≤ 1 {\frac{n}{n+\gamma_{1}} with γ 1 > 0 {\gamma_{1}>0} . In particular, we show that the Riesz transform ∇ ( - Δ + μ ) - 1 2 {\nabla(-\Delta+\mu)^{-\frac{1}{2}}} , which is a special example of the above Calderón–Zygmund operators, is bounded from H ℒ p ⁢ ( ℝ n ) {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} into H p ⁢ ( ℝ n ) {{H}^{p}({\mathbb{R}^{n}})} for n n + γ 1 < p ≤ 1 {\frac{n}{n+\gamma_{1}} with 0 < γ 1 < 1 {0 .

In this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author [L. Ni, An optimal gap theorem, Invent. Math. 189 2012, 3, 737–761]. We also prove a Liouville theorem for plurisubharmonic functions on such a manifold, which generalizes a previous result of L.-F. Tam and the first author [L. Ni and L.-F. Tam, Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature, J. Differential Geom. 64 2003, 3, 457–524] and complements a recent result of Liu [G. Liu, Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds, Duke Math. J. 165 2016, 15, 2899–2919].

Let $n\ge2$ be an integer, $\mathcal{K}_n$ the Weyl algebra over the Laurent polynomial algebra $A_n=\mathbb{C} [x_1^{\pm1}, x_2^{\pm1}, ..., x_n^{\pm1}]$, and $\mathbb{S}_n$ the Lie algebra of divergence zero vector fields on an $n$-dimensional torus. For any $\mathfrak{sl}_n$-module $V$ and any module $P$ over $\mathcal{K}_n$, we define an $\mathbb{S}_n$-module structure on the tensor product $P\otimes V$. In this paper, necessary and sufficient conditions for the $\mathbb{S}_n$-modules $P\otimes V$ to be simple are given, and an isomorphism criterion for nonminuscule $\mathbb{S}_n$-modules is provided. More precisely, all nonminuscule $\mathbb{S}_n$-modules are simple, and pairwise nonisomorphic. For minuscule $\mathbb{S}_n$-modules, minimal and maximal submodules are concretely constructed., Comment: 20 pages

The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally, we show that ultralimits and asymptotic cones of spaces with quadratic Dehn function also have quadratic Dehn function. The proofs of our results rely on recent existence and regularity results for area-minimizing Sobolev mappings in metric spaces.

In this paper, we introduce super Weyl groups, their distinguished elements and properties for basic classical Lie superalgebras. Then we formulate Jantzen filtration for baby Verma modules in graded restricted module categories of basic classical Lie superalgebras over an algebraically closed field of odd characteristic, and prove a sum formula in the corresponding Grothendieck groups. We finally obtain a strong linkage principle in such categories., Comment: 31pages

The main purpose of this paper is to investigate the non-archimedean counterpart of the classical result stating that the dual of a real or complex metrizable locally convex space, equipped with the locally convex topology of uniform convergence on compact sets, belongs to the topological category formed by the k-spaces. We prove that this counterpart holds when the non-archimedean valued base field 𝕂 {\mathbb{K}} is locally compact, but fails for any non-locally compact 𝕂 {\mathbb{K}} . Here we deal with a topological subcategory, the one formed by the k 0 {k_{0}} -spaces, the adequate non-archimedean substitutes for k-spaces. As a product, we complete some of the achievements on the non-archimedean Banach–Dieudonné Theorem presented in [C. Perez-Garcia and W. H. Schikhof, The p-adic Banach–Dieudonné theorem and semi-compact inductive limits, p-adic Functional Analysis (Poznań 1998), Lecture Notes Pure Appl. Math. 207, Dekker, New York 1999, 295–307]. Also, we use our results to construct in a simple way natural examples of k-spaces (which are also k 0 {k_{0}} -spaces) whose products are not k 0 {k_{0}} -spaces. This in turn improves the, rather involved, example given in [C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces over non-Archimedean Valued Fields, Cambridge Stud. Adv. Math. 119, Cambridge University Press, Cambridge, 2010] of two k 0 {k_{0}} -spaces whose product is not a k 0 {k_{0}} -space. Our theory covers an important class of non-archimedean Fréchet spaces, the Köthe sequence spaces, which have a relevant influence on applications such as the definition of a non-archimedean Laplace and Fourier transform.

Given a vector bundle, its (stable) order is the smallest positive integer t such that the t-fold self-Whitney sum is (stably) trivial. So far, the order and the stable order of the canonical vector bundle over configuration spaces of Euclidean spaces have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54], [F. R. Cohen, M. E. Mahowald and R. J. Milgram, The stable decomposition for the double loop space of a sphere, Algebraic and Geometric Topology (Stanford 1976), Proc. Sympos. Pure Math. 32 Part 2, American Mathematical Society, Providence 1978, 225–228], and [S.-W. Yang, Order of the Canonical Vector Bundle on C n ⁢ ( k ) / Σ k {C_{n}(k)/\Sigma_{k}} , ProQuest LLC, Ann Arbor, 1978]. Moreover, the order and the stable order of the canonical vector bundle over configuration spaces of closed orientable Riemann surfaces with genus greater than or equal to one have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54]. In this paper, we mainly study the order and the stable order of the canonical vector bundle over configuration spaces of spheres and disjoint unions of spheres.

In the present paper, right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras will be studied. For this purpose, first the structure of the set of all right 2-Engel elements of a Lie algebra will be examined and then, by taking advantage of it, a number of interesting results about central and commuting automorphisms of Lie algebras will be presented. Finally, a characterization of Lie algebras for which the set of central automorphisms is trivial or the set of commuting automorphisms is trivial will be given.

In this paper, we estimate the shifted convolution sum ∑ n ⩾ 1 λ 1 ⁢ ( 1 , n ) ⁢ λ 2 ⁢ ( n + h ) ⁢ V ⁢ ( n X ) , \sum_{n\geqslant 1}\lambda_{1}(1,n)\lambda_{2}(n+h)V\Big{(}\frac{n}{X}\Big{)}, where V is a smooth function with support in [ 1 , 2 ] {[1,2]} , 1 ⩽ | h | ⩽ X {1\leqslant|h|\leqslant X} , and λ 1 ⁢ ( 1 , n ) {\lambda_{1}(1,n)} and λ 2 ⁢ ( n ) {\lambda_{2}(n)} are the n-th Fourier coefficients of SL ⁢ ( 3 , 𝐙 ) {\mathrm{SL}(3,\mathbf{Z})} and SL ⁢ ( 2 , 𝐙 ) {\mathrm{SL}(2,\mathbf{Z})} Hecke–Maass cusp forms, respectively. We prove an upper bound O ⁢ ( X 21 22 + ε ) {O(X^{\frac{21}{22}+\varepsilon})} , updating a recent result of Munshi.