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In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

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.

In this paper, we continue to investigate the exceptional sets for sums of five and six almost equal cubes of primes. We would also like to establish that almost all natural numbers n, subjected to certain congruence conditions, can be written as n = p 1 3 + ⋯ + p s 3 {n=p_{1}^{3}+\cdots+p_{s}^{3}} ( s = 5 , 6 {s=5,6} ) with | p j - ( n / s ) 1 / 3 | ≤ n θ s / 3 + ε {|p_{j}-(n/s)^{1/3}|\leq n^{\theta_{s}/3+\varepsilon}} ( 1 ≤ j ≤ s {1\leq j\leq s} ), where θ s {\theta_{s}} is as small as possible. The main result of this paper is to improve θ 6 = 5 / 6 + ε {\theta_{6}=5/6+\varepsilon} , which is proven in [M. Wang, Exceptional sets for sums of five and six almost equal prime cubes, Acta Math. Hungar. 156 2018, 2, 424–434], to θ 6 = 9 / 11 + ε {\theta_{6}=9/11+\varepsilon} , as well as prove θ 5 = 8 / 9 + ε {\theta_{5}=8/9+\varepsilon} in another way.

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

In this paper, we first establish a kind of weighted space-time L 2 {L^{2}} estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time L 2 L^{2} estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in H 3 × H 2 {H^{3}\times H^{2}} .

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

In 1874, Franz Mertens proved the so-called Mertens’ theorem, and in 1974, Kenneth S. Williams showed Mertens’ theorem associated with a character. In a previous paper, we presented a graph-theoretic analogue to Williams’ theorem. In this paper, we generalize our previous work in the sense that a character is extended to a representation. To our knowledge, a number-theoretic analogue to our result is not yet known. So, we expect that, by using our methods, it can be proven in the future.

Let F be a totally real number field and let π be a cuspidal automorphic representation of GSp 4 ⁢ ( 𝔸 F ) {\mathrm{GSp_{4}}(\mathbb{A}_{F})} , which contributes irreducibly to coherent cohomology. If π has a Bessel model, we may attach a period p ⁢ ( π ) {p(\pi)} to this datum. In the present paper, which is Part I in a series of two, we establish a relation of these Bessel periods p ⁢ ( π ) {p(\pi)} and all of their twists p ⁢ ( π ⊗ ξ ) {p(\pi\otimes\xi)} under arbitrary algebraic Hecke characters ξ. In the appendix, we show that ( 𝔤 , K ) {(\mathfrak{g},K)} -cohomological cusp forms of GSp 4 ⁢ ( 𝔸 F ) {\mathrm{GSp_{4}}(\mathbb{A}_{F})} all qualify to be of the above type – providing a large source of examples. We expect that these period relations for GSp 4 ⁢ ( 𝔸 F ) {\mathrm{GSp_{4}}(\mathbb{A}_{F})} will allow a conceptual, fine treatment of rationality relations of special values of the spin L-function, which we hope to report on in Part II of this paper.

In this paper we apply a group action approach to the study of Erdős–Falconer-type problems in vector spaces over finite fields and use it to obtain non-trivial exponents for the distribution of simplices. We prove that there exists s 0 ⁢ ( d ) < d ${s_{0}(d) such that if E ⊂ 𝔽 q d ${E\subset{\mathbb{F}}_{q}^{d}}$ , d ≥ 2 ${d\geq 2}$ , with | E | ≥ C ⁢ q s 0 ${|E|\geq Cq^{s_{0}}}$ , then | T d d ⁢ ( E ) | ≥ C ′ ⁢ q ( d + 1 2 ) ${|T^{d}_{d}(E)|\geq C^{\prime}q^{d+1\choose 2}}$ , where T k d ⁢ ( E ) ${T^{d}_{k}(E)}$ denotes the set of congruence classes of k-dimensional simplices determined by k + 1 ${k+1}$ -tuples of points from E. Non-trivial exponents were previously obtained by Chapman, Erdogan, Hart, Iosevich and Koh [4] for T k d ⁢ ( E ) ${T^{d}_{k}(E)}$ with 2 ≤ k ≤ d - 1 ${2\leq k\leq d-1}$ . A non-trivial result for T 2 2 ⁢ ( E ) ${T^{2}_{2}(E)}$ in the plane was obtained by Bennett, Iosevich and Pakianathan [2]. These results are significantly generalized and improved in this paper. In particular, we establish the Wolff exponent 4 3 ${\frac{4}{3}}$ , previously established in [4] for the q ≡ 3 ⁢ mod ⁢ 4 ${q\equiv 3\textup{ mod }4}$ case to the case q ≡ 1 ⁢ mod ⁢ 4 ${q\equiv 1\textup{ mod }4}$ , and this results in a new sum-product type inequality. We also obtain non-trivial results for subsets of the sphere in 𝔽 q d ${{\mathbb{F}}_{q}^{d}}$ , where previous methods have yielded nothing. The key to our approach is a group action perspective which quickly leads to natural and effective formulae in the style of the classical Mattila integral from geometric measure theory.

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

This paper is devoted to the existence of solutions for a class of nonlinear boundary value problems with integral boundary conditions and generalized 𝑝-Laplacian on the positive half-line. We establish sufficient conditions to guarantee the existence of solutions in a special function space by using Leray–Schauder-type arguments. Examples are also given to illustrate the main results.

In this paper, we establish the endpoint estimate ( 0 < p ≤ 1 {0 ) for a trilinear pseudo-differential operator, where the symbol involved is given by the product of two standard symbols from the bilinear Hörmander class B ⁢ S 1 , 0 0 {BS^{0}_{1,0}} . The study of this operator is motivated from the L p {L^{p}} ( 1 < p < ∞ {1 ) estimates for the trilinear Fourier multiplier operator with flag singularities considered in [C. Muscalu, Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoam. 23 2007, 2, 705–742] and Hardy space estimates in [A. Miyachi and N. Tomita, Estimates for trilinear flag paraproducts on L ∞ L^{\infty} and Hardy spaces, Math. Z. 282 2016, 1–2, 577–613], and the L p {L^{p}} ( 1 < p < ∞ {1 ) estimates for the trilinear pseudo-differential operator with flag symbols in [G. Lu and L. Zhang, L p L^{p} -estimates for a trilinear pseudo-differential operator with flag symbols, Indiana Univ. Math. J. 66 2017, 3, 877–900]. More precisely, we will show that the trilinear pseudo-differential operator with flag symbols defined in (1.3) maps from the product of local Hardy spaces to the Lebesgue space, i.e., h p 1 × h p 2 × h p 3 → L p {h^{p_{1}}\times h^{p_{2}}\times h^{p_{3}}\rightarrow L^{p}} with 1 p 1 + 1 p 2 + 1 p 3 = 1 p {\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=\frac{1}{p}} with 0 < p < ∞ {0 (see Theorem 1.6 and Theorem 1.7).

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.

The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

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.

In this paper, our first goal is to rigorously define a Lévy process pinned at random time. Our second task is to establish the Markov property with respect to its completed natural filtration and thus with respect to the usual augmentation of the latter. The resulting conclusion is the right-continuity of completed natural filtration. Certain examples of such process are considered.

This paper studies the Prasad conjecture for the special orthogonal group SO 3 , 3 \mathrm{SO}_{3,3} . Then we use the local theta correspondence between Sp 4 \mathrm{Sp}_{4} and O ⁢ ( V ) \mathrm{O}(V) to study the Sp 4 \mathrm{Sp}_{4} -distinction problems over a quadratic field extension E / F E/F and dim ⁡ V = 4 \dim V=4 or 6. Thus we can verify the Prasad conjecture for a square-integrable representation of Sp 4 ⁢ ( E ) \mathrm{Sp}_{4}(E) .

The spectrum of the Laplacian operator on the positive theta line bundle over the quasi-torus reduces to eigenvalues π ⁢ ℓ {\pi\ell} , ℓ = 0 , 1 , … {\ell=0,1,\ldots{}} , which are called Landau levels. This paper discusses the coherent state transform for each eigenspace associated with a Landau level. We construct a unitary transform valid for each eigenspace. A concrete form of the inverse formula for the proposed transform is also obtained.

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 prove a genuine analogue of the Wiener Tauberian theorem for L p , 1 ⁢ ( G ) {L^{p,1}(G)} ( 1 ≤ p < 2 {1\leq p ), with G = SL ⁢ ( 2 , ℝ ) {G=\mathrm{SL}(2,\mathbb{R})} .

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} .

The Cauchy problem of the 2D Zakharov–Kuznetsov equation ∂ t ⁡ u + ∂ x ⁡ ( ∂ x ⁢ x + ∂ y ⁢ y ) ⁡ u + u ⁢ u x = 0 {\partial_{t}u+\partial_{x}(\partial_{xx}+\partial_{yy})u+uu_{x}=0} is considered. It is shown that the 2D Z-K equation is locally well-posed in the endpoint Sobolev space H - 1 / 4 {H^{-1/4}} , and it is globally well-posed in H - 1 / 4 {H^{-1/4}} with small initial data. In this paper, we mainly establish some new dyadic bilinear estimates to obtain the results, where the main novelty is to parametrize the singularity of the resonance function in terms of a univariate polynomial.

In 1977, Colville, Davis, and Keimel [Positive derivations on f-rings, J. Aust. Math. Soc. Ser. A 23 1977, 3, 371–375] proved that a positive derivation on an Archimedean f-algebra A has its range in the set of nilpotent elements of A. The main objective of this paper is to obtain a generalization of the above Colville, Davis and Keimel result to general derivations. Moreover, we give a new version of the Singer–Wermer conjecture for the class of second-order derivations acting on uniformly complete almost f-algebras.

In this paper, we study the L q {L^{q}} - L r {L^{r}} boundedness of bi-parameter Fourier integral operators defined by general rough Hörmander class amplitudes.

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].

This paper studies the two-weight estimates of variation and oscillation operators for commutators of singular integrals with weighted BMO {\mathrm{BMO}} functions. A new characterization of weighted BMO {\mathrm{BMO}} spaces via the boundedness of variation and oscillation operators for the iterated commutators of Calderón–Zygmund singular integrals in the two-weight setting is given.

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].

In this paper, we consider the long time behavior of solutions for 3D incompressible MHD equations with fractional Laplacian. Firstly, in a periodic bounded domain, we prove the existence of a global attractor. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and magnetic fields. Finally, in the whole space ℝ 3 {\mathbb{R}^{3}} , we establish the sharp algebraic decay rate of solutions to the generalized MHD system provided that the parameters satisfy α , β ∈ ( 0 , 2 ] {\alpha,\beta\in(0,2]} .

The isotropy projection establishes a correspondence between curves in the Lorentz–Minkowski space 𝐄 1 3 {\mathbf{E}_{1}^{3}} and families of cycles in the Euclidean plane (i.e., curves in the Laguerre plane ℒ 2 {\mathcal{L}^{2}} ). In this paper, we shall give necessary and sufficient conditions for two given families of cycles to be related by a (extended) Laguerre transformation in terms of the well known Lorentzian invariants for smooth curves in 𝐄 1 3 {\mathbf{E}_{1}^{3}} . We shall discuss the causal character of the second derivative of unit speed spacelike curves in 𝐄 1 3 {\mathbf{E}_{1}^{3}} in terms of the geometry of the corresponding families of oriented circles and their envelopes. Several families of circles whose envelopes are well-known plane curves are investigated and their Laguerre invariants computed.

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 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.

In this paper, the well-posedness of the higher-order Benjamin–Ono equation u t + ℋ ⁢ ( u x ⁢ x ) + u x ⁢ x ⁢ x = u ⁢ u x - ∂ x ⁡ ( u ⁢ ℋ ⁢ ∂ x ⁡ u + ℋ ⁢ ( u ⁢ ∂ x ⁡ u ) ) u_{t}+\mathcal{H}(u_{xx})+u_{xxx}=uu_{x}-\partial_{x}(u\mathcal{H}\partial_{x}% u+\mathcal{H}(u\partial_{x}u)) is considered. The modified energy method is introduced to consider the equation. It is shown that the Cauchy problem of the higher-order Benjamin–Ono equation is locally well-posed in H 3 / 4 {H^{3/4}} without using the gauge transformation. Moreover, the well-posedness of the higher-order intermediate long wave equation u t + 𝒢 δ ⁢ ( u x ⁢ x ) + u x ⁢ x ⁢ x = u ⁢ u x - ∂ x ⁡ ( u ⁢ 𝒢 δ ⁢ ∂ x ⁡ u + 𝒢 δ ⁢ ( u ⁢ ∂ x ⁡ u ) ) , 𝒢 δ = ℱ x - 1 ⁢ i ⁢ ( coth ⁡ ( δ ⁢ ξ ) ) ⁢ ℱ x , u_{t}+\mathcal{G}_{\delta}(u_{xx})+u_{xxx}=uu_{x}-\partial_{x}(u\mathcal{G}_{% \delta}\partial_{x}u+\mathcal{G}_{\delta}(u\partial_{x}u)),\quad\mathcal{G}_{% \delta}=\mathcal{F}_{x}^{-1}i(\coth(\delta\xi))\mathcal{F}_{x}, is considered. It is shown that the Cauchy problem of the higher-order intermediate long wave equation is locally well-posed in H 3 / 4 {H^{3/4}} .

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.

For the completed Riemann zeta function ξ ⁢ ( s ) {\xi(s)} , it is known that the Riemann hypothesis for ξ ⁢ ( s ) {\xi(s)} implies the Riemann hypothesis for ξ ( m ) ⁢ ( s ) {\xi^{(m)}(s)} , where m is any positive integer. In this paper, we investigate the distribution of the fractional parts of the sequence ( α ⁢ γ m ) {(\alpha\gamma_{m})} , where α is any fixed non-zero real number and γ m {\gamma_{m}} runs over the imaginary parts of the zeros of ξ ( m ) ⁢ ( s ) {\xi^{(m)}(s)} . We also obtain a zero density estimate and an explicit formula for the zeros of ξ ( m ) ⁢ ( s ) {\xi^{(m)}(s)} . In particular, all our results hold uniformly for 0 ≤ m ≤ g ⁢ ( T ) {0\leq m\leq g(T)} , where the function g ⁢ ( T ) {g(T)} tends to infinity with T and g ⁢ ( T ) = o ⁢ ( log ⁡ log ⁡ T ) {g(T)=o(\log\log T)} .

In this paper, we refine the method of Chowla and Erdős on the irrationality of Lambert series and study a necessary condition for the infinite series ∑ θ ⁢ ( n ) / q n {\sum\theta(n)/q^{n}} to be a rational number, where q is an integer with | q | > 1 {|q|>1} and θ is an arithmetic function with suitable divisibility and growth conditions. As applications of our main theorem, we give linear independence results for various kinds of Lambert series.

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.