Hecke algebras for {{\rm GL}_n} over local fields

We study the local Hecke algebra \({\mathcal{H}_{G}(K)}\) for \({G = {\rm GL}_{n}}\) and K a non-archimedean local field of characteristic zero. We show that for \({G = {\rm GL}_{2}}\) and any two such fields K and L, there is a Morita equivalence \({\mathcal{H}_{G}(K) \sim_{M} \mathcal{H}_{G}(L)}\), by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for \({G = {\rm GL}_{n}}\), there is an algebra isomorphism \({\mathcal{H}_{G}(K) \cong \mathcal{H}_{G}(L)}\) which is an isometry for the induced \({L^1}\)-norm if and only if there is a field isomorphism \({K \cong L}\).     

Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators

Let \({L_{w}}{:=-w^{-1}{\rm div}(A\nabla)}\) be the degenerate elliptic operator on the Euclidean space \({{\mathbb{R}^{n}}}\), where w is a Muckenhoupt \({A_{2}({\mathbb{R}^{n}})}\) weight. In this article, the authors establish the Riesz transform characterization of the Hardy space \({H^{p}_{L_{w}}({\mathbb{R}}^{n})}\) associated with L_w, for \({w \in A_{q}({\mathbb{R}}^{n}) \cap RH_{\frac{n}{n-2}}({\mathbb{R}^{n}})}\) with \({n \geq 3}\), \({q \in [1,2]}\) and \({p \in (q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n})^{-1},1]}\) if, for some \({r \in (1,\,2]}\), \({{\{tL_w e^{-tL_w}\}}_{t\geq 0}}\) satisfies the weighted \({L^{r}-L^{2}}\) full off-diagonal estimates.     

Characterizations of the Vertex Operator Algebras ${V_{L}^{T}}$ and VLO${V_{L}^{O}}$

In this paper, we give characterizations of the rational vertex operator algebras \({V_{L}^{T}}\) and \({V_{L}^{O}}\), where L is the root lattice of type A ₁, T is the tetrahedral group, and O is the octahedral group. By these two characterizations, the classification of rational VOAs of central charge 1 is reduced to the characterization of \({V_{L}^{I}}\) where I is the icosahedral group.     

On generalized Stanley sequences

Let \({\mathbb{N}}\) denote the set of all nonnegative integers. Let \({k \ge 3}\) be an integer and \({A_{0} = \{a_{1}, \dots, a_{t}\} (a_{1} < \cdots < a_{t})}\) be a nonnegative set which does not contain an arithmetic progression of length k. We denote \({A = \{a_{1}, a_{2}, \ldots{}\}}\) defined by the following greedy algorithm: if \({l \ge t}\) and \({a_{1}, \dots{}, a_{l}}\) have already been defined, then \({a_{l+1}}\) is the smallest integer \({a > a_{l}}\) such that \({\{a_{1}, \dots, a_{l}\} \cup \{a\}}\) also does not contain a k-term arithmetic progression. This sequence A is called the Stanley sequence of order k generated by A₀. We prove some results about various generalizations of the Stanley sequence.     

{L^{p}} stability for the Boltzmann equation near vacuum

The paper is concerned with the uniform time stability in the Lebesgue space \({L^{p}(\mathbb{R}^{3} \times \mathbb{R}^{3})}\) of solutions to the Boltzmann equation near vacuum. Precisely, for the soft potential case \({-2 < \gamma < 0}\), there exists \(p_{\gamma} > 1\) such that the nonnegative solution with algebraic decay rate in x, v at infinity is stable with respect to small initial data uniformly in time in \({L^{p}}\) with \({1 \leq p < p_{\gamma}}\).     

Arens regularity of projective tensor products

For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product \({A\widehat{\otimes}B}\) (respectively the Haagerup tensor product \({A\otimes^{h}B}\)) to be Arens regular are obtained. Using the non-commutative Grothendieck inequality, we show that, for C*-algebras A and B, \({A\otimes^{\gamma} B}\) is Arens regular if \({A\widehat{\otimes}B}\) and \({A\widehat{\otimes}B^{op}}\) are Arens regular whereas \({A\widehat{\otimes}B}\) is Arens regular if and only if \({A\otimes^{h}B}\) and \({B\otimes^{h}A}\) are, where \({\otimes^h}\), \({\otimes^{\gamma}}\), and \({\widehat{\otimes}}\) are the Haagerup, the Banach space projective tensor norm, and the operator space projective tensor norm, respectively.     

Abelian covers of alternating groups

Let \({C={\rm inf} (k/n)\sum_{i=1}^n x_i(x_{i+1}+\cdots+x_{i+k})^{-1}}\), where the infimum is taken over all pairs of integers \({n\geq k\geq 1}\) and all positive \({x_1,\ldots,x_n}\), \({x_{n+i}=x_i}\). We prove that \({\ln 2 \leq C < 0.9305}\). In the definition of the constant C, the operation \({{\rm inf}_{k}\, {\rm inf}_{n}\, {\rm inf}_{x}}\) can be replaced by \({{\rm lim}_{k \to \infty}\, {\rm lim}_{n \to \infty} {\rm inf}_{x}}\).     

Interlacing of zeros of Gegenbauer polynomials of non-consecutive degree from different sequences

A theorem due to Stieltjes’ states that if \({\{p_n\}_{n=0}^\infty}\) is any orthogonal sequence then, between any two consecutive zeros of p _k , there is at least one zero of p _n whenever k < n, a property called Stieltjes interlacing. We show that Stieltjes interlacing extends to the zeros of Gegenbauer polynomials \({C_{n+1}^{\lambda}}\) and \({C_{n-1}^{\lambda+t}}\), \({\lambda > -\frac 12}\), if 0 < t ≤ k + 1, and also to the zeros of \({C_{n+1}^{\lambda}}\) and \({C_{n-2}^{\lambda +k}}\) if \({k\in\{1,2,3\}}\). More generally, we prove that Stieltjes interlacing holds between the zeros of the kth derivative of \({C_{n}^{\lambda}}\) and the zeros of \({C_{n+1}^{\lambda}}\), \({k\in\{1,2,\dots,n-1\}}\) and we derive associated polynomials that play an analogous role to the de Boor–Saff polynomials in completing the interlacing process of the zeros.     

The rate of decay of stable periodic solutions for Duffing equation with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conditions

For a new class of g(t, x), the existence, uniqueness and stability of \({2\pi}\)-periodic solution of Duffing equation \({x'' + cx' + g(t, x) = h(t)}\) are presented. Moreover, the unique \({2\pi}\)-periodic solution is (exponentially asymptotically stable) and its rate of exponential decay c/2 is sharp. The new criterion characterizes \({g_{x}^{\prime}(t, x) - c^2/4}\) with L ^p -norms \({(p \in [1, \infty])}\), and the classical criterion employs the \({L^{\infty}}\)-norm. The advantage is that we can deal with the case that \({g_{x}^{\prime}(t, x) - c^2/4}\) is beyond the optimal bounds of the \({L^{\infty}}\)-norm, because of the difference between the L ^p -norm and the \({L^{\infty}}\)-norm.     

Generalized commutativity of lattice-ordered groups II

Commutative \({\ell}\)-groups G (in which for all \({x, y \in G, xy = yx}\)) were studied long ago. This was then generalized to the study of \({\ell}\)-groups G in which for a given integer n and for all \({x, y \in G, x^{n}y^{n} = y^{n}x^{n}}\). It was then discovered that if for all \({x, y \in G}\), both \({x^{n}y^{n} = y^{n}x^{n}}\) and \({x^{m}y^{m} = y^{m}x^{m}}\) for two different integers m, n, then also \({x^{d}y^{d} = y^{d}x^{d}}\), where d is the greatest common divisor of m, n.     

Remarks on Complex Symmetric Operators

An operator \({T\in{\mathcal{L}}({\mathcal{H}})}\) is said to be complex symmetric if there exists a conjugation C on \({{\mathcal H}}\) such that \({T= CT^{\ast}C}\). In this paper, we study the spectral radius algebras for complex symmetric operators. In particular, we prove that if A is a complex symmetric operator, then the spectral radius algebra \({{\mathcal B}_{A}}\) associated with A has a nontrivial invariant subspace under some conditions. Finally, we give some relations between \({P_{\tilde{A}}}\) and \({P_{\widetilde{A^{\ast}}}}\) (defined below) when A is complex symmetric.     

Toeplitz Operators from One Fock Space to Another

In this paper, we study Toeplitz operators T _μ from one Fock space \({F^{p}_{\alpha}}\) to another \({F^{q}_{\alpha}}\) for 1 < p, q < ∞ with positive Borel measures μ as symbols. We characterize the boundedness (and compactness) of \({T_\mu: F^{p}_{\alpha} \to F^{q}_{\alpha}}\) in terms of the averaging function \({\widehat{\mu}_r}\) and the t-Berezin transform \({\widetilde{\mu}_t}\) respectively. Quite differently from the Bergman space case, we show that T _μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for some p ≤ q if and only if T _μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for all p ≤ q. In order to prove our main results on T _μ , we introduce and characterize (vanishing) (p, q)-Fock Carleson measures on C ⁿ .     

Schatten-Herz Operators,Berezin Transform and Mixed Norm Spaces

For p, q > 0 we study operators T on the Bergman space \({A_{2}(\mathbb{D)}}\) in the disk such that \({\left(\sum_{j}\Vert T\Delta_{j}\Vert_{p}^{q}\right)^{1/q}<\infty,}\) where the norms \({\Vert\cdot\Vert_{p}}\) are in the Schatten class S _p (A ₂), the projection \({\Delta_{j}f=\sum_{n\in I_{j}}a_{n}z^{n}}\) for \({f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}}\) and \({I_{j}=[2^{j}-1,2^{j+1} )\cap(\mathbb{N}\cup\{0\})}\) for \({j\in\mathbb{N}\cup\{0\}.}\) We consider the relation of this property with mixed norms of the Berezin transform of T and of the related function \({f_{T}(z)={\Vert}T(k_{z})\Vert}\) where k _z is the normalized Bergman kernel. These classes of operators denoted by S(p, q) are closely related when assumed to be positive with other sets of operators, like the class of positive operators on A ₂ for which \({\left(\sum_{j\geq0}(\sum_{n\in I_{j}}|\left\langle T^pe_{n},e_{n}\right\rangle |)^{q/p}\right)^{1/q}<\infty}\) , where \({\{e_{n}\}_{n\geq0}}\) is the canonical basis of A ₂; also we study the relation of Toeplitz operators in S(p, q) with the Schatten-Herz classes, where the decomposition is through dyadic annuli of the domain \({\mathbb{D}}\) .     

On complete sequences

A sequence A of nonnegative integers is called complete if all sufficiently large integers can be represented as the sum of distinct terms taken form A. For a sequence \({S=\{s_{1}, s_{2}, \dots\}}\) of positive integers and a positive real number α, let S _α denote the sequence \({\{\lfloor\alpha s_{1}\rfloor, \lfloor\alpha s_{2}\rfloor, \dots\}}\), where \({\lfloor x \rfloor}\) denotes the greatest integer not greater than x. Let \({{U_S = \{\alpha \mid S_\alpha} \, is complete\}}\). Hegyvári [6] proved that if \({\lim_{n\to\infty} (s_{n+1}-s_{n})=+ \infty}\), \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma < 2}\), and \({U_{S}\ne\emptyset}\), then \({\mu(U_{S}) > 0}\), where \({\mu(U_{S})}\) is the Lebesgue measure of U _S . Yong-Gao Chen and the first author [4] proved that, if \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma \leqq 7/4=1.75}\), then \({\mu(U_{S}) > 0}\). In this paper, we prove that the conclusion holds for \({1 < \gamma \leqq \sqrt[4]{13}=1.898\dots\;}\).     

Applications of strongly convergent sequences to Fourier series by means of modulus functions

The aim of this work is to estimate sums involving P(n), the largest prime factor of an integer \({n \geqq 2}\) under digital constraints \({{f(P(n)) \equiv a}{\rm mod} b}\), for every \({a \in \mathbb{Z}}\) and an integer \({b \geqq 2}\) where f is a strongly q-additive function with integer values (i.e. \({f(aq^j + b) = f(a) + f(b)}\), with \({(a, b, j) \in \mathbb{N}^3}\), \({{0 \leqq b} < q^j}\)). We also estimate the cardinality of the set \({\{{n \leqq x, f(P(n) + c)} \equiv {a {\rm mod} b}, P(n) \equiv l {\rm mod} k\}}\), where \({c \in \mathbb{Z}}\), \({k \geqq 2}\).     

Asymptotic behavior of the p-torsion functions as p goes to 1

Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of u_p, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\).     

Complex homomorphisms on self-conjugate algebras of functions

Let X be a non-void set and A be a subalgebra of \({\mathbb{C}^{X}}\) . We call a \({\mathbb{C}}\) -linear functional \({\varphi}\) on A a 1-evaluation if \({\varphi(f) \in f(X) }\) for all \({f\in A}\) . From the classical Gleason–Kahane–?elazko theorem, it follows that if X in addition is a compact Hausdorff space then a mapping \({\varphi}\) of \({C_{\mathbb{C}}(X) }\) into \({\mathbb{C}}\) is a 1-evaluation if and only if \({\varphi}\) is a \({\mathbb{C}}\) -homomorphism. In this paper, we aim to investigate the extent to which this equivalence between 1-evaluations and \({\mathbb{C}}\) -homomorphisms can be generalized to a wider class of self-conjugate subalgebras of \({\mathbb{C}^{X}}\) . In this regards, we prove that a \({\mathbb{C}}\) -linear functional on a self-conjugate subalgebra A of \({\mathbb{C}^{X}}\) is a positive \({\mathbb{C}}\) -homomorphism if and only if \({\varphi}\) is a \({\overline{1}}\) -evaluation, that is, \({\varphi(f) \in\overline{f\left(X\right)}}\) for all \({f\in A}\) . As consequences of our general study, we prove that 1-evaluations and \({\mathbb{C}}\) -homomorphisms on \({C_{\mathbb{C}}\left( X\right)}\) coincide for any topological space X and we get a new characterization of realcompact topological spaces.     

The Schur functor on tensor powers

Let M be a left module for the Schur algebra S(n, r), and let \({s \in \mathbb{Z}^+}\) . Then \({M^{\otimes s}}\) is a \({(S(n,\,rs), F{\mathfrak{S}_{s}})}\) -bimodule, where the symmetric group \({{\mathfrak{S}_s}}\) on s letters acts on the right by place permutations. We show that the Schur functor f _rs sends \({M^{\otimes s}}\) to the \({(F{\mathfrak{S}_{rs}},F{\mathfrak{S}_s})}\) -bimodule \({F\mathfrak{S}_{rs}\otimes_{F(\mathfrak{S}_{r}\wr{\mathfrak{S}_s})} ((f_rM)^{\otimes s}\otimes_{F} F{\mathfrak{S}_s})}\) . As a corollary, we obtain the image under the Schur functor of the Lie power L ^s (M), exterior power \({\bigwedge^s(M)}\) of M and symmetric power S ^s (M).     

Finite groups with small automizers of {\mathcal {F}}-subgroups

Let \({ \mathcal {F}}\) be a saturated formation and G a finite group such that \({N_{G} (H^{\mathcal {F}})/C_{G} (H^{\mathcal {F}})\cong Inn(H^{\mathcal {F}})}\) for every subgroup H of G. If the minimal non-\({ \mathcal {F}}\)-group is soluble, then \({G \in \mathcal {F}}\).     

Spaces with normal weights and Hadamard gap series

We give a sufficient and necessary condition for an analytic function f(z) on the unit disc \({\mathbb{D}}\) with Hadamard gaps, that is, for \({f(z)=\sum_{k=1}^{\infty}a_kz^{n_k}}\) where \({n_{k+1}/n_k\geq\lambda >1 }\) for all \({k\in \mathbb{N}}\), to belong to the weighted-type space \({ H_\mu^{\infty}}\), under some condition posed on the weight function μ. We can define the corresponding little weighted-type space \({H_{\mu,0}^{\infty}}\) and give a criterion for functions to belong to it.