The existence of eigenvalues for operators acting in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We present conditions that allow us to prove the existence of eigenvalues and characteristic values for operator F(D) ? C(λ): L ²(R ^m ) → L ²(R ^m ), where F(D) is a pseudo-differential operator with a symbol F(iξ) and C(λ): L ²(R ^m ) → L ²(R ^m ) is a linear continuous operator.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-norm of partial sums of Fourier-Legendre series

Asymptotic estimates of L ^q -norms of partial sums of Fourier-Legendre series of functions from L ^p [?1; 1] are obtained.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-consistency of wavelet estimators

This paper deals with the L ^p -consistency of wavelet estimators for a density function based on size-biased random samples. More precisely, we firstly show the L ^p -consistency of wavelet estimators for independent and identically distributed random vectors in R ^d . Then a similar result is obtained for negatively associated samples under the additional assumptions d = 1 and the monotonicity of the weight function.     

Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">P</Emphasis></Superscript>

Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in L² spaces and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of L ^p spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator to prove that they have a bounded holomorphic functional calculus in those L ^p spaces. We also obtain functional calculus results for restrictions to certain subspaces, for a larger range of p. This provides a framework for obtaining L ^p results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator L with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and L ^p bounds on the square-root of L by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in L² extends to L ^p for all p ∈ (1,∞), while the restrictions in p come from the operator-theoretic part of the L² proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces and about the relationship between conical and vertical square functions.     

The strong <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-<Emphasis Type="Italic">greedy</Emphasis> property of the Walsh system

For any 0 < ? < 1 one can find a measurable set E ? [0, 1] with the measure |E| > 1 ? ? such that for each function f(x) ε L ¹ (0, 1) a function g(x) ε L ¹ (0, 1) exists such that it coincides with f (x) on E, its Fourier—Walsh series converges to it in the metric of L ¹ (0, 1), and all nonzero terms of the sequence of Fourier coefficients of the new function obtained by the Walsh system have the modulo decreasing order; consequently, the greedy algorithm for this function converges to it in the L ¹ (0, 1)-norm.     

Non-homogeneous <Emphasis Type="Italic">Tb</Emphasis> Theorem for Bi-parameter <Emphasis Type="Italic">g</Emphasis>-function

The main result of this paper is a bi-parameter Tb theorem for Littlewood–Paley g-function, where b is a tensor product of two pseudo-accretive function. Instead of the doubling measure, we work with a product measure μ = μ_n × μ_m, where the measures μ_n and μ_m are only assumed to be upper doubling. The main techniques of the proof include a bi-parameter b-adapted Haar function decomposition and an averaging identity over good double Whitney regions. Moreover, the non-homogeneous analysis and probabilistic methods are used again.     

Processes of <Emphasis Type="Italic">r</Emphasis><Superscript><Emphasis Type="Italic">t</Emphasis><Emphasis Type="Italic">h</Emphasis></Superscript> largest

For integers n ≥ r, we treat the rth largest of a sample of size n as an \(\mathbb {R}^{\infty }\)-valued stochastic process in r which we denote as M^(r). We show that the sequence regarded in this way satisfies the Markov property. We go on to study the asymptotic behavior of M^(r) as r → ∞, and, borrowing from classical extreme value theory, show that left-tail domain of attraction conditions on the underlying distribution of the sample guarantee weak limits for both the range of M^(r) and M^(r) itself, after norming and centering. In continuous time, an analogous process Y^(r) based on a two-dimensional Poisson process on \(\mathbb {R}_{+}\times \mathbb {R}\) is treated similarly, but we note that the continuous time problems have a distinctive additional feature: there are always infinitely many points below the rth highest point up to time t for any t >?0. This necessitates a different approach to the asymptotics in this case.     

The dual space of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of a vector measure

For a vector measure ν having values in a real or complex Banach space and \({p \in}\) [1, ∞), we consider L ^p (ν) and \({L_{w}^{p}(\nu)}\), the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E _q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L ^p (ν) ^×  = E _q (μ) and \({E_q(\mu)^\times = L_w^p(\nu)}\). It follows that \({L_p (\nu) ^{**} = L_w^p (\nu)}\). We also show that L ¹ (ν) ^×  may be equal or not to E _∞ (μ).     

Intersections of Primary Subgroups in Nonsoluble Finite Groups Isomorphic to <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>)

Given a finite group G with socle isomorphic to L _n (2 ^m ), we describe (up to conjugacy) all ordered pairs of primary subgroups A and B in G such that A ∩ B ^g ≠ 1 for all g ∈ g.     

Some <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Rigidity Results for Complete Manifolds with Harmonic Curvature

Let (M ⁿ , g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R m? the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R m? goes to zero uniformly at infinity if for \(p\geq \frac n2\), the L ^p -norm of R m? is finite. Moreover, If R is positive, then (M ⁿ , g) is compact. As applications, we prove that (M ⁿ , g) is isometric to a spherical space form if for \(p\geq \frac n2\), R is positive and the L ^p -norm of R m? is pinched in [0, C ₁), where C ₁ is an explicit positive constant depending only on n, p, R and the Yamabe constant. We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian n-manifolds with constant positive scalar curvature, which extends Theorem 1 of Hebey and M. Vaugon (J. Geom. Anal. 6, 531–553, 1996). This result is sharp, and we can precisely characterize the case of equality. In particular, when n = 4, we recover results by Gursky (Indiana Univ. Math. J. 43, 747–774, 1994; Ann. Math. 148, 315–337, 1998).     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic"> p</Emphasis></Superscript>-multipliers for the Hilbert space valued functions on the Heisenberg group

It is well known that if m is an L ^p -multiplier for the Fourier transform on \({\mathbb{R}^n}\) , (1 < p < ∞) then there exists a pseudomeasure σ such that T _m f = σ * f . A similar problem is discussed for the L ^p ?Fourier multipliers for \({\mathcal{H}}\) -valued functions on the Heisenberg group, where \({\mathcal{H}}\) is a separable Hilbert space.     

Periodic Groups Saturated with the Groups <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Given an indexing set I and a finite field K_α for each α ∈ I, let ? = {L₂(K_α) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L₂(P) (respectively SL₂(P)) for a suitable locally finite field P.     

<Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">X</Emphasis></Superscript>/<Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">c</Emphasis> Queue with catastrophes and state-dependent control at idle time

We consider an M ^X /M/c queue with catastrophes and state-dependent control at idle time. Properties of the queues which terminate when the servers become idle are first studied. Recurrence, equilibrium distribution, and equilibrium queue-size structure are studied for the case of resurrection and no catastrophes. All of these properties and the first effective catastrophe occurrence time are then investigated for the case of resurrection and catastrophes. In particular, we obtain the Laplace transform of the transition probability for the absorbing M ^X /M/c queue.     

Some Limit Theorems on the Increments of l^p-valued Multi-Parameter Gaussian Processes

In this paper, we establish some limit theorems on the increments of an l^p-valued multiparameter Gaussian process under weaker conditions than those of Csoergoe-Shao theorems published in Ann. Probab. (1993).     

The Center of <Emphasis Type="Italic">Dist</Emphasis> (<Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">m</Emphasis>|<Emphasis Type="Italic">n</Emphasis>)) in Positive Characteristic

The purpose of this paper is to investigate central elements in distribution algebras D i s t(G) of general linear supergroups G = G L(m|n). As an application, we compute explicitly the center of D i s t(G L(1|1)) and its image under Harish-Chandra homomorphism.     

Equivariant Classification of <Emphasis Type="Italic">b</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>-symplectic Surfaces

Inspired by Arnold’s classification of local Poisson structures [1] in the plane using the hierarchy of singularities of smooth functions, we consider the problem of global classification of Poisson structures on surfaces. Among the wide class of Poisson structures, we consider the class of b^m-Poisson structures which can be also visualized using differential forms with singularities as b^m-symplectic structures. In this paper we extend the classification scheme in [24] for b^m-symplectic surfaces to the equivariant setting. When the compact group is the group of deck-transformations of an orientable covering, this yields the classification of these objects for nonorientable surfaces. The paper also includes recipes to construct b^m-symplectic structures on surfaces. The feasibility of such constructions depends on orientability and on the colorability of an associated graph. The desingularization technique in [10] is revisited for surfaces and the compatibility with this classification scheme is analyzed in detail.     

2-Recognizability by prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and let Γ(G) be the prime graph of G. Assume p prime. We determine the finite groups G such that Γ(G) = Γ(PSL(2, p ²)) and prove that if p ≠ 2, 3, 7 is a prime then k(Γ(PSL(2, p ²))) = 2. We infer that if G is a finite group satisfying |G| = |PSL(2, p ²)| and Γ(G) = Γ(PSL(2, p ²)) then G ? PSL(2, p ²). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications are also considered of this result to the problem of recognition of finite groups by element orders.     

<Emphasis Type="Italic">l</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis>,<Emphasis Type="Italic">s</Emphasis></Superscript>-Singular values and spectral radius of partially symmetric rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties of l ^k,s -singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest l ^k,s -singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest l ^k,s -singular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.     

Lebesgue weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-algebra on locally compact groups

Let G be a locally compact group, ω a weight function on G, and 1<p<∞. We introduce the Lebesgue weighted L ^p -space \({\mathcal{L}}_{\omega}^{1,p}(G)= L^{p}(G,\omega)\cap L^{1}(G)\) as a Banach space and introduce its dual. Furthermore, we consider this space as a Banach algebra with respect to the usual convolution and show that \({\mathcal{L}}_{\omega}^{1,p}(G)\) admits a bounded approximate identity if and only if G is discrete. In addition, we prove that amenability of this algebra implies that G is discrete and amenable. Moreover, we discuss the converse of this result.     

Decomposition theorems for<Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> spaces

We study the Möbius invariant spacesQ _p andQ _{p, 0} of analytic functions. These scales of spaces include BMOA=Q₁, VMOA=Q_{1, 0} and the Dirichlet space=Q₀. Using the Bergman metric, we establish decomposition theorems for these spaces. We obtain also a fractional derivative characterization for bothQ _p andQ _{p, 0} .