On a Generalization of Szász–Mirakjan–Kantorovich Operators

In this paper we introduce and study a sequence of positive linear operators acting on suitable spaces of measurable functions on [0,+∞[, including L ^p ([0,+∞[) spaces, 1 ≤ p < +∞, as well as continuous function spaces with polynomial weights. These operators generalize the Szász–Mirakjan–Kantorovich operators and they allow to approximate (or to reconstruct) suitable measurable functions by knowing their mean values on a sequence of subintervals of [0,+∞[ that do not constitute a subdivision of it. We also give some estimates of the rates of convergence by means of suitable moduli of smoothness.     

Bounds for the weighted Lp discrepancy and tractability of integration

Quite recently Sloan and Woźniakowski (J. Complexity 14 (1998) 1) introduced a new notion of discrepancy, the so-called weighted L^p discrepancy of points in the d-dimensional unit cube for a sequence γ=(γ₁,γ₂,…) of weights. In this paper we prove a nice formula for the weighted L^p discrepancy for even p. We use this formula to derive an upper bound for the average weighted L^p discrepancy. This bound enables us to give conditions on the sequence of weights γ such that there exists N points in [0,1)^d for which the weighted L^p discrepancy is uniformly bounded in d and goes to zero polynomially in N⁻¹.Finally we use these facts to generalize some results from Sloan and Woźniakowski (1998) on (strong) QMC-tractability of integration in weighted Sobolev spaces.     

Disjoint hypercyclic powers of weighted translations on groups

Let G be a locally compact group and let 1 ≤ p < 1. Recently, Chen et al. characterized hypercyclic, supercyclic and chaotic weighted translations on locally compact groups and their homogeneous spaces. There has been an increasing interest in studying the disjoint hypercyclicity acting on various spaces of holomorphic functions. In this note, we will study disjoint hypercyclic and disjoint supercyclic powers of weighted translation operators on the Lebesgue space L ^p(G) in terms of the weights. Sufficient and necessary conditions for disjoint hypercyclic and disjoint supercyclic powers of weighted translations generated by aperiodic elements on groups will be given.     

Local Hardy–Littlewood maximal operator

In this article we define and investigate a local Hardy–Littlewood maximal operator in Euclidean spaces. It is proved that this operator satisfies weighted L ^p , p > 1, and weighted weak type (1,1) estimates with weight function ${w \in A^p_{\rm{loc}}}In this article we define and investigate a local Hardy–Littlewood maximal operator in Euclidean spaces. It is proved that this operator satisfies weighted L ^p , p > 1, and weighted weak type (1,1) estimates with weight function w ? A^p_loc{w \in A^p_{\rm{loc}}}, the class of local A _p weights which is larger than the Muckenhoupt A _p class. Also, the condition w ? A^p_loc{w \in A^p_{\rm{loc}}} turns out to be necessary for the weighted weak type (p,p), p ≥ 1, inequality to hold.     

Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction

For second order linear equations and inequalities which are degenerate elliptic but which possess a uniformly elliptic direction, we formulate and prove weak maximum principles which are compatible with a solvability theory in suitably weighted versions of L²-based Sobolev spaces. The operators are not necessarily in divergence form, have terms of lower order, and have low regularity assumptions on the coefficients. The needed weighted Sobolev spaces are, in general, anisotropic spaces defined by a non-negative continuous matrix weight. As preparation, we prove a Poincaré inequality with respect to such matrix weights and analyze the elementary properties of the weighted spaces. Comparisons to known results and examples of operators which are elliptic away from a hyperplane of arbitrary codimension are given. Finally, in the important special case of operators whose principal part is of Grushin type, we apply these results to obtain some spectral theory results such as the existence of a principal eigenvalue.     

Hölder type inequalities in Lorentz spaces

We study optimal Hölder type inequalities for the Lorentz spaces L ^p,s (R, μ), in the range 1 < p < ∞, 1 ≤ s ≤ ∞, for both the maximal and the dual norms. These estimates also give sharp results for the corresponding associate norms.     

Global solutions of the Navier–Stokes equations for isentropic flow with large external potential force

We prove the global-in-time existence of weak solutions to the Navier–Stokes equations of compressible isentropic flow in three space dimensions with adiabatic exponent γ ≥ 1. Initial data and solutions are small in L ² around a non-constant steady state with densities being positive and essentially bounded. No smallness assumption is imposed on the external forces when γ = 1. A great deal of information about partial regularity and large-time behavior is obtained.     

Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in L1‐spaces

We investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) L¹‐spaces. We deal with both the cases of hard and soft potentials (with angular cut‐off). For hard potentials, we provide a new proof of the fact that, in weighted L¹‐spaces with exponential or algebraic weights, the solutions converge exponentially fast towards equilibrium. Our approach uses weak‐compactness arguments combined with recent results of the second author on positive semigroups in L¹‐spaces. For soft potentials, in L¹‐spaces, we exploit the convergence to ergodic projection for perturbed substochastic semigroup to show that, for very general initial datum, solutions to the linear Boltzmann equation converges to equilibrium in large time. Moreover, for a large class of initial data, we also prove that the convergence rate is at least algebraic. Notice that, for soft potentials, no exponential rate of convergence is expected because of the absence of spectral gap.     

Some new and short proofs for a class of Caffarelli-Kohn-Nirenberg type inequalities

In this note we provide simple and short proofs for a class of inequalities of Caffarelli-Kohn-Nirenberg type with sharp constants. Our approach suggests some definitions of weighted Sobolev spaces and their embedding into weighted L² spaces. These may be useful in studying solvability of problems involving new singular PDEs.     

Elliptic equations in weighted Sobolev spaces of infinite order with L1 data

We deal with the existence and uniqueness of weak solutions for a class of strongly nonlinear boundary value problems of higher order with L¹ data in anisotropic‐weighted Sobolev spaces of infinite order. Copyright © 2009 John Wiley & Sons, Ltd.     

A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.     

Extrapolation for Classes of Weights Related to a Family of Operators and Applications

In this work we give extrapolation results on weighted Lebesgue spaces for weights associated to a family of operators. The starting point for the extrapolation can be the knowledge of boundedness on a particular Lebesgue space as well as the boundedness on the extremal case L ^∞. This analysis can be applied to a variety of operators appearing in the context of a Schrödinger operator (??Δ?+?V) where V satisfies a reverse Hölder inequality. In that case the weights involved are a localized version of Muckenhoupt weights.     

Exact Multiplicity and Stability of Periodic Solutions for a Duffing Equation

The purpose of this paper is to give new criteria for the exact multiplicity and stability of 2π-periodic solutions for Duffing equation x′′ + cx′ + g(t, x) = h(t). The proof is based on the connections between degree theory and local index of periodic solutions due to Ortega [17], and the new L ^p estimates (1 ≤ p ≤ ∞) for periodic and anti-periodic eigenvalues of Hill’s equation due to Zhang and Li [26]. The class of g(t, x) has been greatly generalized.     

Tractability results for weighted Banach spaces of smooth functions

We study the L_∞-approximation problem for weighted Banach spaces of smooth d-variate functions, where d can be arbitrarily large. We consider the worst case error for algorithms that use finitely many pieces of information from different classes. Adaptive algorithms are also allowed. For a scale of Banach spaces we prove necessary and sufficient conditions for tractability in the case of product weights. Furthermore, we show the equivalence of weak tractability with the fact that the problem does not suffer from the curse of dimensionality.     

Best approximation with wavelets in weighted Orlicz spaces

Democracy functions of wavelet admissible bases are computed for weighted Orlicz Spaces L ^??(w) in terms of the fundamental function of L ^??(w). In particular, we prove that these bases are greedy in L ^??(w) if and only if L ^??(w) =?L ^p (w), 1?<?p?<???. Also, sharp embeddings for the approximation spaces are given in terms of weighted discrete Lorentz spaces. For L ^p (w) the approximation spaces are identified with weighted Besov spaces.     

Bloom’s inequality: commutators in a two-weight setting

In 1985, Bloom characterized the boundedness of the commutator [b, H] as a map between a pair of weighted L^p spaces, where both weights are in A_p. The characterization is in terms of a novel BMO condition. We give a ‘modern’ proof of this result, in the case of p = 2. In a subsequent paper, this argument will be used to generalize Bloom’s result to all Calderón–Zygmund operators and dimensions.     

Minimization of Eigenvalues of One-Dimensional p-Laplacian with Integrable Potentials

In this paper, we will use the variational method and limiting approach to solve the minimization problems of the Dirichlet/Neumann eigenvalues of the one-dimensional p-Laplacian when the L ¹ norm of integrable potentials is given. Combining with the results for the corresponding maximization problems, we have obtained the explicit results for these eigenvalues.     

Eigenvalue estimates and L 1 energy on closed manifolds

In this paper, we study Lichnerowicz type estimate for eigenvalues of drifting Laplacian operator and the decay rates of L1 and L2 energy for drifting heat equation on closed Riemannian manifolds with weighted measure.