Weighted Lp Spaces Research Articles

Persistence properties for a two-component b-family system with higher-order nonlinearity in weighted Lp spaces

In this paper, we focus on a two-component b-family system with higher-order nonlinearity, which contains some integrable Camassa–Holm type models. By virtue of moderate weight functions, we demonstrate the persistence properties for solutions to this system in weighted Lp spaces. Our persistence results cover and improve the related results of some Camassa–Holm type models.

Generalized Carleson embeddings of Müntz spaces

This paper establishes Carleson embeddings of Müntz spaces MΛq into weighted Lebesgue spaces Lp(dμ), where μ is a Borel regular measure on [0,1] satisfying μ([1−ε])≲εβ. In the case β⩾1 we show that such measures are exactly the ones for which Carleson embeddings Lpβ↪Lp(dμ) hold. The case β∈(0,1) is more intricate, but we characterize such measures μ in terms of a summability condition on their moments. Our proof relies on a generalization of Lp estimates à la Gurariy-Macaev in the weighted Lp spaces setting, which we think can be of interest in other contexts.

A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind

In this note, the hybrid method (combination of the homotopy perturbation method (HPM) and the Gauss elimination method (GEM)) is developed as a semi-analytical solution for the first kind system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. Before applying the HPM, we have to first reduce the system of CSIEs into a triangle system of algebraic equations using GEM, which is then carried out using the HPM. Using the theory of the bounded, unbounded and semi-bounded solutions of CSIEs, we are able to find inverse operators for the system of CSIEs of the first kind. A stability analysis and convergent of the proposed method has been conducted in the weighted Lp space. Moreover, the proposed method is proven to be exact in the Holder class of functions for the system of characteristic SIEs for any type of initial guess. For each of the four cases, several examples are provided and examined to demonstrate the proposed method’s validity and accuracy. Obtained results are compared with the Chebyshev collocation method and modified HPM (MHPM). Example 3 reveals that the error term of the MHPM is slightly superior to that of the HPM. One of the features of the proposed method is that it can be solved as a complex-valued system of CSIEs. Numerical results revealed that the hybrid method dominates others.

Peakons and Persistence Properties of Solution for the Interacting System of Popowicz

This paper focuses on a two-component interacting system introduced by Popowicz, which has the coupling form of the Camassa–Holm and Degasperis–Procesi equations. Using distribution theory, single peakon solutions and several double peakon solutions of the system are described in an explicit expression. Moreover, dynamic behaviors of several types of double peakon solutions are illustrated through figures. In addition, the persistence properties of the solutions to the Popowicz system in weighted Lp spaces is considered via a large class of moderate weights.

Weighted [formula omitted] Markov factors with doubling weights on the ball

Let Lp,w,1≤p<∞, denote the weighted Lp space of functions on the unit ball Bd with a doubling weight w on Bd. The Markov factor for Lp,w of a polynomial P is defined by ‖|∇P|‖p,w‖P‖p,w, where ∇P is the gradient of P. We investigate the worst case Markov factors for Lp,w and prove that the degree of these factors is at most 2. In particular, for the Gegenbauer weight wμ(x)=(1−|x|2)μ−1/2,μ≥0, the exponent 2 is sharp. We also study the average case Markov factor for L2,w on random polynomials with independent N(0,σ2) coefficients and obtain that the upper bound of the average (expected) Markov factor is order degree to the 3/2, as compared to the degree squared worst case upper bound.

On the Cyclicity of Dilated Systems in Lattices: Multiplicative Sequences, Polynomials, Dirichlet-type Spaces and Algebras

Abstract The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice X, including weighted ℓ p spaces. In particular, general multiplicative and completely multiplicative sequences are treated. After the Fourier–Bohr transformation, we deal with the cyclicity property in function spaces on the corresponding infinite dimensional Reinhardt domain 𝔻 X ∞ \mathbb{D}_X^\infty . Functions with (weakly) dominating free term and (in particular) linearly factorable functions are considered. The most attention is paid to the cases of the polydiscs 𝔻 X ∞ , | ℂ N = 𝔻 N \mathbb{D}_X^\infty ,|{\mathbb{C}^N} = {\mathbb{D}^N} and the ℓ p-unit balls 𝔻 X ∞ , | ℂ N = 𝔹 p N \mathbb{D}_X^\infty ,|{\mathbb{C}^N} = \mathbb{B}_p^N , in particular to Dirichlet-type and Dirichlet–Drury–Arveson-type spaces and algebras, as X = ℓ p ( ℤ + N , ( 1 + α ) s ) X = {\ell ^p}\left( {_ + ^N,{{\left( {1 + \alpha } \right)}^s}} \right)) , s = (s 1, s 2, … ) and X = ℓ p ( ℤ + N , ( α ! | α | ! ) t ( 1 + | α | ) s ) X = {\ell ^p}\left( {\mathbb{Z}_ + ^N,\,\,{{\left( {{{\alpha !} \over {\left| \alpha \right|!}}} \right)}^t}{{\left( {1 + \left| \alpha \right|} \right)}^s}} \right) , s,t ≥ 0, as well as to their infinite variables analogues. We priviledged the largest possible scale of spaces and the most elementary instruments used.

Hardy–Littlewood-type theorems for Fourier transforms in [formula omitted

We obtain Fourier inequalities in the weighted Lp spaces for any 1<p<∞ involving the Hardy–Cesàro and Hardy–Bellman operators. We extend these results to product Hardy spaces for p⩽1. Moreover, boundedness of the Hardy-Cesàro and Hardy-Bellman operators in various spaces (Lebesgue, Hardy, BMO) is discussed. One of our main tools is an appropriate version of the Hardy–Littlewood–Paley inequality ‖fˆ‖Lp′,q≲‖f‖Lp,q.

Piecewise Monotone Approximation of Unbounded Functions In Weighted Space LP,w([-,

In this paper, investigate the approximation of unbounded functions in weighted space, by using trigonometric polynomials considered. We introduced type of polynomials piecewise monotone having same local monotonicity as unbounded functions without affecting the order of huge error have a finite number of max. and min. unbounded functions that amount. In addition, we established not included any of extreme points of this functions, of and closed subset γ on closed intervals then there exist class of polynomials such that the best of approximation has high or order of and such that for sufficiently great of the polynomials and functions have the same monotonicity at each of γ.

Cesàro Means of Weighted Orthogonal Expansions on Regular Domains

In this paper, we investigate Cesàro means for the weighted orthogonal polynomial expansions on spheres with weights being invariant under a general finite reflection group on Rd. Our theorems extend previous results only for specific reflection groups. Precisely, we consider the weight function hκ(x):=∏ν∈R+|x,ν|κν,x∈Rd on the unit sphere; the upper estimates of the Cesàro kernels and Cesàro means are obtained and used to prove the convergence of the Cesàro (C,δ) means in the weighted Lp space for δ above the corresponding index. We also establish similar results for the corresponding estimates on the unit ball and the simplex.

APPROXIMATION PROPERTIES OF MODIFIED GAUSS-WEIERSTRASS INTEGRAL OPERATORS IN EXPONENTIALLY WEIGHTED Lp SPACES

In here, we use modi ed Gauss-Weierstrass operators and givesome approximation results in the exponential weighted Lp spaces. Theseoperators are reproduce not only 1 but also a certain exponential functions.Forthis purpose, rstly we consider modi ed Gauss-Weierstrass integral operatorsfrom exponentially weighted Lp;a (R) into Lp;2a (R) spaces. Then, we give rate of convergence of the operators in Lp;2a (R) : Also, we prove the convergence of operators in the exponential weighted Lp;2a (R) spaces using the Korovkin type theorem. Finally, we give pointwise convergence of the operators at a generalized Lebesgue point.

The Cauchy problem for a generalized Riemann-type hydrodynamical equation

In this work, we investigate the Cauchy problem for a generalized Riemann-type hydrodynamical equation. The local well-posedness of the equation in Besov spaces is derived by using Littlewood–Paley decomposition and transport equation theory. Then, we show that a finite maximal life span for a solution necessarily implies wave breaking for this solution and give a condition on the initial data to ensure wave breaking for this equation by making use of the method of characteristics; otherwise, the equation has a global smooth solution. In addition, we establish persistence results for solutions of the equation in weighted Lp spaces for a large class of moderate weights.

A Weyl pseudodifferential calculus associated with exponential weights on Rd

We construct a Weyl pseudodifferential calculus tailored to studying boundedness of operators on weighted Lp spaces over Rd with weights of the form exp(−ϕ(x)), for ϕ a C2 function, a setting in which the operator associated to the weighted Dirichlet form typically has only holomorphic functional calculus. A symbol class giving rise to bounded operators on Lp is determined, and its properties are analyzed. This theory is used to calculate an upper bounded on the H∞ angle of relevant operators and deduces known optimal results in some cases. Finally, the symbol class is enriched and studied under an algebraic viewpoint.

Sample numbers and optimal Lagrange interpolation in Sobolev spaces

This paper investigates the optimal recovery of Sobolev spaces W∞r[−1,1],r∈ℕ in space L∞[−1,1] and weighted spaces Lp,ω[−1,1],1≤p<∞ with ω a continuous integrable weight function in (−1,1). We obtain the values of the sampling numbers of W∞r[−1,1] in L∞[−1,1] and Lp,ω[−1,1],1≤p<∞. We prove that the Lagrange interpolation algorithms based on the Chebyshev nodes of the first kind are optimal for p=∞. Meanwhile, we prove that the Lagrange interpolation algorithms based on the zeros of polynomial of degree r with the leading coefficient 1 of the least deviation from zero in Lp,ω[−1,1] are optimal for 1≤p<∞. We also give the optimal Lagrange interpolation algorithms when we ask the endpoints to be included in the nodes.

Approximation by Haar type polynomials in weighted rearrangement invariant spaces

We investigate basis and approximation properties of Haar type systems introduced by Vilenkin. The direct approximation theorem by polynomials with respect to a such system is proved in weighted nontrivial rearrangement invariant space under some conditions on weight. Also a direct approximation theorem by double Haar type polynomials is obtained in weighted Lp spaces with special weights.

The heat equation with rough boundary conditions and holomorphic functional calculus

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded H∞-calculus on weighted Lp-spaces for power weights which fall outside the classical class of Ap-weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data.

Disjoint hypercyclic weighted pseudoshift operators generated by different shifts

Let I be a countably infinite index set, and let X be a Banach sequence space over I. In this article, we characterize the disjoint hypercyclic and supercyclic weighted pseudoshift operators on X in terms of the weights, the OP-basis, and the shift mappings on I. Also, the shifts on weighted Lp spaces of a directed tree and the operator weighted shifts on ℓ2(Z,K) are investigated as special cases.

Weighted approximation by modified Picard operators

Herein, the aim is to further investigate the properties of the generalized Picard operators introduced in Agratini et al. (Positivity 3(21):1189–1199, 2017). The motivation is based on with the purpose of furnishing appropriate positive approximation processes in the setting of large classes of exponential weighted Lp spaces via different type theorems. For this propose, firstly we give the boundness of the operators, acting from an exponential weighted $$L_{p}$$ space into itself. Also, using an exponential weighted modulus of continuity a quantitative type theorem as well as the global smoothness property of the operators are presented. Then, we give pointwise approximation property of the operators at a generalized Lebesgue point. Finally under a certain condition, again the weighted Lp approximation is formulated without using Korovkin type theorem.

Disjoint hypercyclic weighted translations on locally compact hausdorff spaces

Abstract Let G be a locally compact second countable Hausdorff space with a positive regular Borel measure λ, where λ is invariant under a continuous injective mapping φ : G → G. We characterize the disjoint hypercyclicity of finite weighted translations generated by φ acting on the weighted space Lp (G, ω) (1 ≤ p &lt; ∞).

Weighted [formula omitted] estimates and Fujita exponent for a nonlocal equation

We investigate a nonlocal equation ∂tu=∫RnJ(x−y)u(y,t)dy−‖J‖L1u(x,t)+a(x,t)up in Rn, where a is unbounded and J belongs to a weighted space. Crucial weighted Lp and interpolation estimates for the Green operator are established by a new method based on the sharp Young’s inequality, the asymptotic behavior of a regular varying coefficients exponential series, and the properties of auxiliary functions Γ=(1+|x|2∕η)b∕2 that −Γ∕η≲J∗Γ−Γ≲Γ∕η and η−b+∕2≲Γ∕xb≲η−b−∕2. Blow-up behaviors are investigated by employing test functions ϕR=Γ (η=R) instead of principal eigenfunctions. Global well-posedness in weighted Lp spaces for the Cauchy problem is proved. When a∼xσ the Fujita exponent is shown to be 1+(σ+2)∕n. Our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations.

Spectra of Generalized Bochner–Riesz Means on Weighted Spaces

In this paper, we introduce a family of generalized Bochner–Riesz means and show that their spectra on weighted Lp spaces, , are either the whole complex plane or the interval for some . Also we extend the above spectral property to weighted Triebel–Lizorkin spaces, weighted Besov spaces, and weighted Herz spaces.