Weighted Lebesgue Spaces Research Articles

Neumann inhomogeneous initial-boundary value problem for the 2D nonlinear Schrödinger equation

This paper is the first attempt to give a rigorous mathematical study of Neumann initial boundary value problems for the multidimensional dispersive evolution equations considering as example famous nonlinear Schrodinger equation. We consider the inhomogeneous initial-boundary value problem for the nonlinear Schrodinger equation, formulated on upper right-quarter plane with initial data $$u({\mathbf {x}},t)\left| _{t=0}\right. =u_{0}({\mathbf {x}})$$ and Neumann boundary data $$u_{x_{1}}\left| _{\partial _{1}{\mathbf {D}}}\right. =h_{1}(x_{2},t),u_{x_{2}}\left| _{\partial _{2}{\mathbf {D}}}\right. =h_{2}(x_{1},t)$$ given in a suitable weighted Lebesgue spaces. We are interested in the study of the influence of the Neumann boundary data on the asymptotic behavior of solutions for large time. We show that problem admits global solutions whose long-time behavior essentially depends on the boundary data. To get a nonlinear theory for the multidimensional model. we propose general method based on Riemann–Hilbert approach and theory Cauchy type integral equations. The advantage of this method is that it can also be applied to non-integrable equations with general inhomogeneous boundary data.

Fejér Sums and Chebyshev Polynomials

Let $$L^p_w[-1,1]$$ be the weighted Lebesgue space on $$[-1,1]$$ with $$w(t)=(1-t^2)^{-1/2}$$. We prove that the rate of convergence in $$L^p_w[-1,1]$$ of the Fejer sums is equivalent to a fractional modulus of smoothness of order 1/2.

Compactness Criteria for Fractional Integral Operators

Abstract We establish necessary and sufficient conditions for the compactness of fractional integral operators from Lp (X, μ) to Lq (X, μ) with 1 < p < q < ∞, where μ is a measure on a quasi-metric measure space X. As an application we obtain criteria for the compactness of fractional integral operators defined in weighted Lebesgue spaces over bounded domains of the Euclidean space ℝ n with the Lebesgue measure, and also for the fractional integral operator associated to rectifiable curves of the complex plane.

The Boas Problem on Hankel Transforms

Norm equivalences between a function and its Hankel transform are studied both in the context of weighted Lebesgue spaces with power weights, and in Lorentz spaces. Boas-type results involving real-valued general monotone functions are obtained. Corresponding results for the Fourier transform are also given.

Multidimensional bilinear Hardy inequalities

A characterization of n-dimensional bilinear Hardy inequalities in weighted Lebesgue spaces is given.

The Basis Property of Ultraspherical Jacobi Polynomials in a Weighted Lebesgue Space with Variable Exponent

The problem of the basis property of ultraspherical Jacobi polynomials in a Lebesgue space with variable exponent is studied. We obtain sufficient conditions on the variable exponent p(x) > 1 that guarantee the uniform boundedness of the sequence S n α,α (f), n = 0,1,..., of Fourier sums with respect to the ultraspherical Jacobi polynomials P k α,α (x) in the weighted Lebesgue space L μ p( x) ([-1, 1]) with weight μ = μ(x) = (1 - x2)α, where α >-1/2. The case α = -1/2 is studied separately. It is shown that, for the uniform boundedness of the sequence Sn-1/2, -1/2 (f), n = 0,1,..., of Fourier—Chebyshev sums in the space L μ p( x) ([-1,1]) with μ(x) = (1 - x2)-1/2, it suffices and, in a certain sense, necessary that the variable exponent p satisfy the Dini-Lipschitz condition of the form $$\left| {p(x) - p(y)} \right| \leq \frac{d}{{ - \ln \left| {x - y} \right|}},\;\;\;\text{where}\;\left| {x - y} \right| \leq \frac{1}{2},\;\;x,y \in [ - 1,1],\;\;d > 0,$$ and the condition p(x) > 1 for all x ∈ [-1,1].

Bessel type Kolmogorov inequalities on weighted Lebesgue spaces

ABSTRACT In this paper, we obtained Bessel type Kolmogorov inequalities for Riesz–Bessel transforms related to generalized shift operator in weighted Lebesgue spaces.

Boundedness of Differential Transforms for One-Sided Fractional Poisson-Type Operator Sequence

In this paper, we analyze the convergence speed of a series related with $\mathcal{P}_\tau^\alpha f$ by discussing the behavior of the family of operators \begin{equation*} T_N^\alpha f(t) = \sum_{j=N_1}^{N_2} v_j(\mathcal{P}_{a_{j+1}}^\alpha f(t)-\mathcal{P}_{a_j}^\alpha f(t)),\quad ~N=(N_1,N_2)\in \mathbb{Z}^2\quad \hbox{with} \quad N_1<N_2, \end{equation*} where $\{v_j\}_{j\in \mathbb Z}$ is a bounded number sequence, and $\{a_j\}_{j\in \mathbb{Z}}$ is a $\rho$-lacunary sequence of positive numbers, that is, $1<\rho \leq a_{j+1}/a_j, \text{for all}\ j\in \mathbb{Z}.$ We shall show the boundedness of the maximal operator \begin{equation*}T^*f(t)=\sup_N |T_N^\alpha f(t)|, \quad t\in\mathbb{R}, \end{equation*} in the one-sided weighted Lebesgue spaces $L^p(\mathbb{R},\omega)(\omega \in A_p^-$), $1< p < \infty$. As a consequence we infer the existence of the limit, in norm and almost everywhere, of the family $T_N^\alpha f$ for functions in $L^p(\mathbb{R},\omega)$. Results for $L^1(\mathbb{R},\omega)(\omega \in A_1^-)$, $L^\infty(\mathbb{R})$ and $BMO(\mathbb{R})$ are also obtained. It is also shown that the local size of $T^*f$, for functions $f$ having local support, is the same with the order of a singular integral. Moreover, if $\{v_j\}_{j\in \mathbb Z}\in \ell^p(\mathbb Z)$, we get an intermediate size between the local size of singular integrals and Hardy-Littlewood maximal operator.

Vertical square functions and other operators associated with an elliptic operator

We study the vertical and conical square functions defined via elliptic operators in divergence form. In general, vertical and conical square functions are equivalent operators just in L2. But when this square functions are defined through the heat or Poisson semigroups that arise from an elliptic operator, we are able to find open intervals containing 2 where the equivalence holds. The intervals in question depend ultimately on the range where the semigroup is uniformly bounded or has off-diagonal estimates. As a consequence we obtain new boundedness results for some square functions. Besides, we consider a non-tangential maximal function associated with the Poisson semigroup and extend the known range where that operator is bounded. Our methods are based on the use of extrapolation for Muckenhoupt weights and change of angle estimates. All our results are obtained in the general setting of a degenerate elliptic operator, where the degeneracy is given by an A2 weight, in weighted Lebesgue spaces. Of course they are valid in the unweighted and/or non-degenerate situations, which can be seen as special cases, and they provide new results even in those particular settings.We also consider the square root of a degenerate elliptic operator in divergence form Lw and improve the lower bound of the interval where this operator is known to be bounded on Lp(vdw). Finally, we give unweighted boundedness results for the degenerate operators under consideration.

Characterizations of Some Properties on Weighted Modulation and Wiener Amalgam Spaces

In this paper, we characterize some properties on weighted modulation and Wiener amalgam spaces by the corresponding properties on weighted Lebesgue spaces. As applications, we obtain sharp conditions for product inequalities, convolution inequalities, and embedding on weighted modulation and Wiener amalgam spaces. By a unified approach different from others we give a complete answer to the question of finding sharp conditions of certain relations on weighted modulation and Wiener amalgam spaces.

Multidimensional Bilinear Hardy Inequalities

A characterization of the n-dimensional bilinear Hardy inequality in weighted Lebesgue spaces is given.

On derivative of trigonometric polynomials and characterizations of modulus of smoothness in weighted Lebesgue space with variable exponent

In this paper we investigate some properties of approximation polynomials in particular de la Vallee-Poussin means, Fejer means and partial sums of Fourier series in weighted Lebesgue spaces with variable exponent. In addition to these we prove a simultaneous type theorem and some theorems on the equivalence of modulus of smoothness and the K-functional in weighted Lebesgue space with variable exponent.

On bilinear weighted inequalities with Volterra integral operators

Necessary and sufficient conditions on the boundedness in weighted Lebesgue spaces on the semiaxis for bilinear inequalities with Volterra integral operators are given.

Compactness of commutators of pseudo-differential operators with smooth symbols on weighted Lebesgue spaces

In this paper, we obtain the compactness of commutators of pseudo-differential operators with smooth symbols on weighted $$L^{p}$$ spaces where the weight functions belong to a class of new weights which includes the classical Muckenhoupt weights and the results are new even on weighted $$L^{p}$$ spaces where the weight functions belong to Muckenhoupt weights. In addition, a strong sufficient condition for compactness in new weighted $$L^{p}(1<p<\infty )$$ spaces is given, which has interest in its own right.

Approximation by Nörlund and Riesz means in weighted Lebesgue space with variable exponent

We investigate the approximation properties of Nörlund and Riesz means of trigonometric Fourier series are investigated in the subset of weighted Lebesgue space with variable exponent.

Exponential Sampling Series: Convergence in Mellin–Lebesgue Spaces

In this paper we study norm-convergence to a function f of its generalized exponential sampling series in weighted Lebesgue spaces. Key roles are taken by a result on the norm-density of the test functions and the notion of bounded coarse variation. Some examples are described.

Neumann eigenvalue problems on the exterior domains

For p∈(1,∞), we consider the following weighted Neumann eigenvalue problem on B1c, the exterior of the closed unit ball in RN: (0.1)−Δpϕ=λg|ϕ|p−2ϕinB1c,∂ϕ∂ν=0on∂B1,where Δp is the p-Laplace operator and g∈Lloc1(B1c) is an indefinite weight function. Depending on the values of p and the dimension N, we take g in certain Lorentz spaces or weighted Lebesgue spaces and show that (0.1) admits an unbounded sequence of positive eigenvalues that includes a unique principal eigenvalue. For this purpose, we establish the compact embeddings of W1,p(B1c) into Lp(B1c,|g|) for g in certain weighted Lebesgue spaces. For N>p, we also provide an alternate proof for the embedding of W1,p(B1c) into the Lorentz space Lp∗,p(B1c). Further, we show that the set of all eigenvalues of (0.1) is closed.

Bilinear square spectral multipliers on stratified groups

In this paper, on stratified groups, we give a reasonable definition of bilinear square spectral multiplier and its commutator. Then we study their boundedness on weighted Lebesgue spaces under the assumption of its weak boundedness and a sum-type Hormander condition. Besides, we show that, in Euclidean spaces, the smoothness conditions applied to ensure the boundedness of bilinear square Fourier multiplier can be replaced by the sum-type Hormander condition.

A note on maximal commutators with rough kernels

This paper gives a characterization of compactness for maximal commutators with rough kernels in weighted Lebesgue spaces, which is new and interesting even in un-weighted cases. Meanwhile, a new characterization of weighted boundedness for such operators is also established.

Bilinear Weighted Inequalities with Volterra Integral Operators

Necessary and sufficient conditions for the boundedness of a class of bilinear inequalities with Volterra integral operators in weighted Lebesgue spaces on the real half-line are given.