Lebesgue Spaces Lp Research Articles

Variable Lebesgue Space over Weighted Homogeneous Tree

An infinite homogeneous tree is a special type of graph that has a completely symmetrical structure in all directions. For an infinite homogeneous tree T=(V,E) with the natural distance d defined on graphs and a weighted measure μ of exponential growth, the authors introduce the variable Lebesgue space Lp(·)(μ) over (V,d,μ) and investigate it under the global Hölder continuity condition for p(·). As an application, the strong and weak boundedness of the maximal operator relevant to admissible trapezoids on Lp(·)(μ) is obtained, and an unbounded example is presented.

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.

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Optimal non-absolute domains for the Hardy operator minus identity

We characterize the optimal non-absolute domain for the Hardy operator minus the identity, in the Lebesgue space Lp(0,∞), 1≤p≤∞. We also address a similar question for the dual Hardy operator minus the identity.

The semi-discrete diffusion convection equation with decay

In this paper we find the fundamental solution of the semi-discrete diffusion convection equation with decay, and we show that under a certain combination of the parameters of the equation said fundamental solution constitutes a uniformly continuous semigroup of operators in the Lebesgue spaces ℓp(Z),1≤p≤∞.

EXTRAPOLATION IN SOME MIXED-NORMED FUNCTION SPACES

Abstract. From the beginning of this century non-standard function spaces attracted a considerable attention of researchers. One of such spaces is variable exponent Lebesgue space (Nakano space) 𝐿𝑝(.); another one is grand Lebesgue space 𝐿𝑝0). The main reason to study these spaces was to solve a number of contemporary problems arising naturally in Applied Mathematics and PDEs (see, e.g., [1-5]). Mixed norm function spaces play an important role in Harmonic Analysis and PDEs (see, e.g. the survey by L. Huang and D. Yang (2019), papers by N. V. Krylov.

Generation of semigroups associated to strongly coupled elliptic operators in [formula omitted

A class of vector-valued elliptic operators with unbounded coefficients, coupled up to the second-order is investigated in the Lebesgue space Lp(Rd;Rm) with p∈(1,∞), providing sufficient conditions for the generation of an analytic C0-semigroup T(t). Under further assumptions, a characterization of the domain of the infinitesimal generator is given.

Examining Nonlinear Fredholm Equations in Lebesgue Spaces with Variable Exponents

We investigate the existence of solutions for the Fredholm integral equation Φ(ϑ)=G(ϑ,Φ(ϑ))+∫01F(ϑ,ζ,Φ(ζ))dζ, for ϑ∈[0,1], in the setting of the modular function spaces Lρ. We also derive an application of this research within the framework of variable exponent Lebesgue spaces Lp(·) subject to specific conditions imposed on the exponent function p(·) and the functions F and G.

On Erdélyi–Kober Fractional Operator and Quadratic Integral Equations in Orlicz Spaces

We provide and prove some new fundamental properties of the Erdélyi–Kober (EK) fractional operator, including monotonicity, boundedness, acting, and continuity in both Lebesgue spaces (Lp) and Orlicz spaces (Lφ). We employ these properties with the concept of the measure of noncompactness (MNC) associated with the fixed-point hypothesis (FPT) in solving a quadratic integral equation of fractional order in Lp,p≥1 and Lφ. Finally, we provide a few examples to support our findings. Our suppositions can be successfully applied to various fractional problems.

When are the norms of the Riesz projection and the backward shift operator equal to one?

The lower estimate by Gohberg and Krupnik (1968) and the upper estimate by Hollenbeck and Verbitsky (2000) for the norm of the Riesz projection P on the Lebesgue space Lp lead to ‖P‖Lp→Lp=1/sin⁡(π/p) for every p∈(1,∞). Hence L2 is the only space among all Lebesgue spaces Lp for which the norm of the Riesz projection P is equal to one. Banach function spaces X are far-reaching generalisations of Lebesgue spaces Lp. We prove that the norm of P is equal to one on the space X if and only if X coincides with L2 and there exists a constant C∈(0,∞) such that ‖f‖X=C‖f‖L2 for all functions f∈X. Independently from this, we also show that the norm of P on X is equal to one if and only if the norm of the backward shift operator S on the abstract Hardy space H[X] built upon X is equal to one.

Heat kernels and Hardy spaces on non-tangentially accessible domains with applications to global regularity of inhomogeneous Dirichlet problems

Let n ≥ 2 n\ge 2 and Ω \Omega be a bounded non-tangentially accessible domain (for short, NTA domain) of R n \mathbb {R}^n . Assume that L D L_D is a second-order divergence form elliptic operator having real-valued, bounded, measurable coefficients on L 2 ( Ω ) L^2(\Omega ) with the Dirichlet boundary condition. The main aim of this article is threefold. First, the authors prove that the heat kernels { K t L D } t > 0 \{K_t^{L_D}\}_{t>0} generated by L D L_D are Hölder continuous. Second, for any p ∈ ( 0 , 1 ] p\in (0,1] , the authors introduce the ‘geometrical’ Hardy space H r p ( Ω ) H^p_r(\Omega ) by restricting any element of the Hardy space H p ( R n ) H^p(\mathbb {R}^n) to Ω \Omega and the authors show that, when p ∈ ( n n + δ 0 , 1 ] p\in (\frac {n}{n+\delta _0},1] , H r p ( Ω ) = H p ( Ω ) = H L D p ( Ω ) H^p_r(\Omega )=H^p(\Omega )=H^p_{L_D}(\Omega ) with equivalent quasi-norms, where H p ( Ω ) H^p(\Omega ) and H L D p ( Ω ) H^p_{L_D}(\Omega ) respectively denote the Hardy space on Ω \Omega and the Hardy space associated with L D L_D and where δ 0 ∈ ( 0 , 1 ] \delta _0\in (0,1] is the critical index of the Hölder continuity for the kernels { K t L D } t > 0 \{K_t^{L_D}\}_{t>0} . Third, as applications, the authors obtain the global gradient estimates in both L p ( Ω ) L^p(\Omega ) , with p ∈ ( 1 , p 0 ) p\in (1,p_0) , and H z p ( Ω ) H^p_z(\Omega ) , with p ∈ ( n n + 1 , 1 ] p\in (\frac {n}{n+1},1] , for the inhomogeneous Dirichlet problem of second-order divergence form elliptic equations on bounded NTA domains, where p 0 ∈ ( 2 , ∞ ) p_0\in (2,\infty ) is a constant depending only on n n , Ω \Omega , and the coefficient matrix of L D L_D . Here, the ‘geometrical’ Hardy space H z p ( Ω ) H^p_z(\Omega ) is defined by restricting any element of the Hardy space H p ( R n ) H^p(\mathbb {R}^n) supported in Ω ¯ \overline {\Omega } to Ω \Omega , where Ω ¯ \overline {\Omega } denotes the closure of Ω \Omega in R n \mathbb {R}^n . It is worth pointing out that the range p ∈ ( 1 , p 0 ) p\in (1,p_0) for the global gradient estimate in the scale of Lebesgue spaces L p ( Ω ) L^p(\Omega ) is sharp and the above results are established without any additional assumptions on both the coefficient matrix of L D L_D and the domain Ω \Omega .

On approximation of double Fourier series and its conjugate series for functions in mixed Lebesgue space

In this paper, we study the approximation of double Fourier series and its conjugate series for functions in mixed Lebesgue space Lp, p ∈ [1,∞]2 using double Karamata Kλ,μ means.

Weighted Jump and Variational Inequalities for Hypersingular Integrals with Rough Kernels

In this paper, we study the weighted jump function and variation of hypersingular integral operators with rough kernels which are defined as $${T_{\Omega,\alpha,\varepsilon}}f\left(x \right) = \int_{\left| y \right|>\varepsilon} {{{\Omega ({y^\prime})} \over {{{\left| y \right|}^{n + \alpha}}}}f(x - y)dy,} $$ where α ≥ 0, Ω is an integrable function on the unit sphere $${\mathbb S^{n - 1}}$$ satisfying certain cancellation conditions. More precisely, we show that for 1 < p < ∞, the jump function and variation of the family of truncated hypersingular integrals {TΩ,α, ε}ε>0 extends to bounded operators from the weighted Sobolev space L (w) to the weighted Lebesgue space Lp(w) with $$\Omega \,\,{L^q}({\mathbb S^{n - 1}})$$ where q > 1.

Stability of p(·)-Integrable Solutions for Fractional Boundary Value Problem via Piecewise Constant Functions

The goal of this work is to study a multi-term boundary value problem (BVP) for fractional differential equations in the variable exponent Lebesgue space (Lp(·)). Both the existence, uniqueness, and the stability in the sense of Ulam–Hyers are established. Our results are obtained using two fixed-point theorems, then illustrating the results with a comprehensive example.

On the weighted maximal operators of Marcinkiewicz type Cesàro means of two-dimensional Walsh-Fourier series

In this paper we investigate the behaviour of the weighted maximal operators of Marcinkiewicz type (C,?)-means ??,* p (f) := supn?P |??n(f)|/ n2/p?(2+?) in the Hardy space Hp(G2) (0 &lt; ? &lt; 1 and p &lt; 2/(2 + ?)). It is showed that the maximal operators ??,* p (f) are bounded from the dyadic Hardy space Hp(G2) to the Lebesgue space Lp(G2), and that this is in a sense sharp. It was also proved a strong convergence theorem for the Marcinkiewicz type (C, ?) means of Walsh-Fourier series in Hp(G2).

Approximation of functions in the Lp.w (X)-space and application

The purpose of this research is to study the approximation of functions in the space Lp.w which is an expansion of the Lebesgue space Lp using the Fejer operator in terms of the average modulus of the second-order with the application of two of the theorems.

Caputo Fractional Evolution Equations in Discrete Sequences Spaces

In this paper, we treat some fractional differential equations on the sequence Lebesgue spaces ℓp(N0) with p≥1. The Caputo fractional calculus extends the usual derivation. The operator, associated to the Cauchy problem, is defined by a convolution with a sequence of compact support and belongs to the Banach algebra ℓ1(Z). We treat in detail some of these compact support sequences. We use techniques from Banach algebras and a Functional Analysis to explicity check the solution of the problem.

Codimension 1 minimal projections onto the lines in [formula omitted

Let X denote a real Banach space and V a subspace of X; we say that projection Pmin:X→V is minimal if ‖Pmin‖≤‖P‖ for every projection P from X to V. Considered as a subspace of Lebesgue space Lp[−1,1] (p≥1), let Y≔[1,t] denote the subspace of lines. In this paper we consider several natural codimension 1 overspaces X⊃Y (within Lp) and characterize minimal projections Pmin:X→Y for p=2n. The characterization utilizes the (so called) Chalmers–Metcalf operator and as such makes heavy use of extremal pairs associated with projections. We include results that show minimal projections in this setting are unique and include a section where algorithms and numerical results for ‖Pmin‖ are given.

ON THE DENSITY OF LAGUERRE FUNCTIONS IN SOME BANACH FUNCTION SPACES

Let λ &gt; 0 and Φλ := {ϕ1,λ, ϕ2,λ, . . . } be the system of dilated Laguerre functions. We show that if L1 (R+) ∩ L∞(R+) is embedded into a separable Banach function space X(R+), then the linear span of Φλ is dense in X(R+). This implies that the linear span of Φλ is dense in every separable rearrangement-invariant space X(R+) and in every separable variable Lebesgue space Lp(·) (R+)

Boundary value problem of Riemann-Liouville fractional differential equations in the variable exponent Lebesgue spaces Lp(.)

This manuscript deals with the existence, uniqueness and stability of solutions to the boundary value problem (BVP) of Riemann-Liouville (RL) fractional differential equations (FDEs) in the variable exponent Lebesgue spaces (Lp(.)). The generalized intervals and piece-wise constant functions are utilized to extract the aims of current paper. The variable exponent Lebesgue spaces (Lp(.)) are converted to the classical Lebesgue spaces. Further, the Banach contraction principle is used, the Ulam-Hyers-stability is examined and finally, an illustrative example is given to the validity of the observed results.