Classical Lebesgue Research Articles

Boundedness of Bessel–Riesz Operator in Variable Lebesgue Measure Spaces

In this manuscript, we establish the boundedness of the Bessel–Riesz operator Iα,γf in variable Lebesgue spaces Lp(·). We prove that Iα,γf is bounded from Lp(·) to Lp(·) and from Lp(·) to Lq(·). We explore various scenarios for the boundedness of Iα,γf under general conditions, including constraints on the Hardy–Littlewood maximal operator M. To prove these results, we employ the boundedness of M, along with Hölder’s inequality and classical dyadic decomposition techniques. Our findings unify and generalize previous results in classical Lebesgue spaces. In some cases, the results are new even for constant exponents in Lebesgue spaces.

Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓq(·)(Mp(·),v(·)) with Variable Exponents

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.

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.

Hardy Type Inequalities in Classical and Grand Lebesgue Spaces $L_{p)}$, $0<p\le 1$, for Quasi-Monotone Functions

In 2020 Rovshan A. Bandaliev et al. proved the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces $L_{p)} (0,1)$, $0<p\leq1$. In particular, they established similar results for the Hardy operator in classical weighted Lebesgue spaces. Moreover, it is proved that the grand Lebesgue space $L_{p) } (0,1)$ is a quasi-Banach function space. In this work, we are interested in Hardy inequalities applied to quasi-monotonic functions in classical Lebesgue spaces and grand Lebesgue spaces. We establish the boundedness of Hardy operator for quasi-monotone functions in grand Lebesgue spaces $L_{p)}$, $w(0,1),$ $0<p\leq1$. In addition some integral inequalities for the Hardy operator are proved in classical weighted Lebesgue spaces $L_{p,w}(0,1)$, $0<p<1,$ for quasi-monotone functions. All inequalities are proved with sharp constants. Some results of Rovshan A. Bandaliev et al. are deduced as particular cases. Also other estimates are obtained in classical Lebesgue spaces for Hardy's operator and its dual.

On Weighted Cauchy-Type Problem of Riemann-Liouville Fractional Differential Equations in Lebesgue Spaces with Variable Exponent

This paper aims to investigate the existence, uniqueness, and stability properties for a class of fractional weighted Cauchy-type problem in the variable exponent Lebesgue space $L^{p(.)}$. The obtained results are set up by employing generalized intervals and piece-wise constant functions so that the $L^{p(.)}$ is transformed into the classical Lebesgue spaces. Moreover, the usual Banach Contraction Principle is utilized, and the Ulam-Hyers (UH) stability is studied. At the final stage, we provide an example to support the accuracy of the obtained results.

Decomposition Integrals of Set-Valued Functions Based on Fuzzy Measures

The decomposition integrals of set-valued functions with regards to fuzzy measures are introduced in a natural way. These integrals are an extension of the decomposition integral for real-valued functions and include several types of set-valued integrals, such as the Aumann integral based on the classical Lebesgue integral, the set-valued Choquet, pan-, concave and Shilkret integrals of set-valued functions with regard to capacity, etc. Some basic properties are presented and the monotonicity of the integrals in the sense of different types of the preorder relations are shown. By means of the monotonicity, the Chebyshev inequalities of decomposition integrals for set-valued functions are established. As a special case, we show the linearity of concave integrals of set-valued functions in terms of the equivalence relation based on a kind of preorder. The coincidences among the set-valued Choquet, the set-valued pan-integral and the set-valued concave integral are presented.

Maximal ergodic inequalities for some positive operators on noncommutative Lp$L_p$‐spaces

AbstractIn this paper, we push forward the conjecture on a noncommutative version of the maximal ergodic theorem for positive contractions due to Ackoglu. That is, we establish the one‐sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative ‐spaces for a fixed , which particularly applies to positive isometries, the convex hull of positive Lamperti contractions, and also the power‐bounded doubly Lamperti operators; moreover, it is known that this subclass recovers all positive contractions on the classical Lebesgue spaces . As an unexpected consequence, we deduce a completely bounded version of Ackoglu's theorem by making use of Ackoglu's original dilation. We also observe that the concrete examples of positive contractions without Akcoglu's dilation, which were constructed by Junge–Le Merdy [J. Funct. Anal. 249 (2007), no. 1, 220–252.], still satisfy the maximal ergodic inequalities. Together with the class of power‐bounded positive invertible operators considered by Hong–Liao–Wang [Duke Math. J. (2020). In press], we thereby verify the noncommutative Ackoglu ergodic theorem for all the positive operators that have naturally appeared in the literature up to the moment of writing. Based on the noncommutative Calderón transference principle established in [Duke Math. J. (2020). In press], the general pattern of the demonstration follows from the classical one due to Kan, but several new ideas are absolutely necessary in the noncommutative setting. Let us just mention three of them: the argument for the maximal ergodic theorem for positive isometries is highly nontrivial compared to the classical case; the novel simultaneous dilation property is indispensable to get the ergodic theorem for the convex hull of positive Lamperti contractions; numerous adjustments are truly needed in this new setting to conclude a desired structural theorem for positive doubly Lamperti operators since the orthogonal relations of operator algebras are completely different from those in classical measure theory.

On Soft Lebesgue Measure

In this paper, we introduce the concept of soft intervals, soft ordering and sequences of soft real numbers, and some of their structural properties are studied. The notion of soft Lebesgue measure on the soft real numbers has been introduced. Also, a correspondence relationship has been established between the soft Lebesgue measure and the classical Lebesgue measure. Furthermore, we have studied some exciting results and relations between the soft Lebesgue measure and the Lebesgue measure of soft real sets.

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.

Growth Envelopes of Some Variable and Mixed Function Spaces

We study unboundedness properties of functions belonging to Lebesgue and Lorentz spaces with variable and mixed norms using growth envelopes. Our results extend the ones for the corresponding classical spaces in a natural way. In the case of spaces with mixed norms, it turns out that the unboundedness in the worst direction, i.e., in the direction where p_{i} is the smallest, is crucial. More precisely, the growth envelope is given by {mathfrak {E}}_{{mathsf {G}}}(L_{overrightarrow{p}}(varOmega )) = (t^{-1/min {p_{1}, ldots , p_{d} }},min {p_{1}, ldots , p_{d} }) for mixed Lebesgue and {mathfrak {E}}_{{mathsf {G}}}(L_{overrightarrow{p},q}(varOmega )) = (t^{-1/min {p_{1}, ldots , p_{d} }},q) for mixed Lorentz spaces, respectively. For the variable Lebesgue spaces, we obtain {mathfrak {E}}_{{mathsf {G}}}(L_{p(cdot )}(varOmega )) = (t^{-1/p_{-}},p_{-}), where p_{-} is the essential infimum of p(cdot ), subject to some further assumptions. Similarly, for the variable Lorentz space, it holds{mathfrak {E}}_{{mathsf {G}}}(L_{p(cdot ),q}(varOmega )) = (t^{-1/p_{-}},q). The growth envelope is used for Hardy-type inequalities and limiting embeddings. In particular, as a by-product, we determine the smallest classical Lebesgue (Lorentz) space which contains a fixed mixed or variable Lebesgue (Lorentz) space, respectively.

A Review on the Lebesgue Spaces

In this article, we begin with classical Lebesgue spaces Lp with p being constant and review the various properties such as completeness and duality of the space. To this end, we also discuss the boundedness of Hardy-Littlewood maximal function and interpolation on such spaces. Finally, we focus our attention on variable exponent Lebesgue spaces and review various results on it. Moreover, we also see the differences in between these Lebesgue spaces.

Density topologies for strictly positive Borel measures

The aim of this paper is to describe density topologies generated by Borel complete regular measures. For any such measure μ a generalization of the classical Lebesgue differentiation theorem is proved here. Using it, we define a topology Tμ which is some kind of abstract density topology and, simultaneously, a natural generalization of the classical density topology Td.We focus on separation axioms of topologies Tμ. We show that considered topologies are regular and not normal. In some cases we prove that Tμ gives a Tychonoff space. We also consider homeomorphisms between topologies Tμ. It is shown that for any atomless measure ν the topology Tν is homeomorphic to Td. However, there are measures μ generating topologies quite different than Td, for example separable and not connected. Some cases deliver us examples of spaces which are not Lindelöf but they are separable.

Linear methods of approximation in weighted Lebesgue spaces with variable exponent

Some estimations in below for the deviations conducted by the Zygmund means and by the Abel-Poisson sums in the weighted Lebesgue spaces with variable exponent are obtained. In the classical Lebesgue spaces these estimations were proved by M. F. Timan. The considered weight functions satisfy the well known Muckenhout condition. For the proofs of main results some estimations obtained in the classical weighted Lebesgue spaces and also an extrapolation theorem proved in the weighted variable exponent Lebesgue spaces are used. Main results are new even in the nonweighted variable exponent Lebesgue spaces.

Unrestricted Cesàro summability of $d$-dimensional Fourier series and Lebesgue points

We generalize the classical Lebesgue's theorem to multi-dimensional functions. We prove that the Cesàro means of the Fourier series of the multi-dimensional function $f\in L_1(\log L)^{d-1}(\mathbb{T}^d)\supset L_p(\mathbb{T}^d) (1<p<\infty)$ converge to $f$ at each strong Lebesgue point.

Lebesgue points of ell _1-Ces\xe0ro summability of d-dimensional Fourier series

We generalize the classical Lebesgue’s theorem and prove that the ell _1-Cesàro means of the Fourier series of the multi-dimensional function fin L_1({{mathbb {T}}}^d) converge to f at each strong omega -Lebesgue point.

Fixed point properties and reflexivity in variable Lebesgue spaces

In this paper the weak fixed point property (w-FPP) and the fixed point property (FPP) in Variable Lebesgue Spaces are studied. Given (Ω,Σ,μ) a σ-finite measure and p(⋅) a variable exponent function, the w-FPP is completely characterized for the variable Lebesgue space Lp(⋅)(Ω) in terms of the exponent function p(⋅) and the absence of an isometric copy of L1[0,1]. In particular, every reflexive Lp(⋅)(Ω) has the FPP and our results bring to light the existence of some nonreflexive variable Lebesgue spaces satisfying the w-FPP, in sharp contrast with the classic Lebesgue Lp-spaces. In connection with the FPP, we prove that Maurey's result for L1-spaces can be extended to the larger class of variable Lp(⋅)(Ω) spaces with order continuous norm, that is, every reflexive subspace of Lp(⋅)(Ω) has the FPP. Nevertheless, Maurey's converse does not longer hold in the variable setting, since some nonreflexive subspaces of Lp(⋅)(Ω) satisfying the FPP can be found. As a consequence, we discover that several nonreflexive Nakano sequence spaces ℓpn do have the FPP endowed with the Luxemburg norm. As far as the authors are concerned, this family of sequence spaces gives rise to the first known nonreflexive classic Banach spaces enjoying the FPP without requiring of any renorming procedure. The failure of asymptotically isometric copies of ℓ1 in Lp(⋅)(Ω) is also analyzed.

Noncommutative maximal ergodic inequalities associated with doubling conditions

In this paper, we establish the one-sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative $L_p$-spaces for a fixed $1<p<\infty$, which particularly applies to positive isometries and general positive Lamperti contractions or power bounded doubly Lamperti operators; moreover, it is known that this subclass recovers all positive contractions on the classical Lebesgue spaces $L_p([0,1])$. Our study falls into neither the category of positive contractions considered by Junge-Xu nor the class of power bounded positive invertible operators considered by Hong-Liao-Wang. Our strategy essentially relies on various structural characterizations and dilation properties associated with Lamperti operators, which are of independent interest. More precisely, we give a structural description of Lamperti operators in the noncommutative setting, and obtain a simultaneous dilation theorem for Lamperti contractions. As a consequence we establish the maximal ergodic theorem for the strong closure of the convex hull of corresponding family of positive contractions. Moreover, in conjunction with a newly-built structural theorem, we also obtain the maximal ergodic inequalities for positive power bounded doubly Lamperti operators. We also observe that the concrete examples of positive contractions without Akcoglu's dilation, which were constructed by Junge-Le Merdy, still satisfy the maximal ergodic inequalities. We also discuss some other examples, showing sharp contrast to the classical situation.

Weighted norm inequalities for the bilinear maximal operator on variable Lebesgue spaces

We extend the theory of weighted norm inequalities on variable Lebesgue spaces to the case of bilinear operators. We introduce a bilinear version of the variable Ap(·) condition and show that it is necessary and sufficient for the bilinear maximal operator to satisfy a weighted norm inequality. Our work generalizes the linear results of the first author, Fiorenza, and Neugebauer [7] in the variable Lebesgue spaces and the bilinear results of Lerner et al. [22] in the classical Lebesgue spaces. As an application we prove weighted norm inequalities for bilinear singular integral operators in the variable Lebesgue spaces.

Certain results for a class of nonlinear functional spaces

In this article, we study properties of a class of functional spaces, so-called pn-spaces, which arise from investigation of nonlinear differential equations. We establish some integral inequalities to analyse the structures of the pn-spaces with the constant and variable exponent. We prove embedding theorems, which indicate the relation of these spaces with the well known classical Lebesgue and Sobolev spaces with the constant and variable exponents.

A REVERSE HÖLDER INEQUALITY IN L^p(x)(Ω)

In this study, at first we provide a general overview of L^p(x)(Ω) spaces, also known as variable exponent Lebesgue spaces. They are a generalization of classical Lebesgue spaces L^p in the sense that constant exponent replaced by a measurable function. Then, based on classical Lebesgue space approach we prove a reverse of Hölder inequality in L^p(x)(Ω). Therefore, our proof in variable exponent Lebesgue space is very similar to that in classical Lebesgue space.