Nonnegative Measurable Function Research Articles

A Radon-Nikodym theorem for monotone measures

A version of Radon-Nikodym theorem for the Choquet integral w.r.t. monotone measures is proved. Without any presumptive condition, we obtain a necessary and sufficient condition for the ordered pair (μ,ν) of finite monotone measures to have the so-called Radon-Nikodym property related to a nonnegative measurable function f. If ν is null-continuous and weakly null-additive, then f is uniquely determined almost everywhere by ν and thus is called the Radon-Nikodym derivative of μ w.r.t. ν. For σ-finite monotone measures, a Radon-Nikodym type theorem is also obtained under the assumption that the monotone measures are lower continuous and null-additive.

The Pólya-Szegő inequality for smoothing rearrangements

A basic version of the Pólya-Szegő inequality states that if Φ is a Young function, the Φ-Dirichlet energy—the integral of Φ(‖∇f‖)—of a suitable function f∈V(Rn), the class of nonnegative measurable functions on Rn that vanish at infinity, does not increase under symmetric decreasing rearrangement. This fact, along with variants that apply to polarizations and to Steiner and certain other rearrangements, has numerous applications. Very general versions of the inequality are proved that hold for all smoothing rearrangements, those that do not increase the modulus of continuity of functions. The results cover all the main classes of functions previously considered: Lipschitz functions f∈V(Rn), functions f∈W1,p(Rn)∩V(Rn) (when 1≤p<∞ and Φ(t)=tp), and functions f∈Wloc1,1(Rn)∩V(Rn). In addition, anisotropic versions of these results, in which the role of the unit ball is played by a convex body containing the origin in its interior, are established. Taken together, the results bring together all the basic versions of the Pólya-Szegő inequality previously available under a common and very general framework.

Integral of Ordered Banach Algebra Valued Measurable Functions

This research is concerned with the set of functions in ordered Banach algebra values that links up between functional analysis and measure theory. We generalized the concept of integration by using the measure space and the measurable function where is an ordered Banach algebra by using the integration of a simple measurable function with values ​​in an ordered Banach algebra space (represented by an indicator function that has values ​​in an ordered Banach algebra) and the integral of a non-negative measurable function that has values ​​in an ordered Banach algebra. The aim of this research is to define the integration of functions by using the measure in the ordered Banach algebra space. This study generalized the definition of integration for the measurable function with values in the ordered Banach algebra space.

On the finiteness of strong maximal functions associated to functions whose integrals are strongly differentiable

Besicovitch proved that if f is an integrable function on R2 whose associated strong maximal function MSf is finite a.e., then the integral of f is strongly differentiable. On the other hand, Papoulis proved the existence of an integrable function on R2 (taking on both positive and negative values) whose integral is strongly differentiable but whose associated strong maximal function is infinite on a set of positive measure. In this paper, we prove that if n≥2 and if f is a measurable nonnegative function on Rn whose integral is strongly differentiable and moreover such that f(1+log+⁡f)n−2 is integrable, then MSf is finite a.e. We also show this result is sharp by proving that, if φ is a continuous increasing function on [0,∞) such that φ(0)=0 and with φ(u)=o(u(1+log+⁡u)n−2)(u→∞), then there exists a nonnegative measurable function f on Rn such that φ(f) is integrable on Rn and the integral of f is strongly differentiable, although MSf is infinite almost everywhere.

Quantitative description on initial trace for the solvability to heat equation with exponential boundary in unbounded domain

In this paper, we study the solvability of heat equation in R+N with flux ∂∂νu=exp⁡(pu) on ∂R+N for any constant p>0. The initial trace μ(x) is a Radon measure in RN with suppμ⊂R+N‾ or a nonnegative measurable function in R+N. We at first show the necessary conditions on the initial trace for the existence of solutions for every positive p, and then we divide into three cases to give sufficient conditions on the solvability of solutions, where all of the descriptions on the initial trace are determined quantitatively. For some cases, we identify the strongest singularity on the initial trace to guarantee the solvability of this nonlinear parabolic problem.

Weighted inequalities involving Hardy and Copson operators

We characterize a four-weight inequality involving the Hardy operator and the Copson operator. More precisely, given p1,p2,q1,q2∈(0,∞), we find necessary and sufficient conditions on non-negative measurable functions u1,u2,v1,v2 on (0,∞) for which there exists a positive constant c such that the inequality(∫0∞(∫0tf(s)p2v2(s)p2ds)q2p2u2(t)q2dt)1q2≤c(∫0∞(∫t∞f(s)p1v1(s)p1ds)q1p1u1(t)q1dt)1q1 holds for every non-negative measurable function f on (0,∞). The proof is based on discretizing and antidiscretizing techniques. The principal innovation consists in development of a new method which carefully avoids duality techniques and therefore enables us to obtain the characterization in previously unavailable situations, solving thereby a long-standing open problem. We then apply the characterization of the inequality to the establishing of criteria for embeddings between weighted Copson spaces Copp1,q1(u1,v1) and weighted Cesàro spaces Cesp2,q2(u2,v2), and also between spaces Sq(w) equipped with the norm ‖f‖Sq(w)=(∫0∞[f⁎⁎(t)−f⁎(t)]qw(t)dt)1/q and classical Lorentz spaces of type Λ.

Optimal singularities of initial functions for solvability of a semilinear parabolic system

Let $(u, v)$ be a nonnegative solution to the semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_{t} u = D_{1} \Delta u + v^{p}, \quad x \in \mathbf{R}^{N}, \ t > 0,\\ \partial_{t} v = D_{2} \Delta v + u^{q}, \quad x \in \mathbf{R}^{N}, \ t > 0,\\ (u(\cdot,0), v(\cdot,0)) = (\mu, \nu), \quad x \in \mathbf{R}^{N}, \end{array} \right. $$ where $D_{1}$, $D_{2} > 0$, $0 < p \leq q$ with $pq > 1$ and $(\mu, \nu)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in $\mathbf{R}^{N}$. In this paper we study sufficient conditions on the initial data for the solvability of problem (P) and clarify optimal singularities of the initial functions for the solvability.

Weighted Inequalities for a Superposition of the Copson Operator and the Hardy Operator

We study a three-weight inequality for the superposition of the Hardy operator and the Copson operator, namely \begin{equation*} \bigg(\int_a^b \bigg(\int_t^b \bigg(\int_a^s f(\tau)^p v(\tau) \,d\tau \bigg)^\frac{q}{p} u(s) \,ds \bigg)^{\frac{r}{q}} w(t) \,dt \bigg)^{\frac{1}{r}} \leq C \int_a^b f(t)\,dt, \end{equation*} in which $(a,b)$ is any nontrivial interval, $q,r$ are positive real parameters and $p\in(0,1]$. A simple change of variables can be used to obtain any weighted $L^p$-norm with $p\ge1$ on the right-hand side. Another simple change of variables can be used to equivalently turn this inequality into the one in which the Hardy and Copson operators swap their positions. We focus on characterizing those triples of weight functions $(u,v,w)$ for which this inequality holds for all nonnegative measurable functions $f$ with a constant independent of $f$. We use a new type of approach based on an innovative method of discretization which enables us to avoid duality techniques and therefore to remove various restrictions that appear in earlier work. This paper is dedicated to Professor Stefan Samko on the occasion of his 80th birthday.

A Coq Formalization of Lebesgue Integration of Nonnegative Functions

Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical concept of integration and the associated results in a formal proof assistant helps in providing the highest confidence on the correctness of numerical programs involving the use of integration, directly or indirectly. By its capability to extend the (Riemann) integral to a wide class of irregular functions, and to functions defined on more general spaces than the real line, the Lebesgue integral is perfectly suited for use in mathematical fields such as probability theory, numerical mathematics, and real analysis. In this article, we present the Coq formalization of \(\sigma \)-algebras, measures, simple functions, and integration of nonnegative measurable functions, up to the full formal proofs of the Beppo Levi (monotone convergence) theorem and Fatou’s lemma. More than a plain formalization of the known literature, we present several design choices made to balance the harmony between mathematical readability and usability of Coq theorems. These results are a first milestone toward the formalization of \(L^p\) spaces such as Banach spaces.

Existence of global solutions of a nonautonomous semilinear equation with varying reaction

Let be a generating function, where are nonnegative measurable functions, and let be a continuous function. We prove that reaction-diffusion equations of the prototype possess nontrivial positive global solutions under suitable assumptions on and

Fourier restriction in low fractal dimensions

AbstractLet$S \subset \mathbb {R}^{n}$be a smooth compact hypersurface with a strictly positive second fundamental form,$E$be the Fourier extension operator on$S$, and$X$be a Lebesgue measurable subset of$\mathbb {R}^{n}$. If$X$contains a ball of each radius, then the problem of determining the range of exponents$(p,q)$for which the estimate$\| Ef \|_{L^{q}(X)} \lesssim \| f \|_{L^{p}(S)}$holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set$X$: there is a number$0 &lt; \alpha \leq n$such that$|X \cap B_R| \lesssim R^{\alpha }$for all balls$B_R$in$\mathbb {R}^{n}$of radius$R \geq 1$. On the left-hand side of this estimate, we are integrating the function$|Ef(x)|^{q}$against the measure$\chi _X \,{\textrm {d}}x$. Our approach consists of replacing the characteristic function$\chi _X$of$X$by an appropriate weight function$H$, and studying the resulting estimate in three different regimes: small values of$\alpha$, intermediate values of$\alpha$, and large values of$\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted Hölder-type inequality that holds for general non-negative Lebesgue measurable functions on$\mathbb {R}^{n}$and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du–Zhang theorem in the range$0 &lt; \alpha &lt; n/2$.

Tamped functions: A rearrangement in dimension 1

We define a new rearrangement, called rearrangement by tamping, for non-negative measurable functions defined on R+. This rearrangement has many properties in common with the well-known Schwarz non-increasing rearrangement such as the Pólya–Szegő inequality. Contrary to the Schwarz rearrangement, the tamping also preserves the homogeneous Dirichlet boundary condition of a function.

An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family

Let X be a topological space equipped with a complete positive σ -finite measure and T a subset of the reals with 0 as an accumulation point. Let a t x , y be a nonnegative measurable function on X × X which integrates to 1 in each variable. For a function f ∈ L 2 X and t ∈ T , define A t f x ≡ ∫ a t x , y f y d y . We assume that A t f converges to f in L 2 , as t ⟶ 0 in T . For example, A t is a diffusion semigroup (with T = 0 , ∞ ). For W a finite measure space and w ∈ W , select real-valued h w ∈ L 2 X , defined everywhere, with h w L 2 X ≤ 1 . Define the distance D by D x , y ≡ h w x − h w y L 2 W . Our main result is an equivalence between the smoothness of an L 2 X function f (as measured by an L 2 -Lipschitz condition involving a t · , · and the distance D ) and the rate of convergence of A t f to f .

A note on the coincidence of decomposition integrals and superdecomposition integrals

It is known that the decomposition integral IH and the superdecomposition integral IH related to the system H of all finite chains from A⧹{∅}, where A is a σ-algebra of the subsets of a nonvoid set X, coincide with each other for each monotone measure and for each nonnegative measurable function, and they are equal to the Choquet integral. In this note, we show that the converse is also true. That is, for a complete system H of finite set systems from A⧹{∅}, if IH=IH holds for each monotone measure and for each nonnegative measurable function, then the system H and the system of all finite chains from A⧹{∅} are equivalent in some sense and the integrals involved are the Choquet integral.

Convergence in measure theorems of nonlinear integrals of functions integrable to the pth power

One can find a subadditive measure μ and a sequence {fn}n∈N of nonnegative measurable functions converging in μ-measure to such a function f with the property that for any p>1, fnp no longer converges in μ-measure to fp. Therefore, in nonadditive measure theory an additional and intrinsic discussion is needed to prove integral convergence theorems for functions integrable to the pth power.The purpose of the paper is to propose a method for automatically deriving pth power versions of convergence in measure theorems of nonlinear integrals from already established ones. The proposed method is based on the robustness of some properties of integral functional under the operation f↦fp raising a nonnegative function f to the pth power.

Quasilinear elliptic equations with sub-natural growth terms and nonlinear potential theory

We discuss recent advances in the theory of quasilinear equations of the type $ -\Delta_{p} u = \sigma u^{q} \; \; \text{in} \;\; \mathbb{R}^n, $ in the case $0<q< p-1$, where $\sigma$ is a nonnegative measurable function, or measure, for the $p$-Laplacian $\Delta_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$, as well as more general quasilinear, fractional Laplacian, and Hessian operators. Within this context, we obtain some new results, in particular, necessary and sufficient conditions for the existence of solutions $u \in \text{BMO}(\mathbb{R}^n)$, $u \in L^r_{{\rm loc}}(\mathbb{R}^n)$, etc., and prove an enhanced version of Wolff's inequality for intrinsic nonlinear potentials associated with such problems.

Isoperimetric Inequalities for Non-Local Dirichlet Forms

Let $(E,{\mathscr{E}} F,\mu )$ be a σ-finite measure space. For a non-negative symmetric measurable function J(x,y) on E × E, consider the quadratic form $$ \mathscr{E}(f,f):= \frac{1}{2}{\int}_{E\times E} (f(x)-f(y))^{2} J(x,y) \mu(\text{\text{d}} x) \mu(\text{\text{d}} y) $$ in L2(μ). We characterize the relationship between the isoperimetric inequality and the super Poincare inequality associated with ${\mathscr{E}}$ . In particular, sharp Orlicz-Sobolev type and Poincare type isoperimetric inequalities are derived for stable-like Dirichlet forms on $\mathbb {R}^{n}$ , which include the existing fractional isoperimetric inequality as a special example.

Estimate for the Entropy Numbers of the Weighted Hardy Operators that Act from Banach Space to q-Banach Space

We study weighted Hardy operators from the Banach space Lp to the q-Banach space Lq and obtain estimates for the entropy numbers of one- and two-weighted Hardy operators with nonnegative measurable weight functions.

Wolff’s inequality for intrinsic nonlinear potentials and quasilinear elliptic equations

We prove an analogue of Wolff’s inequality for the so-called intrinsic nonlinear potentials associated with the quasilinear elliptic equation −Δpu=σuqinRn,in the case 0<q<p−1. Here Δpu=div(|∇u|p−2∇u) is the p-Laplacian, and σ is a nonnegative measurable function (or measure). As an application, we give a necessary and sufficient condition for the existence of a positive solution u∈Lr(Rn) (0<r<∞) to this problem, which was open even in the case p=2.Our version of Wolff’s inequality for intrinsic nonlinear potentials relies on a new characterization of discrete Littlewood–Paley spaces fp,q(σ) defined in terms of characteristic functions of dyadic cubes in Rn.

On the Generalized Ideal Asymptotically Statistical Equivalent of Order α for Functions

We introduce new definitions related to the notions of asymptotically Iλ -statistical equivalent of order α to multiple L and strongly Iλ -asymptotically equivalent of order α to multiple L by using two nonnegative real-valued Lebesque measurable functions in the interval (1, ∞) instead of sequences. In addition, we also present some inclusion theorems.