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Let $E$ be a rearrangement invariant (r.i.) function space on $[0,1]$, and let $Z_E$ consist of all measurable functions $f$ on $(0,\infty)$ such that $f^*\chi_{[0,1]}\in E$ and $f^*\chi_{[1,\infty)}\in L^2$. We reveal close connections between properties of the generalized Rosenthal's space, corresponding to the space $Z_E$, and the behaviour of independent symmetrically distributed random variables in $E$. The results obtained are applied to consider the problem of the existence of isomorphisms between r.i.\ spaces on $[0,1]$ and $(0,\infty)$. Exploiting particular properties of disjoint sequences, we identify a rather wide new class of r.i.\ spaces on $[0,1]$ ``close'' to $L^\infty$, which fail to be isomorphic to r.i.\ spaces on $(0,\infty)$. In particular, this property is shared by the Lorentz spaces $\Lambda_2(\log^{-\alpha}(e/u))$, with $0, Comment: submitted

The aim of this paper is to study the existence of weak solutions for a class of Hardy–Schrodinger-type equations driven by the fractional N/s-Laplace operator. More precisely we consider $$\begin{aligned} (-\Delta )_{N/s}^{s}u+V(x)|u|^{\frac{N}{s}-2}u=\lambda \frac{ h(x) +|u|^{\frac{N}{s}-2}u}{|x|^{\frac{\sigma (N-s)}{s}}}+ \frac{f(x,u)}{|x|^{\gamma } }\;\;\;\text {in}\;\;{\mathbb {R}}^{N}, \end{aligned}$$ where $$(-\Delta )_{N/s}^{s}$$ is the fractional N/s-Laplace operator, $$N\ge 2$$ , $$0< \sigma0, $$ $$V :{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$$ is the potential function changing signs over $${\mathbb {R}}^{N}$$ . By combining the fixed point theory with the singular fractional Trudinger–Moser inequality we establish the existence of nontrivial weak solutions for these equations.

Let $S \subset \Bbb R^n$ 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 $\Bbb R^n$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^q(X)} \leq C \| f \|_{L^p(S)}$ 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 $0 < \alpha \leq n$ such that $|X \cap B_R| \leq c \, R^\alpha$ for all balls $B_R$ in $\Bbb R^n$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^q$ against the measure $\chi_X dx$. 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 H\"{o}lder-type inequality that holds for general non-negative Lebesgue measurable functions on $\Bbb R^n$, 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 $0 < \alpha < n/2$., Comment: 31 pages. Minor revision

S. Marcus used a fat Cantor set to show that there is a quasicontinuous function from the interval [0,1] to R that is not Lebesgue measurable. However, his article is not very well known. We give a...

We consider couples (XA, YA) of Hardy-type spaces defined for quasi-Banach lattices of measurable functions on 𝕋×Ω. Under certain fairly general assumptions, the following conditions are shown to be equivalent: (XA, YA) is K-closed in (X, Y), this couple is stable with respect to the real interpolation in the sense that (XA, YA)θ,p = (XA + YA) ∩ (X, Y)θ,p, the inclusion (X1−θYθ)A ⊂ (XA, YA)θ,∞ holds true, and lattices (L1, (Xr)′Yr)δ,q are BMO-regular for some values of the parameters. The last property is weaker than the BMO-regularity of (X, Y), and it requires further study. Some new (compared to the main article) results are given concerning the characterization of this property in terms of the boundedness of the standard harmonic analysis operators such as the Hilbert transform and the Hardy-Littlewood maximal operator.

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.

We introduce the ℒ p spaces of measurable functions whose p th power is summable with respect to the uniform measure over the Levi-Civita field. These spaces are the counterparts of the real L p spaces based upon the Lebesgue measure. Nevertheless, they lack some properties of the L p spaces: for instance, the ℒ p spaces are not sequentially complete with respect to the p -norm. This motivates the study of the completions of the ℒ p spaces with respect to strong convergence, denoted by ℒ s p . It turns out that the ℒ s p spaces are Banach spaces and that it is possible to define an inner product over ℒ s 2 , thus making it a Hilbert space. Despite these positive results, these spaces are still not rich enough to represent every real continuous function. For this reason, we settle upon the representation of real measurable functions as sequences of measurable functions in ℛ that weakly converge in measure. We also define a duality between measurable functions and representatives of continuous functions. This duality enables the study of some measurable functions that represent real distributions. We focus our discussion on the representatives of the Dirac distribution and on the well-known problem of the product between the Dirac and the Heaviside distribution, and we show that the solution obtained with measurable functions over ℛ is consistent with the result obtained with other nonlinear generalized functions.

In present paper, an integrable function is constructed, which is universal for the class of Lebesgue measurable functions, with almost everywhere convergence, with respect to the trigonometric system in the quasi-usual sense.

Let m, p, q ∈ (0, ∞) and let u, v, w be nonnegative weights. We characterize validity of the inequality $$\left(\int_{0}^{\infty} w(t)\left(f^{*}(t)\right)^{q} \mathrm{d} t\right)^{\frac{1}{q}} \leq C\left(\int_{0}^{\infty} v(t)\left(\int_{t}^{\infty} u(s)\left(f^{*}(s)\right)^{m} \mathrm{d} s\right)^{\frac{p}{m}} \mathrm{d} t\right)^{\frac{1}{p}}$$ for all measurable functions f defined on ℝn and provide equivalent estimates of the optimal constant C > 0 in terms of the weights and exponents. The obtained conditions characterize the embedding of the Copson—Lorentz space CLm,p(u, v), generated by the functional $$\|f\|_{C L^{m, p}(u, v)}:=\left(\int_{0}^{\infty} v(t)\left(\int_{t}^{\infty} u(s)\left(f^{*}(s)\right)^{m} \ \mathrm{d} s\right)^{\frac{p}{m}} \mathrm{d} t\right)^{\frac{1}{p}},$$ into the Lorentz space Λq(w). Moreover, the results are applied to describe the associated space of the Copson—Lorentz space CLm,p(u, v) for the full range of exponents m, p ∈ (0, ∞).

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in $\RN$, when $N\geq2$, as \begin{equation} \Lp u+u^{p-1}=\lambda\hspace{0.2mm}k(x)u^{r-1}-h(x)u^{q-1}.\nonumber \end{equation} Here, $h(x),k(x)>0$ are Lebesgue measurable functions, $10$ is a parameter.

We investigate the integrability of Banach space valued strongly measurable functions defined on [0, 1]. In the case of functions given by $$f=\sum _{n=1}^{\infty }x_n \chi _{E_n}$$ where $$x_n\in X$$ and the sets $$E_n$$ are Lebesgue measurable and pairwise disjoint subsets of [0, 1], there are well known characterizations for Bochner and McShane integrability of f. The absolute (resp. unconditional) convergence of $$\sum _{n=1}^{\infty }x_n m(E_n)$$ is equivalent to Bochner (resp. McShane) integrability of f. We give some conditions for scalar McShane and weakly McShane integrability of f and their relation with unconditionally converging operators. We also study the space of vector valued multipliers of strongly McShane $$(\mathcal {SM})-$$ integrable functions. We prove that if X is a commutative Banach algebra, with identity e of norm one, satisfying Radon–Nikodym property and $$M: \mathcal {SM}\rightarrow \mathcal {SM}$$ is a bounded linear operator, then there exists $$g\in L^\infty ([0,1],X)$$ such that $$M(f)= fg$$ for all $$f\in \mathcal {SM}$$ . Some results on multipliers of McShane $$({\mathcal {M}})-$$ integrable functions are also derived.

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.

In control theory, problems occur regarding the behavior of solutions of some abstract Cauchy problems like that 0.1$$\begin{aligned} \begin{array}{ll} u'(t) =&{} -A(u(t))-f(t)b,\quad t\in \mathbb {R} \\ u(\infty )=&{}\lim \nolimits _{t\rightarrow \infty }u(t)=0. \end{array} \end{aligned}$$Here A generates a strongly continuous semigroup $$\mathbf{T}=\{T(t)\}$$ acting on a complex Banach space X, f is a complex valued measurable function defined on $$\mathbb {R}_+$$ verifying a certain integral condition (as in Theorem 4.1 below), $$b\in X$$ is a randomly chosen vector and the limit is considered in the norm of X. We prove that the Cauchy Problem (0.1) has at least one solution (that is unique when X is a complex Hilbert space) provided the semigroup $$\mathbf{T}$$ is $$\Phi $$-weakly stable, that is, for every $$x\in X$$ and $$x'\in X'$$ of norms less than or equal to 1 the map $$t\mapsto \Phi (|\langle e^{tA}x, x'\rangle |$$ belongs to $$L^1(\mathbb {R}_+)$$. Concrete examples and even the expression of solutions are also provided in this paper. Here $$\Phi $$ is a given N-function, $$X'$$ denotes the strong dual of X and $$\langle \cdot , \cdot \rangle $$ denotes the duality map between X and $$X'$$. It is known (Storozhuk in Sib Math J 51:330–337, 2010) that the uniform spectral bound $$s_0(A)$$ is negative whenever the semigroup $$\mathbf{T}$$ generated of A is $$\Phi $$-weakly stable for the above $$\Phi $$. We complete this result by proving that if the semigroup is $$\Phi $$-weakly stable then there exists a positive number $$\nu $$ such that $$s_0(A)\le -\nu $$. An implicit expression of $$\nu $$, in terms of $$\Phi $$ and $$\mathbf{T}$$, is also given. The condition that $$\Phi $$ is positive near to 0 is necessary in the proofs. A counterexample showing this is provided in the last section of the paper.

This paper considers the problems on the existence and uniqueness, the pth(p ≥ 1)-moment and the almost sure stability with a general decay for the global solution of stochastic delay differential equations with Markovian switching, when the drift term and the diffusion term satisfy the locally Lipschitz condition and the monotonicity condition. By using the Lyapunov function approach, the Barbalat Lemma and the nonnegative semi-martingale convergence theorem, some sufficient conditions are proposed to guarantee the existence and uniqueness as well as the stability with a general decay for the global solution of such equations. It is mentioned that, in this paper, the time-varying delay is a bounded measurable function. The derived stability results are more general, which not only include the exponential stability but also the polynomial stability as well as the logarithmic one. At last, two examples are given to show the effectiveness of the theoretical results obtained.

In this work we are interested in the periodic solutions of the singular problem involving variable exponent with a homogeneous Dirichlet boundary conditions modeled as $$\begin{aligned} {\partial _t u}-\varDelta u =\displaystyle \frac{f}{u^{\gamma (t,x)}}\text { in }]0,T[\times \varOmega \end{aligned}$$ Where $$\varOmega $$ is an open regular bounded subset of $${\mathbb {R}}^{N}$$ , $$T>0$$ is the period, $$\gamma (t,x)$$ is a nonnegative periodic function belonging in $${\mathcal {C}}(\overline{Q_{T}})$$ and f is a nonnegative measurable function periodic in time with period T and belonging to a certain Lebesgue space. Under suitable assumptions on $$\gamma $$ and f, we prove an existence result of a nonnegative weak time periodic solution to the considered problem.

Since if we have a function f: R R continuous and a function g: R R Lebesgue measurable does not necessarily result that the function g o f is Lebesgue measurable, with the help of Cantor’s set we will constructed a Lebesgue measurable function h: R -h> R and a continuous function u: R -h> R, such that the function h o u: R —» R is not Lebesgue measurable.

If f,g,h:Rn⟶R≥0 are non-negative measurable functions such that h(x+y) is greater than or equal to the p-sum of f(x) and g(y), where −1/n≤p≤∞, p≠0, then the Borell–Brascamp–Lieb inequality asserts that the integral of h is not smaller than the q-sum of the integrals of f and g, for q=p/(np+1). In this paper we obtain a discrete analog for the sum over finite subsets of the integer lattice Zn: under the same assumption as before, for A,B⊂Zn}, then ∑A+Bh≥[(∑rf(A)f)q+(∑Bg)q]1/q, where rf(A) is obtained by removing points from A in a particular way, and depending on f. We also prove that the classical Borell–Brascamp–Lieb inequality for Riemann integrable functions can be obtained as a consequence of this new discrete version.

The aim of this paper is to give some sufficient conditions, called generalized good-λ conditions, to obtain the weighted Lorentz comparisons between two measurable functions. Moreover, we also present several applications of these results for the gradient estimates of solutions to quasilinear elliptic problems in weighted Lorentz spaces.

In this paper, we consider a continuous-time semi-Markov process (SMP) in Polish spaces. We first introduce the semi-additive functional in semi-Markov cases, a natural generalization of the additive functional of Markov process (MP). The main contribution of this paper is the properties of semi-additive functional aforementioned. First, semi-additive functionals of SMPs are characterized in terms of a cadlag function with zero initial value and a measurable function. Second, the necessary and sufficient conditions are investigated under which a semi-additive functional of SMP is a semimartingale, a local martingale, or a special semimartingale respectively. By the way, the Ito type formula is given. Finally, we study the expected cumulative discounted value of the semi-additive functional of an SMP. We prove that it solves a measure-valued Poisson equation and give the uniqueness conditions.

In the present manuscript, we define a non-negative parametric variant of Baskakov–Durrmeyer operators to study the convergence of Lebesgue measurable functions and introduce these as α-Baskakov–Durrmeyer operators. We study the uniform convergence of these operators in weighted spaces.

We consider a Schrodinger operator L = − d 2 / d x 2 + V ( x ) on R , where V is a real-valued measurable function, and give an explicit and simple characterization of intrinsic ultracontractivity (IU) of the Schrodinger semigroup generated by L for a wide class of potentials. By making use of it, we also give new examples of potentials for which the semigroups satisfy (IU) or non-(IU).

Differential equations and control systems with impulse action and random parameters are considered. These objects are characterized by stochastic behavior: the lengths θk of the intervals between the times of the impulses τk, k = 0,1,…, are random variables and the magnitudes of the impulses also depend on random actions. The basic object of research is the control system $$\dot x = f(t,x,u),\;\;\;t\neq\tau_k,\\\triangle{x}\mid_{t=\tau_k}=g(x,w_k,v_k),$$ which depends on random parameters θk = τk+1 — τk and vk, k = 0,1,…. A probability measure μ is defined on the set Σ of all possible sequences ((θ0, v0),…, (θk, vk),… ). Admissible controls u = u(t) are bounded measurable functions with values in a compact set U ⊂ Rm, and the vector wk is also a control affecting the behavior of the system at the times τk. We consider the set $$\mathfrak{M}=\{(t,x):t\in[0,+\infty),x\in{M}(t)\}$$ defined by the function t ↦ M(t), which is continuous in the Hausdorff metric. The main result of the paper is sufficient conditions for the Lyapunov stability and asymptotic stability of the set M with probability 1. It is shown that the stability analysis of a set by means of the method of Lyapunov functions can be reduced to studying the stability of the zero solution of the corresponding differential equation. We also study the asymptotic behavior of solutions of differential equations with impulse action and random parameters. Conditions are obtained under which the solutions possess the Lyapunov stability and asymptotic stability for all values of the random parameter and with probability 1. The results are illustrated by a probability model of a population subject to harvesting and by a model of two-species competition with impulse action.

Let $\varphi:\mathbb{R}^{n}\times[0,\infty)\to[0,\infty)$ satisfy that $\varphi(x,\cdot)$ , for any given $x\in\mathbb{R}^{n}$ , is an Orlicz function and that $\varphi(\cdot ,t)$ is a Muckenhoupt $A_{\infty}$ weight uniformly in $t\in(0,\infty)$ . The weak Musielak–Orlicz Hardy space $\mathit{WH}^{\varphi}(\mathbb{R}^{n})$ is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak–Orlicz space $\mathit{WL}^{\varphi}(\mathbb{R}^{n})$ . For parameter $\rho\in(0,\infty)$ and measurable function $f$ on $\mathbb{R}^{n}$ , the parametric Marcinkiewicz integral $\mu _{\Omega}^{\rho}$ related to the Littlewood–Paley $g$ -function is defined by setting, for all $x\in\mathbb{R}^{n}$ , ¶ \[\mu^{\rho}_{\Omega}(f)(x):=(\int_{0}^{\infty}\vert\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}f(y){dy}\vert^{2}\frac{dt}{t^{2\rho+1}})^{1/2},\] where $\Omega$ is homogeneous of degree zero satisfying the cancellation condition. ¶ In this article, we discuss the boundedness of the parametric Marcinkiewicz integral $\mu_{\Omega}^{\rho}$ with rough kernel from weak Musielak–Orlicz Hardy space $\mathit{WH}^{\varphi}(\mathbb{R}^{n})$ to weak Musielak–Orlicz space $\mathit{WL}^{\varphi}(\mathbb{R}^{n})$ . These results are new even for the classical weighted weak Hardy space of Quek and Yang, and probably new for the classical weak Hardy space of Fefferman and Soria.

Given some regular weight $$\omega $$ on the unit disk $$\mathbb {D}$$, let $$L^p_\omega $$ be the space of all Lebesgue measurable functions on $$\mathbb {D}$$ for which $$\begin{aligned} \Vert f\Vert _{L^p_\omega }=\left( \int _{\mathbb {D}} |f(z)|^p \omega (z)\mathrm{d}A(z)\right) ^{\frac{1}{p}}

Atar, Chowdhary, and Dupuis have recently exhibited a variational formula for exponential integrals of bounded measurable functions in terms of Renyi divergences. We show that a variational characterization of the Renyi divergences between two probability distributions on a measurable space in terms of relative entropies, when combined with the elementary variational formula for exponential integrals of bounded measurable functions in terms of relative entropy, yields the variational formula of Atar, Chowdhary, and Dupuis as a corollary. We then develop an analogous variational characterization of the Renyi divergence rates between two stationary finite state Markov chains in terms of relative entropy rates. When combined with Varadhan’s variational characterization of the spectral radius of square matrices with nonnegative entries in terms of relative entropy, this yields an analog of the variational formula of Atar, Chowdary, and Dupuis in the framework of stationary finite state Markov chains.

In this paper we study the Dirichlet problem for two related equations involving the 1-Laplacian and a total variation term as reaction, namely: with homogeneous Dirichlet boundary conditions on $$\partial \varOmega $$, where $$\varOmega $$ is a regular, bounded domain in $$\mathbb {R}^N$$. Here f is a measurable function belonging to some suitable Lebesgue space, while g(u) is a continuous function having the same sign as u and such that $$g(\pm \infty ) = \pm \infty $$. As far as Eq. (1) is concerned, we show that a bounded solution exists if the datum f belongs to $$L^N(\varOmega )$$. When the absorption term g(u) is missing, i.e. in the case of Eq. (2), we show that if $$f\in L^N(\varOmega )$$, and its norm is small, then the only solution of (2) is $$u\equiv 0$$. In the case where the norm of f is not small, several cases may happen. Depending on $$\varOmega $$ and f, we show examples where no solution of (2) exists, other examples where $$u\equiv 0$$ is still a solution, and finally examples with nontrivial solutions. Some of these results can be viewed as a translation to the 1-Laplacian operator of known results by Ferone and Murat.

In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle $\mathbb T,$ there is a function f holomorphic in the unit disc, having $\psi $ as radial limit a.e. on $\mathbb T.$ We consider an analogous problem for solutions f of homogenous elliptic equations $Pf=0$ and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.

In this paper, we introduce the concepts of Lebesgue–Stieltjes $$\Diamond _\alpha $$ -measure, $$\Diamond _\alpha $$ -measurable function and $$\Diamond _\alpha $$ -integral. Based on the theory of combined measurability on time scales, some basic properties including the relationships, the extensions and the composition theorems are established. Particularly, through the switch coefficient of combined theory on time scales, one can obtain the Lebesgue–Stieltjes $$\varDelta $$ -measure and Lebesgue–Stieltjes $$\nabla $$ -measure if by taking $$\alpha =1$$ and $$\alpha =0$$ , respectively. In addition, several examples are provided to demonstrate the effectiveness of the obtained results in each section.

In this article we have studied bicomplex valued measurable functions on an arbitrary measurable space. We have established the bicomplex version of Lebesgue's dominated convergence theorem and some other results related to this theorem. Also we have proved the bicomplex version of Lebesgue-Radon-Nikodym theorem. Finally we have introduced the idea of hyperbolic version of invariant measure.AMS Subject Classification (2010) : 28E99, 30G35.

A function $$F:\mathbb {R}^2\rightarrow \mathbb {R}$$ is sup-measurable if, for each (Lebesgue) measurable function $$f:\mathbb {R}\rightarrow \mathbb {R}$$ , the Caratheodory superposition $$F_f:\mathbb {R}\rightarrow \mathbb {R}$$ given by $$F_f: x\mapsto F(x,f(x))$$ is measurable. The existence of non-measurable sup-measurable functions is independent of ZFC. We prove, assuming CH, that the family of all non-measurable sup-measurable functions $$F:\mathbb {R}^2\rightarrow \mathbb {R}$$ (plus the zero function) contains a linear vector space of dimension $$2^\mathfrak {c}$$ . A function $$F:\mathbb {R}^2\rightarrow \mathbb {R}$$ is separately measurable if all its vertical and horizontal sections are measurable. In the second part of this note we show that the family of non-measurable separately measurable functions is $$2^\mathfrak {c}$$ -lineable.

In this paper we introduce the concepts of strongly deferred Cesaro summable and $$\mu $$ -deferred statistical convergence of real-valued functions $$x=x(t)$$ which are measurable (in the Lebesgue sense) in the interval $$[1,\infty )$$ . In addition, the relations between the set of strong deferred Cesaro summable and $$\mu $$ -deferred statistical convergent of functions have been examined under some restrictions.

We improve the discretization technique for weighted Lorentz norms by eliminating all "non-degeneracy" restrictions on the involved weights. We use the new method to provide equivalent estimates on the optimal constant $C$ such that the inequality $$\left( \int_0^L (f^*(t))^{p_2} w(t)\,\mathrm{d}t \right)^\frac 1{p_2} \le C \left( \int_0^L \left( \int_0^t u(s)\,\mathrm{d}s \right)^{-\frac {p_1}\alpha} \left( \int_0^t (f^*(s))^\alpha u(s) \,\mathrm{d}s \right)^\frac {p_1}\alpha v(t) \,\mathrm{d}t \right)^\frac 1{p_1}$$ holds for all relevant measurable functions, where $L\in(0,\infty]$, $\alpha, p_1, p_2 \in (0,\infty)$ and $u$, $v$, $w$ are locally integrable weights, $u$ being strictly positive. It the case of weights that would be otherwise excluded by the restrictions, it is shown that additional limit terms naturally appear in the characterizations of the optimal $C$. A weak analogue for $p_1=\infty$ is also presented., Comment: 23 pages

Let ( X , M , &mu, ) be a &sigma, finite measure space and denote by P ( X ) the &mu, measurable functions f : X &rarr, [ 0 , &infin, ] , f &lt, &infin, &mu, ae. Suppose K : X &times, X &rarr, ) is &mu, &times, measurable and define the mutually transposed operators T and T &prime, on P ( X ) by ( T f ) ( x ) = &int, X K ( x , y ) f ( y ) d &mu, ( y ) and ( T &prime, g ) ( y ) = &int, X K ( x , y ) g ( x ) d &mu, ( x ) , f , g &isin, P ( X ) , x , y &isin, X . Our interest is in inequalities involving a fixed (weight) function w &isin, P ( X ) and an index p &isin, ( 1 , &infin, ) such that: (*): &int, X [ w ( x ) ( T f ) ( x ) ] p d &mu, ( x ) ≲ C &int, X [ w ( y ) f ( y ) ] p d &mu, ( y ) . The constant C &gt, 1 is to be independent of f &isin, P ( X ) . We wish to construct all w for which (*) holds. Considerations concerning Schur&rsquo, s Lemma ensure that every such w is within constant multiples of expressions of the form ϕ 1 1 / p &minus, 1 ϕ 2 1 / p , where ϕ 1 , ϕ 2 &isin, P ( X ) satisfy T ϕ 1 &le, C 1 ϕ 1 and T &prime, ϕ 2 &le, C 2 ϕ 2 . Our fundamental result shows that the ϕ 1 and ϕ 2 above are within constant multiples of (**): &psi, 1 + &sum, j = 1 &infin, E &minus, j T ( j ) &psi, 1 and &psi, 2 + &sum, j T &prime, ( j ) &psi, 2 respectively, here &psi, 1 , &psi, 2 &isin, P ( X ) , E &gt, 1 and T ( j ) , T &prime, ( j ) are the jth iterates of T and T &prime, This result is explored in the context of Poisson, Bessel and Gauss&ndash, Weierstrass means and of Hardy averaging operators. All but the Hardy averaging operators are defined through symmetric kernels K ( x , y ) = K ( y , x ) , so that T &prime, = T . This means that only the first series in (**) needs to be studied.

The statements on the BIBO stability of continuous-time convolution systems found in engineering textbooks are often either too vague (because of lack of hypotheses) or mathematically incorrect. What is more troubling is that they usually exclude the identity operator. The purpose of this note is to clarify the issue while presenting some fixes. In particular, we show that a linear shift-invariant system is BIBO-stable in the $L_\infty$ -sense if and only if its impulse response is included in the space of bounded Radon measures, which is a superset of $L_1(\mathbb {R})$ (Lebesgue's space of absolutely integrable functions). As we restrict the scope of this characterization to the convolution operators whose impulse response is a measurable function, we recover the classical statement.

In this chapter, we consider the asymptotic behavior, as T → +∞, of some functionals of the form \(I_T(t)=F_T(\xi _T(t))+\int \limits _{0}^{t} g_T(\xi _T(s))\,dW_T(s)\), t ≥ 0. Here ξT(t) is the solution to the time-inhomogeneous Ito stochastic differential equation $$\displaystyle d\xi _T (t)=a_T(t,\,\xi _T(t))\,dt+dW_T(t),\;\; t\ge 0,\;\; \xi _T (0)=x_0, $$ where T > 0 is a parameter, \(a_T (t,x), x\in \mathbb R\) are measurable functions, \(\left |a_T(t,x)\right |\leq L_T\) for all \(x\in \mathbb R\) and t ≥ 0, WT are standard Wiener processes, \(F_T(x),\, x\in \mathbb R \) are continuous functions, and \(g_T(x),\, x\in \mathbb R \) are measurable locally bounded functions. Section 6.1 contains some preliminary remarks, notations, and basic definitions. The asymptotic behavior of the integral functionals of the Lebesgue integral type is investigated in Sect. 6.3. Section 6.4 contains some results about the weak convergence of the martingale type functionals and the mixed functionals. Section 6.5 includes several examples. Auxiliary results are collected in Sect. 6.6.

In their 1976 paper, Nagel and Rudin characterize the closed unitarily and M\"obius invariant spaces of continuous and $L^p$-functions on a sphere, for $1\leq p, Comment: 16 pages, submitted for publication

Let $\MM$ be the space of finite measures on a Locally compact Polish space, and let $\BG$ be the Gamma distribution on $\MM$ with intensity measure $\nu\in \MM$. Let $\nn^{ext}$ be the extrinsic derivative with tangent bundle $T\MM= \cup_{\eta\in\MM} L^2(\eta)$, and let $\GA: T\MM\to T\MM$ be measurable such that $\GA_\eta$ is a positive definite linear operator on $L^2(\eta)$ for every $\eta\in \MM$. Moreover, for a measurable function $V$ on $\MM$, let $\d\BG^V= \e^V\d\BG$. We investigate the Poincar\'e, weak Poincar\'e and super Poincar\'e inequalities for the Dirichlet form $$\EE_{\GA,V}(F,G):= \int_\MM \_{L^2(\eta)}\, \d\BG^V(\eta),$$ which characterize various properties of the associated Markov semigroup. The main results are extended to the space of finite signed measures., Comment: 34 pages

In this chapter, we consider homogeneous one-dimensional stochastic differential equations with non-regular dependence on a parameter. The asymptotic behavior of the mixed functionals of the form \(I_T(t)=F_T(\xi _T(t))+\int \limits _{0}^{t} g_T(\xi _T(s))\,d\xi _T(s)\), t ≥ 0 is studied as T → +∞. Here ξT is a strong solution to the stochastic differential equation dξT(t) = aT(ξT(t)) dt + dWT(t), T > 0 is a parameter, aT = aT(x) is a set of measurable functions, \(\left |a_T(x)\right |\leq L_T\) for all \(x\in \mathbb R\), WT = WT(t) are standard Wiener processes, \( F_T= F_T(x), x\in \mathbb R\) are continuous functions, and \( g_T= g_T(x), x\in {\mathbb R} \) are locally bounded real-valued functions. Section 5.1 contains some preliminary remarks. We prove a theorem about the weak compactness of the family of some processes in Sect. 5.2. Section 5.3 includes a theorem concerning the weak convergence of some stochastic processes to the solutions of Ito SDEs. In Sect. 5.4 we consider the asymptotic behavior of integral functionals of Lebesgue integral type. Section 5.5 is devoted to asymptotic behavior of the integral functionals of martingale type. The explicit form of the limiting processes for IT(t) is established in Sect. 5.6 under very non-regular dependence of gT and aT on the parameter T. This section summarizes the main results and their proofs. Section 5.7 contains several examples. Auxiliary results are collected in Sect. 5.8.

To determine the area of the region below the graph of a map, what one usually does is divide the base, i.e. the interval of integration, in a suitable number of subintervals. Then for each interval we choose one of the infinitely many values that f(x) attains on it. The roundoff errors made (both up and down) in doing so tend to become smaller, until they vanish, as the subintervals shrink to 0. This is the typical logic of a definition of integral such as the one adopted by Cauchy, which was justified under the assumption that the map f to be integrated is continuous. Riemann wanted to set up a general theory to include totally discontinuous maps as well and, although his aim was different from Cauchy’s, he adopted a similar construction. In this way he managed to capture the integrability of only certain discontinuous maps, certainly not the majority. Indeed Henri Lebesgue, in the 1926 paper “Sur le developpment de la notion d’integrale”, wrote that when Riemann’s method would work for some discontinuous functions it was by sheer accident. Lebesgue therefore scrapped Riemann’s method altogether, considering it inadequate in view of generalisations, and adopted instead Cauchy’s guideline that close values of f(x) should be gathered and multiplied by the measure of the corresponding set along the x-axis. For this purpose Lebesgue, in order to group the values f(x) by proximity, divided into a finite number of subintervals not the interval of integration, but the interval of range values between the infimum and supremum of f. Thus over each subinterval the values f(x) are certainly near one another, and we can make them as near as we want by refining the partition, i.e. by increasing the number of subintervals. Given the discontinuities of f, the pre-image of each subinterval in the codomain may not be one interval. On the contrary it might be scattered over the entire domain of integration, to form a complicated linear set of points. In this way the subintervals \(I_i\) appearing in Riemann’s definition, whose measure is elementary, are replaced by sets \(A_i\), which may be rather complicated. Roughly, we may say that a map f is Lebesgue integrable if every set that f projects on the horizontal axis is measurable. Within this framework the study we have made of so-called measurable functions is fully justified.

Del Junco--Lema\'nczyk showed that a generic measure preserving transformation satisfies a certain orthogonality conditions. More precisely, there is a dense $G_\delta$ subset of measure preserving transformations such that for every $T\in G$ and $k(1), k(2), \dots, k(l)\in \mathbb{Z}^+$, $k'(1), k'(2), \dots, k'(l')\in \mathbb{Z}^+$, the convolutions \[ \sigma_{T^{k(1)}} \ast\cdots\ast \sigma_{T^{k(l)}} \ \text{and} \ \sigma_{T^{k'(1)}} \ast\cdots \ast\sigma_{T^{k'(l')}} \] are mutually singular, provided that $(k(1), k(2), \dots, k(l))$ is not a rearrangement of $(k'(1), k'(2), \dots, k'(l'))$. We will introduce an analogous orthogonality conditions for continuous unitary representations of $L^0(\mu,\mathbb{T})$ which we denote by DL--condition. We connect the DL--condition with a result of Solecki which states that every continuous unitary representations of $L^0(\mu,\mathbb{T})$ is a direct sum of action by pointwise multiplication on measure spaces $(X^{|\kappa|},\lambda_\kappa)$ where $\kappa$ is an increasing finite sequence of non-zero integers. In particular, we show that the "probabilistic" DL-condition translates to "deterministic" orthogonality conditions on the measures $\lambda_\kappa$., Comment: 19 pages

We provide extension procedures for nonlinear expectations to the space of all bounded measurable functions. We first discuss a maximal extension for convex expectations which have a representation in terms of finitely additive measures. One of the main results of this article is an extension procedure for convex expectations which are continuous from above and therefore admit a representation in terms of countably additive measures. This can be seen as a nonlinear version of the Daniell–Stone theorem. From this, we deduce a robust Kolmogorov extension theorem which is then used to extend nonlinear kernels to an infinite-dimensional path space. We then apply this theorem to construct nonlinear Markov processes with a given family of nonlinear transition kernels.

In this paper we introduce a new class of operators on vector lattices. We say that a linear or nonlinear operator $T$ from a vector lattice $E$ to a vector lattice $F$ is atomic if there exists a Boolean homomorphism $\Phi$ from the Boolean algebra $\mathfrak{B}(E)$ of all order projections on $E$ to $\mathfrak{B}(F)$ such that $T\pi=\Phi(\pi)T$ for every order projection $\pi\in\mathfrak{B}(E)$. We show that the set of all atomic operators defined on a vector lattice $E$ with the principal projection property and taking values in a Dedekind complete vector lattice $F$, is a band in the vector lattice of all regular orthogonally additive operators from $E$ to $F$. We give the formula for the order projection onto this band, and we obtain an analytic representation for atomic operators between spaces of measurable functions. Finally, we consider the procedure of the extension of an atomic map from a lateral ideal to the whole space., Comment: 22 pages

Linear delayed differential systems $$\begin{aligned} \dot{x}_i(t)=\sum _{j=1}^m \sum _{k=1}^{r_{ij}}a_{ij}^{k}(t)x_j\left( h_{ij}^{k}(t)\right) ,\quad i=1,\dots ,m \end{aligned}$$are considered on a half-infinity interval $$t\ge 0$$. It is assumed that m and $$r_{ij}$$, $$i,j=1,\dots ,m$$ are natural numbers and the coefficients $$a_{ij}^{k}:[0,\infty )\rightarrow \mathbb {R}$$ and delays $$h_{ij}^{k}:[0,\infty )\rightarrow {\mathbb {R}}$$ are Lebesgue measurable functions. New explicit results on uniform exponential stability, depending on all delays, are derived. The conditions obtained do not require the dominance of diagonal terms over the off-diagonal terms as most of the existing stability tests for non-autonomous delay differential systems do.

Let $$p(\cdot )$$ be a measurable function on $${\mathbb R}^n$$ with $$0

The structuralism is a powerful toll for ordering and classifying knowledge of fundamental mathematical objects. Such an important structure is the Lebesgue measurable sets or Lebesgue non-measurable sets (such a set exists, according to Vitali construction), as well as Lebesgue measurable functions or Lebesgue non-measurable functions. In the following we show that f : (ℝ, £,) → ℝ Lebesgue measurable ⇎ {x ∈ X | f(x) = c} ∈ £, for any c ∈ ℝ, and we will give some important examples of Lebegue non-measurable functions.

In this paper, we study the speed of mixing of absorbing Markov processes with state space E , that is, the rate of the convergence in the following limits: lim t → ∞ E x [ f ( X p t ) g ( X t ) | T > t ] = ∫ E f ( y ) β ( d y ) ∫ E g ( y ) α ( d y ) , lim t → ∞ E x [ f ( X p t ) g ( X q t ) | T > t ] = ∫ E f ( y ) β ( d y ) ∫ E g ( y ) β ( d y ) , where f , g are bounded and measurable functions defined on E , x ∈ E , 0 p q 1 , T is the absorption time of the process, α is a quasi-stationary distribution and β is a quasi-ergodic distribution. Under general conditions, we show that the convergence rates of the limits are exponential. As an application, we apply our result to the logistic Feller diffusion process and the birth–death process with catastrophes.

In this paper, we consider the two-dimensional Hausdorff operators on the power weighted Hardy space $$H_{{{\left| X \right|}^\alpha }}^1({R^2})( - 1 \leqslant \alpha \leqslant 0)$$ , defined by $${H_{\Phi ,A}}f(x) = \int {_{{R^2}}} \Phi (u)f(A(u)x)du,$$ , where Φ ∈ L loc 1(R 2), A(u) = (a ij (u)) i,j=1 2 is a 2 × 2 matrix, and each a i,j is a measurable function. We obtain that H Φ,A is bounded from $$H_{{{\left| X \right|}^\alpha }}^1({R^2})( - 1 \leqslant \alpha \leqslant 0)$$ to itself, if $$\int {_{{R^2}}} \left| {\Phi (u)} \right|\left| {\det \;{A^{ - 1}}(u)} \right|{\left\| {A(u)} \right\|^{ - \alpha }}\;\ln \;(1 + \frac{{{{\left\| {{A^{ - 1}}(u)} \right\|}^2}}}{{\left| {\det \;{A^{ - 1}}(u)} \right|}})du < \infty .$$ . This result improves some known theorems, and in some sense it is sharp.

In this paper we prove the Poincare-type weighted inequality $$\begin{aligned} \Vert v^{1/q} f \Vert _{L^q(\Omega )} \le C \Vert \omega ^{1/p} \nabla f \Vert _{L^p(\Omega )}, \quad q\ge p>1, \end{aligned}$$ for a locally Lipschitz function f with a weighted mean equal to zero over a convex bounded domain $$\Omega $$ ; here the weights v, $$\omega $$ are positive measurable functions which satisfy a certain compatibility condition. This result is a generalization of the well-known weighted Poincare inequality to the case of more general weights in the sense that we do not use the traditional conditions of high summability $$v,\, \omega ^{-\frac{1}{p-1}}\in L^{r,loc}$$ with $$r>1$$ for $$q=p$$ or the reverse doubling condition on the function v for $$q>p$$ . In other words, a Sawyer type sufficient condition on weight functions is established.

Let Ω be a Lebesgue-measurable set in ℝ n of finite positive Lebesgue measure. In this note we calculate the lack of equi-measurability of the set Kc (Ω), c > 0, of all Lebesgue-measurable functions f : Ω → ℝ such that 0 ≤ f ≤ c, a.e. on Ω. From our result we repair a gap in the Example 2.3 of the paper [APPELL, J.—DE PASCALE, E.: Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili, Boll. Un. Mat. Ital. B (6) 3 (1984), 497–515].

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 Polya–Szegő inequality. Contrary to the Schwarz rearrangement, the tamping also preserves the homogeneous Dirichlet boundary condition of a function.