Integrating with respect to functions which are constant on intervals whose bounds are discontinuity points (of those functions) is frequent in many branches of Mathematics, specially in stochastic processes. For such functions and alike extension, a comparison between Riemann-Stieltjes and Lebesgue-Stieltjes integration and the integrals formulas leads to interesting facts for students (as complements of Measure Theory and Integrations) and for practitioners and and researchers. We undergone conditions of existence the Riemann-Stieltjes integrals related to that type of function and compare the results with what should be expected for Lebesgue-Stieltjes theory.
We show how two change-of-variables formulae for Lebesgue-Stieltjes integrals generalize when all continuity hypotheses on the integrators are dropped. We find that a sort of "mass splitting phenomenon" arises.
For $μ>β>0$, the generalized Stieltjes operators $$ \mathcal{S}_{β,μ} f(t):={t^{μ-β}}\int_0^\infty {s^{β-1}\over (s+t)^μ}f(s)ds, \qquad t>0, $$ defined on Sobolev spaces $\mathcal{T}_p^{(α)}(t^α)$ (where $α\ge 0$ is the fractional order of derivation and these spaces are embedded in $L^p(\RR^+)$ for $p\ge 1$) are studied in detail. If $0 < β- \pp < μ$, then operators $\mathcal{S}_{β,μ}$ are bounded (and we compute their operator norms which depend on $p$); commute and factorize with generalized Cesáro operator on $\mathcal{T}_p^{(α)}(t^α)$ . We calculate and represent explicitly their spectrum…
We prove an inequality of the Loéve-Young type for the Riemann-Stieltjes integrals driven by irregular signals attaining their values in Banach spaces and, as a result, we derive a new theorem on the existence of the Riemann-Stieltjes integrals driven by such signals. Also, for any $p\ge1$ we introduce the space of regulated signals $f:[a,b] \rightarrow W$ ($a<b$ are real numbers and $W$ is a Banach space), which may be uniformly approximated with accuracy $δ>0$ by signals whose total variation is of order $δ^{1-p}$ as $δ\rightarrow 0+$ and prove that they satisfy the assumptions of the theore…
H. Cartan in his book on differential calculus proved a theorem generalizing a Cauchy's mean-value theorem to the case of functions taking values in a Banach space. Cartan used this theorem in a masterful way to develop the entire theory of differential calculus and theory of differential equations in finite and infinite dimensional Banach spaces. The author proves a generalization of this theorem to the case when the inequality involving the derivatives holds everywhere with exception of a set of Lebesgue measure zero, and the derivatives are replaced by weaker derivatives. Namely the right-s…
This note completely describes the bounded or compact Riemann-Stieltjes integral operators $T_g$ acting between the weighted Bergman space pairs $(A^p_α,A^q_β)$ in terms of particular regularities of the holomorphic symbols $g$ on the open unit ball of $\Bbb C^n$.
In this paper, we define two types of partitions of an hyperbolic interval: weak and strong. Strong partitions enables us to define, in a natural way, a notion of hyperbolic valued functions of bounded variation and hyperbolic analogue of Riemann-Stieltjes integral. We prove a deep relation between both concepts like it occurs in the context of real analysis.
This paper is devoted to characterizing the Riemann-Stieltjes operators and pointwise multipliers acting on M${\rm \ddot{o}}$bius invariant spaces $Q_p$, which unify BMOA and Bloch space in the scale of $p$. The boundedness and compactness of these operators on $Q_p$ spaces are determined by means of an embedding theorem, i.e. $Q_p$ spaces boundedly embedded in the non-isotropic tent type spaces $T_q^\infty$.
Let $μ$ be a nonnegative Borel measure on the unit disk of the complex plane. We characterize those measures $μ$ such that the general family of spaces of analytic functions, $F(p,q,s)$, which contain many classical function spaces, including the Bloch space, $BMOA$ and the $Q_s$ spaces, are embedded boundedly or compactly into the tent-type spaces $T^{\infty}_{p,s}(μ)$. The results are applied to characterize boundedness and compactness of Riemann-Stieltjes operators and multiplication operators on $F(p,q,s)$.
We study Riemann-Lebesgue integrability of a vector function relative to an arbitrary non-negative set function. We obtain some classical integral properties. Results regarding the continuity properties of the integral and relationships among Riemann-Lebesgue, Birkhoff simple and Gould integrabilities are also established.
We derive a sharp Grand Lebesgue Space norm estimations for normalized eigen functions for the Laplace-Beltrami operator defined on the compact smooth Riemann manifold. These estimates allow us to deduce in particular the exponential decreasing tail of distribution for these eigen functions.
This paper contains a development of the Theory of Lebesgue and Bochner spaces of summable functions. It represents a synthesis of the results due to H. Lebesgue, S. Banach, S. Bochner, G. Fubini, S. Saks, F. Riesz, N. Dunford, P. Halmos, and other contributors to this theory. The construction of the theory is based on the notion of a measure on a prering of sets in any abstract space X. No topological structure of the space X is required for the development of the theory. Measures on prerings generalize the notion of abstract Lebesgue measures. These measures are readily available and it is n…
In this work we study the Riemann-Liouville fractional integral of order $α\in(0,1/p)$ as an operator from $L^p(I;X)$ into $L^{q}(I;X)$, with $1\leq q\leq p/(1-pα)$, whether $I=[t_0,t_1]$ or $I=[t_0,\infty)$ and $X$ is a Banach space. Our main result give necessary and sufficient conditions to ensure the compactness of the Riemann-Liouville fractional integral from $L^p(t_0,t_1;X)$ into $L^{q}(t_0,t_1;X)$, when $1\leq q< p/(1-pα)$.
In this paper we study the Riemann-Liouville fractional integral of order $α>0$ as a linear operator from $L^p(I,X)$ into itself, when $1\leq p\leq \infty$, $I=[t_0,t_1]$ (or $I=[t_0,\infty)$) and $X$ is a Banach space. In particular, when $I=[t_0,t_1]$, we obtain necessary and sufficient conditions to ensure its compactness. We also prove that Riemann-Liouville fractional integral defines a $C_0-$semigroup but does not defines a uniformly continuous semigroup. We close this study by presenting lower and higher bounds to the norm of this operator.
In classical analysis, Lebesgue first proved that $\mathbb{R}$ has the property that each Riemann integrable function from $[a,b]$ into $\mathbb{R}$ is continuous almost everywhere. This property is named as the Lebesgue property. Though the Lebesgue property may be breakdown in many infinite dimensional spaces including Banach or quasi Banach spaces, to determine spaces having this property is still an interesting problem. In this paper, we study Riemann integration for vector-value functions in metrizable vector spaces and prove the fundamental theorems of calculus and primitives for continu…
Using truncated variation techniques we obtain an improved version of the Loeve-Young inequality for the Riemann-Stieltjes integrals driven by rough paths. This allowed us to strenghten some result on the existence of solutions of integral equations driven by moderately irregular signals. We introduce also a new Banach space, containing as a proper subspace the paths with finite $p$-variation, and develop, in a systematic way, several parallel results for the paths from this space, obtained so far for the paths with finite $p$-variation. We start the paper with a general theorem on the existen…
This paper presents Stieltjes-type integration for operator-valued functions with respect to spectral families. The relation between Riemann-Stieltjes integrals associated with some classes of spectral families including, in particular, those that arise in the context of the Stone representation theorem for Banach spaces is established. The developed approach is applied to the structural analysis of the operators commuting with a one-parameter representation satisfying the Generalized Stone theorem.
In this paper we focus on the relation between Riemann integrability and weak continuity. A Banach space $X$ is said to have the weak Lebesgue property if every Riemann integrable function from $[0,1]$ into $X$ is weakly continuous almost everywhere. We prove that the weak Lebesgue property is stable under $\ell_1$-sums and obtain new examples of Banach spaces with and without this property. Furthermore, we characterize Dunford-Pettis operators in terms of Riemann integrability and provide a quantitative result about the size of the set of $τ$-continuous non Riemann integrable functions, with …
The process of integration was a subject of significant development during the last century. Despite that the Lebesgue integral is complete and has many good properties, its inability to integrate all derivatives prompted the introduction of new approaches - Denjoy, Perron and others introduced new ways of integration aimed at preserving the good properties of the Lebesgue integral but extending the set of functions to which it could be applied. The goal was achieved but neither of the new approaches was elegant or simple or transparent. In the 50s a new integral was introduced, independently …
In this paper one-weight inequalities with general weights for Riemann-Liouville transform and $ n-$ dimensional fractional integral operator in variable exponent Lebesgue spaces defined on $\mathbb{R}^{n}$ are investigated. In particular, we derive necessary and sufficient conditions governing one-weight inequalities for these operators on the cone of non-negative decreasing functions in $L^{p(x)}$ spaces.
It is shown that if $$ \sum_{n=1}^{N}n\left|c_{n}\right|=O(N), $$ then Lebesgue summability, $(\mathrm{C},β)$ summability ($β>0$), Abel summability, Riemann summability, and $(γ,κ)$ summability ($κ\geq 1$) of the series $\sum_{n=0}^{\infty}c_{n}$ are all equivalent to one another.
The aim of this paper is to provide characterizations of the Lebesgue-almost everywhere continuity of a function f : [a, b] $\rightarrow$ R. These characterizations permit to obtain necessary and sufficient conditions for the Riemann integrability of f .
We show that the set of Lebesgue integrable functions in $[0,1]$ which are nowhere essentially bounded is spaceable, improving a result from [F. J. García-Pacheco, M. Martín, and J. B. Seoane-Sepúlveda. \textit{Lineability, spaceability, and algebrability of certain subsets of function spaces,} Taiwanese J. Math., \textbf{13} (2009), no. 4, 1257--1269], and that it is strongly $\mathfrak{c}$-algebrable. We prove strong $\mathfrak{c}$-algebrability and non-separable spaceability of the set of functions of bounded variation which have a dense set of jump discontinuities. Applications to sets of …
In the standard theory of delay equations, the fundamental solution does not 'live' in the state space. To eliminate this age-old anomaly, we enlarge the state space. As a consequence, we lose the strong continuity of the solution operators and this, in turn, has as a consequence that the Riemann integral no longer suffices for giving meaning to the variation-of-constants formula. To compensate, we develop the Stieltjes-Pettis integral in the setting of a norming dual pair of spaces. Part I provides general theory, Part II deals with "retarded" equations, and in Part III we show how the Stielt…
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 fi…
