Riemann-Type Definition of the Improper Integrals

Riemann-type definitions of the Riemann improper integral and of the Lebesgue improper integral are obtained from McShane's definition of the Lebesgue integral by imposing a Kurzweil-Henstock's condition on McShane's partitions.

Classes of regular Sobolev mappings

We prove that a slight modification of the notion of α-absolute continuity introduced in [D. Bongiorno, Absolutely continuous functions in Rn, J. Math. Anal. Appl. 303 (2005) 119–134] is equivalent to the notion of n, λ-absolute continuity given by S. Hencl in [S. Hencl, On the notions of absolute continuity for functions of several variables, Fund. Math. 173 (2002) 175–189].

Derivatives not first return integrable on a fractal set

We extend to s-dimensional fractal sets the notion of first return integral (Definition 5) and we prove that there are s-derivatives not s-first return integrable.

Regular subclasses in the Sobolev space

Abstract We study some slight modifications of the class α - A C n ( Ω , R m ) introduced in [D. Bongiorno, Absolutely continuous functions in R n , J. Math. Anal. and Appl. 303 (2005) 119–134]. In particular we prove that the classes α - A C λ n ( Ω , R m ) , 0 λ 1 , introduced in [C. Di Bari, C. Vetro, A remark on absolutely continuous functions in R n , Rend. Circ. Matem. Palermo 55 (2006) 296–304] are independent by λ and contain properly the class α - A C n ( Ω , R m ) . Moreover we prove that α - A C n ( Ω , R m ) = ( α - A C λ n ( Ω , R m ) ) ∩ ( α - A C n , λ ( Ω , R m ) ) , where α - A C n , λ ( Ω , R m ) is the symmetric class of α - A C λ n ( Ω , R m ) , 0 λ 1 .

RADEMACHER'S THEOREM IN BANACH SPACES WITHOUT RNP

Abstract We improve a Duda’s theorem concerning metric and w *-Gâteaux differentiability of Lipschitz mappings, by replacing the σ-ideal 𝓐 of Aronszajn null sets [ARONSZAJN, N.: Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), 147–190], with the smaller σ-ideal 𝓐 of Preiss-Zajíček null sets [PREISS, D.—ZAJÍČEK, L.: Directional derivatives of Lipschitz functions, Israel J. Math. 125 (2001), 1–27]. We also prove the inclusion C̃ o ⊂ 𝓐, where C̃ o is the σ-ideal of Preiss null sets [PREISS, D.: Gâteaux differentiability of cone-monotone and pointwise Lipschitz functions, Israel J. Math. 203 (2014), 501–534].

Regular subclasses in the Sobolev space W_{loc}^{1,n}

Westudy some slight modifications of the class AC^n introduced in [D. Bongiorno, Absolutely continuous functions in Rn, J. Math. Anal. and Appl. 303 (2005) 119 134]. In particular we prove that the classes introduced in [C. Di Bari, C. Vetro, A remark on absolutely continuous functions in R^n, Rend. Circ. Matem. Palermo 55 (2006) 296 304] are independent by and contain properly the class AC^n.

On the problem of nearly derivatives

Regularity in the Sobolev space $W_{loc}^{1,n}(R^n,R^m)$ and $alpha$-absolute continuity

A first return examination of vector valued integrals.

We prove that for each Bochner integrable function f there exists a trajectory yielding the Bochner integral of f, and that on infinite dimensional Banach spaces there exist Pettis integrable functions f such that no trajectory yields the Pettis integral of f.

An Integral on a Complete Metric Measure Space

We study a Henstock-Kurzweil type integral defined on a complete metric measure space \(X\) endowed with a Radon measure \(\mu\) and with a family of “cells” \(\mathcal{F}\) that satisfies the Vitali covering theorem with respect to \(\mu\). This integral encloses, in particular, the classical Henstock-Kurzweil integral on the real line, the dyadic Henstock-Kurzweil integral, the Mawhin’s integral [19], and the \(s\)-HK integral [4]. The main result of this paper is the extension of the usual descriptive characterizations of the Henstock-Kurzweil integral on the real line, in terms of \(ACG^*\) functions (Main Theorem 1) and in terms of variational measures (Main Theorem 2).

On the Hencl's notion of absolute continuity

Abstract We prove that a slight modification of the notion of α-absolute continuity introduced in [D. Bongiorno, Absolutely continuous functions in R n , J. Math. Anal. Appl. 303 (2005) 119–134] is equivalent to the notion of n, λ-absolute continuity given by S. Hencl in [S. Hencl, On the notions of absolute continuity for functions of several variables, Fund. Math. 173 (2002) 175–189].

ON THE FUNDAMENTAL THEOREM OF CALCULUS FOR FRACTAL SETS

The aim of this paper is to formulate the best version of the Fundamental theorem of Calculus for real functions on a fractal subset of the real line. In order to do that an integral of Henstock–Kurzweil type is introduced.

Absolutely continuous functions in Rn

Abstract For each 0 α 1 we consider a natural n-dimensional extension of the classical notion of absolute continuous function. We compare it with the Malý's and Hencl's definitions. It follows that each α-absolute continuous function is continuous, weak differentiable with gradient in L n , differentiable almost everywhere and satisfies the formula on change of variables.

“Regularity conditions in the Sobolev space W 1,n”

Absolutely continuous functions and differentiability in Rn

Abstract We relativize the notion of absolute continuity of functions in R n , due to Rado, Reichelderfer and Malý, to subsets of R n and use it to characterize functions (possibly vector valued) differentiable almost everywhere.

METRIC DIFFERENTIABILITY OF LIPSCHITZ MAPS

AbstractAn extension of Rademacher’s theorem is proved for Lipschitz mappings between Banach spaces without the Radon–Nikodým property.

Variational measures related to local systems and the Ward propery of P-adic path bases

Some properties of absolutely continuous variational measures associated with local systems of sets are established. The classes of functions generating such measures are described. It is shown by constructing an example that there exists a P-adic path system that defines a differentiation basis which does not possess Ward property.

Absolutely continuous functions with values in a Banach space

Abstract Let Ω be an open subset of R n , n > 1 , and let X be a Banach space. We prove that α-absolutely continuous functions f : Ω → X are continuous and differentiable (in some sense) almost everywhere in Ω.

The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line

The aim of this paper is to introduce an Henstock-Kurzweil type integration process for real functions on a fractal subset E of the real line.

On the problem of regularity in the Sobolev space Wloc1,n

Abstract We prove that a variant of the Hencl's notion of A C λ n -mapping (see [S. Hencl, On the notions of absolute continuity for functions of several variables, Fund. Math. 173 (2002) 175–189]), in which λ is not a constant, produces a new solution to the problem of regularity in the Sobolev space W loc 1 , n .

Linear dynamics induced by odometers

Weighted shifts are an important concrete class of operators in linear dynamics. In particular, they are an essential tool in distinguishing variety dynamical properties. Recently, a systematic study of dynamical properties of composition operators on $L^p$ spaces has been initiated. This class of operators includes weighted shifts and also allows flexibility in construction of other concrete examples. In this article, we study one such concrete class of operators, namely composition operators induced by measures on odometers. In particular, we study measures on odometers which induce mixing and transitive linear operators on $L^p$ spaces.

A Riemann sums method in the Theory of vector Integration

Rolewicz-type chaotic operators

In this article we introduce a new class of Rolewicz-type operators in l_p, $1 \le p < \infty$. We exhibit a collection F of cardinality continuum of operators of this type which are chaotic and remain so under almost all finite linear combinations, provided that the linear combination has sufficiently large norm. As a corollary to our main result we also obtain that there exists a countable collection of such operators whose all finite linear combinations are chaotic provided that they have sufficiently large norm.