For any $\alpha ,\beta >0$ with $\alpha +\beta <1$ we provide a simple construction of an $\alpha$-Hölde function $f:[0,1]\to ℝ$ and a $\beta$-Hölder function $g:[0,1]\to ℝ$ such that the integral ${\int }_0^{1}f\mathrm{d}g$ fails to exist even in the Kurzweil-Stieltjes sense.

We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by ${\sum }_{n=1}^{\infty }{x}_n{\chi }_{E_n}$, where $x_n$ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for Bochner and Pettis integrability of $f$. The function $f$ is Bochner integrable if and only if the series ${\sum }_{n=1}^{\infty }{x}_n\left|E_n\right|$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability...

The space $\mathrm{\mathscr{H}𝒦}$ of Henstock-Kurzweil integrable functions on $[a,b]$ is the uncountable union of Fréchet spaces $\mathrm{\mathscr{H}𝒦}\left(X\right)$. In this paper, on each Fréchet space $\mathrm{\mathscr{H}𝒦}\left(X\right)$, an $F$-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space. It is known that every control-convergent sequence in the $\mathrm{\mathscr{H}𝒦}$ space always belongs to a $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space for some $X$. We illustrate how...

It is known that there is no natural Banach norm on the space $\mathrm{\mathscr{H}𝒦}$ of $n$-dimensional Henstock-Kurzweil integrable functions on $[a,b]$. We show that the $\mathrm{\mathscr{H}𝒦}$ space is the uncountable union of Fréchet spaces $\mathrm{\mathscr{H}𝒦}\left(X\right)$. On each $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space, an $F$-norm ${\parallel ·\parallel }^X$ is defined. A ${\parallel ·\parallel }^X$-convergent sequence is equivalent to a control-convergent sequence. Furthermore, an $F$-norm is also defined for a ${\parallel ·\parallel }^X$-continuous linear operator. Hence, many important results in functional analysis hold for the $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space. It is well-known that every...

We present a variation-of-constants formula for functional differential equations of the form $$\dot{y}=ℒ\left(t\right)y_t+f(y_t,t),y_{t_0}=\varphi ,$$ where $ℒ$ is a bounded linear operator and $\varphi$ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application $t\mapsto f(y_t,t)$ is Kurzweil integrable with $t$ in an interval of $ℝ$, for each regulated function $y$. This means that $t\mapsto f(y_t,t)$ may admit not only many discontinuities, but it can also be highly oscillating and yet, we are...

R. Deville and J. Rodríguez proved that, for every Hilbert generated space $X$, every Pettis integrable function $f:[0,1]\to X$ is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space $X$ and a scalarly null (hence Pettis integrable) function from $[0,1]$ into $X$, which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from $[0,1]$ (mostly) into $C\left(K\right)$ spaces. We focus in more detail on...

We present the full descriptive characterizations of the strong McShane integral (or the variational McShane integral) of a Banach space valued function $f:W\to X$ defined on a non-degenerate closed subinterval $W$ of ${ℝ}^m$ in terms of strong absolute continuity or, equivalently, in terms of McShane variational measure $V_{ℳ}F$ generated by the primitive $F:{ℐ}_W\to X$ of $f$, where ${ℐ}_W$ is the family of all closed non-degenerate subintervals of $W$.

In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators ${ℳ}^{+}$ and ${ℳ}^{-}$. More precisely, we prove that ${ℳ}^{+}$ and ${ℳ}^{-}$ map $W^{1,p}\left(ℝ\right)\to W^{1,p}\left(ℝ\right)$ with $1<p<\infty$, boundedly and continuously. In addition, we show that the discrete versions $M^{+}$ and $M^{-}$ map $\mathrm{BV}\left(\mathbb{Z}\right)\to \mathrm{BV}\left(\mathbb{Z}\right)$ boundedly and map $l^1\left(\mathbb{Z}\right)\to \mathrm{BV}\left(\mathbb{Z}\right)$ continuously. Specially, we obtain the sharp variation inequalities of $M^{+}$ and $M^{-}$, that is, $$\mathrm{Var}\left(M^{+}\left(f\right)\right)\le \mathrm{Var}\left(f\right)\text{and}\mathrm{Var}\left(M^{-}\left(f\right)\right)\le \mathrm{Var}\left(f\right)$$ if $f\in \mathrm{BV}\left(\mathbb{Z}\right)$, where $\mathrm{Var}\left(f\right)$ is the total variation of $f$ on $\mathbb{Z}$ and $\mathrm{BV}\left(\mathbb{Z}\right)$ is the set of all functions $f:\mathbb{Z}\to ℝ$ satisfying $\mathrm{Var}\left(f\right)<\infty$.

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem $$\mathrm{d}x=\mathrm{d}A\left(t\right)·f(t,x),h\left(x\right)=0$$ is established, where $f:[a,b]\times {ℝ}^n\to {ℝ}^n$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A:[a,b]\to {ℝ}^{n\times n}$ with bounded total variation components, and $h:{BV}_s([a,b],{ℝ}^n)\to {ℝ}^n$ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition $x\left(t_1\left(x\right)\right)=ℬ\left(x\right)·x\left(t_2\left(x\right)\right)+c_0,$ where $t_i:{BV}_s([a,b],{ℝ}^n)\to [a,b]$ $(i=1,2)$ and $ℬ:{BV}_s([a,b],{ℝ}^n)\to {ℝ}^n$ are continuous...

Equiintegrability in a compact interval $E$ may be defined as a uniform integrability property that involves both the integrand $f_n$ and the corresponding primitive $F_n$. The pointwise convergence of the integrands $f_n$ to some $f$ and the equiintegrability of the functions $f_n$ together imply that $f$ is also integrable with primitive $F$ and that the primitives $F_n$ converge uniformly to $F$. In this paper, another uniform integrability property called uniform double Lusin condition introduced in the papers...

Let ${}_s$ be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis ${}_s$ differentiates the integral of f if s ∉ S, and $D{̅}_{s}f\left(x\right)=limsu{p}_{diam\left(R\right)\to 0,x\in R{\in }_s}{\left|R\right|}^{-1}{\int }_{R}f=\infty$ almost everywhere if s ∈ S. If the condition $D{̅}_{s}f\left(x\right)=\infty$ holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a $G_{\delta }$ (resp. a $G_{\delta \sigma }$).

We prove the $L^p$ boundedness of the Marcinkiewicz integral operators ${\mu }_{\Omega }$ on ${ℝ}^{n₁}\times \cdots \times {ℝ}^{n_k}$ under the condition that $\Omega \in L{\left(logL\right)}^{k/2}{(}^{n₁-1}\times \cdots {\times }^{n_k-1})$. The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.