Continuity in the Alexiewicz norm

Erik Talvila

Displaying similar documents to “Continuity in the Alexiewicz norm”

Henstock-Kurzweil integral on $BV$ sets

Jan Malý, Washek Frank Pfeffer (2016)

Mathematica Bohemica

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The generalized Riemann integral of Pfeffer (1991) is defined on all bounded $BV$ subsets of $ℝ^{n}$ , but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of $σ$ -finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of $BV$ sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect...

Variational Henstock integrability of Banach space valued functions

Luisa Di Piazza, Valeria Marraffa, Kazimierz Musiał (2016)

Mathematica Bohemica

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We study the integrability of Banach space valued strongly measurable functions defined on $[0, 1]$ . In the case of functions $f$ given by $\sum_{n = 1}^{\infty} x_{n} χ_{E_{n}}$ , 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 $\sum_{n = 1}^{\infty} x_{n} | E_{n} |$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability...

On coincidence of Pettis and McShane integrability

Marián J. Fabián (2015)

Czechoslovak Mathematical Journal

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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 $C (K)$ spaces. We focus in more detail on...

On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals

Abraham Racca, Emmanuel Cabral (2016)

Mathematica Bohemica

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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...

Compact operators and integral equations in the $ℋ𝒦$ space

Varayu Boonpogkrong (2022)

Czechoslovak Mathematical Journal

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The space $ℋ𝒦$ of Henstock-Kurzweil integrable functions on $[a, b]$ is the uncountable union of Fréchet spaces $ℋ𝒦 (X)$ . In this paper, on each Fréchet space $ℋ𝒦 (X)$ , 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 $ℋ𝒦 (X)$ space. It is known that every control-convergent sequence in the $ℋ𝒦$ space always belongs to a $ℋ𝒦 (X)$ space for some $X$ . We illustrate how...

The $L^{r}$ Henstock-Kurzweil integral

Paul M. Musial, Yoram Sagher (2004)

Studia Mathematica

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We present a method of integration along the lines of the Henstock-Kurzweil integral. All $L^{r}$ -derivatives are integrable in this method.

Large structures made of nowhere $L^{q}$ functions

Szymon Głąb, Pedro L. Kaufmann, Leonardo Pellegrini (2014)

Studia Mathematica

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We say that a real-valued function f defined on a positive Borel measure space (X,μ) is nowhere q-integrable if, for each nonvoid open subset U of X, the restriction ${f |}_{U}$ is not in $L^{q} (U)$ . When (X,μ) has some natural properties, we show that certain sets of functions defined in X which are p-integrable for some p’s but nowhere q-integrable for some other q’s (0 < p,q < ∞) admit a variety of large linear and algebraic structures within them. The presented results answer a question of Bernal-González,...

Cauchy's residue theorem for a class of real valued functions

Branko Sarić (2010)

Czechoslovak Mathematical Journal

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Let $[a, b]$ be an interval in $ℝ$ and let $F$ be a real valued function defined at the endpoints of $[a, b]$ and with a certain number of discontinuities within $[a, b]$ . Assuming $F$ to be differentiable on a set $[a, b] ∖ E$ to the derivative $f$ , where $E$ is a subset of $[a, b]$ at whose points $F$ can take values $\pm \infty$ or not be defined at all, we adopt the convention that $F$ and $f$ are equal to $0$ at all points of $E$ and show that $𝒦ℋ -vt \int_{a}^{b} f = F (b) - F (a)$ , where $𝒦ℋ -vt$ denotes the total value of the integral. The paper ends with a few examples that illustrate the...

A complete characterization of R-sets in the theory of differentiation of integrals

G. A. Karagulyan (2007)

Studia Mathematica

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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 $D ̅_{s} f (x) = l i m s u p_{d i a m (R) \to 0, x \in R \in_{s}} {| R |}^{- 1} \int_{R} f = \infty$ almost everywhere if s ∈ S. If the condition $D ̅_{s} f (x) = \infty$ 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 $G_{δ}$ (resp. a $G_{δ σ}$ ).

On the regularity of the one-sided Hardy-Littlewood maximal functions

Feng Liu, Suzhen Mao (2017)

Czechoslovak Mathematical Journal

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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 $W^{1, p} (ℝ) \to W^{1, p} (ℝ)$ with $1 < p < \infty$ , boundedly and continuously. In addition, we show that the discrete versions $M^{+}$ and $M^{-}$ map $BV (ℤ) \to BV (ℤ)$ boundedly and map $l^{1} (ℤ) \to BV (ℤ)$ continuously. Specially, we obtain the sharp variation inequalities of $M^{+}$ and $M^{-}$ , that is, $Var (M^{+} (f)) \leq Var (f) and Var (M^{-} (f)) \leq Var (f)$ if $f \in BV (ℤ)$ , where $Var (f)$ is the total variation of $f$ on $ℤ$ and $BV (ℤ)$ is the set of all functions $f : ℤ \to ℝ$ satisfying $Var (f) < \infty$ .

Existence of discontinuous absolute minima for certain multiple integrals without growth properties

Lamberto Cesari (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In the present paper the author discusses certain multiple integrals $I(u)$ of the calculus of variations satisfying convexity conditions, and no growth property, and the corresponding Serrin integrals $\mathfrak{I}(u)$ , to which the existence theorems in [3,4,5] do not apply. However, in the present paper, the integrals $I(u)$ and $\mathfrak{I}(u)$ are reduced to simpler form $H(v)$ and $\mathcal{H}(v)$ to which the existence theorems above apply. Thus, we derive that $I(u)\leq\mathfrak{I}(u)$ , $H(v)\leq\mathcal{H}(v)$ , we obtain the existence of the absolute minimum for the Serrin forms $\mathfrak{I}(u)$ ...

Generalized Lebesgue points for Sobolev functions

Nijjwal Karak (2017)

Czechoslovak Mathematical Journal

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In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X, d, μ)$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B (x, r)$ converge to $f (x)$ when $r$ converges to $0$ . We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

Marcinkiewicz integrals on product spaces

H. Al-Qassem, A. Al-Salman, L. C. Cheng, Y. Pan (2005)

Studia Mathematica

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We prove the $L^{p}$ boundedness of the Marcinkiewicz integral operators $μ_{Ω}$ on $ℝ^{n ₁} \times \dots \times ℝ^{n_{k}}$ under the condition that $Ω \in L {(l o g L)}^{k / 2} (^{n ₁ - 1} \times \dots \times^{n_{k} - 1})$ . The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.