Continuity in the Alexiewicz norm

Talvila, Erik. "Continuity in the Alexiewicz norm." Mathematica Bohemica 131.2 (2006): 189-196. <http://eudml.org/doc/249914>.

@article{Talvila2006,
abstract = {If $f$ is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of $f$ is $\Vert f\Vert =\sup _I|\int _I f|$ where the supremum is taken over all intervals $I\subset \{\mathbb \{R\}\}$. Define the translation $\tau _x$ by $\tau _xf(y)=f(y-x)$. Then $\Vert \tau _xf-f\Vert $ tends to $0$ as $x$ tends to $0$, i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $\Vert \tau _xf-f\Vert $ can tend to 0 arbitrarily slowly. In general, $\Vert \tau _xf-f\Vert \ge \mathop \{\text\{osc\}\}f|x|$ as $x\rightarrow 0$, where $ \mathop \{\text\{osc\}\}f$ is the oscillation of $f$. It is shown that if $F$ is a primitive of $f$ then $\Vert \tau _xF-F\Vert \le \Vert f\Vert |x|$. An example shows that the function $y\mapsto \tau _xF(y)-F(y)$ need not be in $L^1$. However, if $f\in L^1$ then $\Vert \tau _xF-F\Vert _1\le \Vert f\Vert _1|x|$. For a positive weight function $w$ on the real line, necessary and sufficient conditions on $w$ are given so that $\Vert (\tau _xf-f)w\Vert \rightarrow 0$ as $x\rightarrow 0$ whenever $fw$ is Henstock-Kurzweil integrable. Applications are made to the Poisson integral on the disc and half-plane. All of the results also hold with the distributional Denjoy integral, which arises from the completion of the space of Henstock-Kurzweil integrable functions as a subspace of Schwartz distributions.},
author = {Talvila, Erik},
journal = {Mathematica Bohemica},
keywords = {Henstock-Kurzweil integral; Alexiewicz norm; distributional Denjoy integral; Poisson integral; Henstock-Kurzweil integral; distributional Denjoy integral; Poisson integral},
language = {eng},
number = {2},
pages = {189-196},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Continuity in the Alexiewicz norm},
url = {http://eudml.org/doc/249914},
volume = {131},
year = {2006},
}

TY - JOUR
AU - Talvila, Erik
TI - Continuity in the Alexiewicz norm
JO - Mathematica Bohemica
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 131
IS - 2
SP - 189
EP - 196
AB - If $f$ is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of $f$ is $\Vert f\Vert =\sup _I|\int _I f|$ where the supremum is taken over all intervals $I\subset {\mathbb {R}}$. Define the translation $\tau _x$ by $\tau _xf(y)=f(y-x)$. Then $\Vert \tau _xf-f\Vert $ tends to $0$ as $x$ tends to $0$, i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $\Vert \tau _xf-f\Vert $ can tend to 0 arbitrarily slowly. In general, $\Vert \tau _xf-f\Vert \ge \mathop {\text{osc}}f|x|$ as $x\rightarrow 0$, where $ \mathop {\text{osc}}f$ is the oscillation of $f$. It is shown that if $F$ is a primitive of $f$ then $\Vert \tau _xF-F\Vert \le \Vert f\Vert |x|$. An example shows that the function $y\mapsto \tau _xF(y)-F(y)$ need not be in $L^1$. However, if $f\in L^1$ then $\Vert \tau _xF-F\Vert _1\le \Vert f\Vert _1|x|$. For a positive weight function $w$ on the real line, necessary and sufficient conditions on $w$ are given so that $\Vert (\tau _xf-f)w\Vert \rightarrow 0$ as $x\rightarrow 0$ whenever $fw$ is Henstock-Kurzweil integrable. Applications are made to the Poisson integral on the disc and half-plane. All of the results also hold with the distributional Denjoy integral, which arises from the completion of the space of Henstock-Kurzweil integrable functions as a subspace of Schwartz distributions.
LA - eng
KW - Henstock-Kurzweil integral; Alexiewicz norm; distributional Denjoy integral; Poisson integral; Henstock-Kurzweil integral; distributional Denjoy integral; Poisson integral
UR - http://eudml.org/doc/249914
ER -