Session S05 - Advances on Spaces of Non-absolutely Integrable Functions and Related Applications

Monday, July 12, 16:40 ~ 17:10 UTC-3

The L p primitive integral on the real line

Erik Talvila

University of the Fraser Valley, Canada   -   The $L^p$ primitive integral on the real line

For each $1\leq p<\infty$ a space of integrable Schwartz distributions, $L{\!}'^{\,p}$, is defined by taking the distributional derivative of all functions in $L^p$. Here, $L^p$ is with respect to Lebesgue measure on the real line. If $f\in L{\!}'^{\,p}$ such that $f$ is the distributional derivative of $F\in L^p$ then the integral is defined as $\int^\infty_{-\infty} fG=-\int^\infty_{-\infty} F(x)g(x)\,dx$, where $g\in L^q$, $G(x)= \int_0^x g(t)\,dt$ and $1/p+1/q=1$. A norm is $\lVert f\rVert'_p=\lVert F\rVert_p$. The spaces $L{\!}'^{\,p}$ and $L^p$ are isometrically isomorphic. Functions and distributions in $L{\!}'^{\,p}$ share many properties with functions in $L^p$. For example, $L{\!}'^{\,p}$ is reflexive, its dual space is identified with $L^q$, and there is a type of H\"older inequality. The $L^p$ primitive integral is able to integrate some functions with local singularities that are not locally integrable in the Lebesgue or Henstock--Kurzweil sense. Some applications are considered.

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