Is well known in the literature the integration by parts formula for dealing with integrals under the form $$ \int_{X} f_{x_i} g dx, $$ where $X\subset \mathbb{R}^n$.
I was wondering what is the situation now if we have a similar but not equal integral:
$$ \int_{X} |f_{x_i}| \cdot | g| dx, $$ where $X\subset \mathbb{R}^n$.
Do you know if the same integration by parts formula follows ?
When I was writing this question I saw that the product could be considered in the Lebesgue space $L^1$, thus, in the same spirit I was wondering what is the situation if now we consider the product in $L^p$, i.e. if we consider an integral given by: $$ \int_{X} |f_{x_i} g|^p dx, $$
is there some known formula for dealing with this case ?