How to handle a scalar product in an integral?

Question

I am having a problem with a certain inequality I try to understand. I think it's just a basic idia (/trick) I'm missing, but I can't seem to find it.

Here's a simplification of the problem:

$ \int_\Omega f(\nabla g(x))$ $\cdot \nabla h(x) dx$ $\overset{?}{\leq}$ $|| \nabla g|| _{L^p}^{p-1}$ $||\nabla h|| _{L^p}^{p-1} C $

where $x \in \Omega \subset R^n$ and $f:R^n \to R^n$, $g:R^n\to R$, $h:R^n\to R$. Also, I have that for all $v \in R^n$ $|f(v)| \leq C |v|^{p-1}$.

So the problem is, that I don't know how to handle the scalar product and how to make the step from the scalar product (and the absolute value of $f(v)$) to the $L^p$ norms. Is there a basic idea I'm missing?

[The simplification I posted may be incomplete, as it is part of a larger equation - but I am still thankful for every suggestion on how to handle integrals of such type]

EDIT: As I look at the other questions posted here, this one may be too low-level. I'm sorry :) Any help is still appreciated.

Answer 1

As it's stated the inequality seems wrong. If you scale $h->\lambda h$ then the integral in the left hand side (LHS) scales as $\lambda$ but the RHS scales as $\lambda^{p-1}$.

Anyway, this kind of estimates are classical and follow from the Holder inequality: $$ |\int f(x)g(x)dx|\leq\int |f(x)g(x)|dx=\|fg\|_{L^1}\leq \|f\|{L^p}\|g\|{L^q} $$ where $$ \frac{1}{p}+\frac{1}{q}=1. $$ I hope that this helps you