How to show that $[|u(x)|+|h(x)|]^{p-1}|h(x)|\in L^1(\Omega)$?

Question

Let $u(x),h(x)\in L^p(\Omega)$,and $\Omega$ is an open subset of $R^N$. How to show that $[|u(x)|+|h(x)|]^{p-1}|h(x)|\in L^1(\Omega)$?

I think some inequality should be used ,but I failed by using Holder's inequality.

Thanks for detail answer or hint.

Answer 1

By Holder's inequality (using conjugate exponents $p/(p-1)$ and $p$),

$$\|(|u| + |h|)^{p-1}|h|\|_1 \le \|(|u| + |h|)^{p-1}\|_{p/(p-1)}\|h\|_p = \||u| + |h|\|_p^{p-1} \|h\|_p \le (\|u\|_p + \|h\|_p)^{p-1}\|h\|_p.$$

Since $\|u\|_p$ and $\|h\|_p$ are finite, then the above inequalities show that $(|u| + |h|)^{p-1}|h| \in L^1(\Omega)$.

Answer 2

Hints: Try to prove the following:

• $|u| + |h| \in L^{p}$
• $f^{p-1} \in L^{p/(p-1)}$ if $f \in L^p$
• $g \, f \in L^1$ if $g \in L^p$ and $f \in L^{p/(p-1)}$