Equivalence of norms in $L_p$ spaces.

Question

Let $(f,g) \in {W_0}^{1,p}(0,1) \times {L^p}(0,1), p\in[1,\infty]$. I want to prove the equivalence of the two norms $${\left\| {{{{f_x} - g} \over 2}} \right\|_p} + {\left\| {{{{f_x} + g} \over 2}} \right\|_p}$$ and the usual one $${\left\| {{f_x}} \right\|_p} + {\left\| g \right\|_p}.$$ The proof is straighforward if $p=2$. Wht about an arbitrary $p$. Is that seems right?. Thank you.

Answer 1

Let $M=\left\|\dfrac{a+b}{2}\right\|_{L^{p}}+\left\|\dfrac{a-b}{2}\right\|_{L^{p}}$, then Minkowski's inequality gives $M\leq\left\|\dfrac{a}{2}\right\|_{L^{p}}+\left\|\dfrac{b}{2}\right\|_{L^{p}}+\left\|\dfrac{a}{2}\right\|_{L^{p}}+\left\|\dfrac{-b}{2}\right\|_{L^{p}}=\|a\|_{L^{p}}+\|b\|_{L^{p}}$.

On the other hand, $\|a\|_{L^{p}}=\left\|\dfrac{a+b}{2}+\dfrac{a-b}{2}\right\|_{L^{p}}\leq\left\|\dfrac{a+b}{2}\right\|_{L^{p}}+\left\|\dfrac{a-b}{2}\right\|_{L^{p}}$, and $\|b\|_{L^{p}}=\left\|\dfrac{a+b}{2}-\dfrac{a-b}{2}\right\|_{L^{p}}\leq\left\|\dfrac{a+b}{2}\right\|_{L^{p}}+\left\|\dfrac{a-b}{2}\right\|_{L^{p}}$, so $\|a\|_{L^{p}}+\|b\|_{L^{p}}\leq 2\left(\left\|\dfrac{a+b}{2}\right\|_{L^{p}}+\left\|\dfrac{a-b}{2}\right\|_{L^{p}}\right)$.