Consider $x(t)\in L_{p_{[a,b]}}$ and $a(t)\in L_{q_{[a,b]}}$. Then it is >defined the following operator $D:L_{p_{[a,b]}}\to L_{p_{{[a,b]}}}$ in which >$Dx(t)=a(t)x(t)$ Let's prove that $D$ is bounded:
$||Dx||_p=(\int_a^b|Dx(t)|^p dt)^{\frac{1}{p}}=(\int_a^b |a(t)|^p|x(t)|^p dt)^{\frac{1}{p}}\leqslant(\int_a^b |a(t)|^q dt)^{\frac{1}{q}}(\int_a^b|x(t)|^p dt)^{\frac{1}{p}}\leqslant \max_{a\leqslant t\leqslant b}|a(t)||x||_p$.
Question:
1) What did the author applied on this passage $(\int_a^b |a(t)|^p|x(t)|^p dt)^{\frac{1}{p}}\leqslant(\int_a^b |a(t)|^q dt)^{\frac{1}{q}}(\int_a^b|x(t)|^p dt)^{\frac{1}{p}}$? Hölder inequality(I do not see how)?
2) What about the last $ (\int_a^b |a(t)|^q dt)^{\frac{1}{q}}(\int_a^b|x(t)|^p dt)^{\frac{1}{p}}\leqslant \max_{a\leqslant t\leqslant b}|a(t)||x||_p$. ? Was it a mistake?