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For a bounded domain $A$ in $\mathbb{R}^n$ my book says that for $1 \leq p<q$ $$ W^{1, q}(A) \subset W^{1, p}(A) $$ and $$ W_0^{1, q}(A) \subset W_0^{1, p}(A), $$ where $W^{1, q}(\Omega)$ is the sobolev space and the $0$ indicates the closure. The book says that this follows by the Hölder Inequality but I do not really understand how? Does it have to do with $L^p$ and $L^q$?

Any help would be great. I am sorry for my little input.

Lonaldin
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    Note that for the same reason, on a bounded domain you always have $L^p(A) \subset L^q(A)$ iff $1\le p<q$. – DominikS Oct 22 '23 at 10:55

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Because A is a bounded domain, then by $\int_{A}u^{p}dx\leq (\int_{A}u^{p\times\frac{q}{p}}dx)^{\frac{p}{q}}(\int_{A}1dx)^{1-\frac{p}{q}}$, then you have $\|u\|_{L^{p}}\leq |A|^{1-\frac{p}{q}}\|u\|_{L^{q}}$, similar to $\|\nabla u\|_{L^{p}}\leq |A|^{1-\frac{p}{q}}\|\nabla u\|_{L^{q}}$.

monotone operator
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