This is the generalized Wirtinger's inequality for higher dimensions, better known as Sobolev inequality.
Let $f$ be a $C^1$ and of compact support function then it holds that:
$$\left ( \int_{Q}\left | f(x) \right |^{p^*}\,{\rm d}x \right )^{1/p^*}\leq K\left ( \int_{Q}\left | \nabla f(x) \right |^p \,{\rm d}x \right )^{1/p}$$
where $Q$ is a compact interval (the domain of $f$) and $\displaystyle p^*=\frac{pn}{n-p}$, whereas $p<n$ and $n$ denoting the dimension of the space.
Proof:
The proof is based on the fundamental theorem of calculus plus some manipulation of the integrals. Basing on this if $n\geq 3$ , setting $p\geq 2$ then $2^*>2$ and by Holder's inequality it holds:
$$\sqrt{\int_Q |f(x)|^2 dx} \leq \bigg( \int_Q |f(x)|^{2^*} dx\bigg)^{1/2^*} \bigg(\int 1 dx\bigg)^{1/q} \leq Vol(Q)^{1/q} K \sqrt{\int_Q |\nabla f(x)|^2 dx}$$
whereas $\displaystyle \frac{1}{p}+\frac{1}{q}=1$.
Squaring both sides we get the desired inequality and the problem comes to an end.
Remark: The constant $K$ is called Sobolev's constant and is closely related to $Q$. The interested reader can get more information at the article Isoperimetric Inequalities.
Let $f$ be a $C^1$ and of compact support function then it holds that:
$$\left ( \int_{Q}\left | f(x) \right |^{p^*}\,{\rm d}x \right )^{1/p^*}\leq K\left ( \int_{Q}\left | \nabla f(x) \right |^p \,{\rm d}x \right )^{1/p}$$
where $Q$ is a compact interval (the domain of $f$) and $\displaystyle p^*=\frac{pn}{n-p}$, whereas $p<n$ and $n$ denoting the dimension of the space.
Proof:
The proof is based on the fundamental theorem of calculus plus some manipulation of the integrals. Basing on this if $n\geq 3$ , setting $p\geq 2$ then $2^*>2$ and by Holder's inequality it holds:
$$\sqrt{\int_Q |f(x)|^2 dx} \leq \bigg( \int_Q |f(x)|^{2^*} dx\bigg)^{1/2^*} \bigg(\int 1 dx\bigg)^{1/q} \leq Vol(Q)^{1/q} K \sqrt{\int_Q |\nabla f(x)|^2 dx}$$
whereas $\displaystyle \frac{1}{p}+\frac{1}{q}=1$.
Squaring both sides we get the desired inequality and the problem comes to an end.
Remark: The constant $K$ is called Sobolev's constant and is closely related to $Q$. The interested reader can get more information at the article Isoperimetric Inequalities.
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