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In the book Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis

The Sobolev space $W^{1,p}(I)$ is defined to be $$W^{1,p}(I)=\{u\in L^{p}(I);\exists g\in L^p(I)\text{ such that } \int_{I}{u\phi^{'}}=-\int_{I}g\phi\ \forall \phi\in C^{1}_c(I)\}$$ where $C^{k}_c(I)$ means continuous, k-time differentiable function on a compact subset of $I.$

For a extension operator $P:W^{1,p}(I)\to W^{1,p}(\mathbb{R}),$ satisfying the following properties: $$(i)\ Pu|_{I}=u,\ \forall u\in W^{1,p}(I),$$ $$(ii)\ ||Pu||_{L^{p}(\mathbb{R})}\le C||u||_{L^{p}(\mathbb{R})},\ \forall u\in W^{1,p}(I),$$ $$(iii)\ ||Pu||_{W^{1,p}(\mathbb{R})}\le C||u||_{W^{1,p}(\mathbb{R})},\ \forall u\in W^{1,p}(I),$$ where $C$ depends only on $|I|\le\infty.$

Let $\mathcal{H}$ be the unit ball in $W^{1,1}(I).$ Let $P$ be the extension operator as above and set $\mathcal{F}=P(\mathcal{H}),$ so that $\mathcal{H}=\mathcal{F}|_{I}.$ We prove that $\mathcal{H}$ has a compact closure in $L^{q}(I)$(for all $1\le q<\infty$) by applying Theorem 4.26(Kolmogorov-M.Riesz-Frechet).Now $\mathcal{F}$ is bounded in $W^{1,1}(\mathbb{R});$ therefore $\mathcal{F}$ is also bounded in $L^{q}(\mathbb{R}),$ since it is bounded both in $L^{1}(\mathbb{R})$ and in $L^{\infty}(\mathbb{R}).$ We now check the condition for Theorem 4.26 i.e. $$\lim_{h\to 0}||\tau_{h}f-f||_q=0,\text{ uniformly in }f\in\mathcal{F}.$$ where $\tau_hf(x)=f(x+h).$By Proposition 8.5 we have, for every $f\in\mathcal{F}\subset W^{1,p}(\mathbb{R}),$ $$||\tau_h f-f||_{L^1(\mathbb{R})}\le |h|||f^{'}||_{L^{1}(\mathbb{R})}\le C|h|,$$ thus $$||\tau_h f-f||^{q}_{L^{q}(\mathbb{R})}\le (2||f||_{L^{\infty}(\mathbb{R})})^{q-1}||\tau_h f-f||_{L^1(\mathbb{R})}\le C|h|,$$ and consequently $$||\tau_h f-f||_{L^q(\mathbb{R})}\le C|h|^{\frac{1}{q}}$$ where $C$ is independent of $f.$ My question is how do I obtain this inequality: $$||\tau_h f-f||^{q}_{L^{q}(\mathbb{R})}\le (2||f||_{L^{\infty}(\mathbb{R})})^{q-1}||\tau_h f-f||_{L^1(\mathbb{R})}$$

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Short version: Hölder's inequality ...


Longer version: This is just the fact that since $f(x) \leq \|f\|_{L^\infty}$, $$ \|f\|_{L^q}^q = \int_{\Bbb R} |f(x)|^q\,\mathrm d x \leq \|f\|_{L^\infty}^{q-1} \int_{\Bbb R} |f(x)|\,\mathrm d x $$ which is an endpoint of Hölder's inequality, and so $$ \|\tau_h f -f\|_{L^q}^q \leq \|\tau_h f -f\|_{L^\infty}^{q-1}\,\|\tau_h f -f\|_{L^1} \leq (2\|f\|_{L^\infty})^{q-1}\,\|\tau_h f -f\|_{L^1} $$ where the last inequality follows by the triangle inequality and the fact that $\|\tau_hf\|_{L^\infty} = \|f\|_{L^\infty}$.

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