For $r<s<t$, showing $||u||_{H^s} \leq \epsilon ||u||_{H^t} +C ||u||_{H^r}$?

Question

The Problem:

Let $H^s(\mathbb{R})$ be the Sobolev Space of order $s$ on the real line, with norm $$ ||u||_{H^s} = \bigg(\int_\mathbb{R}(1+|\xi|^2)^s|\hat{u}(\xi)|^2\,d\xi\bigg)^\frac{1}{2}, $$ where $\hat{u}$ denotes the Fourier transform of $u$, given by $\hat{u}(\xi) = \frac{1}{2\pi}\int_\mathbb{R} u(x)e^{-i\xi x}\,dx$.

Give $r<s<t$, all real, and $\epsilon > 0$ show that there exists a constant $C>0$ such that $$ ||u||_{H^s} \leq \epsilon ||u||_{H^t} +C ||u||_{H^r},$$

for all $u\in H^t(\mathbb{R})$.

Where I'm at:

I've really hit a wall with this analysis prelim problem. None of the standard Sobolev inequalities I've studied (such as Gagliardo-Nirenberg, Poincaré,etc.) seem relevant. I've tried several algebraic manipulations on the integrand as well that have led nowhere. The only thing I can think of is trying to write the integral over $\mathbb{R}$ as the sum of the integral over two regions, such as a ball of radius $\epsilon$ and it's complement. However that has also gone nowhere so far.

Any help would be appreciated, even if it's only the first few steps or just explains what the key manipulation should be.

Answer 1

I would suggest a good-old cut-off argument. We have that$\newcommand{\bbR}{\mathbb{R}}$ $$\begin{align*}\lVert u \rVert_{H^s}^2 =& \int_\bbR (1 + |\xi|^2)^s |\hat u(\xi)|^2\;d\xi \\=& \int_{|\xi| \leq R} (1 + |\xi|^2)^s |\hat u(\xi)|^2\;d\xi + \int_{|\xi| > R} (1 + |\xi|^2)^s |\hat u(\xi)|^2\;d\xi\\ \leq& (1+R^2)^{r-s}\int_{|\xi| \leq R} (1 + |\xi|^2)^r |\hat u(\xi)|^2\;d\xi + \frac{1}{(1+R^2)^{t-s}}\int_{\bbR} (1 + |\xi|^2)^t |\hat u(\xi)|^2\;d\xi\\ \leq & (1+R^2)^{r-s}\int_{|\xi| \leq R} (1 + |\xi|^2)^r |\hat u(\xi)|^2\;d\xi + \frac{1}{(1+R^2)^{t-s}}\int_{\bbR} (1 + |\xi|^2)^t |\hat u(\xi)|^2\;d\xi\\ \leq& (1+R^2)^{r-s} \lVert u \rVert_{H^r}^2 + (1+R^2)^{s-t}\lVert u \rVert_{H^t}^2 \end{align*}$$ Thus, $$\lVert u \rVert_{H^s} \leq (1+R^2)^\frac{r-s}{2} \lVert u \rVert_{H^r} + (1 + R^2)^\frac{s-t}{2} \lVert u \rVert_{H^t}$$ By choosing $R$ such that $(1 + R^2)^\frac{s-t}{2} \leq \epsilon$, we obtain the result.

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Notice that this proof has a conceptual interpretation: for any function $u$, we can consider it's high frequency components and low frequency components. For the high frequency bits, the $H^t$ norm is much larger than the $H^s$ norm, so we can control $H^s$ with just a little $H^t$. On the other hand, the low frequency bits are easy to control, so any $H^r$ will do.

Answer 2

Let $\alpha\in (0,1)$ be such that $s=\alpha t+(1-\alpha)r$. Let $p=1/\alpha$ and $q=1/(1-\alpha)$. Use the inequality $$ ab\leq \frac{a^p}p+\frac{b^q}q $$ where $a=\varepsilon(1+\left\vert \xi\right\rvert ^2)^{\alpha t}$ and $b=(1+\left\vert \xi\right\rvert ^2)^{s-\alpha t}/\varepsilon$.