How to show that $\|x^\alpha \delta^\beta f\|^p \leq C \sum _{|\gamma|\leq d+1}\|x^{\alpha+\gamma}\delta^\beta f\|_\infty$?

Question

I am trying to understand Harmonic Analysis. There was a chapter "The Fourier transformation"

The Schwartz Space $S(R^d)$ is defined as the collection of all functions in $C^{\infty}(R^d)$ that decay rapidly together with all derivatives .In other wards $f\in S(R^d)$ if $f\in C^\infty$ and $x^{\alpha}\partial ^{\beta} f(x)\in L^{\infty}(R^d)$ $\forall, \alpha, \beta \in Z_0^d$

Let $f\in S(R^d)$ and let $1\leq p \leq \infty$ .

Then my question

How to show that $\|x^\alpha \partial^\beta f\|^p \leq C \sum _{|\gamma|\leq d+1}\|x^{\alpha+\gamma}\partial^\beta f\|_\infty$?

Answer 1

Let $d=1$. Note $$\| (1+x^2) g\|_\infty \le \|g\|_\infty+\|x^2 g\|_\infty$$ Thus for $p \ge 1$ $$\|g\|_p^p = \int_{-\infty}^\infty \frac{|(1+x^2)g(x)|^p}{(1+x^2)^p} dx \le \int_{-\infty}^\infty \frac{(\|g\|_\infty+\|x^2 g\|_\infty)^p}{1+x^2} dx= \pi\ (\|g\|_\infty+\|x^2 g\|_\infty)^p$$