I want to show the inequality of Necas on the unit ball, knowing that I proved it for $\mathbb R^d$ using Fourier transform, here it is:

Let $B$ be the unit ball of $\mathbb R^d$. Then there is $C>0$ that depends only on $B$ such that: $$\left \| P \right \| _{L_2(B)} \leqslant C \left \| P \right \| _{\chi(B)} \quad\forall P \in L_2^{0}(B). $$

with the notation

$\\ L_2^{0}(B)=\{P\in L^2(B;\; \int_{B}^{}P(x)dx=0\}$

$\chi(B):=\{P \in H^{-1} ,\nabla P \in (H^{-1}(B))^N\}$ and $\left \| P \right \| _{\chi(B)} = \left \| P \right \|_{H^{-1}(B)}+\left \| \nabla P \right \| _{(H^{-1}(B))^N}$ .

There is the idea: let $\eta \in \mathcal D (B)$ be such that $\eta(x) = 1$ for $|x| \leq 1/3$ and $\eta(x) = 0$ for $|x| \geq 1/2$. Then: $$\left \| \eta P \right \| _{L_2(B)} = \left \| \eta P \right \| _{L_2(\mathbb R^d)}\leqslant C \left \| \eta P \right \| _{\chi(\mathbb R^d)},\;\;\forall P \in L_2^{0}(B).$$

My question is: How can I prove that: $\left \| \eta P \right \| _{\chi(\mathbb R^d)}\leqslant\left \| \eta P \right \| _{\chi(B)}$?

thank you for your help.


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