Suppose that $\Omega \subset \Bbb R^d$ is a sufficiently nice domain. From the examples of orthogonal bases in Hilbert space cases (or looking at a wavelets basis), it seems natural to me that one may expect the elements of a (normalized) Schauder basis $\{u\_n\}\_{n=1}^\infty$ of $W^{1,p}\_0(\Omega)$ (for $p>1$) to be more and more "oscillatory" as $n\to\infty$. I wonder if that is true in the sense that
$$
\lim\_{n\to\infty} \mathscr L^d(\{ x\in\Omega : |\nabla u\_n(x)| \le \varepsilon \}) = 0
$$
for any fixed $\varepsilon>0 $?


If the above is not true, is it possible to construct such a basis that satisfies the above property?


**Edit:** Thanks to a comment, I realized that the proposed notion of oscillation is not true (almost trivially). I want to modify it a little bit to capture what I originally had in mind. For any fixed $\varepsilon>0$, is it true that
$$
\lim\_{n\to\infty} \int\_{\{|\nabla u\_n| \le \varepsilon \}} |\nabla u\_n|^p \,dx = 0 \ ?
$$



Must a Schauder basis for $W^{1,p}_0(\Omega)$ be oscillatory?
Even the modified question does not hold.


Let $u\_n$ be a basis such that $\mathcal{L}^d(\operatorname{spt} u\_n) \to 0$, e.g. a wavelet basis and let $\phi \in C\_0^\infty(\Omega)$ a function such that $$\int\_{\{|\nabla \phi| \leq \varepsilon\}} |\nabla \phi|^p dx > 0$$ for all $\varepsilon > 0$, e.g. a smooth bump.


Now construct a modified basis
$$\tilde{u}\_n = a\_n( \phi + u\_n), $$
where the $a\_n >0$ are chosen in such a way that this is again normalized. By the triangle-inequality
$$ 1 = \| \tilde{u\_n} \|\_{W^{1,p}} \leq a\_n( \| u\_n \|\_{W^{1,p}} + \| \phi \|\_{W^{1,p}}) = a\_n(1 + \| \phi \|\_{W^{1,p}}), $$
so the $a\_n$ are uniformly bounded from below by $a\_-:= \frac{1}{1+\| \phi \|\_{W^{1,p}}}$. On the other hand, since $\int\_{\operatorname{spt}u\_n} |\nabla \phi|^p dx \to 0$, it is easy to show that $a\_n \to a\_-$ and that in particular $a\_n$ is bounded from above by some $a\_+$.


But then
$$ \int\_{\{|\nabla \tilde{u}\_n|\} \leq \varepsilon} |\nabla\tilde{u}\_n|^p dx \geq \int\_{\{|\nabla \tilde{u}\_n|\leq \varepsilon\} \setminus \operatorname{spt} u\_n } |\nabla\tilde{u}\_n|^p dx = a\_n^p \int\_{\{|\nabla (a\_n \phi)|\leq \varepsilon\} \setminus \operatorname{spt} u\_n } |\nabla\phi|^p dx \\\geq a\_-^p \int\_{\{|\nabla (a\_+\phi)|\leq \varepsilon\} \setminus \operatorname{spt} u\_n } |\nabla\phi|^p dx \to a\_-^p \int\_{\{|\nabla \phi|\leq \varepsilon/a\_+\}} |\nabla\phi|^p dx > 0.$$


