Suppose that sequence $v_n\in {\rm H}^2(0,1)$ satisfies two conditions:
(i) ${\rm sup}_{n\in\mathbf{N}}||v_n||_{{\rm L}^{\infty}(0,1)}\leq C_0$;
(ii) the "bounded oscillation property" (BOP) on $(0,1)$:
for every $0\leq a<b<+\infty$ and every $n\in\mathbf{N}$ there exists at most finitely many (possibly none at all) pairwise disjoint non-empty open sub-intervals $(J^n_i)_{i=1}^{N_n}$ in $(0,1)$ such that $\inf_{J^n_i}|v_n'|\leq a$ and $\sup_{J^n_i}|v_n'|\geq b$, where for every $n$ we have $N_n=N_n(a,b)\leq N_0(a,b)$ for some $N_0(a,b)\in\mathbf{N}$.
Does it follows that we have:
(1) (weak assertion) for every $0<\eta<1$ there exists a measurable set $K_{\eta}\subseteq (0,1)$ such that $\lambda((0,1)\backslash K_{\eta})\leq\eta$ and such that $v_n'\vert_{K_{\eta}}$ satisfies the so-called Ball's condition (or tightness condititon) on $K_{\eta}$, which reads: there exists $n_1=n_1(\eta)\in\mathbf{N}$ such that $\lim_{R\rightarrow+\infty}\sup_{n\geq n_1}\lambda\{s\in K_{\eta}: |v_n'(s)|\geq R\}=0$?
(2) (strong assertion) for every $0<\eta<1$ there exists a measurable set $K_{\eta}\subseteq (0,1)$ such that $\lambda((0,1)\backslash K_{\eta})\leq\eta$ and such that there exists $n_2=n_2(\eta)\in\mathbf{N}$ and $C_{\eta}>0$ such that $\sup_{n\geq n_2}||v_n'||_{{\rm L}^{\infty}(K_{\eta})}\leq C_{\eta}$.
If the weak version is true, then BOP is a sufficient condition for pre-compactness of Dirac Young measures $s\mapsto \delta_{v_n'(s)}$ in Young measure space ${\rm L}^{\infty}_{w*}(K_{\eta};{\cal P}(\mathbf{R}))$ (as follows from the statement of the fundamental theorem of Young measures). I guess that weak and strong assertion might be equivalent, so the point is can we prove the weak version? Thanks in advance. Remark: condition (i) excludes the case when $v_n'(s):=c_n$, with $c_n\rightarrow +\infty$, and BOP condition should exclude the case of rapidly oscillating and bounded functions $v_n$. Hint: I think in the first step we should consider piecewise affine and continuous functions $v_n$ (such functions do not belong to ${\rm H}^2(0,1)$). In the second step we can consider piecewise affine and continuous functions $v_n'$, and then use density of piecewise affine and continuous functions in ${\rm H}^1(0,1)$. If it helps, we can add boundary and zero-average conditions $v_n(0)=v_n(1)=0$, $v_n'(0)=v_n'(1)=0$, $\int_0^1v_n=0$, $\int_0^1v_n'=0$. One final remark: ${\rm L}^{\infty}_{w*}(K_{\eta};{\cal P}(\mathbf{R}))$ is not pre-compact: if $v_n'(s):=c_n$, with $c_n\rightarrow +\infty$, then $\delta_{v_n'}\rightarrow 0$ weak-star in ${\rm L}^{\infty}_{w*}((0,1);{\cal M}_b(\mathbf{R}))$, so the limiting measure is a zero-measure, which is not a probability measure. However, Ball's condition prevents such leakage to infinity, and provides (necessary and suffiicient) condition for pre-compactness of a sequence $(\delta_{v_n'})$ in ${\rm L}^{\infty}_{w*}(K_{\eta};{\cal M}_b(\mathbf{R}))$. This question may be a difficult one for readers not familiar with the concept of Young measure.