One defines D_i^{+h} u(x)=\frac{1}{h}(u(x+h\cdot e_i)-u(x)).
D_i^+ u is defined on the maximal subregion \Omega_0=\Omega_0(+h,i).

Now let \Omega=(-1,1) and u\in L^1(\Omega).

How to prove, that the following implication is not true in general:

\left \| D_1^+ u \right \|_1\leq  c\quad  \forall\ h >0 \ \ \Rightarrow\ \ u \in H^{1,1}(\Omega)

In a book about functional analysis I read the following lemma:
"Let 1< p\leq \infty. If the inequality \left \| D_i^{+h} u \right \|_{p;\Omega_0}\leq c holds for all \Omega_0 \subset \subset \Omega and for all 0< h\leq h_0(\Omega_0) then u is weakly differentiable in x_i with \left \| D_i u \right \|_{p; \Omega} \leq c."

Here the problem is p=1, so the lemma can not be used.

Do you have a counter example?

Kind regards,