Problem in functional analysis

Hello,

One defines $\displaystyle D_i^{+h} u(x)=\frac{1}{h}(u(x+h\cdot e_i)-u(x))$.

$\displaystyle D_i^+ u$ is defined on the maximal subregion $\displaystyle \Omega_0=\Omega_0(+h,i)$.

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

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

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

In a book about functional analysis I read the following lemma:

"Let $\displaystyle 1< p\leq \infty$. If the inequality $\displaystyle \left \| D_i^{+h} u \right \|_{p;\Omega_0}\leq c$ holds for all $\displaystyle \Omega_0 \subset \subset \Omega$ and for all $\displaystyle 0< h\leq h_0(\Omega_0)$ then $\displaystyle u$ is weakly differentiable in $\displaystyle x_i$ with $\displaystyle \left \| D_i u \right \|_{p; \Omega} \leq c$."

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

Do you have a counter example?

Kind regards,

Alexander

Re: Problem in functional analysis

I think that you even have this in the classical sense - a function with directional derivatives at a point may not be differentiable there.