# Thread: Problem in functional analysis

1. ## Problem in functional analysis

Hello,

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:

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

"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,
Alexander

2. ## 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.