Qusetion
Suppose that $f\in L^1(R)$ , and there is a constant $\delta>0$,such that $\int_{R}|f(x+t)-f(x)|dx\leq |t|^{1+\delta}$ for all t , then $f=0$ almost everywhere.
Attempt
I think the following lemma is useful.
Lemma:If $f\in L^1(R)$ , let $\hat{f}(\lambda)=\int_{-\infty}^{\infty}f(x)e^{-ix\lambda}dx$ , if$\hat{f} \equiv 0 $ ,then $f=0$ almost everywhere.
Notice that , for $\lambda \neq 0$ , we can show that : $$\hat{f}(\lambda)=\int_{-\infty}^{\infty}f(x)e^{-ix\lambda}dx=-\int_{-\infty}^{\infty}f(x+\frac{\pi}{\lambda})e^{-ix\lambda}dx$$ $$|\hat{f}(\lambda)|\leq\frac{1}{2}\int_{-\infty}^{\infty}|f(x+\frac{\pi}{\lambda})-f(x)|dx\leq \frac{1}{2}|\frac{\pi}{\lambda}|^{1+\delta}$$ Then $\hat{f} \in L^1（R)$. I believe that we can show that $\hat{f} \equiv 0 $ ,but I have no idea to figure it out.
Can anyone give me some hint？Thank you.
Here's an answer!