For $p \ge 1,$ let $L^p[0,2\pi]$ be the space of $2\pi$-periodic measurable real valued functions. The norm in $L^p[0,2\pi]$ is defined as
$$\|f\|_p = \left(\frac{1}{2\pi}\int_0^{2\pi}|f(x)|^pdx\right)^{1/p}.$$
The $W(L_p[0,2\pi],\beta), \beta >0$ spaces are defined to contain $2\pi$-periodic real valued functions such that
$$\frac{1}{2\pi}\int_0^{2\pi}\left|f(x) \sin^\beta\left(\frac{x}{2}\right)\right|^pdx<\infty.$$
The norm in $W(L_p[0,2\pi],\beta)$ is defined as
$$\|f\|_{p,\beta} = \left(\frac{1}{2\pi}\int_0^{2\pi}\left|f(x) \sin^\beta\left(\frac{x}{2}\right)\right|^pdx\right)^{1/p}.$$
Now I have to find an example of a function $f\in W(L_p[0,2\pi],\beta)$ such that $f\notin L_p[0,2\pi]$ and $$\|f(x+t)-f(x)\|_{p,\beta} \le M t^\alpha, \quad0 < \alpha \le 1.$$ An example for $p=\beta=\alpha = 1$ will also be helpful. For $p=\beta=\alpha = 1,$ I tried with
$$f(x)=\frac{x}{\sin(x/2)}, x \in [0, 2\pi]$$
but was unsuccessful. I was unable to control $\sin((x+t)/2)$ in the denominator of the first term of the integral in the weighted norm.
Fix $p \geq 1, \beta \geq0, \epsilon >0$. Suppose that $f(\cdot + t) \in W(L^p, \beta)$ for all $0<t<\epsilon$. We prove that $f \in L^p$.
Indeed, it is enough to prove that for some $\delta>0$ and any $a$
$$\int_a^{a + \delta} |f(x)|^p dx < +\infty $$
(we count $a$, $a+\delta$ modulo $2\pi$). Let us choose $\delta$ small enough (w.r.t. $\epsilon)$ such that for any $a$ there exists a $0<t_a < \epsilon$ such that
$$ sin(x -t_a) \neq 0, \ \ \forall x \in [a, a+\delta].$$
Then we can estimate for a constant $c>0$:
$$ \int_a^{a + \delta} |f(x)|^p dx \leq c \int_a^{a + \delta} |f(x) sin^\beta (x -t_a)|^p dx \leq c \int_0^{2\pi} |f(x+t_a) sin^\beta (x)|^p < +\infty.$$