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For $p ge 1,$ let $L^p[0,2pi]$ be the space of $2pi$-periodic measurable real valued functions. The norm in $L^p[0,2pi]$ is defined as

$$|f|_p = left(frac12piint_0^2pi|f(x)|^pdxright)^1/p.$$

The $W(L_p[0,2pi],beta), beta >0$ spaces are defined to contain $2pi$-periodic real valued functions such that

$$frac12piint_0^2pileft|f(x) sin^betaleft(fracx2right)right|^pdx<infty.$$

The norm in $W(L_p[0,2pi],beta)$ is defined as

$$|f|_p,beta = left(frac12piint_0^2pileft|f(x) sin^betaleft(fracx2right)right|^pdxright)^1/p.$$

Now I have to find an example of a function $fin W(L_p[0,2pi],beta)$ such that $fnotin L_p[0,2pi]$ 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

$$f(x)=fracxsin(x/2), x in [0, 2pi]$$

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 $2pi$). 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 $c>0$:

$$ int_a^a + delta |f(x)|^p dx leq c int_a^a + delta |f(x) sin^beta (x -t_a)|^p dx leq c int_0^2pi |f(x+t_a) sin^beta (x)|^p < +infty.$$