Another simple approach is to take the lacunary series $f(x)=\sum_k a_k e^{i(m_k,x)}$ where $m_k\in \mathbb Z^d$ and $|m_{k+1}\gg |m_k|$. For any modulus of continuity $\omega$, \omega$such that$\Omega(t)=\omega(t)/t\to \infty$as$t\to 0$, the condition that$f$has this modulus of continuity is equivalent to the condition that$|a_k|\le C\omega(|m_k|^{-1})$if the spectrum is sparce enough to ensure that$\Omega(|m_{k+1}|^{-1})\ge 2\Omega(|m_k|^{-1})$and$\omega(|m_k|^{-1})\ge\sum_{\ell>k}\omega(|m_\ell|^{-1})$. If$f\in W^{1.p}$, then $ \left|\int f\nabla\psi\right|\le C\|\psi\|_{L^q}$ for smooth$\psi$. Plugging$e^{-i(m_k,x)}$, we see that unless$a_k=O(|m_k|^{-1})$, we have no chance. Thus, nothing short of Lipschitzness will force$f$to be in$f\in W^{1.p}$. This formally works only on the torus but you can take any smooth partition of unity$g_j$on the torus and notice that one of the functions$g_j f$is also bad. But any of them can be replanted to$\mathbb R^d$if the supports are small enough. 1 Another simple approach is to take the lacunary series$f(x)=\sum_k a_k e^{i(m_k,x)}$where$m_k\in \mathbb Z^d$and$|m_{k+1}\gg |m_k|$. For any modulus of continuity$\omega$, the condition that$f$has this modulus of continuity is equivalent to the condition that$|a_k|\le C\omega(|m_k|^{-1})$if the spectrum is sparce enough to ensure that$\omega(|m_k|^{-1})\ge\sum_{\ell>k}\omega(|m_\ell|^{-1})$. If$f\in W^{1.p}$, then $ \left|\int f\nabla\psi\right|\le C\|\psi\|_{L^q}$ for smooth$\psi$. Plugging$e^{-i(m_k,x)}$, we see that unless$a_k=O(|m_k|^{-1})$, we have no chance. Thus, nothing short of Lipschitzness will force$f$to be in$f\in W^{1.p}$. This formally works only on the torus but you can take any smooth partition of unity$g_j$on the torus and notice that one of the functions$g_j f$is also bad. But any of them can be replanted to$\mathbb R^d\$ if the supports are small enough.