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Hint for the revised question: for $\frac12<s<1$, $p\geq 1$. For $s=m+\sigma$ ($0<\sigma<1$, $m\geq 1$ as above), $p\geq m+1$.

The "if" part is straightforward using the double integral definition of $H^s$. The "only if" part will better be dealt with using Fourier series $i\ \sum \pm|n|^{-\alpha} e^{int}$ ($\pm :=$ sign of $n$) or $\sum |n|^{-\alpha} (e^{int}-1)$ that vanish at $0$ with a singularity.

Hint at a partial answer to the revised question, extended to $p\in (0,\infty)$: for $\frac12<s<1$,   $p\geq 1$ is necessary and sufficient.

The "if" part is straightforward using the double integral definition of $H^s$. The "only if" part will better be dealt with using Fourier series $i\ \sum \pm|n|^{-\alpha} e^{int}$ ($\pm :=$ sign of $n$) or $\sum |n|^{-\alpha} (e^{int}-1)$ that vanish at $0$ with a singularity.

Hint for the revised question: for $\frac12<s<1$, $p\geq 1$. For $s=m+\sigma$ ($0<\sigma<1$, $m\geq 1$ as above), $p\geq m+1$.

The "if" part is straightforward using the double integral definition of $H^s$. The "only if" part will better be dealt with using Fourier series $Ci\ \sum \pm|n|^{-\alpha} e^{int}$ ($\pm :=$ sign of $n$) that have a singularity at $0$.

Hint for the revised question: for $\frac12<s<1$, $p\geq 1$. For $s=m+\sigma$ ($0<\sigma<1$, $m\geq 1$ as above), $p\geq m+1$.

The "if" part is straightforward using the double integral definition of $H^s$. The "only if" part will better be dealt with using Fourier series $i\ \sum \pm|n|^{-\alpha} e^{int}$ ($\pm :=$ sign of $n$) or $\sum |n|^{-\alpha} (e^{int}-1)$ that vanish at $0$ with a singularity.

Hint for the revised question: for $\frac12<s<1$, $p\geq 1$. For $s=m+\sigma$ ($0<\sigma<1$, $m\geq 1$ as above), $p\geq m+1$.

The "if" part is straightforward using the double integral definition of $H^s$. The "only if" part will better be dealt with using Fourier series $C\ \sum |n|^{-\alpha} e^{int}$ or $Ci\ \sum \pm|n|^{-\alpha} e^{int}$ ($\pm :=$ sign of $n$) that have a singularity at $0$.

Hint for the revised question: for $\frac12<s<1$, $p\geq 1$. For $s=m+\sigma$ ($0<\sigma<1$, $m\geq 1$ as above), $p\geq m+1$.

The "if" part is straightforward using the double integral definition of $H^s$. The "only if" part will better be dealt with using Fourier series $Ci\ \sum \pm|n|^{-\alpha} e^{int}$ ($\pm :=$ sign of $n$) that have a singularity at $0$.