Let's consider homogeneous periodic Besov spaces $\dot{B}^s_{p,r}(\mathbb{S}^1)$ defined on the circle. The definition is similar to that defined on the real line, except that we use Fourier expansion of periodic functions on $\mathbb{S}^1$ instead of Fourier transform on the real line. I would like to know whether $\dot{B}^s_{p,r}(\mathbb{S}^1)$ enjoys the same properties as $\dot{B}^s_{p,r}(\mathbb{R}^1)$. For example, $\dot{B}^s_{p_1,r_1}(\mathbb{R}^1)$ can be continuously embedded into $\dot{B}^{s-\frac{1}{p_1}+\frac{1}{p_2}}_{p_2,r_2}(\mathbb{R}^1)$ for $1 \le p_1 \le p_2 \le \infty$ and $1 \le r_1 \le r_2 \le \infty$. Does $\dot{B}^s_{p,r}(\mathbb{S}^1)$ have the same embedding property?



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