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Even more generally, on $\mathbb{R}^d$, for $s>d/2$, we have a continuous embedding $ \dot{H}^s(\mathbb{R}) \rightarrow C^{k,\alpha}$, where the left hand side is homogeneous Sobolev space of degree $s$, and those indices is $k=[s-d/2]$ and $\alpha$ is de 'decimal' part of $s-d/2$. And these orders are exactly compatible with the Sobolev embedding theorem, specified to the C^k function space cases.

And clearly, being in the (homogeneous) Sobolev spaces is a decay condition on the Fourier side.

For more details, see page 37, Theorem 1.50 of the textbook 'Fourier analysis and nonlinear partial differential equations' by Bahouri-Chemin-Dauchin.

Even more generally, on $\mathbb{R}^d$, for $s>d/2$, we have a continuous embedding $ \dot{H}^s(\mathbb{R}) \rightarrow C^{k,\alpha}$, where the left hand side is the homogeneous Sobolev space of degree $s$, and $k=[s-d/2]$ and $\alpha$ is the fractional part of $s-d/2$. These orders are exactly compatible with the Sobolev embedding theorem, specified to the $C^k$ function space cases.

And clearly, being in the (homogeneous) Sobolev spaces is a decay condition on the Fourier side.

For more details, see page 37, Theorem 1.50 of the textbook "Fourier analysis and nonlinear partial differential equations" by H. Bahouri, J.-Y. Chemin, R. Danchin.

Even more generally, on $\mathbb{R}^d$, for $s>d/2$, we have a continuous embedding $ \dot{H}^s(\mathbb{R}) \rightarrow C^{k,\alpha}$, where the right hand side is homogeneous Sobolev space of degree $s$, and those indices is $k=[s-d/2]$ and $\alpha$ is de 'decimal' part of $s-d/2$. And these orders are exactly compatible with the Sobolev embedding theorem, specified to the C^k function space cases.

And clearly, being in the (homogeneous) Sobolev spaces is a decay condition on the Fourier side.

For more details, see page 37, Theorem 1.50 of the textbook 'Fourier analysis and nonlinear partial differential equations' by Bahouri-Chemin-Dauchin.

Even more generally, on $\mathbb{R}^d$, for $s>d/2$, we have a continuous embedding $ \dot{H}^s(\mathbb{R}) \rightarrow C^{k,\alpha}$, where the left hand side is homogeneous Sobolev space of degree $s$, and those indices is $k=[s-d/2]$ and $\alpha$ is de 'decimal' part of $s-d/2$. And these orders are exactly compatible with the Sobolev embedding theorem, specified to the C^k function space cases.

And clearly, being in the (homogeneous) Sobolev spaces is a decay condition on the Fourier side.

For more details, see page 37, Theorem 1.50 of the textbook 'Fourier analysis and nonlinear partial differential equations' by Bahouri-Chemin-Dauchin.

Even more generally, on $\mathbb{R}^d$, for $s>d/2$, we have a continuous embedding $C^{k,\alpha} \rightarrow \dot{H}^s(\mathbb{R})$, where the right hand side is homogeneous Sobolev space of degree $s$, and those indices is $k=[s-d/2]$ and $\alpha$ is de 'decimal' part of $s-d/2$. And these orders are exactly compatible with the Sobolev embedding theorem, specified to the C^k function space cases.

And clearly, being in the (homogeneous) Sobolev spaces is a decay condition on the Fourier side.

For more details, see page 37, Theorem 1.50 of the textbook 'Fourier analysis and nonlinear partial differential equations' by Bahouri-Chemin-Dauchin.

Even more generally, on $\mathbb{R}^d$, for $s>d/2$, we have a continuous embedding $ \dot{H}^s(\mathbb{R}) \rightarrow C^{k,\alpha}$, where the right hand side is homogeneous Sobolev space of degree $s$, and those indices is $k=[s-d/2]$ and $\alpha$ is de 'decimal' part of $s-d/2$. And these orders are exactly compatible with the Sobolev embedding theorem, specified to the C^k function space cases.

And clearly, being in the (homogeneous) Sobolev spaces is a decay condition on the Fourier side.

For more details, see page 37, Theorem 1.50 of the textbook 'Fourier analysis and nonlinear partial differential equations' by Bahouri-Chemin-Dauchin.