The [homogeneous Sobolev space](/page/Homogeneous%20Sobolev%20Space) $\dot{H}^s(\mathbb{R}^n)$ is defined via a [continuous](/page/Continuity) [Fourier transform](/page/Fourier%20Transform), with a norm given by an integral over $\mathbb{R}^n$ weighted by $|\xi|^{2s}$. When one instead works on the torus $\mathbb{T}^n = \mathbb{R}^n/(2\pi\mathbb{Z})^n$, periodicity forces the Fourier transform to become a discrete sum over integer frequencies $k \in \mathbb{Z}^n$. The homogeneous Sobolev norm on the torus is the corresponding sum with weight $|k|^{2s}$, excluding the zero mode — and this page develops that theory from scratch, explains why the torus setting is structurally cleaner than $\mathbb{R}^n$, and connects it to the $\dot{H}^{-1}$ mixing norms that arise in the study of transport equations. [motivation] ## Motivation ### Why the Torus? Many equations in fluid dynamics, mixing theory, and dispersive PDE are naturally posed on periodic domains. The Couette flow, for instance, is studied on $\mathbb{T} \times \mathbb{R}$ or $\mathbb{T}^2$ — the periodicity in the horizontal variable reflects the spatial structure of the problem, not merely a computational convenience. Working on $\mathbb{T}^n$ also has a technical advantage: all Fourier analysis is *automatically* discrete. There is no issue of [functions](/page/Function) failing to be in $L^1$ at infinity, no Schwartz class to manage, and no [distributional](/page/Distribution) subtleties in defining the Fourier transform. Every $L^2(\mathbb{T}^n)$ function has a well-defined, square-summable sequence of Fourier coefficients. ### Why Homogeneous? The [inhomogeneous Sobolev space](/page/Inhomogeneous%20Sobolev%20Space) $H^s(\mathbb{T}^n)$ uses the weight $(1 + |k|^2)^s$, which penalises both high and low frequencies. The *homogeneous* variant $\dot{H}^s(\mathbb{T}^n)$ uses $|k|^{2s}$ with the zero mode excluded, capturing "pure derivative" control. This is the natural norm for measuring *oscillation* — how much a function varies around its mean — without sensitivity to the overall level. It appears throughout the study of incompressible fluids, where the mean of a transported quantity is automatically conserved and only the fluctuations are of interest. ### The $\dot{H}^{-1}$ Mixing Norm A concrete motivation: in the transport equation $\partial_t \theta + u \cdot \nabla\theta = 0$ on $\mathbb{T}^n$ with $\nabla \cdot u = 0$, the $L^2$ norm of $\theta$ is conserved but the flow can create finer and finer spatial oscillations, spreading energy to high-frequency modes. The $\dot{H}^{-1}(\mathbb{T}^n)$ norm, which *downweights* high frequencies by $|k|^{-2}$, decays when energy migrates to large $|k|$. It is therefore the natural quantitative measure of *mixing*: a sequence $\theta_j \rightharpoonup 0$ weakly in $L^2$ if and only if $\|\theta_j\|_{\dot{H}^{-1}} \to 0$ when the $\theta_j$ are mean-zero. The transition from the $\mathbb{R}^n$ integral formula to the torus sum formula for this norm is a key Fourier-methods step; the present page fills that gap. [/motivation] ## [Fourier Series](/page/Fourier%20Series) on $\mathbb{T}^n$ We work on the $n$-torus $\mathbb{T}^n = \mathbb{R}^n/(2\pi\mathbb{Z})^n$, identified with $[-\pi,\pi)^n$ with opposite faces identified. The natural $L^2$ space is $L^2(\mathbb{T}^n)$ with inner product \begin{align*} \langle f, g \rangle_{L^2(\mathbb{T}^n)} = \frac{1}{(2\pi)^n}\int_{\mathbb{T}^n} f(x)\,\overline{g(x)}\,d\mathcal{L}^n(x). \end{align*} The functions $e_k(x) = e^{ik \cdot x}$ for $k \in \mathbb{Z}^n$ form a complete orthonormal system in $L^2(\mathbb{T}^n)$ — this is precisely the content of the [Fourier Series (Trigonometric)](/page/Fourier%20Series%20(Trigonometric)) page in dimension $n = 1$, extended to $n$ dimensions by taking tensor products. Completeness means every $f \in L^2(\mathbb{T}^n)$ decomposes as \begin{align*} f = \sum_{k \in \mathbb{Z}^n} \hat{f}(k)\,e^{ik \cdot x} \quad \text{in } L^2(\mathbb{T}^n), \end{align*} and the Fourier coefficients are \begin{align*} \hat{f}(k) = \frac{1}{(2\pi)^n}\int_{\mathbb{T}^n} f(x)\,e^{-ik \cdot x}\,d\mathcal{L}^n(x), \quad k \in \mathbb{Z}^n. \end{align*} Parseval's identity gives the isometric identification $\ell^2(\mathbb{Z}^n) \cong L^2(\mathbb{T}^n)$: \begin{align*} \|f\|_{L^2(\mathbb{T}^n)}^2 = \sum_{k \in \mathbb{Z}^n} |\hat{f}(k)|^2. \end{align*} The key algebraic property, parallel to the $\mathbb{R}^n$ case, is that [differentiation](/page/Derivative) acts diagonally: $\widehat{\partial_{x_j} f}(k) = ik_j\,\hat{f}(k)$. Controlling derivatives in $L^2(\mathbb{T}^n)$ is therefore equivalent to controlling polynomial weights of the Fourier coefficients. ## Definition The Fourier series setup of the previous section makes the definition essentially forced. We want a space that measures "having $s$ derivatives in $L^2$" for functions on $\mathbb{T}^n$, using the Fourier characterisation: $\partial^{\alpha} f$ corresponds to multiplication by $(ik)^\alpha$ in frequency, so $s$ derivatives in $L^2$ corresponds to $|k|^{2s}|\hat{f}(k)|^2$ being summable. The only subtlety is the zero mode: the weight $|k|^{2s}$ vanishes at $k=0$, so the sum carries no information about the mean $\hat{f}(0)$. Rather than work modulo equivalence classes as in the $\mathbb{R}^n$ theory, we simply require the mean to be zero — a single clean linear constraint that pins down a canonical representative. [definition: Homogeneous Sobolev Space on the Torus] Let $s \in \mathbb{R}$ and $n \ge 1$. The **homogeneous Sobolev space** of order $s$ on $\mathbb{T}^n$ is \begin{align*} \dot{H}^s(\mathbb{T}^n) := \left\{ f \in L^2(\mathbb{T}^n) \;\middle|\; \hat{f}(0) = 0 \text{ and } \sum_{k \in \mathbb{Z}^n \setminus \{0\}} |k|^{2s}|\hat{f}(k)|^2 < \infty \right\}, \end{align*} equipped with the norm and inner product \begin{align*} \|f\|_{\dot{H}^s(\mathbb{T}^n)}^2 &:= \sum_{k \in \mathbb{Z}^n \setminus \{0\}} |k|^{2s}|\hat{f}(k)|^2, \\ \langle f, g \rangle_{\dot{H}^s(\mathbb{T}^n)} &:= \sum_{k \in \mathbb{Z}^n \setminus \{0\}} |k|^{2s}\,\hat{f}(k)\,\overline{\hat{g}(k)}. \end{align*} [/definition] Several features of this definition deserve comment. **The zero-mode condition.** The condition $\hat{f}(0) = 0$ requires $f$ to have zero mean: $\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n} f\,d\mathcal{L}^n = 0$. This is the torus analogue of quotienting out polynomials in the $\mathbb{R}^n$ theory. On $\mathbb{R}^n$, the weight $|\xi|^{2s}$ vanishes at $\xi = 0$, so $\dot{H}^s(\mathbb{R}^n)$ cannot control the zero frequency, forcing one to work in $\mathcal{S}'(\mathbb{R}^n)/\mathcal{P}$ ([tempered distributions](/page/Tempered%20Distributions) modulo polynomials). On $\mathbb{T}^n$, the "polynomial" of lowest degree is the constant function — with Fourier coefficient only at $k = 0$ — so the quotient simplifies to a single excluded mode. **Completeness.** That $\dot{H}^s(\mathbb{T}^n)$ is a genuine Hilbert space is not immediate from the definition: one must verify that the sesquilinear form is non-degenerate (positive definiteness requires the standing condition $\hat{f}(0) = 0$) and that the space is complete. Both follow from an explicit isometric isomorphism with $\ell^2(\mathbb{Z}^n \setminus \{0\})$. [quotetheorem:660] The key move in the proof is that the map $\Psi_s(f)_k = |k|^s\hat{f}(k)$ absorbs the weight $|k|^{2s}$ into the [sequence](/page/Sequence), converting the $\dot{H}^s$ norm into the standard $\ell^2$ norm. Completeness of $\dot{H}^s(\mathbb{T}^n)$ then follows from completeness of $\ell^2$ with no separate [Cauchy sequence](/page/Cauchy%20Sequence) argument. The theorem applies for all $s \in \mathbb{R}$ without sign restriction, so the negative-order space $\dot{H}^{-1}(\mathbb{T}^n)$ — the mixing norm central to the applications below — is on equal [Hilbert space](/page/Hilbert%20Space) footing with the positive-order spaces. **Range of $s$.** For $s > 0$, the weight $|k|^{2s}$ penalises high frequencies, so $\dot{H}^s(\mathbb{T}^n)$ consists of functions with enhanced smoothness. For $s < 0$, the weight $|k|^{2s}$ amplifies high frequencies, making $\dot{H}^s(\mathbb{T}^n)$ a *larger* space containing rougher functions. The space $\dot{H}^0(\mathbb{T}^n)$ is simply $L^2_0(\mathbb{T}^n) := \{f \in L^2(\mathbb{T}^n) : \hat{f}(0) = 0\}$, with norm equal to the $L^2$ norm. ## From Inhomogeneous to Homogeneous: Decomposition and Equivalence The homogeneous space $\dot{H}^s(\mathbb{T}^n)$ does not exist in isolation — it is the mean-zero component of the inhomogeneous space $H^s(\mathbb{T}^n)$. Understanding this relationship requires unpacking what the two spaces actually measure. ### The Inhomogeneous Space and the Orthogonal Decomposition The **inhomogeneous Sobolev space** on the torus is \begin{align*} H^s(\mathbb{T}^n) := \left\{f \in L^2(\mathbb{T}^n) \;\middle|\; \sum_{k \in \mathbb{Z}^n} (1+|k|^2)^s |\hat{f}(k)|^2 < \infty\right\}, \end{align*} with norm $\|f\|_{H^s}^2 = \sum_{k \in \mathbb{Z}^n}(1+|k|^2)^s|\hat{f}(k)|^2$. The weight $(1+|k|^2)^s$ is strictly positive at every $k \in \mathbb{Z}^n$ — including $k = 0$, where it equals $1$ — so $H^s$ controls all Fourier modes simultaneously without any mean-zero constraint. The norm decomposes cleanly by isolating the $k = 0$ mode: \begin{align*} \|f\|_{H^s(\mathbb{T}^n)}^2 = |\hat{f}(0)|^2 + \sum_{k \neq 0} (1+|k|^2)^s|\hat{f}(k)|^2. \end{align*} Correspondingly, every $f \in H^s(\mathbb{T}^n)$ splits $L^2$-orthogonally as \begin{align*} f = \underbrace{\hat{f}(0) \cdot \mathbf{1}}_{\text{constant part}} + \underbrace{f_0}_{\text{mean-zero part}}, \qquad f_0 := f - \hat{f}(0)\cdot\mathbf{1}, \end{align*} and the two parts have disjoint Fourier support ($k = 0$ vs.\ $k \neq 0$), so they are $L^2$-orthogonal. The norm identity $\|f\|_{H^s}^2 = |\hat{f}(0)|^2 + \|f_0\|_{H^s}^2$ exhibits \begin{align*} H^s(\mathbb{T}^n) \cong \mathbb{C}\cdot\mathbf{1} \oplus H^s_0(\mathbb{T}^n) \qquad \text{(orthogonal direct sum)}, \end{align*} where $H^s_0(\mathbb{T}^n) := \{f \in H^s(\mathbb{T}^n) : \hat{f}(0) = 0\}$ is the mean-zero subspace. ### What the Two Spaces Measure The conceptual distinction is now precise. The inhomogeneous norm $\|f\|_{H^s}$ controls both the mean $\hat{f}(0)$ and the oscillating component $f_0$; it is sensitive to adding constants to $f$. The homogeneous norm $\|f\|_{\dot{H}^s}$ discards the mean entirely and measures only $f_0$ — it is invariant under $f \mapsto f + c$ for any constant $c$. Correspondingly, $\dot{H}^s(\mathbb{T}^n)$ is exactly $H^s_0(\mathbb{T}^n)$ as a [set](/page/Set), with the norm $|\cdot|^{2s}$ on the nonzero frequencies replacing $(1+|\cdot|^2)^s$. In physical applications — advection–diffusion, Navier–Stokes, mixing — the mean of a conserved quantity is often a trivial invariant, and one works exclusively with mean-zero functions. In this regime one wants to know whether using $\dot{H}^s$ instead of $H^s$ involves any loss. On $\mathbb{R}^n$ it does; on $\mathbb{T}^n$ it does not. ### Norm Equivalence for Mean-Zero Functions On $\mathbb{R}^n$, the ratio $(1+|\xi|^2)^s/|\xi|^{2s}$ is unbounded near $\xi = 0$, so no finite constant relates $\|f\|_{H^s(\mathbb{R}^n)}$ and $\|f\|_{\dot{H}^s(\mathbb{R}^n)}$ for mean-zero $f$. On $\mathbb{T}^n$, the discrete frequency lattice $\mathbb{Z}^n$ provides a hard lower bound $|k| \geq 1$ for all $k \neq 0$, which forces $1 + |k|^2 \leq 2|k|^2$ and closes the gap between the two weights entirely. [quotetheorem:661] The constant $2^{s/2}$ is explicit and sharp. For mean-zero functions and $s \geq 0$, one can therefore freely switch between $\dot{H}^s(\mathbb{T}^n)$ and $H^s(\mathbb{T}^n)$ in estimates, absorbing the factor $2^{s/2}$ into implied constants. For $s < 0$ the norms are no longer equivalent (the inhomogeneous norm becomes weaker than $L^2$ at high frequencies while the homogeneous norm continues to amplify them), and the two spaces diverge as function spaces. [example: Equivalence Fails on $\mathbb{R}^n$: A Computation] Fix $s = 1$ and $n = 1$. For $\epsilon > 0$, define $f_\epsilon \in L^2(\mathbb{R})$ by specifying its Fourier transform: \begin{align*} \hat{f}_\epsilon(\xi) := \epsilon^{-1/2}\bigl(\mathbf{1}_{[\epsilon,2\epsilon]}(\xi) - \mathbf{1}_{[-2\epsilon,-\epsilon]}(\xi)\bigr), \end{align*} so $f_\epsilon$ is mean-zero (its Fourier transform is odd), supported at frequencies $|\xi| \sim \epsilon$, and $\|\hat{f}_\epsilon\|_{L^2}^2 = 2$. Computing the two norms: \begin{align*} \|f_\epsilon\|_{\dot{H}^1(\mathbb{R})}^2 &= \int_\mathbb{R} |\xi|^2|\hat{f}_\epsilon(\xi)|^2\,d\mathcal{L}^1(\xi) \leq (2\epsilon)^2 \cdot \|\hat{f}_\epsilon\|_{L^2}^2 = 8\epsilon^2, \\ \|f_\epsilon\|_{H^1(\mathbb{R})}^2 &= \int_\mathbb{R}(1+|\xi|^2)|\hat{f}_\epsilon(\xi)|^2\,d\mathcal{L}^1(\xi) \geq \|\hat{f}_\epsilon\|_{L^2}^2 = 2. \end{align*} Therefore $\|f_\epsilon\|_{H^1(\mathbb{R})} / \|f_\epsilon\|_{\dot{H}^1(\mathbb{R})} \geq 1/(2\sqrt{2}\epsilon) \to \infty$ as $\epsilon \to 0$. The ratio is unbounded, so no finite constant $C$ satisfies $\|f\|_{H^1(\mathbb{R})} \leq C\|f\|_{\dot{H}^1(\mathbb{R})}$ for all mean-zero $f$. On $\mathbb{T}$, by contrast, this family cannot be replicated: the smallest admissible frequency is $|k| = 1$, so $\epsilon \to 0$ is simply not available. [/example] ## Comparison with $\dot{H}^s(\mathbb{R}^n)$: The Integral-to-Sum Transition The central structural difference between the two settings is the **dual group**: - The dual of $\mathbb{R}^n$ (as an abelian [group](/page/Group)) is $\mathbb{R}^n$ itself: frequencies are continuous, and the Fourier transform is an integral $\hat{f}(\xi) = \int_{\mathbb{R}^n} e^{-ix\cdot\xi}f(x)\,d\mathcal{L}^n(x)$. - The dual of $\mathbb{T}^n = \mathbb{R}^n/(2\pi\mathbb{Z})^n$ is $\mathbb{Z}^n$: periodicity forces only integer frequencies to be compatible with the domain, and the Fourier transform becomes the discrete sum of Fourier coefficients. This directly explains the transition in the Sobolev norm: | Setting | Norm | |---|---| | $\dot{H}^s(\mathbb{R}^n)$ | $\|f\|^2 = \int_{\mathbb{R}^n} |\xi|^{2s}|\hat{f}(\xi)|^2\,d\mathcal{L}^n(\xi)$ | | $\dot{H}^s(\mathbb{T}^n)$ | $\|f\|^2 = \sum_{k \in \mathbb{Z}^n \setminus \{0\}} |k|^{2s}|\hat{f}(k)|^2$ | The integral is replaced by a sum, the continuous frequency $\xi \in \mathbb{R}^n$ is replaced by the discrete frequency $k \in \mathbb{Z}^n$, and the origin is excluded in both cases (for the same reason: the weight $|\cdot|^{2s}$ vanishes there and cannot control constants). The torus version is also structurally simpler in two respects. First, no quotient by polynomials is needed: the equivalence class $[f] \in \mathcal{S}'(\mathbb{R}^n)/\mathcal{P}$ of the $\mathbb{R}^n$ theory is replaced by the single condition $\hat{f}(0) = 0$ — a clean linear constraint. Second, the well-posedness of the norm is immediate from Parseval, requiring no distributional framework: every $L^2(\mathbb{T}^n)$ function has well-defined Fourier coefficients, and the condition $\hat{f}(0) = 0$ is a genuine equation, not an equivalence class. ## Duality: $\dot{H}^s$ and $\dot{H}^{-s}$ The $\dot{H}^s/\dot{H}^{-s}$ duality is the mechanism that makes negative-order spaces useful: every $\dot{H}^{-s}$ function acts as a bounded linear functional on $\dot{H}^s$, and conversely every such functional arises this way. This is the Sobolev analogue of the $\ell^2$ self-duality, transported to frequency space. [quotetheorem:663] The proof is entirely explicit: the Riesz representer for any functional $\Lambda \in (\dot{H}^s)^*$ is constructed by taking $\hat{f}(k) = |k|^{-2s}\overline{(\Psi_s^* \Lambda)_k}$ in frequency space. This concreteness — the dual element is found by dividing Fourier coefficients by $|k|^{2s}$ — is one of the main advantages of the torus setting over $\mathbb{R}^n$. [remark: Why Negative Sobolev Norms Measure Mixing] For a mean-zero function $\theta \in L^2(\mathbb{T}^n)$, the duality formula gives $\|\theta\|_{\dot{H}^{-1}} = \sup_{\|g\|_{\dot{H}^1} \leq 1} \langle \theta, g \rangle_{L^2}$. When a flow pushes energy to high frequencies — Fourier coefficients spreading to large $|k|$ — the [test functions](/page/Test%20Function) $g \in \dot{H}^1$ are penalised for oscillating rapidly (their $\dot{H}^1$ norm counts $|k|^2|\hat{g}(k)|^2$), so they cannot effectively track the fine-scale oscillations of $\theta$. Consequently $\|\theta\|_{\dot{H}^{-1}}$ decreases even while $\|\theta\|_{L^2}$ stays fixed. This is exactly what mixing does: it creates oscillations so fine that they integrate to zero against any smooth test function, a phenomenon precisely captured by decay in $\dot{H}^{-1}$. [/remark] ## Application: The Mixing Norm for Couette Flow To directly connect this definition with Fourier methods for PDE, we verify the $\dot{H}^{-1}(\mathbb{T}^n)$ estimate for Couette flow using the definitions established here. Consider the transport equation $\partial_t\theta + y\partial_x\theta = 0$ on $\mathbb{T}_x \times \mathbb{R}_y$, with initial data $\theta_{\mathrm{in}}$. The solution is $\theta(t,x,y) = \theta_{\mathrm{in}}(x - yt, y)$. Taking the Fourier transform in $x$ (a Fourier series, since $x \in \mathbb{T}$) and in $y$ (a Fourier transform, since $y \in \mathbb{R}$): \begin{align*} \hat{\theta}(t, k, \eta) = \hat{\theta}_{\mathrm{in}}(k, \eta + kt), \quad k \in \mathbb{Z},\; \eta \in \mathbb{R}. \end{align*} The $\dot{H}^{-1}$ norm in $x$ alone (for fixed $y$) uses the discrete sum over $k \in \mathbb{Z} \setminus \{0\}$: \begin{align*} \|\theta(t)\|_{\dot{H}^{-1}_x}^2 = \sum_{k \neq 0} \frac{1}{k^2} \int_\mathbb{R} |\hat{\theta}_{\mathrm{in}}(k,\eta + kt)|^2\,d\mathcal{L}^1(\eta). \end{align*} The substitution $\eta \mapsto \eta - kt$ shows that each $L^2_\eta$ norm is conserved: $\int|\hat{\theta}_{\mathrm{in}}(k,\eta+kt)|^2\,d\eta = \int|\hat{\theta}_{\mathrm{in}}(k,\eta)|^2\,d\eta$. The decay in $\dot{H}^{-1}$ therefore comes not from this conservation, but from a combined $\dot{H}^{-1}_{x,y}$ norm using both the discrete $k$-sum and the continuous $\eta$-integral. In that full norm, the frequency shift $\eta \to \eta + kt$ moves mass to $|\eta| \sim |k|t$, and using $|k|^2 + |\eta + kt|^2 \geq c(1+t^2)(k^2 + \eta^2)$ (which is the key pointwise inequality), one obtains the mixing estimate \begin{align*} \|\theta(t)\|_{\dot{H}^{-1}(\mathbb{T}_x \times \mathbb{R}_y)} \lesssim \frac{1}{\langle t \rangle}\|\theta_{\mathrm{in}}\|_{\dot{H}^1(\mathbb{T}_x \times \mathbb{R}_y)}. \end{align*} This $O(1/t)$ algebraic decay rate is characteristic of shear-driven mixing; it uses both the discrete-$k$ structure of the torus in $x$ and the continuous Fourier transform in $y$, illustrating how the two theories combine seamlessly. [example: The $\dot{H}^{-1}(\mathbb{T})$ Norm of a Sawtooth Wave] On $\mathbb{T} = [-\pi,\pi)$, let $f(x) = x$. This is mean-zero since $\frac{1}{2\pi}\int_{-\pi}^\pi x\,d\mathcal{L}^1(x) = 0$, so $\hat{f}(0) = 0$ and membership in $\dot{H}^s(\mathbb{T})$ is a legitimate question for any $s$. From the [Fourier Series (Trigonometric)](/page/Fourier%20Series%20(Trigonometric)) page, the Fourier coefficients are $\hat{f}(k) = \frac{(-1)^{k+1}}{ik}$ for $k \neq 0$, giving $|\hat{f}(k)|^2 = \frac{1}{k^2}$. Computing three norms directly from the definition: \begin{align*} \|f\|_{\dot{H}^{-1}(\mathbb{T})}^2 &= \sum_{k \neq 0} |k|^{-2}|\hat{f}(k)|^2 = \sum_{k \neq 0} \frac{1}{k^2} \cdot \frac{1}{k^2} = 2\sum_{k=1}^\infty \frac{1}{k^4} = \frac{\pi^4}{45}, \\ \|f\|_{\dot{H}^0(\mathbb{T})}^2 &= \sum_{k \neq 0} |\hat{f}(k)|^2 = 2\sum_{k=1}^\infty \frac{1}{k^2} = \frac{\pi^2}{3}, \\ \|f\|_{\dot{H}^1(\mathbb{T})}^2 &= \sum_{k \neq 0} k^2|\hat{f}(k)|^2 = \sum_{k \neq 0} k^2 \cdot \frac{1}{k^2} = \sum_{k \neq 0} 1 = +\infty. \end{align*} The first two use $\sum_{k=1}^\infty k^{-4} = \pi^4/90$ and [Parseval](/theorems/585) respectively. The third sum diverges, so $f \notin \dot{H}^1(\mathbb{T})$. This is no accident: the derivative $f' = 1$ a.e.\ is a nonzero constant, with $\widehat{f'}(0) = 1 \neq 0$. A constant fails the zero-mean condition, so $f' \notin \dot{H}^0(\mathbb{T})$, and the definition of $\dot{H}^1(\mathbb{T})$ via the Fourier characterisation of derivatives then forces $f \notin \dot{H}^1(\mathbb{T})$. In summary, $f = (\,\cdot\,)$ belongs to $\dot{H}^s(\mathbb{T})$ if and only if $s < 1$, with the threshold at $s = 1$ reflecting precisely the failure of the derivative to be mean-zero. [/example] ## Sobolev Embedding on the Torus For $s > n/2$, elements of $\dot{H}^s(\mathbb{T}^n)$ are not just $L^2$ functions — they are continuous. The condition $s > n/2$ is the same threshold as in the classical Sobolev embedding on $\mathbb{R}^n$, but the torus proof is more direct: one works purely with the discrete Fourier series rather than the Fourier transform, and the summability of $|k|^{-2s}$ over $\mathbb{Z}^n \setminus \{0\}$ plays the role that [integrability](/page/Integral) of $|\xi|^{-2s}$ plays in the $\mathbb{R}^n$ argument. [quotetheorem:662] The constant $C_{s,n} = (\sum_{k \neq 0}|k|^{-2s})^{1/2}$ is explicit and computable. For example, on $\mathbb{T}^1$ with $s = 1$ (so $s > 1/2$), one has $C_{1,1} = (\sum_{k \neq 0}k^{-2})^{1/2} = \pi/\sqrt{3}$, and the embedding $\dot{H}^1(\mathbb{T}) \hookrightarrow C(\mathbb{T})$ holds with this constant. The proof's Step 4 also shows the Fourier series converges *uniformly*, not merely in $L^2$, which is a strictly stronger conclusion than $L^2$ convergence. This is the torus analogue of [Sobolev embedding into continuous functions](/theorems/226). ## References 1. T. Tao, *Nonlinear Dispersive Equations: Local and Global Analysis* (2006). 2. A. Majda and A. Bertozzi, *Vorticity and Incompressible Flow* (2002). 3. P. Constantin and C. Foias, *Navier–Stokes Equations* (1988). 4. Y. Katznelson, *An Introduction to Harmonic Analysis* (3rd ed., 2004).