Interpolation and Fractional Sobolev Spaces

The Sobolev spaces of 4A.10 Sobolev Spaces and 4A.11 Sobolev Embeddings and Critical Exponents count derivatives in whole numbers: $W^{1,p}$, $W^{2,p}$, $H^k$. Three questions force fractional smoothness on us.

• Boundary values. A function in $H^1(\Omega)$ has a trace on $\partial\Omega$, but the trace is not in $H^1(\partial\Omega)$, nor merely in $L^2(\partial\Omega)$: it has exactly half a derivative, $H^{1/2}$. The Dirichlet problem with boundary data cannot be stated sharply without this.
• Nonlocal operators. Diffusion driven by jumps of all sizes, rather than by Brownian motion, is governed by the fractional Laplacian $(-\Delta)^s$, $0 < s < 1$, whose natural energy space is $H^s$.
• Interpolation. If an operator is bounded on $L^1$ and on $L^2$, is it bounded on $L^p$ in between? If a function is in $L^2$ and in $H^2$, how much smoothness does it have "in between"? Interpolation theorems answer both with a convexity statement: norms in between are controlled by the norms at the ends, with exponents.

This chapter proves the Riesz–Thorin interpolation theorem, with the Hausdorff–Young and Young inequalities as corollaries; defines $H^s$ by the Fourier transform and $W^{s,p}$ by difference quotients, and proves that they agree for $p = 2$; proves the trace theorem $H^s(\mathbb{R}^n) \to H^{s - 1/2}(\mathbb{R}^{n - 1})$; proves the interpolation inequalities of Gagliardo–Nirenberg type that the PDE of Books 6 and 11 use constantly; and introduces the fractional Laplacian. The Marcinkiewicz theorem, the "real" counterpart of Riesz–Thorin, is proved in 6B.8 Lᵖ Theory and Strong Solutions, where singular integrals need it.

By the end of this chapter you will be able to:

• prove the three lines lemma and the Riesz–Thorin theorem, and deduce the Hausdorff–Young and Young inequalities;
• define $H^s(\mathbb{R}^n)$ for real $s$ and $W^{s,p}$ for $0 < s < 1$, and prove the equivalence of the Fourier and difference-quotient definitions when $p = 2$;
• prove the interpolation inequality $\|u\|_{H^s} \leq \|u\|_{H^{s_0}}^{1 - \theta}\|u\|_{H^{s_1}}^\theta$ and the one-dimensional inequality $\|u\|_\infty^2 \leq \|u\|_2\|u'\|_2$, and state Gagliardo–Nirenberg;
• prove the trace theorem and explain why the trace space of $H^1$ is $H^{1/2}$;
• define the fractional Laplacian by its symbol and as a singular integral, and explain its probabilistic meaning.

12.1Jumps of all sizes

12.2The Riesz–Thorin theorem

Complex interpolation rests on one lemma about holomorphic functions. Let $S = \{z : 0 \leq \operatorname{Re}z \leq 1\}$, a closed strip.

The lemma says that $\theta \mapsto \log\sup_t|F(\theta + it)|$ is a convex function: the logarithm of a maximum of a holomorphic function is convex across a strip, a cousin of the convexity of $\log$-moments (3A.7 Lᵖ Spaces and Jensen’s Inequality). The factor $e^{\varepsilon z^2}$ handles the unbounded strip, and some growth condition is essential: $F(z) = \exp\big(\exp(-i\pi(z - \frac12))\big)$ has modulus $1$ on both boundary lines (where $\operatorname{Re}\exp(-i\pi(z - \frac12)) = 0$) but modulus $\exp(e^{\pi y})$ at $z = \frac12 + iy$.

The theorem is a convexity statement: $\log\|T\|_{L^{1/a}\to L^{1/b}}$ is a convex function of $(a, b) = (\frac1p, \frac1q)$ on the square $[0, 1]^2$ (the Riesz diagram). Marcel Riesz proved it in 1926 for real scalars with the restriction $p \leq q$, by a variational argument; Olof Thorin's 1939 proof with complex methods removed the restriction and introduced the three lines idea that later became the general theory of complex interpolation of Banach spaces (Calderón, 1964).

Two corollaries.

The constant $1$ is not sharp for $1 < p < 2$: Babenko (1961, for even conjugates) and Beckner (1975) proved that the best constant is $(p^{1/p}/p'^{1/p'})^{n/2}$, attained by Gaussians, a theme that runs from the uncertainty principle (4A.6 The Fourier Transform) to entropy: differentiating Beckner's sharp inequality at $p = 2$ gives the entropic uncertainty principle conjectured by Hirschman, a relative of the logarithmic Sobolev inequality of 6A.12 Entropy, Information and Diffusion. For $p > 2$ there is no inequality at all: the Fourier transform of an $L^p$ function need not be a function (Exercise 12.11).

Young's inequality is how the heat kernel acts on $L^p$: since $\|\Phi_t\|_r = c_rt^{-\frac n2(1 - \frac1r)}$ for the Gaussian kernel (6A.3 The Heat Equation on ℝⁿ), $\|e^{t\Delta}f\|_q \leq Ct^{-\frac n2(\frac1p - \frac1q)}\|f\|_p$: the $L^p$–$L^q$ smoothing estimate that drives semilinear theory (6A.8 Nonlinear Parabolic Equations).

12.3Fractional Sobolev spaces

The Fourier definition. For real $s$, let $H^s(\mathbb{R}^n)$ be the tempered distributions $u$ with $\hat u$ a function and

(the convention of 4A.6 The Fourier Transform, with $\hat u(\xi) = \int ue^{-2\pi ix\cdot\xi}$; other conventions change $|\xi|^2$ to $4\pi^2|\xi|^2$ and alter constants only). For $s = k$ a nonnegative integer this is $H^k = W^{k,2}$ with an equivalent norm, since $\widehat{\partial^\alpha u} = (2\pi i\xi)^\alpha\hat u$ and $(1 + |\xi|^2)^k$ is comparable to $\sum_{|\alpha|\leq k}|\xi^\alpha|^2$. Negative $s$ are allowed: $H^{-s}$ is the dual of $H^s$ under the $L^2$ pairing, and $\delta_0 \in H^{-s}$ exactly when $s > \frac n2$, since $\hat\delta_0 = 1$.

The difference-quotient definition. For $0 < s < 1$ and $1 \leq p < \infty$, the Gagliardo (Slobodeckij) seminorm is

and $W^{s,p} = \{u \in L^p : [u]_{W^{s,p}} < \infty\}$. It asks for a "fractional Hölder" bound in an averaged sense: $|u(x) - u(y)|$ should be small compared with $|x - y|^s$, on average. It makes sense on any domain $\Omega$ (integrate over $\Omega\times\Omega$) and on manifolds, where there is no Fourier transform.

At $s = 1$ the constant blows up, and $(1 - s)[u]_{W^{s,p}}^p \to c\|\nabla u\|_p^p$ as $s \to 1$ (Bourgain, Brezis and Mironescu, 2001): the fractional seminorms connect continuously to the classical ones after renormalisation. At $s = 0$, $s[u]_{W^{s,p}}^p \to c\|u\|_p^p$ (Maz'ya and Shaposhnikova, 2002).

Interpolation of $H^s$. The Fourier definition makes interpolation a matter of Hölder's inequality.

So a function in $L^2$ and $H^2$ is in $H^1$ with $\|u\|_{H^1}^2 \leq \|u\|_{L^2}\|u\|_{H^2}$: one derivative is the geometric mean of none and two. Indeed $H^s$ is the complex interpolation space $[H^{s_0}, H^{s_1}]_\theta$ in Calderón's sense: the three lines argument applied to the multiplier $(1 + |\xi|^2)^{z/2}$.

12.4Interpolation inequalities

The same idea, with $L^p$ norms and derivatives, gives the inequalities used throughout geometric analysis to control intermediate quantities by extreme ones.

The second inequality is the special case $s_0 = 0$, $s = 1$, $s_1 = 2$ of the previous proposition, in its homogeneous form; the first is a borderline case of Sobolev embedding ($H^s \subseteq L^\infty$ for $s > \frac12$, 4A.6 The Fourier Transform) with the scaling-optimal split between $L^2$ and $H^1$.

The exponent relation is forced by scaling $u(x) \mapsto u(\lambda x)$, exactly as for Sobolev's exponent in 4A.11 Sobolev Embeddings and Critical Exponents (Exercise 12.13). Gagliardo and Nirenberg proved the general case independently in 1959. In geometric flows these inequalities are how one passes from $L^2$ bounds on curvature and its high derivatives to bounds on intermediate derivatives: Hamilton's derivative estimates for the Ricci flow on compact manifolds use interpolation inequalities of exactly this kind (11A.5 Shi's Derivative Estimates), as do the regularity theories for mean curvature flow and harmonic map flow.

12.5Traces

On a half-space $\mathbb{R}^n_+ = \{x_n > 0\}$, a function in $H^1$ has no well-defined values on a set of measure zero in general, but its restriction to the hyperplane $\{x_n = 0\}$ makes sense in a weaker space. The Fourier transform identifies it precisely.

So the trace space of $H^1$ is $H^{1/2}$: a function on the boundary is the boundary value of an $H^1$ function exactly when it has half a derivative in $L^2$. On a bounded smooth domain the same holds, with $H^{1/2}(\partial\Omega)$ defined by the Gagliardo seminorm on $\partial\Omega$ (or by local charts), and $H^1_0(\Omega)$ is exactly the kernel of the trace (4A.10 Sobolev Spaces). This completes the weak theory of the Dirichlet problem $\Delta u = 0$, $u = g$ on $\partial\Omega$: it has a unique solution in $H^1(\Omega)$ for every $g \in H^{1/2}(\partial\Omega)$ (extend $g$ to $G \in H^1$, then solve for $u - G \in H^1_0$ by Lax–Milgram, 4A.5 Hilbert Spaces and Lax–Milgram). For $W^{1,p}$ the trace space is $W^{1 - 1/p,p}(\partial\Omega)$ (Gagliardo, 1957), and for $p = 1$ it is all of $L^1(\partial\Omega)$.

12.6The fractional Laplacian

For $0 < s < 1$ define $(-\Delta)^s$ by its symbol: $\widehat{(-\Delta)^su}(\xi) = (2\pi|\xi|)^{2s}\hat u(\xi)$. Its quadratic form is the homogeneous $H^s$ seminorm, which by Proposition 12.5 is the Gagliardo seminorm, and polarising gives the singular integral formula

(Exercise 12.15). The value at $x$ depends on $u$ everywhere: $(-\Delta)^s$ is nonlocal. Probabilistically, $-(-\Delta)^s$ is the generator of the symmetric $2s$-stable Lévy process, whose jumps of length $r$ have intensity proportional to $r^{-n - 2s}$: exactly the kernel in the integral. Its heat kernel decays like $|x|^{-n - 2s}$, polynomially, rather than like a Gaussian, which is the heavy tail of a Lévy flight.

Luis Caffarelli and Luis Silvestre showed in 2007 that $(-\Delta)^s$ is a Dirichlet-to-Neumann map: if $U(x, y)$ on $\mathbb{R}^n\times(0, \infty)$ solves $\operatorname{div}(y^{1 - 2s}\nabla U) = 0$ with $U(x, 0) = u(x)$, then $(-\Delta)^su = -c\lim_{y\to0}y^{1 - 2s}\partial_yU$. This extension turns nonlocal problems into local, degenerate elliptic ones in one more dimension, and opened the regularity theory of nonlocal equations. For $s = \frac12$ it is the classical fact that $(-\Delta)^{1/2}$ is the Dirichlet-to-Neumann map of the half-space, the operator behind the trace theorem: $\|g\|_{\dot H^{1/2}}^2$ is, up to a constant, the minimal Dirichlet energy of an extension of $g$ (Exercise 12.16).

12.7History

Marcel Riesz's interpolation theorem dates from 1926 and Thorin's complex proof from 1939, Marcinkiewicz's real interpolation theorem from the same year (published posthumously, Zygmund 1956). Hausdorff (1923) and Young (1912) proved the Fourier inequalities named after them for Fourier series. Fractional Sobolev spaces were developed by Aronszajn, Gagliardo and Slobodeckij in the 1950s; Gagliardo's trace theorem appeared in 1957, the Gagliardo–Nirenberg inequalities in 1959, and the abstract theories of complex and real interpolation of Banach spaces (Calderón; Lions and Peetre) in the early 1960s. Caffarelli and Silvestre's extension paper is from 2007.

12.8Exercises

Solution

(a) $|f_{it}| = |f|^{p_\theta\operatorname{Re}(\frac{1 - it}{p_0} + \frac{it}{p_1})} = |f|^{p_\theta/p_0}$, so $\int|f_{it}|^{p_0} = \int|f|^{p_\theta} = 1$. (b) With $\frac{1}{p_0} = 0$ the formula gives $|f_{it}| = |f|^{0} = 1$ on the support of $f$: $f_{it} = e^{i\alpha}$ there, of sup norm $1$. So the same formula works with the convention $\frac1\infty = 0$. (c) Write $|f|^{p_\theta} = |f|^{(1 - \theta)p_\theta}|f|^{\theta p_\theta}$ with exponents $\frac{p_0}{(1 - \theta)p_\theta}$ and $\frac{p_1}{\theta p_\theta}$, whose reciprocals sum to $\frac{(1 - \theta)p_\theta}{p_0} + \frac{\theta p_\theta}{p_1} = 1$; Hölder gives $\int|f|^{p_\theta} \leq \|f\|_{p_0}^{(1 - \theta)p_\theta}\|f\|_{p_1}^{\theta p_\theta}$. This is the scalar shadow of Riesz–Thorin.

Solution

(a) $|g_a| = e^{-\pi\varepsilon x^2}$, so $\|g_a\|_p = (p\varepsilon)^{-1/(2p)}$. $\frac1a = \frac{\varepsilon - i}{\varepsilon^2 + 1}$, so $|\hat g_a| = |a|^{-1/2}e^{-\pi\varepsilon\xi^2/(1 + \varepsilon^2)}$, and $\|\hat g_a\|_{p'} \approx \big(p'\varepsilon\big)^{-1/(2p')}$. The ratio behaves like $\varepsilon^{\frac{1}{2p} - \frac{1}{2p'}}$, which tends to $\infty$ exactly when $\frac1p < \frac1{p'}$, that is $p > 2$. (b) For $p < 2$ the same ratio tends to $0$, consistent with the bound; the chirps $e^{i\pi x^2}$ spread a function out in frequency less than in space when $p > 2$.

Solution

Young with $1 + \frac1q = \frac1p + \frac1r$: $\|\Phi_t*f\|_q \leq \|\Phi_t\|_r\|f\|_p$, and $1 - \frac1r = \frac1p - \frac1q$. Scaling: if $f_\lambda(x) = f(\lambda x)$ then $e^{t\Delta}f_\lambda = (e^{\lambda^2t\Delta}f)(\lambda\cdot)$, and $\|g(\lambda\cdot)\|_q = \lambda^{-n/q}\|g\|_q$; matching powers of $\lambda$ on both sides forces the exponent $-\frac n2(\frac1p - \frac1q)$.

Solution

$\|D^ku_\lambda\|_s = \lambda^{k - n/s}\|D^ku\|_s$. Equating exponents: $j - \frac np = a(m - \frac nr) + (1 - a)(-\frac nq)$; divide by $-n$. For the special case: $0 = a(\frac12 - 1) + \frac{1 - a}{2}$, so $a = \frac12$.

Solution

$\int\hat g\langle\xi'\rangle^{-1}\psi(\xi_n/\langle\xi'\rangle)\,d\xi_n = \hat g\int\psi = \hat g$. And $\int(1 + |\xi|^2)^s|\hat u|^2d\xi_n = |\hat g|^2\langle\xi'\rangle^{-2}\int\langle\xi'\rangle^{2s}(1 + t^2)^s\psi(t)^2\langle\xi'\rangle\,dt = C_\psi|\hat g|^2\langle\xi'\rangle^{2s - 1}$, whose integral over $\xi'$ is $C_\psi\|g\|_{H^{s - 1/2}}^2$.

Solution

The transform of $2u(x) - u(x + h) - u(x - h)$ is $(2 - e^{2\pi ih\cdot\xi} - e^{-2\pi ih\cdot\xi})\hat u = |e^{2\pi ih\cdot\xi} - 1|^2\hat u$. Integrating against $|h|^{-n - 2s}$ gives $I(\xi)\hat u = I(e_1)|\xi|^{2s}\hat u$. Near $h = 0$ the second difference is $O(|h|^2)$ by Taylor (the first-order terms cancel), so the integrand is $O(|h|^{2 - n - 2s})$, integrable since $s < 1$.

Solution

$|\widehat{\nabla_{x'}U}|^2 + |\widehat{\partial_yU}|^2 = (4\pi^2|\xi'|^2 + 4\pi^2|\xi'|^2)|\hat g|^2e^{-4\pi|\xi'|y}$; integrating in $y$ gives $8\pi^2|\xi'|^2|\hat g|^2\cdot\frac{1}{4\pi|\xi'|} = 2\pi|\xi'||\hat g|^2$; integrate over $\xi'$ by Plancherel. Harmonic functions minimise Dirichlet energy with given boundary values (Dirichlet's principle, 6A.2 Harmonic Functions): for any other extension $V = U + W$ with $W = 0$ on the boundary, $\int|\nabla V|^2 = \int|\nabla U|^2 + \int|\nabla W|^2$, since the cross term $2\int\nabla U\cdot\nabla W = -2\int W\Delta U = 0$.