Function Space and Operator Theory for Nonlinear Analysis

Abstract

This chapter examines a number of analytical techniques, which will be applied to diverse nonlinear problems in the remaining chapters. For example, we study Sobolev spaces based on L^p, rather than just L². Sections 1 and 2 discuss the definition of Sobolev spaces H^{k, p}, for $k \in {\mathbb{Z}}^{+}$, and inclusions of the form H^{k, p} ⊂ L^q. Estimates based on such inclusions have refined forms, due to E. Gagliardo and L. Nirenberg. We discuss these in § 3, together with results of J. Moser on estimates on nonlinear functions of an element of a Sobolev space, and on commutators of differential operators and multiplication operators. In § 4 we establish some integral estimates of N. Trudinger, on functions in Sobolev spaces for which L^∞-bounds just fail. In these sections we use such basic tools as Hölder’s inequality and integration by parts.

A Variations on complex interpolation

Let X and Y be Banach spaces, assumed to be linear subspaces of a Hausdorff locally convex space V (with continuous inclusions). We say (X, Y, V) is a compatible triple. For θ ∈ (0, 1), the classical complex interpolation space [X, Y]_θ, introduced in Chap. 4 and much used in this chapter, is defined as follows. First, Z = X + Y gets a natural norm; for v ∈ X + Y,

$$\|{v\|}_{Z} =\inf \,\{\| {v{}_{1}\|}_{X} +\| {v{}_{2}\|}_{Y } : v = {v}_{1} + {v}_{2},\,{v}_{1} \in X,\,{v}_{2} \in Y \}.$$

(A.1)

One has X + Y ≈ X ⊕ Y ∕ L, where L = { (v, − v) : x ∈ X ∩ Y } is a closed linear subspace, so X + Y is a Banach space. Let $\Omega =\{ z \in \mathbb{C} : 0 < \text{ Re}\,z < 1\}$, with closure $\overline{\Omega }$. Define ${\mathcal{H}}_{\Omega }(X,Y)$ to be the space of functions $f : \overline{\Omega } \rightarrow Z = X + Y$, continuous on $\overline{\Omega }$, holomorphic on Ω (with values in X + Y), satisfying f: {Im z = 0} → X continuous, f: {Im z = 1} → Y continuous, and

$$\|u{(z)\|}_{Z} \leq C,\quad \|u{(iy)\|}_{X} \leq C,\quad \|u{(1 + iy)\|}_{Y } \leq C,$$

(A.2)

for some C < ∞, independent of $z \in \overline{\Omega }$ and $y \in \mathbb{R}$. Then, for θ ∈ (0, 1),

$${[X,Y ]}_{\theta } =\{ u(\theta) : u \in {\mathcal{H}}_{\Omega }(X,Y)\}.$$

(A.3)

One has

$${[X,Y ]}_{\theta } \approx {\mathcal{H}}_{\Omega }(X,Y)/\{u \in {\mathcal{H}}_{\Omega }(X,Y) : u(\theta) = 0\},$$

(A.4)

giving [X, Y]_θ the sructure of a Banach space. Here

$$\|{u\|}_{{\mathcal{H}}_{\Omega }(X,Y)} {=\sup\limits_{z\in \overline{\Omega }}}\,\|u{(z)\|}_{Z} {+\sup\limits_{y}}\,\|u{(iy)\|}_{X} {+\sup\limits_{y}}\,\|u{(1 + iy)\|}_{Y }.$$

(A.5)

If I is an interval in $\mathbb{R}$, we say a family of Banach spaces X_s, s ∈ I (subspaces of V) forms a complex interpolation scale provided that for s, t ∈ I, θ ∈ (0, 1),

$${[{X}_{s},{X}_{t}]}_{\theta } = {X}_{(1-\theta)s+\theta t}.$$

(A.6)

Examples of such scales include L^p-Sobolev spaces ${X}_{s} = {H}^{s,p}(M),\ s \in \mathbb{R}$, provided p ∈ (1, ∞), as shown in § 6 of this chapter, the case p = 2 having been done in Chap. 4. It turns out that (A.6) fails for Zygmund spaces ${X}_{s} = {C}_{\ast}^{s}(M)$, but an analogous identity holds for some closely related interpolation functors, which we proceed to introduce.

If (X, Y, V) is a compatible triple, as defined in above, we define ${\mathcal{H}}_{\Omega }(X,Y,V)$ to be the space of functions $u : \overline{\Omega } \rightarrow X + Y = Z$ such that

$$u : \Omega \rightarrow Z\ \text{ is holomorphic,}$$

(A.7)

$$\|u{(z)\|}_{Z} \leq C,\quad \|u{(iy)\|}_{X} \leq C,\quad \|u{(1 + iy)\|}_{Y } \leq C,$$

(A.8)

and

$$u : \overline{\Omega }\rightarrow V \ \text{ is continuous}.$$

(A.9)

For such u, we again use the norm (A.5). Note that the only difference with ${\mathcal{H}}_{\Omega }(X,Y)$ is that we are relaxing the continuity hypothesis for u on $\overline{\Omega }$. ${\mathcal{H}}_{\Omega }(X,Y,V)$ is also a Banach space, and we have a natural isometric inclusion

$${\mathcal{H}}_{\Omega }(X,Y)\hookrightarrow {\mathcal{H}}_{\Omega }(X,Y,V).$$

(A.10)

Now for θ ∈ (0, 1) we set

$${[X,Y ]}_{\theta ;V } =\{ u(\theta) : u \in {\mathcal{H}}_{\Omega }(X,Y,V)\}.$$

(A.11)

Again this space gets a Banach space structure, via

$${[X,Y ]}_{\theta ;V } \approx {\mathcal{H}}_{\Omega }(X,Y,V)/\{u \in {\mathcal{H}}_{\Omega }(X,Y,V) : u(\theta) = 0\},$$

(A.12)

and there is a natural continuous injection

$${[X,Y ]}_{\theta }\hookrightarrow {[X,Y ]}_{\theta ;V }.$$

(A.13)

Sometimes this is an isomorphism. In fact, sometimes [X, Y]_θ = [X, Y]_{θ; V} for practically all reasonable choices of V. For example, one can verify this for $X = {L}^{p}({\mathbb{R}}^{n}),\ Y = {H}^{s,p}({\mathbb{R}}^{n})$, the L^p-Sobolev space, with p ∈ (1, ∞), s ∈ (0, ∞). On the other hand, there are cases where equality in (A.10) does not hold, and where [X, Y]_{θ; V} is of greater interest than [X, Y]_θ.

We next define [X, Y]_θ^b. In this case we assume X and Y are Banach spaces and Y ⊂ X (continuously). We take Ω as above, and set $\widetilde{\Omega } =\{ z \in \mathbb{C} : 0 < \text{ Re}\,z \leq 1\}$, i.e., we throw in the right boundary but not the left boundary. We then define ${\mathcal{H}}_{\Omega }^{b}(X,Y)$ to be the space of functions $u :\widetilde{ \Omega } \rightarrow X$ such that

$$\begin{array}{rcl} & u : \Omega \rightarrow X\ \text{ is holomorphic,} & \\ & \|u{(z)\|}_{X} \leq C,\quad \|u{(1 + iy)\|}_{Y } \leq C,& \\ & u :\widetilde{ \Omega }\rightarrow X\ \text{ is continuous}.\end{array}$$

(A.14)

Note that the essential difference between ${\mathcal{H}}_{\Omega }(X,Y)$ and the space we have just introduced is that we have completely dropped any continuity requirement at {{ Re} z = 0}. We also do not require continuity from {{ Re} z = 1} to Y. The space ${\mathcal{H}}_{\Omega }^{b}(X,Y)$ is a Banach space, with norm

$$\|{u\|}_{{\mathcal{H}}_{\Omega }^{b}(X,Y)} {=\sup\limits_{z\in \widetilde{\Omega }}}\,\|u{(z)\|}_{X} {+\sup\limits_{y}}\,\|u{(1 + iy)\|}_{Y }. $$

(A.15)

Now, for θ ∈ (0, 1), we set

$${[X,Y ]}_{\theta }^{b} =\{ u(\theta) : u \in {\mathcal{H}}_{ \Omega }^{b}(X,Y)\},$$

(A.16)

with the same sort of Banach space structure as arose in (A.4) and (A.12). We have continuous injections

$${[X,Y ]}_{\theta }\hookrightarrow {[X,Y ]}_{\theta ;X}\hookrightarrow {[X,Y ]}_{\theta }^{b}.$$

(A.17)

Our next task is to extend the standard result on operator interpolation from the setting of [X, Y]_θ to that of [X, Y]_{θ; V} and ${[X,Y ]}_{\theta }^{b}$.

Proposition A.1.

Let (X_j,Y_j,V_j) be compatible triples, j = 1,2. Assume that T : V₁ → V₂ is continuous and that

$$T : {X}_{1}\rightarrow {X}_{2},\quad T : {Y }_{1}\rightarrow {Y }_{2},$$

(A.18)

continuously. (Continuity is automatic, by the closed graph theorem.) Then, for each θ ∈ (0,1),

$$T : {[{X}_{1},{Y }_{1}]}_{\theta ;{V }_{1}}\rightarrow {[{X}_{2},{Y }_{2}]}_{\theta ;{V }_{2}}.$$

(A.19)

Furthermore, if Y_j ⊂ X_j (continuously) and T is a continuous linear map satisfying (A.18), then for each θ ∈ (0,1),

$$T : {[{X}_{1},{Y }_{1}]}_{\theta }^{b}\rightarrow {[{X}_{ 2},{Y }_{2}]}_{\theta }^{b}.$$

(A.20)

Proof.

Given f ∈ [X₁, Y₂]_{θ; V}, pick $u \in {\mathcal{H}}_{\Omega }({X}_{1},{Y }_{1},{V }_{1})$ such that f = u(θ). Then we have

$$\mathcal{T} : {\mathcal{H}}_{\Omega }({X}_{1},{Y }_{1},{V }_{1}) \rightarrow {\mathcal{H}}_{\Omega }({X}_{2},{Y }_{2},{V }_{2}),\quad (\mathcal{T} u)(z) = Tu(z),$$

(A.21)

and hence

$$Tf = (\mathcal{T} u)(\theta) \in {[{X}_{2},{Y }_{2}]}_{\theta ;{V }_{2}}. $$

(A.22)

This proves (A.19). The proof of (A.20) is similar.

Remark:

In case V = X + Y, with the weak topology, [X, Y]_{θ; V} is what is denoted ${(X,Y)}_{\theta }^{w}$ in (JJ), and called the weak complex interpolation space.

Alternatives to (A.6) for a family X_s of Banach spaces include

$${[{X}_{s},{X}_{t}]}_{\theta ;V } = {X}_{(1-\theta)s+\theta t} $$

(A.23)

and

$${[{X}_{s},{X}_{t}]}_{\theta }^{b} = {X}_{ (1-\theta)s+\theta t}.$$

(A.24)

Here, as before, we take θ ∈ (0, 1). It is an exercise, using results of § 6, to show that both (A.23) and (A.24), as well as (A.6), hold when X_s = H^{s, p}(M), given p ∈ (1, ∞), where M can be ${\mathbb{R}}^{n}$ or a compact Riemannian manifold. We now discuss the situation for Zygmund spaces.

We start with Zygmund spaces on the torus ${\mathbb{T}}^{n}$. We recall from § 8 that the Zygmund space ${C}_{\ast}^{r}({\mathbb{T}}^{n})$ is defined for $r \in \mathbb{R}$, as follows. Take $\varphi \in {C}_{0}^{\infty }({\mathbb{R}}^{n})$, radial, satisfying φ(ξ) = 1 for |ξ| ≤ 1. Set φ_k(ξ) = φ(2^{− k}ξ). Then set ψ₀ = φ, ψ_k = φ_k − φ_{k − 1} for $k \in \mathbb{N}$, so {ψ_k : k ≥ 0} is a Littlewood–Paley partition of unity. We define ${C}_{\ast}^{r}({\mathbb{T}}^{n})$ to consist of $f \in \mathcal{D}^\prime({\mathbb{T}}^{n})$ such that

$$\|{f\|}_{{C}_{\ast}^{r}} {=\sup\limits_{k\geq 0}}\,{2}^{kr}\|{\psi }_{ k}(D){f\|}_{{L}^{\infty }} < \infty. $$

(A.25)

With Λ = (I − Δ)^{1 ∕ 2} and $s,t \in \mathbb{R}$, we have

$${\Lambda }^{s+it} : {C}_{ {\ast}}^{r}({\mathbb{T}}^{n})\rightarrow {C}_{ {\ast}}^{r-s}({\mathbb{T}}^{n}).$$

(A.26)

By material developed in § 8,

$$r \in {\mathbb{R}}^{+} \setminus {\mathbb{Z}}^{+}\Longrightarrow{C}_{ {\ast}}^{r}({\mathbb{T}}^{n}) = {C}^{r}({\mathbb{T}}^{n}),$$

(A.27)

where, if r = k + α with $k \in {\mathbb{Z}}^{+}$ and 0 < α < 1, ${C}^{r}({\mathbb{T}}^{n})$ consists of functions whose derivatives of order ≤ k are Hölder continuous of exponent α.

We aim to show the following.

Proposition A.2.

If r < s < t and 0 < θ < 1, then

$${[{C}_{\ast}^{s}({\mathbb{T}}^{n}),{C}_{ {\ast}}^{t}({\mathbb{T}}^{n})]}_{ \theta ;{C}_{\ast}^{r}({\mathbb{T}}^{n})} = {C}_{\ast}^{(1-\theta)s+\theta t}({\mathbb{T}}^{n}),$$

(A.28)

and

$${[{C}_{\ast}^{s}({\mathbb{T}}^{n}),{C}_{ {\ast}}^{t}({\mathbb{T}}^{n})]}_{ \theta }^{b} = {C}_{ {\ast}}^{(1-\theta)s+\theta t}({\mathbb{T}}^{n}). $$

(A.29)

Proof.

First, suppose $f \in {[{C}_{\ast}^{s},{C}_{\ast}^{t}]}_{\theta ;{C}_{\ast}^{r}}$, so f = u(θ) for some $u \in {\mathcal{H}}_{\Omega }({C}_{\ast}^{s},{C}_{\ast}^{t},{C}_{\ast}^{r})$. Then consider

$$v(z) = {e}^{{z}^{2} }{\Lambda }^{(t-s)z}{\Lambda }^{s}u(z).$$

(A.30)

Bounds of the type (A.8) on u, together with (8.13) in the torus setting, yield

$$\|v{(iy)\|}_{{C}_{\ast}^{0}},\ \|v{(1 + iy)\|}_{{C}_{\ast}^{0}} \leq C,$$

(A.31)

with C independent of $y \in \mathbb{R}$. In other words,

$$\|{\psi }_{k}(D)v{(z)\|}_{{L}^{\infty }} \leq C,\quad \text{ Re}\,z = 0,1,$$

(A.32)

with C independent of Im z and k. Also, for each $k \in {\mathbb{Z}}^{+}$, ${\psi }_{k}(D)v : \overline{\Omega } \rightarrow {L}^{\infty }({\mathbb{T}}^{n})$ continuously, so the maximum principle implies

$$\|{\psi }_{k}(D){\Lambda }^{(t-s)\theta }{\Lambda }^{s}{f\|}_{{ L}^{\infty }} \leq C,$$

(A.33)

independent of $k \in {\mathbb{Z}}^{+}$. This gives ${\Lambda }^{(1-\theta)s+\theta t}f \in {C}_{\ast}^{0}$, hence $f \in {C}_{\ast}^{(1-\theta)s+\theta t}({\mathbb{T}}^{n})$.

Second, suppose $f \in {C}_{\ast}^{(1-\theta)s+\theta t}({\mathbb{T}}^{n})$. Set

$$u(z) = {e}^{{z}^{2} }{\Lambda }^{(\theta -z)(t-s)}f.$$

(A.34)

Then $u(\theta) = {e}^{{\theta }^{2} }f$. We claim that

$$u \in {\mathcal{H}}_{\Omega }({C}_{\ast}^{s},{C}_{ {\ast}}^{t},{C}_{ {\ast}}^{r}),$$

(A.35)

as long as r < s < t. Once we establish this, we will have the reverse containment in (A.28). Bounds of the form

$$\|u{(z)\|}_{{C}_{\ast}^{s}} \leq C,\quad \|u{(1 + iy)\|}_{{C}_{\ast}^{t}} \leq C$$

(A.36)

follow from (8.13), and are more than adequate versions of (A.8). It remains to establish that

$$u : \overline{\Omega }\rightarrow {C}_{\ast}^{r}({\mathbb{T}}^{n}),\ \text{ continuously.}$$

(A.37)

Indeed, we know $u : \overline{\Omega } \rightarrow {C}_{\ast}^{s}({\mathbb{T}}^{n})$ is bounded. It is readily verified that

$$u : \overline{\Omega }\rightarrow \mathcal{D}^\prime({\mathbb{T}}^{n}),\ \text{ continuously,} $$

(A.38)

and that

$$r < s\Longrightarrow{C}_{\ast}^{s}({\mathbb{T}}^{n})\hookrightarrow {C}_{ {\ast}}^{r}({\mathbb{T}}^{n})\ \text{ is compact.}$$

(A.39)

The result (A.37) follows from these observations. Thus the proof of (A.28) is complete.

We turn to the proof of (A.29). If $u \in {\mathcal{H}}_{\Omega }^{b}({C}_{\ast}^{s},{C}_{\ast}^{t})$, form v(z) as in (A.30), and for ε ∈ (0, 1] set

$${v}_{\epsilon }(z) = {e}^{-\epsilon \Lambda }v(z),\quad {v}_{ \epsilon } :\widetilde{ \Omega } \rightarrow {C}_{\ast}^{0}({\mathbb{T}}^{n})\ \text{ bounded and continuous}$$

(A.40)

(with bound that might depend on ε). We have

$${\psi }_{k}(D){v}_{\epsilon }(\epsilon + iy) = {e}^{{(\epsilon +iy)}^{2} }{\psi }_{k}(D){e}^{-\epsilon \Lambda }{\Lambda }^{(t-s)\epsilon }{\Lambda }^{i(t-s)y}{\Lambda }^{s}u(z).$$

(A.41)

Now $\{{\Lambda }^{s}u(z) : z \in \widetilde{ \Omega }\}$ is bounded in ${C}_{\ast}^{0}({\mathbb{T}}^{n})$, and the operator norm of Λ^{i(t − s)y} on ${C}_{\ast}^{0}({\mathbb{T}}^{n})$ is exponentially bounded in |y|. We have

$$\{{e}^{-\epsilon \Lambda }{\Lambda }^{\epsilon (t-s)} : 0 < \epsilon \leq 1\}\ \text{ bounded in }{OPS }_{ 1,0}^{0}({\mathbb{T}}^{n}), $$

(A.42)

hence bounded in operator norm on ${C}_{\ast}^{0}({\mathbb{T}}^{n})$. We deduce that

$$\|{\psi }_{k}(D){v}_{\epsilon }{(\epsilon + iy)\|}_{{L}^{\infty }} \leq C,$$

(A.43)

independent of $y \in \mathbb{R}$ and ε ∈ (0, 1]. The hypothesis on u also implies

$$\|{\psi }_{k}(D){v}_{\epsilon }{(1 + iy)\|}_{{L}^{\infty }} \leq C,$$

(A.44)

independent of $y \in \mathbb{R}$ and ε ∈ (0, 1]. Now the maximum principle applies. Given θ ∈ (0, 1),

$$\|{\psi }_{k}(D){e}^{-\epsilon \Lambda }v{(\theta)\|}_{{ L}^{\infty }} \leq C,$$

(A.45)

independent of ε. Taking ε ↘ 0 yields $v(\theta) \in {C}_{\ast}^{0}({\mathbb{T}}^{n})$, hence $u(\epsilon) \in {C}_{\ast}^{(1-\theta)s+\theta t}({\mathbb{T}}^{n})$.

This proves one inclusion in (A.29). The proof of the reverse inclusion is similar to that for (A.28). Given $f \in {C}_{\ast}^{(1-\theta)s+\theta t}({\mathbb{T}}^{n})$, take u(z) as in (A.34). The claim is that $u \in {\mathcal{H}}_{\Omega }^{b}({C}_{\ast}^{s},{C}_{\ast}^{t})$. We already have (A.36), and the only thing that remains is to check that

$$u :\widetilde{ \Omega }\rightarrow {C}_{\ast}^{s}({\mathbb{T}}^{n})\ \text{ continuously},$$

(A.46)

and this is straightforward. (What fails is continuity of $u : \overline{\Omega } \rightarrow {C}_{\ast}^{s}({\mathbb{T}}^{n})$ at the left boundary of $\overline{\Omega }$.)

Remark:

In contrast to (A.28)–(A.29), one has

$${[{C}_{\ast}^{s}({\mathbb{T}}^{n}),{C}_{ {\ast}}^{t}({\mathbb{T}}^{n})]}_{ \theta } = \text{ closure of }{C}^{\infty }({\mathbb{T}}^{n})\text{ in }{C}_{ {\ast}}^{(1-\theta)s+\theta t}({\mathbb{T}}^{n}).$$

(A.47)

Related results are given in [Tri].

If ${OPS}_{1,0}^{m}({\mathbb{T}}^{n})$ denotes the class of pseudodifferential operators on ${\mathbb{T}}^{n}$ with symbols in ${S}_{1,0}^{m}$, then for all $s,m \in \mathbb{R}$,

$$P \in { OPS}_{1,0}^{m}({\mathbb{T}}^{n})\Longrightarrow P : {C}_{ {\ast}}^{s}({\mathbb{T}}^{n}) \rightarrow {C}_{ {\ast}}^{s-m}({\mathbb{T}}^{n}). $$

(A.48)

Cf. Proposition 8.6. Using coordinate invariance of $OP{S}_{1,0}^{m}$ and of ${C}^{r}({\mathbb{T}}^{n})$ for $r \in {\mathbb{R}}^{+} \setminus {\mathbb{Z}}^{+}$, we deduce invariance of ${C}_{\ast}^{s}({\mathbb{T}}^{n})$ under diffeomorphisms, for all $s \in \mathbb{R}$.

From here, we can develop the spaces ${C}_{\ast}^{s}(M)$ on a compact Riemannian manifold M and the spaces ${C}_{\ast}^{s}(\overline{M})$ on a compact manifold with boundary. These developments are done in § 8.