Functional Analysis: Function Spaces¶
Job: develop Sobolev, Besov, and Triebel–Lizorkin spaces at frontier depth — the Sobolev embedding and Rellich–Kondrachov compactness, the Littlewood–Paley characterization of Besov / TL spaces, the real and complex interpolation functors (Riesz–Thorin, Marcinkiewicz), and the Mihlin–Hörmander multiplier theorem.
Core¶
Sobolev spaces¶
For \(\Omega\subset\mathbb{R}^n\) open and \(k\in\mathbb{N}\), \(1\le p\le\infty\), the Sobolev space \(W^{k,p}(\Omega)=\{u\in L^p:D^\alpha u\in L^p\;\forall|\alpha|\le k\}\) (weak derivatives). For \(p=2\), \(H^k(\Omega)=W^{k,2}(\Omega)\). For fractional \(s\in\mathbb{R}\), \(H^s(\mathbb{R}^n)=(1-\Delta)^{-s/2}L^2=\{u:\langle\xi\rangle^s\hat u\in L^2\}\) via the Bessel potential (the Sobolev–Hilbert scale).
Theorem (Sobolev embedding). Let \(1\le p<\infty\), \(k\in\mathbb{N}\), \(\Omega\) a bounded Lipschitz domain. (i) Subcritical embedding: if \(kp<n\) and \(1/q=1/p-k/n\), then \(W^{k,p}(\Omega)\hookrightarrow L^q(\Omega)\) continuously. (ii) Critical / Morrey embedding: if \(kp>n\) and \(0<\alpha<k-n/p\), then \(W^{k,p}(\Omega)\hookrightarrow C^{0,\alpha}(\overline\Omega)\) continuously (Hölder continuous). (iii) Limit case (\(kp=n\)): \(W^{k,p}\hookrightarrow L^q\) for all \(q<\infty\) but not \(L^\infty\).
The critical exponent is \(p^\ast=np/(n-kp)\) for \(kp<n\); the embedding is \(W^{k,p}\hookrightarrow L^{p^\ast}\).
Theorem (Rellich–Kondrachov). For a bounded Lipschitz domain \(\Omega\) and \(1\le p<n\), \(W^{1,p}_0(\Omega)\hookrightarrow\hookrightarrow L^q(\Omega)\) (compact embedding) for \(1\le q<p^\ast=np/(n-p)\). In particular \(H^1_0(\Omega)\hookrightarrow\hookrightarrow L^2(\Omega)\) for \(n\ge 3\) (the compactness used in the spectral theorem for the Dirichlet Laplacian).
| Embedding | Condition | Type |
|---|---|---|
| \(W^{k,p}\hookrightarrow L^q\) (\(1/q=1/p-k/n\)) | \(kp<n\) | continuous (subcritical) |
| \(W^{k,p}\hookrightarrow C^{0,\alpha}\) (\(\alpha=k-n/p\)) | \(kp>n\) | continuous (Morrey) |
| \(W^{k,p}\hookrightarrow L^q\) for \(q<\infty\) | \(kp=n\) | continuous (limit) |
| \(W^{1,p}_0\hookrightarrow\hookrightarrow L^q\) (\(q<p^\ast\)) | \(1\le p<n\), bounded \(\Omega\) | compact (Rellich–Kondrachov) |
| \(H^1_0\hookrightarrow\hookrightarrow L^2\) | \(n\ge 3\), bounded \(\Omega\) | compact (the spectral-theory input) |
Besov and Triebel–Lizorkin spaces via Littlewood–Paley¶
Let \(\varphi\in C_c^\infty(\mathbb{R}^n)\) with \(\varphi(\xi)=1\) for \(|\xi|\le 1\) and \(\mathrm{supp}\,\varphi\subset B_2\); set \(\varphi_j(\xi)=\varphi(2^{-j}\xi)-\varphi(2^{-(j-1)}\xi)\) for \(j\ge 1\) (a dyadic partition of unity, supported on \(|\xi|\sim 2^j\)) and \(\Delta_j u=\mathcal{F}^{-1}(\varphi_j\hat u)\) the Littlewood–Paley blocks. For \(s\in\mathbb{R}\), \(1\le p,q\le\infty\):
the Besov and Triebel–Lizorkin spaces. The distinction is the order of the norms: Besov takes the \(\ell^q\) norm over frequencies inside the \(L^p\) norm over space; TL takes the \(L^p\) norm outside the \(\ell^q\) sum over frequencies.
| Classical space | Besov / TL identification |
|---|---|
| \(H^s(\mathbb{R}^n)=W^{s,2}\) | \(F^s_{2,2}=B^s_{2,2}\) (Littlewood–Paley equality) |
| \(W^{k,p}(\mathbb{R}^n)\) (\(1<p<\infty\)) | \(F^k_{p,2}\) (Lizorkin representation) |
| \(C^{k,\alpha}(\mathbb{R}^n)\) (\(0<\alpha<1\)) | \(B^{k+\alpha}_{\infty,\infty}\) |
| \(\mathrm{BMO}(\mathbb{R}^n)\) | \(F^0_{\infty,2}\) (up to polynomials) |
| \(\mathrm{BV}(\mathbb{R}^n)\) (bounded variation) | \(B^1_{1,1}\) (embedding; not an equality) |
| \(L^p(\mathbb{R}^n)\) (\(1<p<\infty\)) | \(F^0_{p,2}\) (Littlewood–Paley) |
Interpolation theory¶
Real interpolation (K-method). For a compatible couple \((X_0,X_1)\) of Banach spaces (both embedded in a Hausdorff topological vector space), the real interpolation space \((X_0,X_1)_{\theta,q}\) (\(0<\theta<1\), \(1\le q\le\infty\)) is the space of \(x\) with finite K-functional \(K(t,x)=\inf_{x=x_0+x_1}(\|x_0\|_{X_0}+t\|x_1\|_{X_1})\) and \(\|x\|_{\theta,q}=\bigl(t^{-\theta}K(t,x)\bigr)_{\ell^q(dt/t)}<\infty\).
Complex interpolation (Calderón). The complex interpolation space \([X_0,X_1]_\theta\) is the space of values \(F(\theta)\) of bounded analytic functions \(F:S\to X_0+X_1\) (on the strip \(S=\{0<\mathrm{Re}\,z<1\}\), continuous on \(\overline S\), with \(F(it)\in X_0\), \(F(1+it)\in X_1\)).
Theorem (Riesz–Thorin, complex interpolation of \(L^p\)). For \(1/p=(1-\theta)/p_0+\theta/p_1\) and \(1/q=(1-\theta)/q_0+\theta/q_1\), \([L^{p_0},L^{p_1}]_\theta=L^p\) and \([L^{q_0},L^{q_1}]_\theta=L^q\), and an operator \(T\) bounded \(L^{p_0}\to L^{q_0}\) and \(L^{p_1}\to L^{q_1}\) is bounded \(L^p\to L^q\) with \(\|T\|_{L^p\to L^q}\le\|T\|_{L^{p_0}\to L^{q_0}}^{1-\theta}\|T\|_{L^{p_1}\to L^{q_1}}^\theta\).
Theorem (Marcinkiewicz, real interpolation). If \(T\) is weak-type \((p_0,q_0)\) and \((p_1,q_1)\) (i.e. \(\|Tf\|_{L^{q_i,\infty}}\le C_i\|f\|_{L^{p_i}}\)), then \(T\) is strong-type \((p,q)\) for \(1/p=(1-\theta)/p_0+\theta/p_1\), \(1/q=(1-\theta)/q_0+\theta/q_1\), with \(\|T\|_{L^p\to L^q}\le C\) (the real-interpolation version, allowing weak endpoints).
| Interpolation theorem | Method | Hypotheses | Conclusion |
|---|---|---|---|
| Riesz–Thorin | complex | \(T\) bounded \(L^{p_0}\to L^{q_0}\) and \(L^{p_1}\to L^{q_1}\) (strong) | \(T\) bounded \(L^p\to L^q\) (interpolated) |
| Marcinkiewicz | real | \(T\) weak-type \((p_0,q_0)\) and \((p_1,q_1)\) | \(T\) strong-type \((p,q)\) (interpolated, with \(p_i<q_i\)) |
| Stein interpolation | complex / analytic family | \(T_z\) analytic family, bounded at endpoints | \(T_\theta\) bounded at interior |
| Real interpolation of Besov | real | \((B^{s_0}_{p,q_0},B^{s_1}_{p,q_1})_{\theta,q}=B^s_{p,q}\) | \(s=(1-\theta)s_0+\theta s_1\) |
Fourier multiplier theorems¶
Theorem (Mihlin–Hörmander). Let \(m:\mathbb{R}^n\setminus\{0\}\to\mathbb{C}\) be a bounded function with Mihlin derivative bounds $$ |\partial^\alpha m(\xi)|\le C_\alpha|\xi|^{-|\alpha|}\quad\text{for }|\alpha|\le \lfloor n/2\rfloor+1. $$ Then the Fourier multiplier \(T_m f=\mathcal{F}^{-1}(m\hat f)\) is bounded on \(L^p(\mathbb{R}^n)\) for \(1<p<\infty\), with \(\|T_m\|_{L^p\to L^p}\le C_p\sup_\alpha C_\alpha\). The number of derivatives required is \(\lfloor n/2\rfloor+1\) (strictly more than \(n/2\)); the zeroth-order bound \(|m|\le C\) alone is insufficient.
The Mihlin theorem is the \(L^p\) boundedness of multipliers with controlled derivatives, the foundation of the Calderón–Zygmund / singular-integral theory (the link to harmonic analysis and to the PDE elliptic theory).
See it¶
The Sobolev embedding \(H^1_0(\Omega)\hookrightarrow L^{2^\ast}(\Omega)\) and the Rellich compactness, as the input to the Dirichlet Laplacian spectral theorem. Let \(\Omega\subset\mathbb{R}^n\) bounded Lipschitz, \(n\ge 3\), \(2^\ast=2n/(n-2)\). The Gagliardo–Nirenberg–Sobolev inequality:
proved by the integration-by-parts / scaling argument: for \(u\in C_c^\infty(\Omega)\), the estimate \(|u(x)|\le C\int_{\mathbb{R}^n}\frac{|\nabla u(y)|}{|x-y|^{n-1}}\,dy\) (the Riesz potential / Hardy–Littlewood–Sobolev) gives, after applying the HLS inequality and the critical exponent, the embedding \(H^1_0\hookrightarrow L^{2^\ast}\). The constant \(C\) depends on \(n\) only (the sharp constant is the Aubin–Talenti constant, related to the Yamabe problem).
The Rellich–Kondrachov compactness \(H^1_0(\Omega)\hookrightarrow\hookrightarrow L^2(\Omega)\): a bounded sequence in \(H^1_0\) has a convergent subsequence in \(L^2\). The proof uses the Arzelà–Ascoli / Fréchet–Kolmogorov compactness criterion in \(L^2\) (equi-continuity via the gradient bound + compactness of translations + extension to a larger domain); the embedding is compact because \(H^1_0\) is "smaller" than \(L^2\) by one derivative, and the Rellich compactness is the analog of the Arzelà–Ascoli theorem for Sobolev functions.
The spectral-theory input: the Dirichlet Laplacian \(-\Delta_D\) on \(\Omega\) (the Friedrichs extension, see Spectral Theory) has the Rayleigh quotient \(R[u]=\|\nabla u\|^2/\|u\|^2\) on \(H^1_0(\Omega)\). The compactness \(H^1_0\hookrightarrow\hookrightarrow L^2\) implies the resolvent \((-\Delta_D-\lambda)^{-1}:L^2\to L^2\) is compact (it factors through \(H^1_0\hookrightarrow\hookrightarrow L^2\)), so by the spectral theorem for compact self-adjoint operators the spectrum of \(-\Delta_D\) is discrete: a sequence of eigenvalues \(0<\lambda_1\le\lambda_2\le\cdots\to\infty\) with finite-multiplicity eigenfunctions \(u_k\in H^1_0\cap H^2\). The eigenvalues are the min-max values \(\lambda_k=\min_{\dim V=k}\max_{u\in V\setminus\{0\}}R[u]\) (the Courant–Fischer / Rayleigh–Ritz), and Weyl's law gives the asymptotics \(\lambda_k\sim(2\pi)^2(\omega_n|\Omega|)^{-2/n}k^{2/n}\) (the dimension-dependent growth). The single embedding \(H^1_0\hookrightarrow\hookrightarrow L^2\) is thus the input to the entire spectral theory of the Dirichlet Laplacian: the discreteness of the spectrum (compact resolvent), the eigenbasis, and the Weyl asymptotics — the operational content of "Sobolev embedding + Rellich compactness \(\Rightarrow\) spectral theorem for elliptic operators on bounded domains."
Check¶
Compute the critical exponent \(2^\ast=2n/(n-2)\) for the embedding \(H^1_0\hookrightarrow L^{2^\ast}\), and identify the dimension range where the embedding is compact into \(L^2\).
The scaling: \(H^1_0\) has the scaling \(u_\lambda(x)=u(\lambda x)\) with \(\|\nabla u_\lambda\|_{L^2}=\lambda^{n/2-1}\|\nabla u\|_{L^2}\) and \(\|u_\lambda\|_{L^q}=\lambda^{n/q}\|u\|_{L^q}\); the embedding \(\|u\|_{L^q}\le C\|\nabla u\|_{L^2}\) is scale-invariant when \(n/q=n/2-1\), i.e. \(q=2n/(n-2)=2^\ast\) (for \(n\ge 3\)). For \(n=2\) the critical exponent is \(2^\ast=\infty\) (the embedding \(H^1_0\hookrightarrow L^q\) for all \(q<\infty\), but not \(L^\infty\) by the Moser–Trudinger inequality); for \(n=1\), \(H^1_0\hookrightarrow C^0\) (the Morrey embedding). The compact embedding \(H^1_0\hookrightarrow\hookrightarrow L^2\) holds for \(n\ge 3\) by Rellich–Kondrachov (\(2<2^\ast\), so \(L^2\) is below the critical exponent, giving compactness); for \(n=1,2\) the compactness also holds (\(H^1_0\hookrightarrow\hookrightarrow L^q\) for \(q<2^\ast=\infty\) in \(n=2\), including \(q=2\); for \(n=1\), \(H^1_0\hookrightarrow\hookrightarrow C^0\) and hence \(L^2\)).
Explain the distinction between Besov \(B^s_{p,q}\) and Triebel–Lizorkin \(F^s_{p,q}\) via the ordering of the norms, and identify which classical spaces are realized as both / as one but not the other.
Besov takes the \(\ell^q\) norm over the dyadic frequencies inside the \(L^p\) norm over space: \(\|u\|_{B^s_{p,q}}=\bigl\|\bigl(2^{js}\|\Delta_j u\|_{L^p}\bigr)_{\ell^q}\bigr\|\), i.e. for each frequency \(j\) take the \(L^p\) norm, then sum over \(j\) with the \(\ell^q\) weight. Triebel–Lizorkin takes the \(L^p\) norm outside the \(\ell^q\) sum: \(\|u\|_{F^s_{p,q}}=\bigl\|\bigl(\sum_j 2^{jsq}|\Delta_j u|^q\bigr)^{1/q}\bigr\|_{L^p}\), i.e. at each point \(x\) sum over \(j\) with the \(\ell^q\) weight (a square-function estimate), then take the \(L^p\) norm. For \(p=q=2\) the two coincide: \(B^s_{2,2}=F^s_{2,2}=H^s\) (the Littlewood–Paley equality). For \(p\ne 2\) they differ: \(W^{k,p}=F^k_{p,2}\) for \(1<p<\infty\) (the Lizorkin representation, a square-function characterization) but \(W^{k,p}\ne B^k_{p,q}\) in general (the Besov with \(q\ne 2\) is a strict refinement — the Sobolev space corresponds to \(q=2\) in the TL scale). The Hölder space \(C^{k,\alpha}=B^{k+\alpha}_{\infty,\infty}\) is Besov, not TL (the \(L^\infty\) version of TL with \(q=2\) is BMO, not Hölder). Hence: Besov captures the Hölder / BV scale (extremal \(L^p\) and \(\ell^q\)); TL captures the Sobolev / BMO scale (the \(L^p\) square-function side). They agree at \(p=q=2\) and diverge elsewhere, with the TL side aligned to the classical PDE Sobolev / BMO spaces and the Besov side aligned to the Hölder / BV spaces.
Pitfalls¶
- Besov \(\ne\) Triebel–Lizorkin (different \(\ell^q\) / \(L^p\) ordering): \(B^s_{p,q}\) has \(\ell^q\) inside \(L^p\); \(F^s_{p,q}\) has \(L^p\) outside \(\ell^q\). They coincide for \(p=q=2\) (both \(=H^s\)) but differ in general. Conflating them is the standard error.
- The Mihlin multiplier needs derivative bounds of order \(>n/2\) (not just the zeroth-order bound): \(|\partial^\alpha m|\le C|\xi|^{-|\alpha|}\) for \(|\alpha|\le\lfloor n/2\rfloor+1\); the zeroth-order \(|m|\le C\) alone is insufficient (there are bounded multipliers that are not \(L^p\) multipliers for \(p\ne 2\)). The required order is the Sobolev embedding threshold \(s>n/2\) for \(L^\infty\) control of the derivatives.
- The Rellich compactness requires a bounded domain: on \(\mathbb{R}^n\), \(H^1(\mathbb{R}^n)\hookrightarrow L^{2^\ast}\) is continuous but not compact (the translations escape); the compactness holds on bounded Lipschitz domains (or with weights / decaying potentials).
- The Sobolev critical exponent \(p^\ast\) is the borderline of the embedding: \(W^{k,p}\hookrightarrow L^{p^\ast}\) is continuous but not compact at \(p^\ast\) (compact only for \(q<p^\ast\)); the Moser–Trudinger inequality handles the limit case (\(kp=n\)).
- The complex interpolation \([L^{p_0},L^{p_1}]_\theta=L^p\) is an equality, but the real interpolation \((L^{p_0},L^{p_1})_{\theta,p}=L^p\) is also an equality only for the right \(q=p\); the real interpolation \((L^{p_0},L^{p_1})_{\theta,q}\) for \(q\ne p\) is a Lorentz space \(L^{p,q}\) (a refinement of \(L^p\)), not \(L^p\) itself.
- The Marcinkiewicz interpolation requires weak-type endpoints (not strong): the weak-\(L^{q_i}\) norm \(\|Tf\|_{L^{q_i,\infty}}\le C_i\|f\|_{L^{p_i}}\); the strong-type \((p_i,q_i)\) is a stronger hypothesis, and the Riesz–Thorin theorem applies there. Do not confuse the two interpolation theorems.
Refs¶
- Adams–Fournier, Sobolev Spaces; Evans, PDE (Sobolev embedding, Rellich)
- Runst–Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear PDE; Triebel, Theory of Function Spaces (Besov, TL)
- Bergh–Löfström, Interpolation Spaces: An Introduction (real / complex interpolation)
- Stein, Singular Integrals and Differentiability Properties of Functions; Grafakos, Modern Fourier Analysis (Mihlin, Littlewood–Paley)
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