Sobolev spaces are among the fundamental tools of modern analysis and partial differential equations. This post gives a streamlined introduction to locally convex spaces, test functions, distributions, Fourier analysis, and embedding theorems.
1 Test Functions and Distributions
Notation
Let \(\Omega\) be an open subset of \(\mathbb R^n\), and let \(K\Subset\Omega\) be a relatively compact subset of \(\Omega\).
| Notation | Meaning |
|---|---|
| \(C(\Omega)\) | the space of all continuous functions on \(\Omega\) |
| \(C^k(\Omega)\) | the space of all functions on \(\Omega\) with continuous derivatives up to order \(k\) |
| \(C^\infty(\Omega)\) | the space of all infinitely differentiable functions on \(\Omega\) |
| \(C_c(\Omega)\), \(C_c^k(\Omega)\), \(C_c^\infty(\Omega)\) | the corresponding spaces of functions with compact support |
| \(L_{\mathrm{loc}}^1(\Omega)\) | the space of all locally integrable functions |
| \(W^{m,p}(\Omega)\) | the space of functions with distributional derivatives up to order \(m\) in \(L^p(\Omega)\) |
| \(H^{m,p}(\Omega)\) | the closure of \(C^\infty(\Omega)\cap W^{m,p}(\Omega)\) in \(W^{m,p}(\Omega)\) |
| \(\mathcal D(\Omega)\) | the space \(C_c^\infty(\Omega)\) endowed with the inductive limit topology |
Example 1.1 (The space \(C(\Omega)\))
Choose a sequence of compact sets \(K_0\neq\varnothing\), \(K_n\uparrow\Omega\), and define
\[p_n(f)=\sup\{|f(x)|:x\in K_n\}.\]
This is a seminorm on \(C(\Omega)\). A family \(\mathcal P\) of seminorms on a vector space \(X\) is said to be separating if, for each \(x\neq0\), there is at least one \(p\in\mathcal P\) such that \(p(x)\neq0\).
Since \(\mathcal P\) is a countable separating family of seminorms, the sets
\[V_n=\left\{f\in C(\Omega):p_n(f)<\frac1n\right\}\]
form a convex local base for \(C(\Omega)\). This topology is compatible with the metric
\[d(f,g)=\max_{n\geq1}\frac{2^{-n}p_n(f-g)}{1+p_n(f-g)},\]
which is complete. Hence \(C(\Omega)\) is a Fréchet space.
Lemma 1.1
If \(X\) is a topological vector space with a countable local base, then there is a metric \(d\) on \(X\) such that:
- \(d\) is compatible with the topology of \(X\);
- the open balls centered at \(0\) are balanced;
- \(d\) is invariant: \(d(x+z,y+z)=d(x,y)\) for \(x,y,z\in X\).
If, in addition, \(X\) is locally convex, then \(d\) can be chosen so that all open balls are convex.
Lemma 1.2
Suppose \(\mathcal P\) is a separating family of seminorms on a vector space \(X\). Associate to each \(p\in\mathcal P\) and each positive integer \(n\) the set
\[V(p,n)=\left\{x:p(x)<\frac1n\right\}.\]
Then these sets form a convex balanced local subbase for a topology \(\tau\) on \(X\), which turns \(X\) into a locally convex space such that:
- every \(p\in\mathcal P\) is continuous;
- a set \(E\subseteq X\) is bounded if and only if every \(p\in\mathcal P\) is bounded on \(E\).
Example 1.2 (The spaces \(C^k(\Omega)\), \(C^\infty(\Omega)\), and \(C_K^\infty(\Omega)\))
Choose compact sets \(K_0\neq\varnothing\), \(K_N\uparrow\Omega\), and define seminorms \(p_N\) on \(C^\infty(\Omega)\), for \(N=1,2,3,\ldots\), by
\[p_N(f)=\max\{|D^\alpha f(x)|:x\in K_N, |\alpha|<N\}.\]
For \(C^k(\Omega)\), replace the derivative bound \(N\) by \(k\). These seminorms define a metrizable locally convex topology on \(C^\infty(\Omega)\). For each \(x\in\Omega\), the functional \(f\mapsto f(x)\) is continuous in this topology.
If \(K\) is compact in \(\Omega\), then \(C_K^\infty(\Omega)\) denotes the space of all \(f\in C^\infty(\Omega)\) whose support lies in \(K\). Since \(C_K^\infty(\Omega)\) is the intersection of the null spaces of the evaluation functionals as \(x\) ranges over \(\Omega\setminus K\), it follows that \(C_K^\infty(\Omega)\) is closed in \(C^\infty(\Omega)\).
Theorem 1.1
\(C^\infty(\Omega)\) is a Fréchet space.
Proof
Let \(\{f_i\}\) be a Cauchy sequence in \(C^\infty(\Omega)\), and fix \(N\). Then \(f_i-f_j\in V_N=\{f:p_N(f)<1/N\}\) whenever \(i\) and \(j\) are sufficiently large. Thus \(|D^\alpha f_i-D^\alpha f_j|<1/N\) on \(K_N\) whenever \(|\alpha|<N\). It follows that each \(D^\alpha f_i\) converges uniformly on compact subsets of \(\Omega\) to a function \(g_\alpha\). In particular, \(f_i(x)\to g_0(x)\). It follows that \(g_0\in C^\infty(\Omega)\), that \(g_\alpha=D^\alpha g_0\), and that \(f_i\to g_0\) in the topology of \(C^\infty(\Omega)\).
Theorem 1.2
\(C^\infty(\Omega)\) is a Montel space, that is, a locally convex topological space with the Heine-Borel property.1
Proof
Suppose that \(E\subseteq C^\infty(\Omega)\) is closed and bounded. By Lemma 1.2, the boundedness of \(E\) is equivalent to the existence of numbers \(M_N<\infty\) such that \(p_N(f)<M_N\) for \(N=1,2,3,\ldots\) and all \(f\in E\). The inequalities \(|D^\alpha f|<M_N\), valid on \(K_N\) when \(|\alpha|<N\), imply the equicontinuity of \(\{D^\beta f:f\in E\}\) on \(K_{N-1}\) when \(|\beta|<N-1\). By Ascoli's theorem and Cantor's diagonal process, every sequence in \(E\) contains a subsequence \(\{f_i\}\) for which \(D^\beta f_i\) converges uniformly on compact subsets of \(\Omega\) for every multi-index \(\beta\). Hence \(\{f_i\}\) converges in the topology of \(C^\infty(\Omega)\). This proves that \(E\) is compact.
Corollary 1.1
\(C^\infty(\Omega)\) is not normable.
To make differential calculus more flexible, we enlarge the class of differentiable functions. This program was developed by L. Schwartz. He used the letter \(\mathcal D\) for \(C_0^\infty(\Omega)\) and defined a generalized function to be a linear functional on \(\mathcal D\).
Example 1.3 (The test function space \(\mathcal D(\Omega)\))
For \(\varphi\in\mathcal D(\Omega)\) and \(N=0,1,2,\ldots\), introduce the norms
\[\|\varphi\|_N=\max\{|D^\alpha\varphi(x)|:x\in\Omega,\ |\alpha|<N\}.\]
The restrictions of these norms to any fixed \(\mathcal D_K(\mathbb R^d)\subseteq\mathcal D(\Omega)\) induce the same topology on \(\mathcal D_K(\mathbb R^d)\) as the seminorms in Example 1.2. However, this topology is not complete. For example, let \(d=1\), choose \(\varphi\in\mathcal D(\mathbb R)\) with support in \([0,1]\) and \(\varphi>0\) on \((0,1)\), and define
\[\psi_m(x)=\varphi(x-1)+\frac12\varphi(x-2)+\cdots+\frac1m\varphi(x-m).\]
Then \(\{\psi_m\}\) is a Cauchy sequence in the suggested topology of \(\mathcal D(\mathbb R)\), but \(\lim\psi_m\) does not have compact support and therefore does not belong to \(\mathcal D(\mathbb R)\).
We therefore introduce another topology on \(\mathcal D(\Omega)\). Related results can be found in Köthe.2
Definition 1.1
Given a set \(X\), a family of topological spaces \(\{(Y_i,\tau_i)\}_{i\in I}\), and associated functions \(f_i:Y_i\rightarrow X, i\in I,\) the inductive topology on \(X\) induced by the family \(\{f_i:i\in I\}\) is the finest topology \(\tau\) on \(X\) such that every map \(f_i:(Y_i,\tau_i)\rightarrow(X,\tau)\) is continuous.
Theorem 1.3 (Characteristic property of the final topology)
A function \(g\) from \(X\) to a topological space \(Z\) is continuous if and only if \(g\circ f_i\) is continuous for every \(i\in I\).
\[\require{AMScd} \begin{CD} Y_i @>{f_i}>> X\\ @V{g\circ f_i}VV @VV{g}V\\ Z @= Z \end{CD} \]
Theorem 1.4
Let \(E\) be a vector space and let \((E_n)_{n\geq0}\) be a sequence of linear subspaces such that \(E_n\subseteq E_{n+1}\) for all \(n=0,1,2,\dots\) and \(E=\varinjlim E_n=\bigcup_{n=0}^\infty E_n.\) Suppose that each \(E_n\) is equipped with a locally convex topology \(\tau_n\) such that
- for each \(n\), the topology induced by \(\tau_{n+1}\) on \(E_n\) is \(\tau_n\);
- \(E_n\) is closed in \(E_{n+1}\) with respect to \(\tau_{n+1}\).
Let \(\tau\) be the finest locally convex topology on \(E\) for which all canonical injections \(f_n:E_n\hookrightarrow E\) are continuous. Then \(E\) is complete if and only if all the spaces \(E_n\) are complete.
Definition 1.2 (The test-function space \(\mathcal D(\Omega)\))
The space \(C_c^\infty(\Omega)\) is endowed with the final topology of a sequence of Fréchet spaces \(\bigl(C_{K_n}^\infty(\Omega),\tau_n\bigr)\) with canonical embeddings \(i_n:C_{K_n}^\infty(\Omega)\rightarrow C_c^\infty(\Omega),\) where \(\tau_n\) is defined in Example 1.2. Write \(C_K^\infty(\Omega)=\mathcal D_K(\Omega)\) and define \(\mathcal D(\Omega)=\varinjlim\mathcal D_{K_n}(\Omega).\) This definition is independent of the choice of the sequence \(K_n\).
Definition 1.3
A continuous linear functional on \(\mathcal D(\Omega)\) is called a distribution.
Definition 1.4
A locally integrable function \(f\) is said to have the \(\alpha\)-th distributional derivative \(g\) if \(D^\alpha\Lambda_f=\Lambda_g\) for some locally integrable function \(g\). This derivative is denoted by \(D^\alpha f\).
Theorem 1.5
\(D\Lambda_f=\Lambda_{Df}\) if and only if \(f\) is absolutely continuous.
Theorem 1.6
Let \(\Lambda\) be a linear functional on \(\mathcal D(\Omega)\). Then \(\Lambda\) is a distribution if and only if, for every compact set \(K\subseteq\Omega\), there exist a nonnegative integer \(N\) and a constant \(C<\infty\) such that \(|\Lambda\varphi|\leq C\|\varphi\|_N\) for every \(\varphi\in\mathcal D_K(\Omega)\).
Definition 1.5
Suppose \(\Lambda\in\mathcal D'(\Omega)\). If \(\omega\) is an open subset of \(\Omega\) and \(\Lambda(\varphi)=0\) for every \(\varphi\in\mathcal D(\omega)\), we say that \(\Lambda\) vanishes in \(\omega\).
Let \(W\) be the union of all open sets \(\omega\subseteq\Omega\) in which \(\Lambda\) vanishes. The complement of \(W\) is called the support of \(\Lambda\).
Lemma 1.3
If \(W\) is as above, then \(\Lambda\) vanishes in \(W\).
Theorem 1.7 (Local property as continuous functions)
Suppose \(\Lambda\in\mathcal D'(\Omega)\) and \(K\) is a compact subset of \(\Omega\). Then there exist a continuous function \(f\) on \(\Omega\) and a multi-index \(\alpha\) such that
\[\Lambda\varphi=(-1)^{|\alpha|}\int_\Omega f(x)(D^\alpha\varphi)(x)dx\]
for every \(\varphi\in\mathcal D_K(\Omega)\).
Let \(\mathcal E'(\Omega)\) denote the topological dual of \(C^\infty(\Omega)\) equipped with its natural topology.
Theorem 1.8
Let \(\Omega\) be an open set in \(\mathbb R^n\). Then:
- \(\mathcal E'(\Omega)\subseteq\mathcal D'(\Omega)\), and the identity map is continuous with respect to the strong topologies;
- the elements of \(\mathcal E'(\Omega)\) are distributions with compact support in \(\Omega\).
2 Fourier Transformations
Definition 2.1
A function \(\varphi\in C^\infty(\mathbb R^n)\) is said to be rapidly decreasing at infinity if
\[\lim_{|x|\to\infty}|x^\alpha\partial^p\varphi(x)|=0\]
for every \(\alpha\in\mathbb N\) and every \(p\in\mathbb N\).
The set of all functions in \(C^\infty(\mathbb R^n)\) that are rapidly decreasing at infinity is a complex vector space denoted by \(\mathcal S(\mathbb R^n)\).
Define seminorms on \(\mathcal S(\mathbb R^n)\) by
\[r_{\alpha,p}(\varphi)=\sup_{x\in\mathbb R^n}|x^\alpha\partial^p\varphi(x)|,\quad\alpha,p\in\mathbb N.\]
The countable family of seminorms \((r_{\alpha,p})\) defines a Hausdorff locally convex topology on \(\mathcal S(\mathbb R^n)\). This topology is metrizable and complete. Thus \(\mathcal S(\mathbb R^n)\) is a Fréchet space. In fact, \(\mathcal S(\mathbb R^n)\) is a Montel space.
Theorem 2.1
\(\mathcal S(\mathbb R^n)\) has the Heine–Borel property.
Theorem 2.2
We have the continuous embeddings
\[C_c(\mathbb R^n)\subseteq\mathcal S(\mathbb R^n)\subseteq C^\infty(\mathbb R^n).\]
Moreover, \(C_c(\mathbb R^n)\) is dense in \(\mathcal S(\mathbb R^n)\), and \(\mathcal S(\mathbb R^n)\) is dense in \(C^\infty(\mathbb R^n)\).
Definition 2.2
The elements \(\varphi\in\mathcal S(\mathbb R^n)\) are called tempered distributions.
Theorem 2.3
Every tempered distribution is the derivative of a continuous function that increases slowly at infinity.
Lemma 2.1
If \(\varphi\) is a tempered distribution, then there exists \(k>0\) such that \[(1+|x|^2)^{-k/2}\varphi\in L^\infty.\]
Theorem 2.4 (Plancherel)
There is a linear isometry \(\Psi\) of \(L^2(\mathbb R^n)\) onto \(L^2(\mathbb R^n)\) uniquely determined by the requirement that
\[\Psi f=\widehat f\]
for every \(f\in\mathcal S(\mathbb R^n)\).
3 Imbedding Theorem
Theorem 3.1 (Meyers and Serrin) \(H^{m,p}(\Omega)=W^{m,p}(\Omega).\)
Theorem 3.2 (Sobolev) If either \(mp>n\) or \(m=n\) and \(p=1\), then \(W^{j+m,p}(\Omega)\hookrightarrow C_B^j(\Omega).\)
Theorem 3.3 (Sobolev) If \(mp<n\), then \(W^{m,p}(\Omega)\hookrightarrow L^q(\Omega)\) for \(p<q<p^*=\frac{np}{n-mp}.\)
Theorem 3.4 (Sobolev) If \(mp\geq n\), then \(W^{m,p}(\Omega)\hookrightarrow L^q(\Omega)\) for \(p\leq q\leq\infty.\)
Theorem 3.5 (Morrey) If \(mp>n>(m-1)p,\) then \(W^{j+m,p}(\Omega)\hookrightarrow C_B^{j,\lambda}(\Omega)\) for \(0<\lambda<m-\frac np.\)
Theorem 3.6 (Morrey) If \(n=(m-1)p,\) then \(W^{j+m,p}(\Omega)\hookrightarrow C_B^{j,\lambda}(\Omega)\) for \(0<\lambda<1\). If \(n=m-1\) and \(p=1\), then the result also holds for \(\lambda=1\).
References
- R. A. Adams and John J. F. Fournier. Sobolev Spaces. Elsevier, 2003.
- J. Barros-Neto. An Introduction to the Theory of Distributions. Pure and Applied Mathematics 14. M. Dekker, New York, 1973. 中译本：《广义函数引论》，上海科技出版社，1981，欧阳光中、朱学炎译。
- I. M. Gel'fand and G. E. Shilov. Generalized Functions, Volume 2: Spaces of Fundamental and Generalized Functions. American Mathematical Society, 2016. 中译本：《广义函数 2》，科学出版社，1984，夏道行译。
- W. Rudin. Functional Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, 1991.
- L. Schwartz. Théorie des distributions. Publications de l'Institut de Mathématique de l'Université de Strasbourg. Hermann, 1966. 中译本：《广义函数论》，高等教育出版社，2010，姚家燕译。