Hodge theory lecture 4: Sobolev L 2 -spaces and Rellich lemma NRU - PowerPoint PPT Presentation

Hodge theory, lecture 4 M. Verbitsky Hodge theory lecture 4: Sobolev L 2 -spaces and Rellich lemma NRU HSE, Moscow Misha Verbitsky, February 3, 2018 1

Hodge theory, lecture 4 M. Verbitsky Banach spaces DEFINITION: Let M be a topological space, and � f � := sup M | f | the sup- norm on functions . C 0 -topology , or uniform topology on the space C 0 ( M ) of bounded continuous functions is topology defined by the sup-norm. DEFINITION: A Banach space is a complete normed vector space. THEOREM: A space of bounded continuous functions on M with C 0 - topology is Banach. Proof: A uniform limit of continuous functions is continuous (Weierstrass), and a limit of a Cauchy sequence of functions in C 0 ( M ) exists pointwise because R is complete. 2

Hodge theory, lecture 4 M. Verbitsky Stone-Weierstrass approximation theorem DEFINITION: Let A ⊂ C 0 M be a subspace in the space of continuous functions. We say that A separates the points of M if for all distinct points x, y ∈ M , there exists f ∈ A such that f ( x ) � = f ( y ). THEOREM: ( Stone-Weierstrass approximation theorem ) Let M be a compact manifold and A ⊂ C 0 M be a subring separating points, and A its closure. Then A = C 0 M . Proof: Handouts or the next lecture. 3

Hodge theory, lecture 4 M. Verbitsky Hilbert spaces (reminder) DEFINITION: Hilbert space is a complete, infinite-dimensional Hermitian space which is second countable (that is, has a countable dense set). DEFINITION: Orthonormal basis in a Hilbert space H is a set of pairwise orthogonal vectors { x α } which satisfy | x α | = 1, and such that H is the closure of the subspace generated by the set { x α } . THEOREM: Any Hilbert space has a basis, and all such bases are countable. THEOREM: All Hilbert spaces are isometric . Proof: Each Hilbert space has a countable orthonormal basis. 4

Hodge theory, lecture 4 M. Verbitsky Fourier series CLAIM: (”Fourier series”) Functions e k ( t ) = e 2 π √− 1 kt , k ∈ Z on S 1 = R / Z form an orthonormal basis in the space L 2 ( S 1 ) of square-integrable functions on the circle. S 1 e 2 π √− 1 kt dt = 0 for all k � = 0 (prove Proof: Orthogonality is clear from � To show that the space of Fourier polynomials � n it). i = − n a k e k ( t ) is dense in the space of continuous functions on circle, use the Stone-Weierstrass ap- proximation theorem, applied to the ring R = � sin( mx ) , cos( nx ) � of functions obtained from real and imaginary parts of e 2 π √− 1 kt . DEFINITION: Fourier monomials on a torus are functions F l 1 ,...,l n := exp(2 π √− 1 � n i =1 l i t i ), where l 1 , ..., l n ∈ Z . CLAIM: Fourier monomials form an orthonormal basis in the space L 2 ( T n ) of square-integrable functions on the torus T n . Proof: The same. 5

Hodge theory, lecture 4 M. Verbitsky L 2 -norms on vector spaces THEOREM: Let V be a vector space, and g 1 , g 2 two scalar products. We say that g 1 is bounded by g 2 if for some C > 0, one has g 1 � Cg 2 . EXERCISE: Prove that this is equivalent to the continuity of the map ( V, g 2 ) − → ( V, g 1 ) . REMARK: Let g 1 be bounded by g 2 . Then the identity map extends to a continuous map on the corresponding completion spaces L 2 ( V, g 2 ) − → L 2 ( V, g 1 ) . REMARK: The topology induced by g 1 is equivalent to topology induced by g 2 if and only if C − 1 g 2 � g 1 � Cg 2 . 6

Hodge theory, lecture 4 M. Verbitsky Sobolev’s L 2 -norm on C ∞ c ( R n ) DEFINITION: Denote by C ∞ c ( R n ) the space of smooth functions with com- pact support. For each differential monomial P α = ∂ k 1 ∂ k 2 ... ∂ k n ∂x k 1 ∂x k 2 ∂x k n 1 1 2 consider the corresponding partial derivative P α ( f ) = ∂ k 1 ∂ k 2 ... ∂ k n f. ∂x k 1 ∂x k 2 ∂x k n 1 1 2 Given f ∈ C ∞ c ( R n ), one defines the L 2 p Sobolev’s norm | f | p as follows: � | P α ( f ) | 2 Vol | f | 2 � s = deg P α � p where the sum is taken over all differential monomials P α of degree � p , and Vol = dx 1 ∧ dx 2 ∧ ...dx n - the standard volume form. REMARK: Same formula defines Sobolev’s L 2 -norm L 2 p on the space of smooth functions on a torus T n . 7

Hodge theory, lecture 4 M. Verbitsky Sobolev’s L 2 -norm on a torus CLAIM: The Fourier monomials F l 1 ,...,l n := e 2 π √− 1 � l i t i are eigenvectors for the differential monomials P α = ∂ k 1 ∂ k 2 ... ∂ kn . Moreover, P α ( F l 1 ,...,l n ) = ∂x k 1 ∂x k 2 ∂x kn 1 i =1 (2 π √− 1 k i ) l i . 1 2 � n COROLLARY: The Fourier monomials are orthogonal in the Sobolev’s L 2 p - metric, and p n | F l 1 ,...,l n | 2 (2 πl i ) 2 k i . � � 2 ,p = k 1 + ... + k n =1 i =1 8

Hodge theory, lecture 4 M. Verbitsky Weak convergence (reminder) DEFINITION: Let x i ∈ H be a sequence of points in a Hilbert space H . We say that x i weakly converges to x ∈ H if for any z ∈ H one has lim i g ( x i , z ) = g ( x, z ). REMARK: Let y ( i ) = α j ( i ) e j be a sequence of points in a a Hilbert space with orthonormal basis e i . Then y ( i ) converges to y = � j α j e j if and only if lim i α j ( i ) = α i . CLAIM: For any sequence { y ( i ) = � j α j ( i ) e j } of points in a unit ball, there exists a subsequence { ˜ y ( i ) = ˜ α j ( i ) e i } weakly converging to y ∈ H . Proof: Indeed, | α j ( i ) | � 1, hence there exist a subsequence ˜ y ( i ) = ˜ α j ( i ) x j with ˜ α j ( i ) converging for each j . The limit belongs to the unit ball because � � �� n otherwise j =1 ˜ α j ( i ) e j � > 1, which is impossible. � � REMARK: Note that the function x − → | x | is not continuous in weak topology. Indeed, weak limit of { e i } is 0. The proof above shows that | · | is semicontinuous. 9

Hodge theory, lecture 4 M. Verbitsky Compact operators (reminder) DEFINITION: Precompact set is a set which has compact closure. A compact operator is an operator which maps bounded sets to precompact. THEOREM: Let A : H − → H 1 be an operator on Hilbert spaces. Then A is compact if and only if it maps weakly convergent sequences to convergent ones. 10

Hodge theory, lecture 4 M. Verbitsky Rellich lemma for a torus THEOREM: (Rellich lemma for a torus) The identity map L 2 p ( T n ) − → L 2 p − 1 ( T n ) . is compact. Proof. Step 1: Consider, instead of L 2 p -metric, the metric q p which is orthog- i =1 l p onal in the same basis and satisfies | F l 1 ,...,l n | q p := 1 + (2 π ) p � n i . Clearly, | F l 1 ,...,l n | q p � | F l 1 ,...,l n | 2 ,p and | F l 1 ,...,l n | q p � C − 1 | F l 1 ,...,l n | 2 ,p , where C is a number of differential monomials of degree p . Therefore, q p and L 2 p induce the same topology, and it would suffice to prove the Rellich lemma for the identity map L 2 ( T n , q p ) − → L 2 ( T n , q p − 1 ). Step 2: Now, | F l 1 ,...,l n | 2 i =1 (2 π ) p l 2 p � n n q p i = . � i =1 (2 π ) p − 1 l 2 p − 2 max l 2 | F l 1 ,...,l n | 2 � n q p − 1 i i Step 3: Let x i ∈ L 2 ( T n , q p ) be weakly converging to x , with | x i | q p < 1. Let x i = y i + z i , with y i being the sum of all Fourier terms with max | l i | < N , √ n N | z i − z | q p < 2 √ n and z i the rest. Then | z i − z | q p − 1 < N , and y i converges to y because it is a sum of finitely many terms which all converge. We obtain that lim i | x i − x | q p − 1 = 0 , hence a x i (strongly) converges to x . 11

Hodge theory, lecture 4 M. Verbitsky Franz Rellich (1906-1955) After Weyl’s resignation [from G¨ ottingen], his former assistant, Franz Rellich, became In- stitute Director ... Rellich had only a low-level appointment and ... was not an established figure ... There was need for a prominent mathematical figure who was suitable politically to take over the leadership in Gottingen. Furthermore, in mid-December, Rellich was ordered to report on January 7 for ten weeks to a field-sports camp near Berlin. This was, in fact, a mistake, since Rellich, as an Austrian citizen, was not subject to such forced training reg- imens. When he arrived at the camp, he was not admitted on these grounds. However, on December 27, the Curator had, after some hesitation, replaced Rellich with Werner Weber as acting director of the Mathematical Institute. Rellich himself would lose his position at Gottingen six months later, on June 18. – S. L. Segal, Mathematicians under the Nazis 12

Hodge theory, lecture 4 M. Verbitsky Rellich lemma for C ∞ K ( R n ) K ( R n ) be the space of smooth functions on R n with COROLLARY: Let C ∞ support in a compact set K . Then the identity map L 2 p ( C ∞ K ( R n )) − → L 2 p − 1 ( C ∞ K ( R n )) is compact. Proof: We consider a quotient map R n − → T n which is bijective on K for an This embeds C ∞ K ( R n ) to C ∞ ( T n ), and this appropriate choice of a lattice. embedding is compatible with the L 2 p -norms. 13