Reminder Notes for the Course on Distribution Theory

Reminder Notes for the Course on Distribution Theory T. C. Dorlas Dublin Institute for Advanced Studies School of Theoretical Physics 10 Burlington Road, Dublin 4, Ireland. Email: dorlas@stp.dias.ie March 30, 2011 1 Basic Definitions Definition 1.1 Let Ω R n be open. We define D(Ω) = {φ : Ω C : C (Ω) and supp(φ) compact }. (One defines, in general, the support of a function φ : Ω C, as the complement of the largest open set on which φ = 0.) Functions in D(Ω) are called test functions. We also set, for K Ω, D K = {φ D(Ω) : supp(φ) K}. Definition 1.2 Let K R n be compact. A sequence (φ n ) n N with φ n D K is said to converge to φ D K if φ n φ 0 and p φ n p φ 0 for all multi-indices p N n 0 i.e. p = (p 1,..., p n ) with p i = 0, 1, 2,.... Here ψ = sup x R n ψ(x) is the supremum norm, so φ n φ 0 means that φ n φ uniformly. Notation: φ n DK φ. 1

Definition 1.3 Let Ω R n be open. A distribution on Ω is a linear functional T : D(Ω) C such that T DK is continuous for all K Ω compact, i.e. φ n DK φ = T (φ n ) T (φ). The set of distributions on Ω is denoted D (Ω). Proposition 1.1 If f L 1 loc (Rn ) then T f (φ) = f(x)φ(x) dx (φ D(Ω)) defines a distribution. If µ is a complex-valued Radon measure on Ω R n then T µ (φ) = φ dµ (φ D(Ω)) defines a distribution. Lemma 1 The function λ : R R defined by { e 1/t for t > 0 λ(t) = 0 for t 0 is C. Proposition 1.2 The function ρ : R n R defined by ρ(x) = c λ(1 x 2 ), where c > 0 is a constant, belongs to D(R n ) and has support B(0; 1). We choose c such that ρ(x) d n x = 1. Definition 1.4 For T 1, T 2 D (Ω) we define (T 1 + T 2 )(φ) = T 1 (φ) + T 2 (φ) (φ D(Ω)). For a C and T D (Ω) we set (at )(φ) = a T (φ) if φ D(Ω), and more generally, if α C (Ω), (αt )(φ) = T (αφ) (φ D(Ω)). 2

Proposition 1.3 With the above definitions, D (Ω) is a linear space, and if α C (Ω) and T D (Ω) then αt D (Ω). Proposition 1.4 If f L 1 loc (Rn ) and α C then αt f = T αf. Definition 1.5 If p N n 0 is a multi-index, and T D (Ω) then the p-th derivative of T is defined by ( p T )(φ) = ( 1) p T ( p φ) for φ D(Ω), where p = p 1 + + p n. Proposition 1.5 If f C p (Ω) then p T f = T p f. Proposition 1.6 (Leibniz rule) If T D (Ω) and α C (Ω) then for any multi-index m, m (αt ) = p+q=m m! p! q! ( p α)( q T ), where the sum is over multi-indices p and q and if m = (m 1,..., m n ) then m! = n i=1 m i!. Theorem 1.1 Every distribution T D (R) has a primitive S D (R), i.e. a distribution such that S = T, which is unique up to an additive constant. 2 Regularisation and Partition of Unity Definition 2.1 Let ρ : R n R be the function defined in Prop. 1.2, and set ρ ɛ (x) = 1 ρ ( ) x ɛ n ɛ for ɛ > 0. Given f L 1 loc (R n ), the regularised function f ɛ is defined by f ɛ (x) = (ρ ɛ f)(x) = ρ ɛ (x y)f(y) d n y. Theorem 2.1 If g L 1 loc (Rn ) then g ɛ C (R n ) and supp(g ɛ ) supp(g) + B(0; ɛ). If g : R n C is uniformly continuous (in particular, if g κ C (R n )) then g ɛ g uniformly as ɛ 0. 3

If g C k c (R n ; C) for some k 0 then p g ɛ = ( p g) ɛ p N k 0 : p k and p g ɛ p g 0 as ɛ 0. If f L p (R n ) (p [0, + )) then f ɛ L p (R n ), f ɛ p f p and f ɛ f p 0. Corollary 2.1 If f L 1 loc (Ω) and T f = 0 then f = 0 a.e. If µ is a complexvalued Radon measure on Ω and T µ = 0 then µ = 0. Lemma 1 (Urysohn s C -lemma) Let Ω R n be open and K Ω compact. Then there exists f D(Ω) with 0 f 1 and f K = 1. Definition 2.2 A covering {V j } j J of a topological space X is called a refinement of a covering {U i } i I if for all j J there is i I such that V j U i. A Hausdorff space X is called paracompact if every open cover of X has a locally finite refinement, i.e. one such that for all x X there is an open neighbourhood intersecting only finitely many elements of the refinement. Lemma 2 If Ω R n is an open subset then there exists a sequence (Ω k ) k N of open subsets such that Ω k is compact, and Ω k Ω k+1 for all k N, and Ω = k N Ω k. Theorem 2.2 An open subset Ω R n is paracompact. Remark. It follows from Lemma 2 that every locally finite open cover of Ω is necessarily countable. Definition 2.3 A topological space X is called normal if any two disjoint closed subsets A, B X can be separated, i.e. there exist open sets U, V X such that A U, B V and U V =. Lemma 3 A metric space is normal. Lemma 4 (Shrinking lemma) If {V j } j J is a locally finite open cover of an open set Ω R n such that V j Ω for all j J, then there exists an open cover {W j } j J of Ω such that W j V j for all j J. 4

Definition 2.4 Given an arbitrary open cover {V i } i I of an open subset Ω R n, a partition of unity subordinate to {V i } i I is a collection {α j } j J of functions α j D(Ω) such that 1. For all j J, α j 0, and there exists i I such that supp(α j ) V i ; 2. For all compact sets K Ω, the set {j J : supp(α j ) K } is finite; 3. j J α j = 1 Ω. Theorem 2.3 For every open cover {V i } i I exists a partition of unity subordinate to it. of an open set Ω R n, there 3 Support of a distribution Proposition 3.1 If Ω 1 Ω 2 R n are open subsets, and ϕ D(Ω 1 ), then ϕ : Ω 2 C defined by { ϕ(x) if x Ω 1 ; ϕ(x) = 0 if x Ω 2 \ Ω 1, belongs to D(Ω 2 ). Conversely, if ψ D(Ω 2 ) and supp(ψ) Ω 1, then ψ Ω1 D(Ω 1 ). Remark. D(Ω 2 ). Because of this simple proposition, we can write D(Ω 1 ) Definition 3.1 If Ω 1 Ω 2 R n are open, and T D (Ω 2 ), we write (by abuse of notation) T Ω1 for the restriction of T to D(Ω 1 ). T is called zero in Ω 1 if T Ω1 = 0, and T 1, T 2 D (Ω 2 ) are said to agree on Ω 1 if T 1 T 2 is zero in Ω 1. Lemma 1 If Ω 1, Ω 2 R n are open, and φ D(Ω 1 Ω 2 ) then there exist φ 1 D(Ω 1 ) and φ 2 D(Ω 2 ) such that φ = φ 1 + φ 2. (See the remark above.) Lemma 2 Let Ω 1, Ω 2 Ω be open, and T D (Ω). Suppose that T Ω1 = 0 and T Ω2 = 0. Then T Ω1 Ω 2 = 0. 5

Proposition 3.2 Let Ω i R n (i I) be open, and set Ω = i I Ω i. If T D (Ω) and T i I = 0 for all i I, then T = 0. This proposition justifies the following definition: Definition 3.2 Let Ω R n be open, T D (Ω). If O Ω is the largest open subset of Ω such that T O = 0, then supp(t ) := Ω \ O is called the support of T. Proposition 3.3 If µ is a (possibly complex-valued) Radon measure on Ω R n, then supp(t µ ) = supp(µ). Using the existence of a partition of unity (Theorem 2.3) one can also reconstruct a distribution on Ω from its restrictions to the elements of an open cover: Theorem 3.1 Suppose that Ω = i I Ω i, where Ω i R n (i I) are open. Let T i D (Ω i ) be given such that T i Ωi Ω j = T j Ωi Ω j for all i, j I. Then there exists a unique distribution T D (Ω) such that T i = T Ωi. 4 Distributions of finite order Theorem 4.1 Let Ω R n be open. A linear map T : D(Ω) C is a distribution if and only if for every compact set K Ω, there exist M (0, + ) and m N 0 such that T (ϕ) M ϕ m, where ϕ m = sup sup x R n p N n 0 : p m p ϕ(x). The proof goes via the following statement: If T D (Ω) is a distribution, and K Ω is compact then: ɛ > 0 m N 0, δ > 0 : ϕ D K : ϕ m δ = T (ϕ) ɛ. This statement is proved by contradiction. We now define more spaces of test functions: ( ) 6

Definition 4.1 For Ω R n open and m N 0, we define and if K Ω is compact, D m (Ω) = {φ C m (Ω) : supp(φ) compact }, D m K(Ω) = {φ D m (Ω) : supp(φ) K}. For a sequence (φ n ) n N of functions φ n DK m(ω) we write φ n D m K DK m(ω) and φ n φ m 0. φ if φ Note that DK m(ω) with the norm m is a Banach space. We have immediately: Proposition 4.1 A linear form L : DK m (Ω) C is continuous if and only if it is bounded, i.e. there exists a constant M (0, + ) such that L(ϕ) M ϕ m ϕ D m K(Ω). Definition 4.2 A linear form L : D m (Ω) C is said to be continuous if L D m is continuous for all compact K Ω. The set of such continuous K (Ω) linear forms is denoted D m (Ω). Definition 4.3 A distribution T D (Ω) is said to be of order m if for all compact K Ω there exists M (0, + ) such that T (ϕ) M ϕ m ϕ D K (Ω). T is of order m if it is of order m but not of order m 1. T has finite order if it is of order m for some m N 0. Theorem 4.2 If T D (Ω) is of order m, then there exists a unique T D m (Ω) such that T D(Ω) = T. Conversely, if L D m (Ω), then T = L D(Ω) is a distribution of order m, and L = T. Proposition 4.2 If T D m (Ω) (Ω R n open) then T x i i = 1,..., n. D m+1 (Ω) for 7

Proposition 4.3 If T D m (Ω) and α C m (Ω) then αt is well-defined, αt D m (Ω) and Leibniz rule holds, i.e. r (αt ) = for multi-indices r with r m. p+q=r r! p! q! ( p α)( q T ) Proposition 4.4 If f L 1 loc (Ω) then T f is a distribution of order 0. More generally, if µ is a Radon measure on Ω then T µ is of order 0, and conversely, if T is a distribution of order 0 then there exists a Radon measure µ on Ω such that T = T µ. Definition 4.4 A distribution T D (Ω) is called positive if ϕ D(Ω), ϕ 0 = T (ϕ) 0. Proposition 4.5 A positive distribution is of order 0. Corollary 4.1 If T D (Ω) is a positive distribution then there exists a positive Radon measure µ on Ω such that T = T µ. 5 Distribution with compact support Definition 5.1 Let Ω R n be open. We denote E(Ω) = C (Ω). For K Ω compact and φ E(Ω) we define φ K,m = sup x K sup p: p m p φ(x). A sequence (φ n ) n N of functions φ n E(Ω) is said to converge in the sense of E to φ E(Ω) if φ n φ K,m 0 for all compact K Ω and all m N 0. Notation: φ n E φ. Theorem 5.1 A linear form L : E(Ω) C is continuous if and only if there exist a compact set K Ω, M (0, + ) and m N 0 such that L(φ) M φ K,m φ E(Ω). Notation: L E (Ω). 8

Lemma 1 D(Ω) is dense in E(Ω). Theorem 5.2 The distributions T D (Ω) of compact support are precisely those for which there exists L E (Ω) with T = L D(Ω). Moreover, given T, L is unique. Again, we have statements about derivatives and multiplication by functions: Proposition 5.1 If T D (Ω) has compact support, then T also has compact support, and x i ( ) T φ (φ) = T φ E(Ω). x i x i Proposition 5.2 If T E (Ω) and β E(Ω), then βt has compact support and supp(βt ) supp(t ). Moreover, Leibniz rule holds. Proposition 5.3 Every distribution of compact support has finite order. Corollary 5.1 Every distribution has locally finite order. 6 Convolution of distributions Definition 6.1 If f, g : R n C are two functions, their convolution product f g is defined by (f g)(x) = f(x y)g(y) d n y provided this integral is well-defined. By a simple change of variable we have Proposition 6.1 If the convolution product f g of f, g : R n C exists then g f also exists and g f = f g. Proposition 6.2 If f, g L 1 (R n ) then f g L 1 (R n ) and N 1 (f g) N 1 (f) N 1 (g). 9

Proposition 6.3 If f, g : R n C are continuous, and at least one of f or g has compact support, then f g exists and supp(f g) supp(f) + supp(g). Corollary 6.1 If f, g κ C (R n ) then f g κ C (R n ). Proposition 6.4 If f, g C 1 (R n ) and at least one of f or g has compact support, then (f g) = f g = f g x i x i x i for i = 1,..., n. We can now extend the definition of convolution product to that of a function and a distribution: Definition 6.2 Given a function α : R n C and x R n, we define τ x α and ˇα by (τ x α)(y) = α(y x) and ˇα(y) = α( y). For α D(R n ) and T D (R n ) or α E(R n ) and T E (R n ) we then define the convolution T α by T α = T f with f(x) = T (τ x ˇα). This definition is viable because f L 1 loc (Rn ) as follows from the following theorem: Theorem 6.1 If T D (R n ) and α D(R n ) then T α = T f, where f C (R n ), and for any multi-index p, we have p (T α) = ( p T ) α = T ( p α). The same holds for T E (R n ) and α E(R n ). Proposition 6.5 Let α E(R n ) and T D (R n ) and suppose that either or both have compact support. Then supp(t α) supp(t ) + supp(α). 10

Theorem 6.2 Given T D (R n ) or T E (R n ), define L : D(R n ) E(R n ) respectively L : D(R n ) D(R n ) by (Lϕ)(x) = T (τ x ˇϕ), i.e. T Lϕ = T ϕ. Then L is a continuous linear map satisfying τ x L = L τ x x R n. Conversely, given a continuous linear map L : D(R n ) E(R n ) respectively L : D(R n ) D(R n ) which commutes with translations, there exists a unique distribution T D (R n ) resp. E (R n ) such that T Lϕ = T ϕ for all ϕ D(R n ) resp. E(R n ). We can now extend the definition of convolution to two distributions. First note that if both are functions, T f g (ϕ) = T g ( ˇf ϕ) = T f (ǧ ϕ). Moreover, T ˇf(ϕ) = T f ( ˇϕ), so we define Definition 6.3 If T D (R n ) then we define Ť D (R n ) by Ť (ϕ) = T ( ˇϕ) and Definition 6.4 Let S, T D (R n ) and suppose that at least one of S and T has compact support. Then we define the convolution S T by i.e. Ť ϕ = T α. (S T )(ϕ) = S(α), where α(x) = T (τ x ϕ) = Ť (τ x ˇϕ), Indeed, we verify that Proposition 6.6 If S, T D (R n ) and at least one of S or T has compact support, then S T D (R n ). Moreover, the definition is consistent with the earlier definition: Theorem 6.3 Suppose T D (R) and α D(R n ) or T E (R n ) and α E(R n ). Then, for any ϕ D(R n ), (T α)(ϕ) = (T T α )(ϕ) = (T α T )(ϕ) = T (ˇα ϕ). 11

The associative and commutative law hold: Theorem 6.4 If R, S, T D (R n ) and at least two of these distributions has compact support, then R (S T ) = (R S) T. Theorem 6.5 If S, T D (R n ) and at least one of these has compact support then S T = T S. Moreover, we have the analogue of Theorem 6.1 for differentiation: Theorem 6.6 If S, T D (R n ) and at least one of supp(s) and supp(t ) is compact, then for any multi-index p, p (S T ) = ( p S) T = S ( p T ). 7 Regularisation of distributions Definition 7.1 If (T n ) n N is a sequence of distributions T n D (R n ) then we say that it converges to a distribution T D (R n ) if T n (φ) T (φ) for all φ D(R n ). Notation: T n T. Theorem 7.1 Let α D(R n ) be such that α(x) 0 for all x R n, and α(x) dx = 1. (For example α = ρ as in Prop. 1.2.) Define αɛ (x) = 1 α ( ) x ɛ n ɛ. Then {α ɛ } ɛ>0 is an approximation of δ in the following sense: 1. lim ɛ 0 α ɛ = δ in the sense of distributions, i.e. T αɛ δ; 2. α ɛ φ D φ for all φ D(R n ); 3. T α ɛ T for all T D (R n ). Corollary 7.1 D(R n ) is dense in D (R n ), i.e. for every T D (R n ) there exists a sequence (φ n ) n N of functions φ n D(R n ) such that T φn T. As an application, we have Proposition 7.1 A distribution T D (R n ) is positive (T 0) if and only if T α ɛ 0 for all ɛ > 0. Theorem 7.2 Let T D (R). The following are equivalent: 1. T = T f for a non-decreasing function f : R R; 2. T 0. (This was not proved in the lectures.) 12

8 Tempered distributions Definition 8.1 The Schwartz space of rapidly decreasing functions S(R n ) is the space of functions φ C (R n ) such that for all m N, the expression φ m := sup x R n p m sup (1 + x 2 ) m p φ(x) < +. One says that a sequence (φ n ) n N of functions φ n S(R n ) converges in the sense of S to φ S(R n ) if φ n φ m 0 for all m N. Note the following elementary properties: Proposition 8.1 The maps φ φ m form a directed sequence of seminorms, i.e. the following hold: 1. m 1 m 2 = φ m1 φ m2 for all φ S(R n ); 2. φ 1 + φ 2 m φ 1 m + φ 2 m φ 1, φ 2 S(R n ); 3. λφ m = λ φ m φ S(R n ); λ C. Moreover, the following follows immediately from Leibniz formula: Proposition 8.2 If φ, ψ S(R n ) then φψ S(R n ). Later we also show that φ ψ S. Proposition 8.3 D(R n ) is dense in S(R n ). We now define tempered distributions as continuous linear forms on S: Definition 8.2 A tempered distribution is a linear form T : S(R n ) C which is continuous in the sense that if φ n φ in S then T (φ n ) T (φ). Notation: T S (R n ). Theorem 8.1 If T S (R n ) is a tempered distribution then there exists M > 0 and m N such that T (φ) M φ m φ S(R n ). (8.1) Moreover, if T D (R n ) is a distribution such that (8.1) holds for all φ D(R n ), then there exists a unique tempered distribution T S(R n ) such that T D(R n ) = T. 13

We can thus identify the tempered distributions with the distributions T for which (8.1) holds. Some examples of tempered distributions are: Proposition 8.4 Every distribution T D (R n ) with compact support is tempered. Proposition 8.5 Let µ be a (complex-valued) Radon measure on R n such that there exist C > 0 and N N for which ν(r) := µ (B(0, R)) C(1 + R N ) R > 0. (Here B(0, R) denotes the ball centred at 0 with radius R.) Then T µ is a tempered distribution. Proposition 8.6 Let T S (R n ). Then, for any multi-index p, for any polynomial P, and for any g S(R n ), p T, P T and g T are also tempered distributions. 9 Fourier transformation Definition 9.1 If f L 1 (R n ), then we define the Fourier transform ˆf of f by 1 ˆf(ξ) = (Ff)(ξ) = f(x) e ix ξ dx. (2π) n/2 Proposition 9.1 If f L 1 (R n ) then ˆf C b (R n ) (i.e. it is a bounded continuous function) and ˆf (2π) n/2 f 1. The following properties are easily derived: Proposition 9.2 If f S(R n ) then ˆf S(R n ) and the following hold: 1. p ˆf(ξ) = (2π) n/2 e ix ξ ( ix) p f(x) dx; 2. p f(ξ) = (iξ) p ˆf(ξ); 14

3. 4. (τ h f)(ξ) = e iξ h ˆf(ξ); τ h ˆf = ( e ix h f(x) ). Corollary 9.1 Fourier transformation is a continuous linear map F : S(R n ) S(R n ). Lemma 1 Define the Gauss function G on R n by Then G S(R n ) and Ĝ = G. G(x) = exp[ 1 2 x 2 ]. Theorem 9.1 (Fourier inversion) Fourier transformation is a continuous linear bijection F : S(R n ) S(R n ) with continuous inverse given by (F 1 f)(x) = that is, f = ˇf. Moreover, ˆˇf = ˇˆf. 1 (2π) n/2 f(ξ) eix ξ dξ, Theorem 9.2 If f, g S(R n ) then F g S(R n ) and we have fg = (2π) n/2 ( ˆf ĝ) and f g = (2π) n/2 ˆf ĝ. Theorem 9.3 (Parseval identity) For all f cals(r n ) and g L 1 (R n ) the following identity holds: f(x) g(x) dx = ˆf(ξ) ĝ(ξ) dξ. Moreover, this identity extends to f, g L 2 (R n ). Because of this result, it is natural to define the Fourier transform of a tempered distribution as follows: 15

Definition 9.2 If T S (R n ) then we define the Fourier transform of T by ˆT (φ) = T ( ˆφ) φ S(R n ). By the continuity of the Fourier transform we immediately have that the Fourier transform of a tempered distribution is also a tempered distribution. Moreover all relevant properties extend to distributions: Theorem 9.4 Let T S (R n ). Then the following hold: 1. 2. 3. 4. 5. 6. p ˆT = αp T where α p (x) = ( ix) p ; p T = ˇα p ˆT ; τ h T = ě h ˆT where eh (x) = e ix h ; τ h ˆT = ê h T ; ˆŤ = ˇˆT ; ˆT = Ť. Corollary 9.2 Fourier transformation is a continuous linear bijection F : S (R n ) S(R n ) with continuous inverse. Proposition 9.3 (Riemann-Lebesgue lemma) If f L 1 (R n ) then ˆT f = T ˆf and moreover ˆf is continuous, ˆf (2π) n/2 f 1, and lim ξ ˆf(ξ) = 0. 16

(The latter property follows from the fact that D is dense in L 1 so that we can write f = φ + h with φ D and h 1 small. Then ˆf = ˆφ + ĥ, and ˆφ tends to zero at infinity since it belongs to S.) Definition 9.3 If T S(R n ) and α S(R n ) then we define the convolution product T α by T α = T f where f(x) = T (τ x ˇα). Theorem 9.5 If T S (R n ) and α S(R n ) then T α S (R n ), T α = T f with f C (R n ), and the following identities hold for all multi-indices p: p (T α) = ( p T ) α = T ( p α). Theorem 9.6 Given T S (R n ), define L : S(R n ) E(R n ) by (Lφ)(x) = T (τ x ˇφ), i.e. TLφ = T φ for φ S(R n ). The n L is a continuous linear map which commutes with translations, i.e. τ x L = Lτ x for all x R n. Conversely, if L : S(R n ) E(R n ) is a continuous linear map which commutes with translations, then there exists a unique T S (R n ) such that T Lφ = T φ for all φ S(R n ). Theorem 9.7 If T S (R n ) and φ, ψ S(R n ) then the following hold: 1. 2. 3. (T φ) ψ = T (φ ψ); T φ = (2π) n/2 ˆφ ˆT ; ˆT ˆφ = (2π) n/2 φ T. 17