The Space of Tempered Distributions

2.3 The Class of Tempered Distributions

2.3.3 The Space of Tempered Distributions

Having set down the basic definitions of distributions, we now focus our study on the space of tempered distributions. These distributions are the most useful in harmonic analysis. The main reason for this is that the subject is concerned with boundedness

of translation-invariant operators, and every such bounded operator fromL^p(Rⁿ)to L^q(Rⁿ)is given by convolution with a tempered distribution. This fact is shown in Section 2.5.

Suppose thatf andgare Schwartz functions andαa multi-index. Integrating by parts|α|times, we obtain

Z

Rⁿ

(∂^αf)(x)g(x)dx= (−1)^|α|

Z

Rⁿ

f(x)(∂^αg)(x)dx. (2.3.6) If we wanted to define the derivative of a tempered distributionu, we would have to give a definition that extends the definition of the derivative of the function and that satisfies (2.3.6) forginS⁰and f ∈S if the integrals in (2.3.6) are interpreted as actions of distributions on functions. We simply use equation (2.3.6) to define the derivative of a distribution.

Definition 2.3.6.Letu∈S⁰andα a multi-index. Define ∂^αu,f

= (−1)^|α|

u,∂^αf

. (2.3.7)

Ifuis a function, the derivatives ofuin the sense of distributions are calleddistri- butional derivatives.

In view of Theorem 2.2.14, it is natural to give the following:

Definition 2.3.7.Let u∈S⁰. We define the Fourier transform bu and the inverse Fourier transformu^∨of a tempered distributionuby

u,b f

= u,bf

and

u^∨,f

= u,f^∨

, (2.3.8)

for all f inS.

Example 2.3.8.We observe thatδb₀=1. More generally, for any multi-indexα we have

(∂^αδ₀)b = (2πix)^α. To see this, observe that for all f ∈S we have

(∂^αδ0)b, f

=

∂^αδ0, bf

= (−1)^|α|

δ0,∂^αbf

= (−1)^|α|

δ0,((−2πix)^αf(x))b

= (−1)^|α|((−2πix)^αf(x))b(0)

= (−1)^|α|

Z

Rⁿ

(−2πix)^αf(x)dx

= Z

Rⁿ

(2πix)^αf(x)dx.

This calculation indicates that(∂^αδ0)

bcan be identified with the function(2πix)^α.

Example 2.3.9.Recall that forx₀∈Rⁿ,δx₀(f) = δx₀,f

=f(x₀). Then

δcx₀,h

= δx₀,bh

=bh(x₀) = Z

Rⁿ

h(x)e^−2πix·x0dx, h∈S(Rⁿ), that is,δcx₀ can be identified with the functionx7→e^−2πix·x0. In particular,δb0=1.

Example 2.3.10.The functione^|x|2is not inS⁰(Rⁿ)and therefore its Fourier trans- form is not defined as a distribution. However, the Fourier transform of any locally integrable function with polynomial growth at infinity is defined as a tempered dis- tribution.

Now observe that the following are true whenever f,gare inS. Z

Rⁿ

g(x)f(x−t)dx = Z

Rⁿ

g(x+t)f(x)dx, Z

Rⁿ

g(ax)f(x)dx = Z

Rⁿ

g(x)a⁻ⁿf(a⁻¹x)dx, Z

Rⁿg(x)e f(x)dx = Z

Rⁿ

g(x)fe(x)dx,

(2.3.9)

for allt∈Rⁿanda>0. Recall now the definitions ofτ^t,δ^a, andegiven in (2.2.12).

Motivated by (2.3.9), we give the following:

Definition 2.3.11.Thetranslationτ^t(u), thedilationδ^a(u), and thereflectionueof a tempered distributionuare defined as follows:

τ^t(u),f

=

u,τ^−t(f)

, (2.3.10)

δ^a(u),f

=

u,a⁻ⁿδ^1/a(f)

, (2.3.11)

u,e f

= u,fe

, (2.3.12)

for allt∈Rⁿanda>0. LetAbe an invertible matrix. The composition of a distri- butionuwith an invertible matrixAis the distribution

u^A,ϕ

=|detA|⁻¹ u,ϕ^A

−1

, (2.3.13)

whereϕ^A−1(x) =ϕ(A⁻¹x).

It is easy to see that the operations of translation, dilation, reflection, and differ- entiation are continuous on tempered distributions.

Example 2.3.12.The Dirac mass at the origin δ0 is equal to its reflection, while δ^a(δ₀) =a⁻ⁿδ0. Also,τ^x(δ₀) =δxfor anyx∈Rⁿ.

Now observe that for f,g, andhinS we have Z

Rⁿ

(h∗g)(x)f(x)dx= Z

Rⁿ

g(x)(eh∗f)(x)dx. (2.3.14)

Motivated by (2.3.14), we define the convolution of a function with a tempered distribution as follows:

Definition 2.3.13.Letu∈S⁰andh∈S. Define the convolutionh∗uby h∗u,f

= u,eh∗f

, f ∈S. (2.3.15)

Example 2.3.14.Letu=δx₀ andf∈S. Thenf∗δx₀is the functionx7→f(x−x₀), for whenh∈S, we have

f∗δx₀,h

=

δx₀,ef∗h

= (fe∗h)(x₀) = Z

Rⁿ

f(x−x₀)h(x)dx. It follows that convolution withδ₀is the identity operator.

We now define the product of a function and a distribution.

Definition 2.3.15.Letu∈S⁰and lethbe aC^∞function that has at most polynomial growth at infinity and the same is true for all of its derivatives. This means that it satisfies|(∂^αh)(x)| ≤C(1+|x|)^kα for allα and some k_α>0. Then define the producthuofhanduby

hu,f

= u,h f

, f∈S. (2.3.16)

Note thath f is inS and thus (2.3.16) is well defined. The product of an arbitrary C^∞function with a tempered distribution is not defined.

We observe that if a functiongis supported in a setK, then for all f ∈C₀^∞(K^c) we have

Z

Rⁿ

f(x)g(x)dx=0. (2.3.17) Moreover, the support ofgis the intersection of all closed setsKwith the property (2.3.17) for all f inC₀^∞(K^c). Motivated by the preceding observation we give the following:

Definition 2.3.16.Letube inD⁰(Rⁿ). Thesupportofu(suppu) is the intersection of all closed setsKwith the property

ϕ∈C^∞(Rⁿ), suppϕ⊆K^c =⇒ u,ϕ

=0. (2.3.18) Distributions with compact support are exactly those whose support (as defined in the previous definition) is a compact set. To prove this assertion, we start with a distribution u with compact support as defined in Definition 2.3.3. Then there existC,N,m>0 such that (2.3.4) holds. For a smooth function f whose support is contained inB(0,N)^c, the expression on the right in (2.3.4) vanishes and we must therefore have

u,f

=0. This shows that the support ofuis bounded, and since it is already closed (as an intersection of closed sets), it must be compact. Conversely, if the support ofuas defined in Definition 2.3.16 is a compact set, then there exists anN>0 such that suppuis contained inB(0,N). We take a smooth functionηthat

is equal to 1 onB(0,N)and vanishes offB(0,N+1). Then the support of f(1−η) does not meet the support ofu, and we must have

u,f

= u,fη

+

u,f(1−η)

= u,fη

.

Taking mto be the integer that corresponds to the compact set K=B(0,N+1) in (2.3.2), and using that theL^∞ norm of∂^α(fη)is controlled by a finite sum of seminormsρeα,N+1(f)with|α| ≤m, we obtain the validity of (2.3.4).

Example 2.3.17.The support of Dirac mass atx₀is the set{x₀}.

Along the same lines, we give the following definition:

Definition 2.3.18.We say that a distributionuinD⁰(Rⁿ)coincides with the function hon an open setΩ if

u,f

= Z

Rⁿ

f(x)h(x)dx for all f inC0^∞(Ω). (2.3.19) When (2.3.19) occurs we often say thatuagrees withhaway fromΩ^c.

This definition implies that the support of the distributionu−his contained in the setΩ^c.

Example 2.3.19.The distribution|x|²+δ_a1+δ_a2, wherea₁,a₂are inRⁿ, coincides with the function|x|²on any open set not containing the pointsa₁anda₂. Also, the distribution in Example 2.3.5 (8) coincides with the functionx⁻¹χ|x|≤1away from the origin in the real line.

Having ended the streak of definitions regarding operations with distributions, we now discuss properties of convolutions and Fourier transforms.

Theorem 2.3.20.If u∈S⁰andϕ∈S, thenϕ∗u is aC^∞function. Moreover, for all multi-indicesα there exist constants Cα,kα>0such that

|∂^α(ϕ∗u)(x)| ≤Cα(1+|x|)^kα.

Furthermore, if u has compact support, then f∗u is a Schwartz function.

Proof. Letψbe inS(Rⁿ). We have (ϕ∗u)(ψ) =u(ϕe∗ψ) =u

Z

Rⁿϕ(e · −y)ψ(y)dy

=u Z

Rⁿ

τ^y(ϕ)(e ·)ψ(y)dy

(2.3.20)

= Z

Rⁿ

u(τ^y(ϕ))ψe (y)dy,

where the last step is justified by the continuity ofuand by the fact that the Riemann sums of the integral in (2.3.20) converge to that integral in the topology ofS, a fact proved later. This calculation identifies the functionϕ∗uas

(ϕ∗u)(x) =u τ^x(ϕ)e

. (2.3.21)

We now show that(ϕ∗u)(x)is aC^∞function. Lete_j= (0, . . . ,1, . . . ,0)with 1 in the jth entry and zero elsewhere. Then

τ^−hej(ϕ∗u)(x)−(ϕ∗u)(x)

h =u

τ^−hej(τ^x(ϕ))e −τ^x(ϕ)e h

→u(τ^x(∂_jϕ))e by the continuity ofuand the fact that τ^−hej(τ^x(ϕ))−τe ^x(ϕ)e

/htends toτ^x(∂_jϕ)e in S ash→0. See Exercise 2.3.5(a). The same calculation for higher-order derivatives shows that ϕ∗u∈C^∞ and that∂^γ(ϕ∗u) = (∂^γϕ)∗u for all multi-indices γ. It follows from (2.3.3) that for someC,m, andkwe have

|∂^α(ϕ∗u)(x)| ≤C

∑

|γ|≤m

|β|≤k

sup

y∈Rⁿ

|y^γτ^x(∂^α+βϕ)(y)|e

=C

∑

|γ|≤m

|β|≤k

sup

y∈Rⁿ

|(x+y)^γ(∂^α+βϕ)(y)|e

≤C_m

∑

|β|≤k

sup

y∈Rⁿ

(|x|^m+|y|^m)|(∂^α+βϕe)(y)|,

(2.3.22)

and this clearly implies that∂^α(ϕ∗u)grows at most polynomially at infinity.

We now indicate whyϕ∗uis Schwartz wheneveruhas compact support. Apply- ing estimate (2.3.4) to the functiony7→ϕ(x−y)yields that

u,ϕ(x− ·)

=|(ϕ∗u)(x)| ≤C

∑

|α|≤m

sup

|y|≤N

|∂_y^αϕ(x−y)|

for some constantsC,m,N. Since

|∂_y^αϕ(x−y)| ≤Cα,M(1+|x−y|)^−M≤Cα,M,N(1+|x|)^−M

for|x| ≥2N, it follows thatϕ∗udecays rapidly at infinity. Since∂^γ(ϕ∗u) = (∂^γϕ)∗

u, the same argument yields that all the derivatives ofϕ∗udecay rapidly at infinity;

henceϕ∗uis a Schwartz function. Incidentally, this argument actually shows that any Schwartz seminorm ofϕ∗uis controlled by a finite sum of Schwartz seminorms ofϕ.

We now return to the point left open concerning the convergence of the Riemann sums in (2.3.20) in the topology ofS(Rⁿ). For eachN=1,2, . . ., consider a parti- tion of[−N,N]ⁿinto(2N²)ⁿcubesQ_mof side length 1/Nand lety_mbe the center of eachQ_m. For multi-indicesα,β, we must show that

D_N(x) =

(2N²)ⁿ

∑

m=1

x^α∂_x^βϕ(xe −y_m)ψ(y_m)|Q_m| − Z

Rⁿ

x^α∂_x^βϕe(x−y)ψ(y)dy converges to zero inL^∞(Rⁿ)asN→∞. We have

x^α∂_x^βϕe(x−ym)ψ(y_m)|Q_m| − Z

Qm

x^α∂_x^βϕ(xe −y)ψ(y)dy

= Z

Qm

x^α(y−y_m)·∇y

∂_x^βϕ(xe −y)ψ(y) (ξ)dy for someξ =y+θ(y_m−y), whereθ∈[0,1]. It follows that|y| ≤ |ξ|+√

n/N≤

|ξ|+1 forN≥√

n. It is easy to see that the last integrand is at most C|x|^|α|

√n N

1 (1+|x−ξ|)^M

1 (2+|ξ|)^M forMlarge (pickM>2|α|), which in turn is at most

C⁰|x|^|α|

√n N

1 (1+|x|)^M/2

1

(2+|ξ|)^M/2≤C⁰|x|^|α|

√n N

1 (1+|x|)^M/2

1 (1+|y|)^M/2. Inserting this estimate for the integrand in the last displayed integral, we obtain

|D_N(x)| ≤ C⁰⁰ N

|x|^|α|

(1+|x|)^M/2 Z

[−N,N]ⁿ

dy (1+|y|)^M/2+

Z

([−N,N]ⁿ)^c

|x^α∂_x^βϕ(x−e y)ψ(y)|dy.

But the integrand in the last integral is controlled by C⁰⁰⁰|x|^|α|

(1+|x−y|)^M dy

(1+|y|)^M ≤ C⁰⁰⁰|x|^|α|

(1+|x|)^M/2 dy (1+|y|)^M/2.

Using these estimates it is now easy to see that limN→∞sup_x∈Rn|D_N(x)|=0.

Next we have the following important result regarding distributions with compact support:

Theorem 2.3.21.If u is inE⁰(Rⁿ), thenbu is a real analytic function onRⁿ. More- over,bu has a holomorphic extension onCⁿ. In particular,u is ab C^∞function. Fur- thermore,bu and all of its derivatives have polynomial growth at infinity.

Proof. Given a distribution u with compact support and a polynomial p(ξ), the action ofuon the C^∞ functionξ 7→p(ξ)e^−2πix·ξ is a well defined function ofx, which we denote byu(p(·)e^{−2πix·(·)}). Herexis an element ofRⁿorCⁿ.

It is straightforward to verify that the function ofz= (z₁, . . . ,z_n) F(z) =u e^{−2πi(·)·z}

defined onCⁿis holomorphic, in fact entire. Indeed, the continuity and linearity of uand the fact that(e^−2πiξjh−1)/h→ −2πiξ_jinC^∞(Rⁿ)ash→0 in the complex plane imply thatFis differentiable and its derivative with respect toz_jis the action of the distributionuto theC^∞function

ξ 7→(−2πiξ_j)e^{−2πi∑nj=1ξjzj}. By induction it follows that for all multi-indicesα we have

∂_z^α1

1 · · ·∂_z^α_nⁿF=u (−2πi(·))^αe^{−2πi∑nj=1(·)zj} .

SinceFis entire, its restriction onRⁿ, i.e.,F(x₁, . . . ,x_n), wherex_j=Rez_j, is real analytic. Also, as a consequence of (2.3.4), this restriction and all of its derivatives have polynomial growth at infinity.

Now for f inS(Rⁿ)we have

u,b f

= u,bf

=u Z

Rⁿ

f(x)e^−2πix·ξdx

= Z

Rⁿ

f(x)u(e^{−2πix·(·)})dx, provided we can justify the passage of u inside the integral. The reason for this is that the Riemann sums of the integral of f(x)e^−2πix·ξ overRⁿconverge to it in the topology ofC^∞, and thus the linear functionalucan be interchanged with the integral. We conclude that the tempered distributionubcan be identified with the real analytic functionx7→F(x)whose derivatives have polynomial growth at infinity.

To justify the fact concerning the convergence of the Riemann sums, we argue as in the proof of the previous theorem. For eachN=1,2, . . ., consider a partition of [−N,N]ⁿinto(2N²)ⁿcubesQ_mof side length 1/Nand lety_mbe the center of each Q_m. For a multi-indexαlet

D_N(ξ) =

(2N²)ⁿ

∑

m=1

f(y_m)(−2πiy_m)^αe^{−2πiym·ξ}|Q_m| − Z

Rⁿ

f(x)(−2πix)^αe^−2πx·ξdx.

We must show that for everyM>0, sup_|ξ|≤M|D_N(ξ)|converges to zero asN→∞.

Settingg(x) =f(x)(−2πix)^α, we write

D_N(ξ) =

(2N²)ⁿ m=1

∑

Z

Qm

g(y_m)e^{−2πiym·ξ}−g(x)e^−2πix·ξ dx+

Z

([−N,N]ⁿ)^c

g(x)e^−2πx·ξdx. Using the mean value theorem, we bound the absolute value of the expression inside the square brackets by

|∇g(z_m)|+2π|ξ| |g(z_m)|

√n

N ≤C_K(1+|ξ|) (1+|z_m|)^K

√n N , for some pointzmin the cubeQm. Since

120 2 Maximal Functions, Fourier Transform, and Distributions (2N²)ⁿ

m=1

∑

Qm

C_K(1+|ξ|)

(1+|zm|)^K ≤C_K(1+M)<∞

for|ξ| ≤M, it follows that sup_|ξ|≤M|D_N(ξ)| →0 asN→∞. Next we give a proposition that extends the properties of the Fourier transform to tempered distributions.

Proposition 2.3.22.Given u, v inS(Rⁿ), f∈S, y∈Rⁿ, b a complex scalar,α a multi-index, and a>0, we have

(1) u+v=u+v , (2) bu =bu ,

(3) If u_j→u inS, thenu_j→u in S, (4) (u)= (u),

(5) (τ^y(u))=e^−2πiy·ξu , (6) (e^2πix·yu)=τ^y(u),

(7) (δ^a(u))= (u)a=a⁻ⁿ(δ^a−1(u)), (8) (∂^αu)= (2πiξ)^αu ,

(9) ∂^αu= ((−2πix)^αu), (10) (u)^∨=u ,

(11) f∗u=f u , (12) f u=f∗u ,

(13) (Leibniz’s rule)∂^m_j(f u) =∑^m_k=0m

k

(∂^k_jf)(∂_j^m−ku), m∈Z⁺,

(14) (Leibniz’s rule)∂^α(f u) =∑^α_γ1¹₌0···∑^α_γnⁿ₌0

_α

γ11

···_αn

γn

(∂^γf)(∂^α−γu),

(15) If u_k, u∈L^p(Rⁿ) and u_k→u in L^p (1≤p≤∞), then u_k→u inS(Rⁿ). Therefore, convergence inS implies convergence in L^p, which in turn implies convergence inS(Rⁿ).

Proof. All the statements can be proved easily using duality and the corresponding

statements for Schwartz functions.

We continue with an application of Theorem 2.3.21.

Proposition 2.3.23.Given u∈S(Rⁿ), there exists a sequence ofC₀^∞functions f_k such that f_k→u in the sense of tempered distributions; in particular, C₀^∞(Rⁿ)is dense inS(Rⁿ).

Proof. Fix a function inC₀^∞(Rⁿ)withϕ(x) =1 in a neighborhood of the origin. Let ϕ_k(x) =δ^1/k(ϕ)(x) =ϕ(x/k). It follows from Exercise 2.3.5(b) that foru∈S⁰(Rⁿ), ϕ_ku→uinS⁰. By Proposition 2.3.22 (3), we have that the mapu7→(ϕ_ku)b^∨ is continuous onS⁰(Rⁿ). Now Theorem 2.3.21 gives that(ϕ_ku)b^∨ is aC^∞ function and thereforeϕj(ϕ_ku)b^∨is inC₀^∞(Rⁿ). As observed,ϕj(ϕ_ku)b^∨→(ϕ_ku)b^∨inS⁰when kis fixed and j→∞. Exercise 2.3.5(c) gives that the diagonal sequenceϕ_k(ϕ_kf)

b converges to fbinS as k→∞for all f ∈S. Using duality and Exercise 2.2.2, we conclude that the sequence ofC₀^∞functionsϕ_k(ϕ_ku)b^∨ converges touinS⁰as

k→∞.