ft of distributions

5.1 §5. Fourier transformation of distributions 5.1. Rapidly decreasing functions. In the following we study an important tool in the treatment o f differential operators, the Fourier transform . To begin with, it is useful in the study of differential operators on Rnwith constant coefficients, but it can also be used in the more advanced theory to treat operators on subset s Ω and with variable coefficients. The space D/prime(Rn) is too large to permit a sensible definition of the Fourier transform. We therefore restrict the attention to a somewha t smaller space of distributions, S/prime(Rn), the dual of a space of test functions S(Rn) which is slightly larger than C∞ 0(Rn). (Sresp.S/primeis often called the Schwartz space of test functions resp. distributions, after Laurent Schwa rtz.) It will be convenient to introduce the function /angbracketleftx/angbracketright= (1 + |x|2)1 2, (5.1) and its powers /angbracketleftx/angbracketrights, s∈R. Since |x|//angbracketleftx/angbracketright →1 for |x| → ∞ ,/angbracketleftx/angbracketrightis of the same order of magnitude as |x|, but has the advantage of being a positive C∞function on all of Rn. We note that one has for m∈N0: /angbracketleftx/angbracketright2m= (1 +x2 1+· · ·+x2 n)m=/summationdisplay |α|≤mCm,αx2α/braceleftBigg ≤Cm/summationtext |α|≤mx2α, ≥/summationtext |α|≤mx2α,(5.2) with positive integers Cm,αandCm. This is shown by multiplying out the expression (1+ x2 1+· · ·+x2 n)m; we recall from (A.9) that Cm,α=m! α!(m−|α|)!. Definition 5.1. The vector space S(Rn)(often just denoted S) is defined as the space of C∞functionsϕ(x)onRnsuch thatxαDβϕ(x)is bounded for all multi-indices αandβ∈Nn 0.Sis provided with the family of seminorms pM(ϕ) = sup/braceleftbig /angbracketleftx/angbracketrightM|Dαϕ(x)|/vextendsingle/vextendsinglex∈Rn,|α| ≤M/bracerightbig , M ∈N0. (5.3) The functions ϕ∈ Sare called rapidly decreasing functions . With this system of seminorms, S(Rn) is a Fr´ echet space. The seminorms are in fact norms and form an increasing family, so in particu lar, the system has the max-property (cf. Remark B.6). We could also have tak en the family of seminorms pα,β(ϕ) = sup { |xαDβϕ(x)| |x∈Rn}, (5.4) whereαandβrun through Nn 0; it is seen from the inequalities (5.2) that the family of seminorms (5.4) defines the same topology as the fam ily (5.3). 5.2 As a local neighborhood basis at 0 we can take the sets VM,1 N={ϕ∈ S | /angbracketleftx/angbracketrightM|Dαϕ(x)|<1 Nfor|α| ≤M}, (5.5) forM∈N0,N∈N. The topology on C∞ 0(Rn) is stronger than the topology induced on this space by S(Rn), since the sets VM,1 N∩C∞ 0(Rn) are open, con- vex balanced neighborhoods of 0 in C∞ 0(Rn) =/uniontextC∞ Kj(Rn); this follows since their intersections with C∞ Kj(Rn) are open neighborhoods of 0 in C∞ Kj(Rn). (One may observe that the topology induced from Sis metrizable by Theo- rem B.8, whereas the usual topology on C∞ 0(Rn) is not metrizable.) Lemma 5.2. 1◦For1≤p≤ ∞,S(Rn)is continuously injected in Lp(Rn). 2◦For1≤p≤ ∞,S(Rn)⊂C∞ Lp(Rn). 3◦For1≤p<∞,S(Rn)is dense in Lp(Rn). Proof. 1◦. We have for ϕinS(Rn),MinN0andαinNn 0with|α| ≤Mthat |Dαϕ(x)| ≤pM(ϕ)/angbracketleftx/angbracketright−M for allxinRn. In particular, /bardblϕ/bardblL∞= sup |ϕ(x)|=p0(ϕ). For 1 ≤p<∞, /bardblϕ/bardblp Lp≤pM(ϕ)p/integraldisplay Rn/angbracketleftx/angbracketright−Mpdx; here we note that /integraldisplay Rn/angbracketleftx/angbracketright−Mpdx=/integraldisplay Rn(1 +|x|2)−Mp/2dx<∞whenM >n p. (5.6) Then /bardblϕ/bardblLp≤CMpM(ϕ) forM >n p, completing the proof of 1◦. Now 2◦follows, since Dαf∈ Sfor allαwhenf∈ S. 3◦follows from the fact that the subset C∞ 0(Rn) ofS(Rn) is dense in Lp(Rn) forp<∞, cf. Theorem 2.10. /square It is obvious that multiplication by a polynomial maps SintoS. There are also other C∞functions that define multiplication operators in S. L. Schwartz introduced the following space of functions (“op´ erateurs de multi- plication”) containing the polynomials: Definition 5.3. The vector space OM(Rn)(or just OM)ofslowly increas- ing functions onRnconsists of the functions p(x)∈C∞(Rn)which satify: For anyα∈Nn 0there exists c >0anda∈R(depending on pandα) such that |Dαp(x)| ≤c/angbracketleftx/angbracketrightafor allx∈Rn. 5.3 For example, /angbracketleftx/angbracketrightt∈ OMfor anyt∈R; this is seen by repeated application of the rule ∂j/angbracketleftx/angbracketrights=s/angbracketleftx/angbracketrights−2xj. (5.7) The elements of OMdefine multiplication operators Mp:f/mapsto→pfwhich (by the Leibniz formula) map Scontinuously into S. In particular, since S(Rn)⊂ O M(Rn), we see that ϕψbelongs to S(Rn) whenϕandψbelong toS(Rn). Clearly,∂αandDαarecontinuous operators inS(Rn). Whenf∈L1(Rn), the Fourier transformed function ( Ff)(ξ) is defined by the formula (Ff)(ξ)≡ˆf(ξ) =/integraldisplay Rne−ix·ξf(x)dx. We now recall some of the more elementary rules for the Fourie r transfor- mation of functions (proofs are included for the convenienc e of the reader): Theorem 5.4. 1◦The Fourier transform Fis a continuous linear map of L1(Rn)into CL∞(Rn), such that when f∈L1(Rn), then /bardblˆf/bardblL∞≤ /bardblf/bardblL1,ˆf(ξ)→0for|ξ| → ∞. (5.8) 2◦The Fourier transform is a continuous linear map of S(Rn)intoS(Rn), and one has for f∈ S(Rn),ξ∈Rn: F[xαDβ xf(x)](ξ) = (−Dξ)α(ξβˆf(ξ)), (5.9) for all multi-indices αandβ∈Nn 0. 3◦Defining the co-Fourier transform Fby Ff(ξ) =/integraldisplay Rne+ix·ξf(x)dx,whereby Ff=Ff (it is likewise continuous from StoS), one has for f∈ S(Rn)withˆf=Ff: f(x) = (2π)−n/integraldisplay eiξ·xˆf(ξ)dξ[≡(2π)−nFˆf], (5.10) so the operator Fmaps S(Rn)bijectively onto S(Rn)withF−1= (2π)−nF. Proof. 1◦. The inequality |ˆf(ξ)|=|/integraldisplay eix·ξf(x)|dx≤/integraldisplay |f(x)|dx 5.4 shows the first statement in (5.8), so FmapsL1intoL∞. Whenf∈L1, the functions e−ix·ξf(x) have the integrable majorant |f(x)|for allξ, so the continuity of ˆf(ξ) follows from Lemma 2.8 1◦. That ˆf(ξ)→0 for|ξ| → ∞ will be shown further below. 2◦. Whenf∈L1withxjf(x)∈L1, the functions of x ∂ξj(e−ix·ξf(x)) =−ixje−ix·ξf(x) have the integrable majorant |xjf(x)|, so it follows from Lemma 2.8 2◦that ∂ξjˆf(ξ) exists and equals F(−ixjf(x)). Then also −Dξjˆf=F(xjf(x)). Whenf∈ S, we can apply this rule to all derivatives, obtaining the for mula (−Dξ)αˆf=F(xαf). Whenf∈ S, we find by integration by parts (cf. (A.20)) that /integraldisplay e−ix·ξ∂xjf(x)dx= lim R→∞/integraldisplay |x|≤Re−ix·ξ∂xjf(x)dx= lim R→∞/parenleftBig/integraldisplay |x|≤Riξje−ix·ξf(x)dx+/integraldisplay |x|=Re−ix·ξf(x)xj |x|dx/parenrightBig =iξjˆf(ξ), (sinceRn−1sup{|f(x)| | |x|=R} →0 forR→ ∞), showing the formula iξjˆf=F(∂xjf(x)). Then also F(Dxjf) =ξjˆf. Repeated application gives thatF(Dβf) =ξβˆffor allβ∈Nn 0. Formula (5.9) follows for f∈ Sby combination of the two facts we have shown. Note here that by 1◦, the right hand side is bounded and continuous for allα,β; this implies (in view of the Leibniz formula) that ˆf∈ S. This shows that Fmaps SintoS; the continuity will be shown below. Let us first complete the proof of 1◦: Letf∈L1and letε >0. Since Sis dense inL1(Lemma 5.2), there is a g∈ Swith/bardblf−g/bardblL1< ε/2. Then by 1◦,|ˆf(ξ)−ˆg(ξ)|< ε/2 for allξ. Since ˆg∈ Sby 2◦, we can find an R >0 such that |ˆg(ξ)| ≤ε/2 for|ξ| ≥R. Then |ˆf(ξ)| ≤ |ˆf(ξ)−ˆg(ξ)|+|ˆg(ξ)|<εfor|ξ| ≥R. Now for the continuity: Note that (5.9) implies F((I−∆)f) =F((I−∂2 x1− · · · −∂2 xn)f(x)) = (1 +ξ2 1+· · ·+ξ2 n)ˆf(ξ) =/angbracketleftξ/angbracketright2ˆf.(5.11) Then we have for each k∈N0, takingl=k 2ifkis even,l=k+1 2ifkis odd, 5.5 and recalling (5.6)): p0(ˆf) = sup |ˆf(ξ)| ≤ /bardbl/angbracketleftx/angbracketright−n−1/angbracketleftx/angbracketrightn+1f/bardblL1≤ /bardbl/angbracketleftx/angbracketright−n−1/bardblL1pn+1(f), pk(ˆf) = sup ξ∈Rn,|α|≤k|/angbracketleftξ/angbracketrightkDα ξˆf(ξ)| ≤ sup ξ∈Rn,|α|≤k|/angbracketleftξ/angbracketright2lDα ξˆf(ξ)| = sup ξ∈Rn,|α|≤k|F[(1−∆)l(xαf(x))]| ≤sup |α|≤k/bardbl(1−∆)l(xαf(x))/bardblL1 ≤ /bardbl/angbracketleftx/angbracketright−n−1/bardblL1sup x∈Rn,|α|≤k|/angbracketleftx/angbracketrightn+1(1−∆)l(xαf(x))| ≤Cpk+n+1(f). (5.12) This shows that Fis continuous from StoS. 3◦. Observe first that since Ff=Ff, the operator Fhas properties anal- ogous to those of F. To show (5.10), we need to calculate (2π)−n/integraltext eiξ·x(/integraltext e−iξ·yf(y)dy)dξforf∈ S. The function eiξ·(x−y)f(y) is not integrable on R2n, so we cannot simply change the order of integration. Therefore we introduce an integration factor ψ(ξ)∈ S(Rn), which will be removed later by a passage to the limit. More precisely, we in sert a function ψ(εξ) withψ∈ S(Rn) andε >0. Then we find for each fixed x, by use of the Fubini theorem and the change of variables ( η,z) = (εξ,(y−x)/ε): (2π)−n/integraldisplay Rneiξ·xψ(εξ)ˆf(ξ)dξ= (2π)−n/integraldisplay Rneiξ·xψ(εξ)/parenleftBig/integraldisplay Rne−iξ·yf(y)dy/parenrightBig dξ = (2π)−n/integraldisplay R2ne−iξ·(y−x)ψ(εξ)f(y)dξdy = (2π)−n/integraldisplay R2ne−iη·zψ(η)f(x+εz)dηdz = (2π)−n/integraldisplay Rnˆψ(z)f(x+εz)dz, since the functional determinant is 1. Forε→0, eiξ·xψ(εξ)ˆf(ξ)→ψ(0)eiξ·xˆf(ξ) with |eiξ·xψ(εξ)ˆf(ξ)| ≤C|ˆf(ξ)|, whereC= supξ|ψ(ξ)|. Moreover, ˆψ(z)f(x+εz)→ˆψ(z)f(x) with |ˆψ(z)f(x+εz)| ≤C/prime|ˆψ(z)|, whereC/prime= supy|f(y)|. By the theorem of Lebesgue we then find: (2π)−nψ(0)/integraldisplay eiξ·xˆf(ξ)dξ= (2π)−nf(x)/integraldisplay ˆψ(z)dz. 5.6 One can in particular use ψ(ξ) =e−1 2|ξ|2. In this case, ψ(0) = 1, and ˆψ(z) can be shown to satisfy F(e−1 2|ξ|2) = (2π)n 2e−1 2|z|2; (5.13) then/integraltextˆψ(z)dz= (2π)n 2/integraltext e−1 2|z|2dz= (2π)n. This implies formula (5.10). /square We observe moreover that the map F:L1(Rn)→CL∞(Rn) isinjective . For iff∈L1(Rn) is such that ˆf(ξ) = 0 for all ξ∈Rn, then we have for any ϕ∈C∞ 0(Rn), denoting F−1ϕbyψ, that 0 =/integraldisplay Rnˆf(ξ)ψ(ξ)dξ=/integraldisplay Rn/integraldisplay Rne−ix·ξf(x)ψ(ξ)dxdξ =/integraldisplay Rnf(x)/parenleftbig/integraldisplay Rne−ix·ξψ(ξ)dξ/parenrightbig dx=/integraldisplay Rnf(x)ϕ(x)dx; then it follows by the Du Bois-Reymond Lemma (Lemma 3.2) that f= 0 as an element of L1(Rn). There is the following extension to L2(Rn): Theorem 5.5. (Parseval-Plancherel) 1◦The Fourier transform F:S(Rn)→ S(Rn)extends in a unique way to an isometric isomorphism F2ofL2(Rn,dx)ontoL2(Rn,(2π)−ndx). For f,g∈L2(Rn), /integraldisplay f(x)g(x)dx= (2π)−n/integraldisplay F2f(ξ)F2g(ξ)dξ, /integraldisplay |f(x)|2dx= (2π)−n/integraldisplay |F2f(ξ)|2dξ.(5.14) 2◦One has that F2f=Ffforf∈L2(Rn)∩L1(Rn). (5.15) Moreover, when f∈L2(Rn)and hence 1B(0,N)f∈L1(Rn), then the sequence of continuous functions F(1B(0,N)f)converges in L2(Rn)toF2fforN→ ∞. Proof. 1◦. We first show (5.14) for f,g∈ S. By Theorem 5.4 3◦, g(x) = (2π)−n/integraldisplay eiξ·xˆg(ξ)dξ, so by the Fubini theorem, /integraldisplay f(x)g(x)dx= (2π)−n/integraldisplay f(x)/integraldisplay e−iξ·xˆg(ξ)dξdx = (2π)−n/integraldisplay/parenleftBig/integraldisplay f(x)e−iξ·xdx/parenrightBig ˆg(ξ)dξ= (2π)−n/integraldisplay ˆf(ξ)ˆg(ξ)dξ. 5.7 This shows the first formula, and the second formula follows b y takingf=g. We see from these formulas that F:S(Rn)→L2(Rn,(2π)−ndx) is a linear isometry from S(Rn) considered as a dense subspace of L2(Rn,dx). Since the target space is complete, Fextends in a unique way to a contin- uous linear map F2:L2(Rn,dx)→L2(Rn,(2π)−ndx), which is an isom- etry. The range F2(L2(Rn,dx)) is then also complete, hence is a closed subspace of L2(Rn,(2π)−ndx), but since it contains S(Rn), which is dense in L2(Rn,(2π)−ndx), it must equal L2(Rn,(2π)−ndx). The identities in (5.14) extend by continuity. This shows 1◦. 2◦. Letf∈L2(Rn), thenfN= 1B(0,N)fis clearly in L2(Rn), and it is inL1(Rn) by (A.25). We first show (5.15) for fN. Since S(Rn) is dense in L2(Rn), there is a sequence ϕj∈ S(Rn) withϕj→fNinL2(Rn) forj→ ∞. Then since 1∁B(0,N)is a bounded function, 1∁B(0,N)ϕj→1∁B(0,N)fN= 0 in L2(Rn). Withχ(x) as in (2.3), let ψj=χ(x/N)ϕj; then also ψj→fNin L2(Rn) forj→ ∞. SincefNandψjare supported in B(0,2N), we have by the Cauchy-Schwarz inequality: /bardblfN−ψj/bardblL1≤vol(B(0,2N))1/2/bardblfN−ψj/bardblL2, soψjconverges to fNalso inL1(Rn). Now 1◦above and Theorem 5.4 1◦ give that Fψj→ FfNinCL∞(Rn), Fψj→ F2fNinL2(Rn,(2π)−ndx). Then the limit in CL∞(Rn) is a continuous representative of the limit in L2(Rn,(2π)−ndx). This shows (5.15) for fN. Using the second formula in (5.14) and the Lebesgue theorem w e find: (2π)−n/bardblF2f− F2fN/bardbl2 L2=/bardblf−fN/bardbl2 L2→0 forN→ ∞, showing the convergence statement in 2◦. Finally, we obtain (5.15) in general: When f∈L1(Rn)∩L2(Rn) and we definefNbyfN= 1 B(0,N)f, thenfN→finL1(Rn) as well as in L2(Rn) (by the Lebesgue theorem). Then F2fN→ F2finL2(Rn,(2π)−ndx) andFfN→ FfinCL∞(Rn), so the limits are the same as elements of L2(Rn)./square 2◦shows that the definition of F2onL2is consistent with the definition ofFonL1, so we can drop the index 2, writing Finstead of F2from now on. The isometry property can also be expressed in the way that the operator F= (2π)−n/2F:L2(Rn,dx)→L2(Rn,dx) (5.16) 5.8 is an isometric isomorphism (i.e.,Fis a unitary operator in the Hilbert space L2(Rn)). It is because of this isometry property in connection wit h (5.9) that theL2-theory for distributions is particularly useful for the tr eatment of partial differential equations. Letf=uandg=Fv, then we have as a special case of (5.14) (using (5.10)):/integraldisplay Fuvdx =/integraldisplay uFvdx, foru,v∈L2(Rn,dx). (5.17) The following rules for convolution are known from measure theory: When f∈L1(Rn) andg∈CL∞(Rn), then the convolution (f∗g)(x) =/integraldisplay Rnf(x−y)g(y)dy is defined for all x∈Rn. The function f∗gbelongs toCL∞(Rn), with /bardblf∗g/bardblL∞≤ /bardblf/bardblL1/bardblg/bardblL∞. (5.18) Forf,g∈L1(Rn), the convolution ( f∗g)(x) is defined for almost all x∈Rn and gives an element of L1(Rn) (also denoted f∗g), with /bardblf∗g/bardblL1≤ /bardblf/bardblL1/bardblg/bardblL1. (5.19) The classical result on Fourier transformation of convolut ions is: Theorem 5.6. Whenf,g∈L1(Rn), then F(f∗g) =Ff· Fg. (5.20) Proof. We find by use of the Fubini theorem and a simple change of varia bles: F(f∗g)(ξ) =/integraldisplay Rne−iξ·x/parenleftBig/integraldisplay Rnf(x−y)g(y)dy/parenrightBig dx =/integraldisplay Rng(y)/parenleftBig/integraldisplay Rne−iξ·xf(x−y)dx/parenrightBig dy =/integraldisplay Rng(y)/parenleftBig/integraldisplay Rne−iξ·(x+y)f(x)dx/parenrightBig dy =/integraldisplay Rne−iξ·xf(x)dx/integraldisplay Rne−iξ·yg(y)dy=Ff(ξ)Fg(ξ)./square Observe furthermore: 5.9 Lemma 5.7. Whenϕandψ∈ S(Rn), thenϕ∗ψ∈ S(Rn), andψ/mapsto→ϕ∗ψ is a continuous operator on S(Rn). Proof. Sinceϕandψbelong toL1(Rn)∩CL∞(Rn), the rules (5.18) and (5.19) show that ϕ∗ψ∈L1(Rn)∩CL∞(Rn). Since F(ϕ∗ψ) = (Fϕ)·(Fψ), F(ϕ∗ψ) belongs to S(Rn). Sinceϕ∗ψis continuous and Fis injective from L1(Rn) toCL∞(Rn), mapping S(Rn) onto S(Rn), we find that ϕ∗ψ∈ S(Rn). It is seen from the formula ϕ∗ψ=F−1((Fϕ)·(Fψ)) that the map ψ/mapsto→ϕ∗ψ is continuous. /square 5.2. Temperate distributions. Definition 5.8. S/prime(Rn)(also written as S/prime) is defined as the vector space of continuous linear functionals on S(Rn). The elements of S/prime(Rn)are called temperate distributions .1 According to Lemma B.7, S/primeconsists of the linear functionals Λ on Sfor which there exists an M∈N0and a constant CM(depending on Λ) such that |Λ(ϕ)| ≤CMpM(ϕ),for allϕ∈ S. (5.21) S/prime(Rn) is provided with the weak∗topology (as around (3.28)); this makes S/prime(Rn) a topological vector space. (Its dual space is S(Rn), cf. Section 3.5.) Note that Definition 5.8 does not require introduction of LFspaces etc. (Section B.2), but is based solely on the concept of Fr´ echet spaces. However, it is of interest to set S/prime(Rn) in relation to D/prime(Rn), in particular to justify the use of the word “distribution” in this connection. We firs t show: Lemma 5.9. D(Rn) =C∞ 0(Rn)is a dense subset of S(Rn), with a stronger topology. Proof. As already noted, C∞ 0(Rn)⊂ S(Rn), and the neighborhood basis (5.5) for Sat zero intersects C∞ 0with open neighborhoods of 0 there, so that the topology induced on C∞ 0fromSis weaker than the original toplogy on C∞ 0. To show the denseness, let u∈ S(Rn); then we must show that there is a sequenceuN→uinS(Rn) withuN∈C∞ 0(Rn). For this we take uN(x) =χ(x/N)u(x), (5.22) 1The word “temperate” used for the special distributions allude s to the temperate zone (with moderate temperature), the word can also mean “exercising m oderation and self- restraint”. The word ”tempered” is also often used, but it has more to do with temper (mood), or can indicate a modification of a physical conditio n. The word temperate is used in H¨ ormander’s books [H83], [H85]. 5.10 cf. (2.3). Here uN∈C∞ 0(Rn);uN(x) equalsu(x) for|x| ≤Nand equals 0 for|x| ≥2N. Note that (as used also in the proof of Theorem 4.10) |Dβ xχ(x/N)|=|N−|β|Dβ yχ(y)|y=x/N| ≤CβN−|β|(5.23) for eachβ; hereDβ xχ(x/N) has support in {x|N≤ |x| ≤2N}whenβ/negationslash= 0. For eachM≥0, eachα, we have: sup x∈Rn|/angbracketleftx/angbracketrightMDα[(1−χ(x/N))u(x)]|= sup |x|≥N|/angbracketleftx/angbracketrightMDα[(1−χ(x/N))u(x)]| ≤ /angbracketleftN/angbracketright−1sup |x|≥N|/angbracketleftx/angbracketrightM+1((χ(x/N)−1)Dαu+/summationdisplay 0/negationslash=β≤α/parenleftbigα β/parenrightbig Dβχ(x/N)Dα−βu)| ≤CM,α/angbracketleftN/angbracketright−1→0 forN→ ∞. (5.24) It follows that χ(x/N)u→uinSforN→ ∞./square In particular, a functional Λ ∈ S/prime(Rn) restricts to a continuous functional onD(Rn), also documented by the fact that (5.21) implies, when ϕis supp- ported in a compact set K: |Λ(ϕ)| ≤CMpM(ϕ) =CMsup/braceleftbig /angbracketleftx/angbracketrightM|Dαϕ(x)|/vextendsingle/vextendsinglex∈Rn,|α| ≤M/bracerightbig ≤C/prime Msup/braceleftbig |Dαϕ(x)|/vextendsingle/vextendsinglex∈K,|α| ≤M/bracerightbig , (5.24a) withC/prime M=CMsupx∈K/angbracketleftx/angbracketrightM. Theorem 5.9a. The mapJ: Λ/mapsto→Λ/primefrom S/prime(Rn)toD/prime(Rn)defined by restriction of ΛtoD(Rn), /angbracketleftΛ/prime,ϕ/angbracketright= Λ(ϕ)forϕ∈ D(Rn), is injective, and hence allows an identification of JS/prime(Rn)with a subspace of D/prime(Rn), also called S/prime(Rn): S/prime(Rn)⊂ D/prime(Rn). (5.26) Proof. The mapJ: Λ/mapsto→Λ/primeis injective because of Lemma 5.9, since /angbracketleftΛ/prime,ϕ/angbracketright= 0 for all ϕ∈C∞ 0(Rn) (5.25) implies that Λ is 0 on a dense subset of S, hence equals the 0-functional (since Λ is continuous on S)./square When Λ ∈ S/prime, we now write Λ( ϕ) as/angbracketleftΛ,ϕ/angbracketright, also when ϕ∈ S. Note that the elements of D/prime(Rn) that lie in S/prime(Rn) are exactly those for which there exists anMand aCMsuch that |Λ(ϕ)| ≤CMpM(ϕ) for allϕ∈ D(Rn), (5.26a) independently of the support of ϕ. For, they are continuous on Dwith respect to the topology of S, hence extend (in view of Lemma 5.9) to continuous functionals on S. One may observe that J:S/prime→ D/primeis precisely the adjoint of the continuous injectionι:D → S . 5.11 Lemma 5.10. For1≤p≤ ∞andfinLp(Rn), the mapϕ/mapsto→/integraltext Rnfϕdx , ϕ∈ S(Rn)defines a temperate distribution. In this way one has for each p a continuous injection of Lp(Rn)intoS/prime(Rn). Proof. Denote the map f/mapsto→/integraltext Rnfϕdx by Λ f. Let as usual p/primebe given by 1 p+1 p/prime= 1, with 1/prime=∞and∞/prime= 1. According to the H¨ older inequality, |Λf(ϕ)| ≤ /bardblf/bardblLp/bardblϕ/bardblLp/prime, and here we have by Lemma 5.2: /bardblϕ/bardblLp/prime/braceleftbigg≤CMpM(ϕ),ifp/prime<∞,M >n p/prime, =p0(ϕ),ifp/prime=∞. Hence Λ fis a continuous functional on S, and therefore belongs to S/prime. Then Λfalso defines an element of D/prime, denotedJΛfabove. Since JΛf(ϕ) =/integraltext Rnfϕdx forϕ∈C∞ 0(Rn), we know from the Du Bois-Reymond lemma that the mapf/mapsto→JΛfis injective. Then f/mapsto→Λfmust likewise be injective. /square In particular, S(Rn) is continuously injected into S/prime(Rn). Example 5.11. Here are some examples of elements of S/prime. 1◦Besides the already mentioned functions u∈Lp(Rn),p∈[1,∞],all functionsv∈L1,loc(Rn)with|v(x)| ≤C/angbracketleftx/angbracketrightNfor someNare in S/prime. (Note that these functions include the OM-functions, in particular the polynomials, but they need not be differentiable.) We see this by observing that for such a functionv, |/angbracketleftv,ϕ/angbracketright|=|/integraldisplay vϕdx| ≤C/integraldisplay /angbracketleftx/angbracketright−n−1dxsup{/angbracketleftx/angbracketrightN+n+1|ϕ(x)| |x∈Rn} ≤C/primepN+n+1(ϕ),forϕ∈ S. 2◦Theδ-distribution and its derivatives Dαδare in S/prime, since |/angbracketleftDαδ,ϕ/angbracketright|=|Dαϕ(0)| ≤p|α|(ϕ),forϕ∈ S. 3◦Distributions with compact support. We have shown earlier that a distri- butionuwith compact support satisfies an estimate (3.34); the semin orm in this expression is ≤pNj, so the distribution is a continuous linear functional onS(Rn). Hence E/prime(Rn)⊂ S/prime(Rn)⊂ D/prime(Rn). (5.27) There exist distributions in D/prime(Rn)\S/prime(Rn). An example is ex(forn= 1), cf. Exercise 5.1. In fact it grows too fast for x→ ∞; this illustrates the use of the word “temperate” in the name for S/prime, indicating that the elements of S/primegrow in a controlled way for |x| → ∞ . 5.12 5.3. The Fourier transform on S/prime. The operations DαandMp(multiplication by p) forp∈C∞(Rn) are defined on D/prime(Rn) (Definition 3.5). Concerning their action on S/prime, we have: Lemma 5.12. 1◦Dαmaps S/primecontinuously into S/primefor allα∈Nn 0. 2◦Whenp∈ OM,Mpmaps S/primecontinuously into S/prime, Proof. Forα∈Nn 0,p∈ OM, we set /angbracketleftDαu,ϕ/angbracketright=/angbracketleftu,(−D)αϕ/angbracketright, /angbracketleftpu,ϕ/angbracketright=/angbracketleftu,pϕ/angbracketright,forϕ∈ S(Rn).(5.28) Because of the continuity of the maps in S, it follows that these formulas define distributions in S/prime, which agree with the original definitions on D/prime(Rn). The continuity of the hereby defined maps in S/primeis shown in a similar way as in Theorem 3.8. /square We find, similarly to Theorem 3.15: Lemma 5.13. ForϕinS(Rn)anduinS/prime(Rn), the prescription /angbracketleftϕ∗u,ψ/angbracketright=/angbracketleftu,ˇϕ∗ψ/angbracketright, ψ∈ S(Rn) (5.29) defines a temperate distribution ϕ∗u; andu/mapsto→ϕ∗uis a continuous operator inS/prime(Rn). Moreover, Dα(ϕ∗u) = (Dαϕ)∗u=ϕ∗Dαu,forϕ∈ S, u∈ S/prime, α∈Nn 0, (ϕ∗ψ)∗u=ϕ∗(ψ∗u),forϕ,ψ∈ S, u∈ S/prime.(5.30) Both in multiplication and in convolution formulas we shall henceforth allow the smooth factor to be written to the right, setting u·ϕ=ϕ·uand u∗ϕ=ϕ∗u. Remark 5.14. With the notation of Section 3.5, one finds by a bi-annihila- tor argument (as for DandD/prime) that S(Rn) is a dense subset of S/prime(Rn). An operatorAonS(Rn) therefore has at most one extension to a continuous operator on S/prime(Rn). It is then moreover seen that if a continuous operator AonS(Rn) has a corresponding continuous operator BonS(Rn) such that/integraltext Rn(Aϕ)ψdz=/integraltext Rnϕ(Bψ)dxfor allϕ,ψinS(Rn), thenAhas a unique extension to a continuous operator on S/prime(Rn), namelyB×. Moreover, if the restrictions of AandBtoC∞ 0(Rn) mapC∞ 0(Rn) continuously into C∞ 0(Rn), then the restriction to S/prime(Rn) of the already defined extension of A|C∞ 0(Rn) 5.13 to aσ(D/prime(Rn) ,C∞ 0(Rn)) continuous operator on D/prime(Rn) is precisely the extension of Ato aσ(S/prime(Rn),S(Rn)) continuous operator on S/prime(Rn). The operators of differentiation, multiplication and convo lution introduced onS/primeabove can be considered from this point of view. We shall finally introduce the very important generalization of the Fourier transform and show how it interacts with the other maps: Definition 5.15. Foru∈ S/prime, the prescription /angbracketleftFu,ϕ/angbracketright=/angbracketleftu,Fϕ/angbracketrightfor allϕ∈ S (5.31) defines a temperate distribution Fu(also denoted ˆu); and F:u/mapsto→ Fuis a continuous operator on S/prime(Rn). We also define F= (2π)−n/2F. The definition is of course chosen such that it is consistent w ith the formula (5.17) for the case where u∈ S. It is also consistent with the definition of F2 onL2(Rn), sinceϕk→uinL2(Rn) implies Fϕk→ F2uinS/prime. Similarly, the definition is consistent with the definition on L1(Rn). That Fis acontinuous operator on S/primeis seen as in Theorem 3.8 or by use of Remark 5.14. The operator Fis similarly extended to S/prime, on the basis of the identity /angbracketleftFu,ϕ/angbracketright=/angbracketleftu,Fϕ/angbracketright; (5.32) and since (2π)−nFF= (2π)−nFF=I (5.33) onS, this identity is likewise carried over to S/prime, so we obtain: Theorem 5.16. Fis a homeomorphism of S/primeontoS/prime, with inverse F−1= (2π)−nF. This extension of Fto an operator on S/primegives an enormous freedom in the use of the Fourier transform. We obtain directly from the theorems for FonS, Lemma 5.9 and the definitions of the generalized operators: Theorem 5.17. For allu∈ S/prime, one has when α∈Nn 0andϕ∈ S: (i)F(Dαu) =ξαFu, (ii) F(xαu) = (−Dξ)αFu, (iii) F(ϕ∗u) = (Fϕ)·(Fu), (iv) F(ϕ·u) = (2π)−n(Fϕ)∗(Fu).(5.34) Let us study some special examples. 5.14 Foru=δ, /angbracketleftFu,ϕ/angbracketright=/angbracketleftu,Fϕ/angbracketright= ˆϕ(0) =/integraldisplay ϕ(x)dx=/angbracketleft1,ϕ/angbracketright,forϕ∈ S, hence F[δ] = 1. (5.35) Since clearly also F[δ] = 1 (cf. (5.32)), we get from the inversion formula (5.33) that F[1] = (2π)nδ. (5.36) An application of Theorem 5.17 then gives: F[Dαδ] =ξα F[(−x)α] = (2π)nDα ξδ.(5.37) Remark 5.18. We have shown that Fdefines a homeomorphism of Sonto S, ofL2ontoL2and of S/primeontoS/prime. One can ask for the image by Fof other spaces. For example, F(C∞ 0(Rn)) must be a certain subspace of S; but this is notcontained in C∞ 0(Rn). On the contrary, if ϕ∈C∞ 0(Rn), then ˆϕcan only have compact support if ϕ= 0 ! Forn= 1 we can give a quick explanation of this: When ϕ∈C∞ 0(R), then ˆϕ(ζ) can be defined for allζ∈Cby the formula ˆϕ(ζ) =/integraldisplay supp ϕe−ixζϕ(x)dx, and this function ˆ ϕ(ζ) isholomorphic inζ=ξ+iη∈C, since (∂ξ+i∂η)ˆϕ(ξ+iη) = 0 (the Cauchy-Riemann equation), as is seen by dif- ferentiation under the integral sign. (One could also appea l to Morera’s Theorem.) Now if ˆ ϕ(ζ) is identically 0 on an open, nonempty interval of the real axis, then ˆ ϕ= 0 everywhere. The argument may be extended to n>1. Even for distributions uwith compact support, ˆ u(ζ) is a function ofζ which can be defined for all ζ∈Cn. For, one can show that ˆ ucoincides with the function ˆu(ζ) =/angbracketleftu,ψ(x)e−ix·ζ/angbracketright/bracketleftBig =/angbracketleftu E/prime,e−ix·ζ E/angbracketright/bracketrightBig , (5.38) whereψ(x) is a function ∈C∞ 0(Rn) which is 1 on a neighborhood of supp u. It is seen as in Exercise 3.14 that this function ˆ u(ζ) isC∞as a function of (ξ1,η1,...,ξ n,ηn)∈R2n(ζj=ξj+iηj), with ∂ξjˆu(ζ) =/angbracketleftu,ψ(x)∂ξje−ix·ζ/angbracketright, and similarly for ∂ηj. Sincee−ix·ζsatisfies the Cauchy-Riemann equation in each complex variable ζj, so does ˆu(ζ), so ˆu(ζ) is a holomorphic function of 5.15 ζj∈Cfor eachj. Then it follows also here that the support of ˆ u(ζ) cannot be compact unless u= 0. The spaces of holomorphic functions obtained by applying FtoC∞ 0(Rn) resp.E/prime(Rn) may be characterized by their growth properties in ζ(the Paley- Wiener Theorem, see e.g. the book of W. Rudin [R74], Theorems 7.22, 7.23, or the book of L. H¨ ormander [H63], Theorem 1.7.7). For partial differential operators with constant coefficient s, the Fourier transform gives a remarkable simplification. When P(D) =/summationdisplay |α|≤maαDα(5.39) is a differential operator on Rnwith coefficients aα∈C, the equation P(D)u=f (5.40) (withuandf∈ S/prime) is by Fourier transformation carried over to the multi- plication equation p(ξ)ˆu(ξ) =ˆf(ξ) (5.41) wherep(ξ) is the polynomial p(ξ) =/summationdisplay |α|≤maαξα; (5.42) it is called the symbol of P(D). Them’th order part of P(D) is called the principal part (often denoted Pm(D)), and its associated symbol pmthe principal symbol , i.e., Pm(D) =/summationdisplay |α|=maαDα, p m(ξ) =/summationdisplay |α|=maαξα. (5.43) It is often so that it is the principal part that determines th e solvability properties of (5.40). The operator P(D) is in particular called elliptic if pm(ξ)/negationslash= 0 forξ/negationslash= 0. Note that pm(ξ) is a homogeneous polynomial in ξof degreem. Example 5.19. (“The world’s simplest example” ) Consider the oper- atorP= 1−∆ on Rn. By Fourier transformation, the equation (1−∆)u=fonRn(5.44) 5.16 is carried into the equation (1 +|ξ|2)ˆu=ˆfonRn, (5.45) and this leads by division with 1 + |ξ|2=/angbracketleftξ/angbracketright2to ˆu=/angbracketleftξ/angbracketright−2ˆf . Thus (5.44) has the solution u=F−1(/angbracketleftξ/angbracketright−2Ff). We see that for any fgiven in S/primethere is one and only one solution u∈ S/prime, and iffbelongs to Sthen the solution ubelongs to S. Whenfis given inL2(Rn), we see from (5.45) that (1 + |ξ|2)ˆu(ξ)∈L2. This implies not only thatu∈L2(since ˆu∈L2), but even that DjuandDiDju∈L2for i,j= 1,...,n . Indeed,ξjˆuandξiξjˆuare inL2since |ξj| ≤1 +|ξ|2and |ξiξj| ≤1 2(|ξi|2+|ξj|2)≤ |ξ|2; here we have used the elementary inequality 2ab≤a2+b2fora,b∈R, (5.46) which follows from ( a−b)2≥0. Thus we obtain: u∈ S/prime(Rn) with (1 −∆)u∈L2(Rn) =⇒u∈H2(Rn). (5.47) Conversely, it is clear that u∈H2(Rn) =⇒(1−∆)u∈L2(Rn), so that in fact, u∈H2(Rn)⇐ ⇒(1−∆)u∈L2(Rn). (5.48) In particular, the maximal operator in L2(Rn) forA= 1−∆ hasD(Amax) = H2(Rn). This resembles to some extent what we found for ordinary di ffer- ential operators in Section 4.3, and it demonstrates clearl y the usefulness of the Fourier transform. Note that our estimates show that the graph norm onD(Amax) is equivalent with the H2(Rn)-norm. Since C∞ 0(Rn) is dense in H2(Rn) (Corollary 4.11), we see that Amin=Amaxhere, and the operator is selfadjoint as an unbounded operator in L2(Rn). (More on maximal and minimal operators on Rnin Theorem 6.3ff below.) It should be noted that 1 −∆ is an unusually “nice” operator, since the polynomial 1 + |ξ|2is positive everywhere. As soon as there are zeroes, the theory becomes more complicated. For example, the wave oper ator∂2 t−∆x onRn+1with symbol −τ2+|ξ|2requires rather different techniques. Even 5.17 for the Laplace operator ∆, whose symbol −|ξ|2has just one zero ξ= 0, it is less simple to discuss exact solutions. At any rate, the Laplace operator is elliptic, and it is fairl y easy to show qualitative properties of the solutions of the equation −∆u=fby use of the Fourier transform. We return to this and a further discussio n of differential operators in Chapter 6. First we shall study some properties of the Fourier transform which for example lead to exact results for the equ ation −∆u=f. 5.4. Homogeneity. When calculating the Fourier transform of specific function s, one can sometimes profit from symmetry properties. We shall give som e useful ex- amples. The idea is to use the interaction of the Fourier transform wi th suitable coordinate changes; here we take the orthogonal transformations y=Oxand thedilationsy=λx(=µλ(x)), described in Example 3.21. As mentioned there, the associated maps are given by [T(O)u](y) =u(O−1y) =u(x),wheny=Ox, [T(µλ)u](y) =u(y/λ) =u(x),wheny=λx;(5.49a) they clearly map SintoSandS/primeintoS/prime(where they are interpreted as in (3.58) and (3.56)). For test functions ψ∈ Swe now find (using that (O∗)−1=O): F[T(O)ψ](ξ) =/integraldisplay e−iy·ξψ(O−1y)dy (5.49) =/integraldisplay e−iOx·ξψ(x)dx=/integraldisplay e−ix·O∗ξψ(x)dx =F[ψ](O∗ξ) = [T((O∗)−1)ˆψ](ξ) = [T(O)ˆψ](ξ) ; F[T(µλ)ψ](ξ) =/integraldisplay e−iy·ξψ(y/λ)dy (5.50) =/integraldisplay e−iλx·ξψ(x)|λn|dx=|λn|F[ψ](λξ) = [|λn|T(µ1/λ)ˆψ](ξ). 5.18 This leads to the general rules for u∈ S/prime: /angbracketleftF[T(O)u],ψ/angbracketright=/angbracketleftT(O)u,Fψ/angbracketright =/angbracketleftu,T(O−1)Fψ/angbracketright=/angbracketleftu,F[T(O∗)ψ]/angbracketright =/angbracketleftFu,T(O∗)ψ/angbracketright=/angbracketleftT((O∗)−1)Fu,ψ/angbracketright=/angbracketleftT(O)Fu,ψ/angbracketright; /angbracketleftF[T(µλ)u],ψ/angbracketright=/angbracketleftT(µλ)u,Fψ/angbracketright =/angbracketleftu,|λn|T(µ1/λ)Fψ/angbracketright=/angbracketleftu,F[T(µλ)ψ]/angbracketright =/angbracketleftFu,T(µλ)ψ/angbracketright=/angbracketleft|λn|T(µ1/λ)Fu,ψ/angbracketright. (The rules could also have been obtained from (5.49), (5.50) by extension by continuity, cf. Remark 5.14.) We have shown: Theorem 5.20. LetObe an orthogonal transformation in Rnand letµλbe the multiplication by the scalar λ∈R\{0}. The associated coordinate change mapsT(O)andT(µλ)inS/primeare connected with the Fourier transform in the following way: F[T(O)u] =T((O∗)−1)Fu=T(O)Fu, (5.51) F[T(µλ)u] =|λn|T(µ1/λ)Fu, (5.52) foru∈ S/prime. The theorem is used in the treatment of functions with specia l invariance properties under such coordinate changes. Those functions u(x) which only depend on the distance |x|to 0, may be characterized as the functions that are invariant under all orthogonal transformations , i.e. for which T(O)u=ufor all orthogonal transformations O (5.53) (since the orthogonal transformations are exactly those tr ansformations in Rnwhich preserve |x|). We shall analogously for u∈ S/prime(Rn) say that u depends only on the distance |x|to 0 when (5.53) holds. A function is homogeneous of degree r, whenu(ax) =aru(x) holds for all a>0 and allx∈Rn\ {0}, that is, T(µ1/a)u=aru,for alla>0. (5.54) We say analogously that a distribution u∈ S/prime(Rn) is homogeneous of degree rwhen (5.54) holds. Theorem 5.20 easily implies: 5.19 Corollary 5.21. Letu∈ S/prime(Rn), and letr∈R. 1◦Ifuonly depends on the distance |x|to0, then the same holds for ˆu. 2◦Ifuis homogeneous of degree r, then ˆuis homogeneous of degree −r−n. Proof. 1◦. The identities (5.53) carry over to similar identities for ˆuaccording to (5.51). 2◦. Whenuis homogeneous of degree r, then we have according to (5.52) and (5.54): T(µ1/a)Fu=a−nF[T(µa)u] =a−nF[a−ru] =a−n−rFu, which shows that Fuis homogeneous of degree −n−r./square Let us apply the theorem to the functions u(x) =|x|−r, which have both properties: They are homogeneous of degree −rand depend only on |x|. Letn/2< r < n ; thenucan be integrated into 0 whereas u2can be integrated out to ∞. Then we can write u=χu+ (1−χ)u, whereχu∈L1 and (1 −χ)u∈L2. It follows that ˆ u∈CL∞(Rn) +L2(Rn)⊂L2,loc(Rn)∩ S/prime. We see from Corollary 5.21 that ˆ u(ξ) is a function which only depends on |ξ| and is homogeneous of degree r−n. To determine ˆ umore precisely, we shall consider the function v(ξ) defined by v(ξ) =|ξ|n−rˆu(ξ). It is inL2,loc∩S/prime(sincer<n), and is homogeneous of degree 0 and depends only on |ξ|(i.e., it is invariant under dilations and orthogonal trans forma- tions). Ifvis known to be continuous on Rn\ {0}, the invariance implies that v(ξ) =v(η) for all points ξandη∈Rn\ {0}, sovequals a constant cn,r onRn\ {0}. Sincevis a locally integrable function, it identifies with the constant function cn,ronRn. This gives the formula for ˆ u: F(|x|−r) = ˆu(ξ) =cn,r|ξ|−n+r. (5.55) We want to show this formula, but since we only know on beforeh and thatv∈L2,loc, an extra argument is needed. For example, one can reason as sketched in the following: If a distribution fdefined on a product set Qn=In⊂Rnhas∂x1f= ∂x2f=· · ·=∂xnf= 0 onQn, then it equals a constant, by Exercise 4.14. If a locally integrable function g(x) is invariant under translations in the coordinate directions, we see from Exercise 3.13 that its fir st derivatives in the distribution sense are 0, so by the just mentioned result , it must equal 5.20 a constant. We are almost in this situation with v(ξ), except that vis in- variant not under translations but under dilations and rota tions (orthogonal transformations). But this can be carried over to the rectan gular situation by a change of coordinates: Consider von a conical neighborhood of a point ξ0/negationslash= 0, say, and express it in terms of spherical coordinates the re; then it is invariant under translation in the radial direction as well as in the directions orthogonal to this. Hence it must be constant there, in a neig hborhood of every point, hence constant throughout. Thusvequals a constant function c. Denoting the constant by cn,r, we have obtained (5.55). The constant cn,ris determined by suitable calculations (e.g. integration against exp( −|x|2/2)), and one finds that cn,r= (4π)n/22−rΓ(n−r 2) Γ(r 2)(5.56) forr∈]n/2,n[ (we have this information from [R74], Exercise 8.6). It is s een directly that cn,ris real by observing that F(|x|−r) =F(|−x|−r) =F(|x|−r). Whenr∈]0,n/2[ , thenr/prime=n−rlies in ]n/2,n[ , so we find from (5.55) using that F−1= (2π)−nF: |x|−n+r=|x|−r/prime= (2π)−nF(cn,r/prime|ξ|−n+r/prime) = (2π)−ncn,r/primeF(|ξ|−r); this shows that (5.55) holds for r∈]0,n/2[ , with cn,r= (2π)nc−1 n,n−r= (4π)n/22−rΓ(n−r 2) Γ(r 2). (5.57) So (5.56) is also valid here. Sincecn,rby (5.56) converges to (2 π)n/2forr→n/2, and |x|−ras well as|x|−n+rconverge to |x|−n/2inL1,loc(Rn)∩S/primeforr→n/2, formula (5.55) is extended by passage to the limit to hold also for r=n/2. We have then obtained: Theorem 5.22. Whenr∈]0,n[, then F(|x|−r) =cn,r|ξ|−n+r, (5.58) wherecn,rsatisfies (5.57). Foru(x) =|x|−rwithr≥n, it is not evident how to interpret uas a distribution; there are special theories for this (see e.g. the definition of the “principal value” in [S50], and a general theory of homogene ous distributions in Section 3.2 of [H83]). See also Section 5.6. 5.21 Important special cases of Theorem 5.22 are the formulas |ξ|−2=F/parenleftBig1 4π|x|/parenrightBig forn= 3 ; |ξ|−2=F/parenleftBigΓ(n 2−1) 4πn 2|x|n−2/parenrightBig forn≥3.(5.59) 5.5. Application to the Laplace operator. The preceding results make it possible to treat the equation for the Laplace operator −∆u=f, (5.60) for a reasonable class of functions. Whenuandfare temperate distributions, the equation gives by Fourier transformation: |ξ|2ˆu=ˆf . A solution may then be written (provided that we can give it a m eaning) ˆu(ξ) =|ξ|−2ˆf(ξ). (5.61) Iffis given in S, we can use (5.34 iv) and (5.59) for n≥3, which gives: u(x) =F−1(|ξ|−2)∗f =Γ(n 2−1) 4πn 2/integraldisplayf(y) |x−y|n−2dy=Γ(n 2−1) 4πn 2/integraldisplayf(x−y) |y|n−2dy,(5.62) so this is a solution of (5.60). The solution uis aC∞-function, since dif- ferentiation can be carried under the integral sign. There a re many other solutions, namely all functions u+wwherewruns through the solutions of ∆w= 0, the harmonic functions (which span an infinitely dimensional vector space; already the harmonic polynomials do so). The functionΓ(n 2−1) 4πn 2|x|−n+2is called the Newton potential. Once we have the formula (5.62), we can try to use it for more general fand thereby extend the applicability. For example, if we insert a contin uous function with compact support as f, then we get a function u, which is not always two times differentiable in the classical sense, but still fo r bounded open sets Ω can be shown to belong to H2(Ω) and solve (5.60) in the distribution sense (in fact it solves (5.60) as an H2-function). See also Remark 6.11 later. This solution method may in fact be extended to distribution sf∈ E/prime(Rn), but this requires a generalization of the convolution opera tor which we refrain from including here. 5.22 The operator −∆:u/mapsto→fislocal, in the sense that the shape of fin the neighborhood of a point depends only on the shape of uin a neighborhood of the point (this holds for all differential operators). On t he other hand, the solution operator T:f/mapsto→udefined by (5.62) cannot be expected to be local (we see this explicitly from the expression for Tas an integral operator). Let Ω be a bounded open subset of Rn. By use of Tdefined above, we define the operator TΩas the map that sends ϕ∈C∞ 0(Ω) (extended by 0 in Rn\Ω) into (Tϕ)/vextendsingle/vextendsingle Ω, i.e., TΩ:ϕ/mapsto→(Tϕ)/vextendsingle/vextendsingle Ωforϕ∈C∞ 0(Ω). (5.63) We here have that (−∆TΩϕ)(x) =ϕ(x) forx∈Ω, because ∆ is local. Thus TΩis a right inverse of −∆ on Ω. It is an integral operator TΩϕ=/integraldisplay ΩG(x,y)ϕ(y)dy, (5.64) with the kernel G(x,y) =Γ(n 2−1) 4πn 2|x−y|−n+2,forx,y∈Ω. An interesting question concerning this solution operator is whether it is a Hilbert-Schmidt operator (i.e., an integral operator w hose kernel is in L2(Ω×Ω)). For this we calculate /integraldisplay Ω×Ω|G(x,y)|2dxdy =c/integraldisplay Ω×Ω|x−y|−2n+4dxdy ≤c/prime/integraldisplay |z|,|w|≤R|z|−2n+4dzdw, where we used the coordinate change z=x−y,w=x+y, and chose Rso large that Ω ×Ω⊂ {(x,y)| |x+y| ≤R,|x−y| ≤R}. The integral with respect tozin the last expression (and thereby the full integral) is fini te if and only if −2n+ 4>−n, i.e.,n<4. SoTΩis a Hilbert-Schmidt operator inL2(Ω),whenn= 3 (in particular, a compact operator). One can show more generally that TΩfor bounded Ω is a compact selfad- joint operator in L2(Ω), for which the eigenvalue sequence ( λj(TΩ))j∈Nis in /lscriptpforp > n/ 2 (TΩbelongs to the p’th Schatten class; the Hilbert-Schmidt case is the case p= 2). When Ω is unbounded, TΩis in general not a compact operator in L2(Ω) (unless Ω is very “thin”). 5.23 5.6. Distributions associated with non-integrable functi ons. We shall investigate some more types of distributions on Rand their Fourier transforms. Theorem 5.20 treated homogeneous func tionsuof de- greea, but the desire to have uand ˆuinL1,locput essential restrictions on the values of athat could be covered. Now n= 1, so the calculations before Theorem 5.22 show how the cases a∈]−1,0 [ may be treated. This neither covers the case a= 0 (i.e., functions u=c1H(x) +c2H(−x)) nor the case a=−1 (where the functions u=c1H(x) x+c2H(−x) xare not inL1,locin the neighborhood of 0 if ( c1,c2)/negationslash= (0,0)). We shall now consider these cases. One result is a description of the Fourier transform of the He aviside function (which will allow us to treat the case a= 0), another result is that we give sense to a distribution which outside of 0 behaves like1 x. Whenfis a function on Rwhich is integrable on the intervals ] − ∞,−ε[ and [ε,∞[ for every ε >0, then we define the principal value integral off overRby PV/integraldisplay Rf(x)dx= lim ε→0/integraldisplay R\[−ε,ε]f(x)dx, (5.65) when this limit exists. (It is important in the definition tha t the interval [−ε,ε] issymmetric around 0; when f /∈L1,loc(R) there is the risk of getting another limit by cutting out another interval like for examp le [−ε,2ε].) We now define the distributionen PV1 xby /angbracketleftPV1 x,ϕ/angbracketright= PV/integraldisplay Rϕ(x) xdxforϕ∈C∞ 0(R). (5.66) (In some of the literature, this distribution is denoted vp1 x, for “valeur prin- cipale”.) We have to show that the functional in (5.66) is wel l-defined and continuous on C∞ 0(R). Here we use that by the Taylor formula, ϕ(x) =ϕ(0) +x·ϕ1(x), (5.67) whereϕ1(x) =ϕ(x)−ϕ(0) xis inC∞(R) (the reader should verify this). Moreover, we have for x∈[−R,R], by the mean value theorem, sup |x|≤R|ϕ1(x)|= sup |x|≤R/vextendsingle/vextendsingle/vextendsingleϕ(x)−ϕ(0) x/vextendsingle/vextendsingle/vextendsingle= sup |x|≤R|ϕ/prime(θ(x))| ≤sup |x|≤R|ϕ/prime(x)|, (5.68) 5.24 whereθ(x) is a suitable point between 0 and x. This gives for PV1 x, when suppϕ⊂[−R,R], /angbracketleftPV1 x,ϕ/angbracketright= lim ε→0/integraldisplay |x|>εϕ(x) xdx (5.69) = lim ε→0/bracketleftBig/integraldisplay [−R,−ε]∪[ε,R]ϕ(0) xdx+/integraldisplay [−R,−ε]∪[ε,R]ϕ1(x)dx/bracketrightBig =/integraldisplayR −Rϕ1(x)dx, since the first integral in the bracket is 0 because of the symm etry of1 x. Thus the functional PV1 xis well-defined, and we see from (5.68) that it is a distribution of order 1: |/angbracketleftPV1 x,ϕ/angbracketright| ≤2Rsup |x|≤R|ϕ1(x)| ≤2Rsup |x|≤R|ϕ/prime(x)|, when supp ϕ⊂[−R,R].(5.70) Remark 5.23. One can also associate distributions with the other functio ns1 xm,m∈N; here one uses on one hand the principal value concept, on the other hand a modification of ϕ(x) by a Taylor polynomial at 0; cf. Exercise 5.9. The resulting distributions are called Pf1 xm, where Pf stands for pseudo- function; for m= 1 we have that Pf1 x= PV1 x. Since PV1 x=χPV1 x+ (1−χ)1 xhas its first term in S/prime(R) and second term inL2(R),v= PV1 xbelongs to S/prime, hence has a Fourier transformed ˆ v. We can find it in the following way: Observe that x·PV1 x= 1 (5.71) (using the definitions), so that ˆ vis a solution of the differential equation in S/prime(cf. (5.34) (ii) and (5.36)) i∂ξˆv(ξ) = 2πδ. (5.72) One solution of this equation is −2πiH(ξ) (cf. (3.23)); all other solutions are of the form −2πiH(ξ) +c, (5.72a) 5.25 wherec∈C, cf. Theorem 4.19. We then just have to determine the constan t c. For this we observe that1 xis an odd function and that v= PV1 xis an odd distribution (i.e.,/angbracketleftv,ϕ(−x)/angbracketright=−/angbracketleftv,ϕ(x)/angbracketrightfor allϕ); then the Fourier transform ˆvmust likewise be odd. Let us include a carefully elaborated explanation of what wa s just said: The reflection operator S:ϕ(x)/mapsto→ϕ(−x), ϕ∈C∞ 0(R), (5.73) (for which we have also used the notation ϕ/mapsto→ˇϕ) is a special case of a dilation (3.52), namely with λ=−1. Hence it carries over to distributions in the usual way (cf. (3.56)): /angbracketleftSu,ϕ/angbracketright=/angbracketleftu,Sϕ/angbracketrightfor allϕ∈C∞ 0(R). (5.74) A function uis said to be evenresp.odd, whenSu=uresp.Su=−u; this notation is now extended to distributions. For the conn ection with the Fourier transformation we observe that (FSϕ)(ξ) =/integraldisplay e−ixξϕ(−x)dx =/integraldisplay eiyξϕ(y)dy= (Fϕ)(ξ) = (Fϕ)(−ξ) = (SFϕ)(ξ), or in short: FS=F=SF; (5.75) these formulas are carried over to distributions by use of (5 .74) or (5.51). (The formula (5.33) could be written: F2= (2π)nS.) In particular, we see thatSv=−vimpliesSˆv=−ˆv. The only odd function of the form (5.72a) is the one with c=πi, so: F[PV1 x] =−2πiH(ξ) +πi=−πisignξ; (5.76) cf. (3.25a) for sign ξ. (It is possible to find ˆ vby direct calculations, but then one has to be very careful with the interpretation of converg ences of the occurring integrals of functions not in L1. We have avoided this by building up the Fourier transformation on S/primeby duality from the definition on S.) An application of1 2πF=1 2πFS(cf. (5.75)) to (5.76) gives PV1 x=1 2πFS(−2πiH(ξ) +πi) =1 2πF(2πiH(ξ)−iπ) =iFH(ξ)−i 2F[1] =iFH−πiδ. 5.26 This leads to another interesting formula: FH(x) =−iPV1 ξ+πδ. (5.77) Using this formula, we can find the Fourier transforms of all h omogeneous functions of degree 0. Some further remarks: Corresponding to the decomposition 1 =H(x) +H(−x) we now get the following decomposition of δ(which is used in theoretical physics) δ= (2π)−1F[1] = (2π)−1(FH+FSH) (5.78) =/parenleftbiggδ 2+1 2πiPV1 x/parenrightbigg +/parenleftbiggδ 2−1 2πiPV1 x/parenrightbigg =δ++δ−,where δ±=δ 2±1 2πiPV1 x=1 2πF[H(±x)]. (5.79) Observe also that since H(x) = lim a→0+H(x)e−axinS/prime one has that FH= lim a→0+1 a+iξinS/prime(5.80) (cf. Exercise 5.3), and then δ+=1 2πlim a→0+1 a+ixinS/prime. (5.81) Remark 5.24. To the non-integrable functionH(x) xwe associate the dis- tribution PfH(x) x, defined by /angbracketleftPfH(x) x,ϕ/angbracketright= lim ε→0+/bracketleftBig/integraldisplay∞ εϕ(x) xdx+ϕ(0) logε/bracketrightBig , (5.82) cf. [S61, Exercise II-14, p. 114-115]; note that there is a lo garithmic correc- tion. In this way, every function on Rwhich is homogeneous of degree −1 is included in the distribution theory, namely as a distributi on c1PfH(x) x+c2SPfH(x) x, c1andc2∈C. (5.83) 5.27 In particular, we define Pf1 |x|by Pf1 |x|= PfH(x) x+SPfH(x) x. (5.84) It is shown in Exercise 5.12 that F(Pf1 |x|) =−2 log|ξ|+C, for some constant C. Also for distributions on Rnthere appear logarithmic terms, when one wants to include general homogeneous functions and their Fo urier transforms in the theory. (A complete discussion of homogeneous distri butions may be found in in [H83].) 5.28 Exercises for Chapter 5. 5.1. Letn= 1. Show that ex/∈ S/prime(R), whereasexcos(ex)∈ S/prime(R). (Hint. Find an integral of excos(ex).) 5.2. Show the inequalities (5.2), and show that the systems of sem inorms (5.3) and (5.4) define the same topology. 5.3. Leta>0. WithH(t) denoting the Heaviside function, show that F[H(t)e−at] =1 a+iξ. What is F[H(−t)eat]? 5.4. (a) Show that for n= 1, F−1/bracketleftBig1 1 +ξ2/bracketrightBig =ce−|x|; determinec. (One can use Exercise 5.3.) (b) Show that for n= 3, F−1/bracketleftBig1 1 +|ξ|2/bracketrightBig =c |x|e−|x|, withc=1 4π. (One may observe that the function is the unique solution vin S/primeof (1−∆)v=δ; or one can apply the rotation invariance directly.) 5.5. LetMandnbe integers with 0 <2M <n . Find an integral operator TMonS(Rn) with the following properties: (i) ∆MTMf=fforf∈ S(Rn). (ii) When Ω is a bounded, open subset of Rn, and 2M > n/ 2, then the operator (TM)Ω(defined as in (5.63)) is a Hilbert-Schmidt operator inL2(Ω). 5.6. Show that the differential equation on R3: ∂4u ∂x4 1−∂2u ∂x2 2−∂2u ∂x2∂x3−∂2u ∂x2 3+ 3u=f has one and only one solution u∈ S/primefor eachf∈ S/prime. Determine the values of m∈N0for whichubelongs to the Sobolev space Hm(R3) whenf∈L2(R3). 5.29 5.7. Leta∈C, and show that the distribution u=e−axH(x) is a solution of the differential equation (∂x+a)u=δinD/prime(R). Can we show this by Fourier transformation? 5.8. Show that the Cauchy-Riemann equation /parenleftbigg∂ ∂x+i∂ ∂y/parenrightbigg u(x,y) =f(x,y) onR2has a solution for each f∈ S; describe such a solution. 5.9. Form∈Nandϕ∈C∞ 0(R) we define the functional Λ mby Λm(ϕ) = PV/integraldisplay∞ −∞{x−mϕ(x)−m−2/summationdisplay p=0xp−m p!ϕ(p)(0)}dx. (a) Show that PV ...exists, so that Λ m(ϕ) is well-defined. (b) Show that Λ m(ϕ/prime) =mΛm+1(ϕ). (c) Show that Λ mis a distribution, and that Λm= (−1)m−1(m−1) !dm dxmlog|x|. Λmis also called Pf1 xm, where Pf stands for pseudo-function. 5.10. LetI=RorI=]a,∞[. Show that when uandDmu∈L2(I), thenu∈Hm(I). (One can show this for I=Rby use of the Fourier transformation. Next, one can show it for I=]a,∞[ by use of a cut-off function. This proves the assertion of Remark 4.21.) 5.11. Show that Fsignx=−2iPV1 ξ. 5.12. Consider the locally integrable function log |x|. (a) Letu=H(x) logx. Show that log|x|=u+Su. 5.30 (b) Show that d dxu= PfH(x) x,d dxlog|x|= PV1 x. (c) Show that xPf1 |x|= signx, and that ∂ξF(Pf1 |x|) =−2 PV1 ξ; cf. Exercise 5.11. (d) Show that F(Pf1 |x|) =−2 log|ξ|+C, for some constant C. (e) Show that F(log|x|) =−πPf1 |ξ|+C1δ, for some constant C1. (Information on the constant can be found in [S50, p. 258], [S 61, Exercise V-10], where the Fourier transformation is normalized in a s lightly different way.)