Pseudo-differential Operators and Symmetries

pseudo-differential operators generalized functions and asymptotics, diffusive representation of pseudo differential time operators pseudo differential operators theory and applications
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Published Date:26-07-2017
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INTRODUCTION TO PSEUDO-DIFFERENTIAL OPERATORS M. S. JOSHI 1. Preface These notes cover most of a Part III course on pseudo-differential operators. Theyassumethereaderisfamiliarwithdistributionspartic- ularly the Schwartz kernel theorem - the book by Friedlander provides an excellent introduction to this topic. The point of view taken is somewhere between that of Shubin, Mel- rose’sunpublished notesandthatofChazarainandPirou. Allofwhich provide good places to continue. We assume familiarity with vector bundles and with the theory of Fredholm operators between Banach spaces. For vector bundles one source is Atiyah, 2, and for Fredholm operators the self-contained section 19.1 of Hormander Volume 3 is a good account. 2. Introduction Our purpose in this course is to develop a theory of variable coeffi- cient linear, partial differential operators on manifolds. We will then develop applications of this theory. This will allow us to “solve” ellip- tic operators and in particular the Laplacian acting on k− forms will allow us to prove the Hodge theorem which states that the de Rham cohomology of a compact manifold is just the kernel of the Laplacian which relates the Riemannian geometry of a manifold to its topology - it will also give us an immediate proof that the de Rham cohomol- ogy of a compact manifold is finite dimensional and an easy proof of Poincare duality. A fundamental part of this will be the proof of el- liptic regularity - any distributional solution of an elliptic equation is a smooth function - which will be an easy consequence of our calcu- lus. We will also apply the calculus to the study of the propagation of singularities for non-elliptic equations - in particular we will show that associated to any linear PDE there is a Hamiltonian function on phase space that governs the propagation of singularities of solutions. This 1 arXiv:math/9906155v1 math.AP 23 Jun 19992 M. S. JOSHI theorem gives precise information about where future singularities are, given the knowledge of where they are at some time - this presages the more advanced theories of PDEs which use symplectic geometry more and more which we will touch on, but not do in this course. n We willfirst study operatorsonR andthenlater shift to manifolds. Definition 2.1. A linear partial differential operator of order m on n ∞ n ∞ n R is a map from C (R )→C (R ) of the form X α Pu(x) = f (x)D u(x) α x α≤m ∞ n where f ∈C (R ). α Note α α α 1 n 1 ∂ ∂ α ... D = x i ∂x ∂x 1 n and α =α +···+α . 1 n We put in 1/i as its makes dealing with Fourier transforms easier. α d α ˆ The fact that D f = ξ f(ξ) means that for u of compact support, x we have nZ 1 ix.ξ Pu(x) = e p(x,ξ)uˆ(ξ)dξ, (1) 2π where Definition 2.2. The total symbol of P is X α −ix.ξ ix.ξ p(x,ξ) = f (x)ξ =e P(e ) α α≤m and the principal symbol of P, denoted σ (P), is m X α p (x,ξ) = f (x)ξ . m α α=m n So any differential operator can be thought of as a function onR × x n R - this can be thought of as phase space or the cotangent bundle. ξ To work microlocally means to work on phase space rather than on n R and this will lead to a more coherent theory. The basic idea in micro-localanalysis is that a differential operator is approximated by a multiplication operator on functions on phase space. To make sense of this idea we will introduce the algebra of pseudo-differential operators. Our starting point is (1) -p(x,ξ) is a polynomial inξ and smooth inxINTRODUCTION TO PSEUDO-DIFFERENTIAL OPERATORS 3 so we can think of it a finite sum of homogeneous functions p (x,ξ) m−j where m−j p (x,λξ) =λ p (x,ξ) m−j m−j with j running from 0 to m. We therefore define a (classical) pseudo- differential operator, of order m, to be an operator of the form (1) wherep(x,ξ) is an infinite sum (in a sense, we will later make precise) of terms p (x,ξ) which are smooth in ξ = 6 0 and m−j m−j p (x,λξ) =λ p (x,ξ)λ 0. m−j m−j We do not require smoothness up to 0 because we would be left only with polynomials (see if you can prove this) As before p(x,ξ) is the total symbol and p the principal symbol - often denoted σ (P). m m m n This class is then denoted ΨDO (R ). If we denote the space of cl m homogeneous functions of order m by H we have that m ΨDO cl m σ : →H m m−1 ΨDO cl is an isomorphism. We will show that these classes form a graded algebra under compo- sition i.e. that ′ ′ m n m n m+m n ◦ : ΨDO (R )×ΨDO (R )→ ΨDO (R ). cl cl cl The relationship of the total symbol of the product to the symbols of the original operators is complicated but at the principal level one just takes a product: ′ ′ σ (PQ) =σ (P)σ (Q). m+m m m The identity operator,I, is a pseudo-differential operator with total symbol 1 and principal symbol 1. So if we want to solve PQ=I, we need 1 =σ (PQ) =σ (P)σ (Q). 0 m −m So provided σ (P) is never zero we take Q to be of order −m with m 0 −1 principal symbol σ (P) . We then have that m σ (PQ ) =σ (I) 0 0 0 and thus that −1 PQ −I =R ∈ ΨDO . 0 1 cl4 M. S. JOSHI We now want to solve away the error term - ie we want to findQ such 1 −1 that PQ =−R , putting σ (Q ) =−σ (P) σ (R ) we get 1 1 −m−1 1 m −1 1 −2 P(Q +Q )−I =R ∈ ΨDO . 0 1 2 cl −m−j We can now repeat this argument to get Q ∈ ΨDO such that j cl −m−N P(Q +Q +···+Q )−I ∈ ΨDO . 0 1 N−1 cl If we then sum (in an appropriate sense) to get Q, we have \ −∞ l PQ−I ∈ ΨDO = ΨDO. l Such a Q is called a parametrix and is an inverse at the level of sin- gularities - on a compact manifold the error will be compact and this will imply that P has finite dimensional kernel and cokernel - that is the image is of finite codimension. The above argument can be applied even if σ (P) has some zeros m by working away from the zeros - so one can construct an inverse away from the zero set - so the hard part of the operator is the set where σ (P) = 0. This is called the characteristic variety, char(P) and it m governs much of the operators behaviour. We use these ideas to define a notion of direction and location for a singularity of a distribution u - the wavefront set: \ WF(u) = char(P) ∞ Pu∈C n n - this is a subset ofR ×R −0. Note that ifφ is a smooth function x ξ such thatφu is smooth then we have immediately that (x,ξ)6∈ WF(u) for x with φ(x)6= 0. We later show that singsupp(u) =x:∃ξ, (x,ξ)∈ WF(u) so the wavefront set contains all the information that the singular sup- port does. The theorem of H¨ormander on the propagationof singulari- ∞ ties states that ifPu∈C then WF(u) is a union of complete integral curves of the Hamiltonian associated to the energy function p . One m interpretation of this is that the operator P is a quantised version of the classical system p . When P is the wave operator this means that m the singularities of solutions travel along geodesics in the manifold. Soourfirsttaskinthiscoursewillbetoconstructtheclassofpseudo- differential operators described above and we will then develop the applications.INTRODUCTION TO PSEUDO-DIFFERENTIAL OPERATORS 5 3. Oscillatory integrals Our objective is to make sense of integrals of the form Z iφ(x,θ) e a(x,θ)dθ which are not absolutely convergent. We can think of these as gener- alisations of the Fourier transform. We will need restrictions onφ, the phase, and a, the amplitude, for the integral to make sense. ∞ n k Definition 3.1. φ ∈ C (R ×(R −0)) is a phase function, if φ x θ is homogeneous of degree one in θ, is real-valued and it has no critical points, i.e. ∂φ ∂φ ∂φ ∂φ ′ d φ = ,..., , ,..., x,θ ∂x ∂x ∂θ ∂θ 1 n 1 k is never zero in all coordinates simultaneously. ∞ n Definition 3.2. We shall saya is a symbol of orderm ifa∈C (R × k R ) and β α m−β D D a(x,θ)≤C θ , α,β,K x θ for x in K compact, all θ where 1 2 2 θ= (1+θ ) . m n k m The class is denoted S (R ;R ) or just S . Note that in particular if a(x,θ) is a polynomial in θ then it is a symbol. Note also that the estimates are really only statements about what happens at infinity. In particular, if χ(x,θ) is compactly supported in θ then it is an element of \ −∞ m S = S . m Example 3.3. If a(x,θ) is homogeneous of degree m in θ then if we smooth off near 0, we obtain a symbol of order m. The following proposition is trivial but important. Proposition 3.4. Pointwise multiplication defines a map ′ ′ m m m+m S ×S →S and α β m m−β D D :S →S . x θ6 M. S. JOSHI Sonowletsreturntoourintegral. Forsimplicity, wewillassumethat ais zero in a neighbourhood ofθ = 0 but this is not animportant issue m as we shall see that important effects come from infinity. If a ∈ S and m −k then by the dominated convergence theorem we have that the integral converges to a function which is continuous in x. If we differentiate the integrand with respect to x, we obtain an integral m+1 of the same form except that the amplitude is in S . Hence we l deduce that ifm+l−k then the integral yields a function inC . So −∞ if a∈S then the integral is smooth. What do we do in general? We regularize the integral to obtain a distribution. We want to make sense of the expression: ZZ iφ(x,θ) u,ψ= e a(x,θ)ψ(x)dθdx. ∞ n for any ψ∈C (R ). As this is important we shall do so in two ways. 0 First consider the operator,L defined by n k X X ∂φ ∂ ∂φ ∂ 2 L = + θ . ∂x ∂x ∂θ ∂θ j j l l j=1 l=1 L has the property that   2 2 n k X X ∂φ ∂φ iφ 2 iφ   Le = + θ e . ∂x ∂θ j l j=1 l=1 Note that our hypothesis guarantees that the coefficient will not vanish offθ = 0 andwe have also that is homogeneous of degree 2 from −1 the homogeneity of φ. Call the coefficient χ then putting M = χ L. We have, at least formally, ZZ ZZ iφ(x,θ) r iφ(x,θ) e a(x,θ)ψ(x)dθdx = M e a(x,θ)ψ(x)dθdx. Formally, we can integrate by parts to obtain ZZ iφ(x,θ) t r e (M ) (a(x,θ)ψ(x))dθdx (2) and n k X X ∂ ∂ t M = b + c +d j l ∂x ∂θ j l j=1 l=1INTRODUCTION TO PSEUDO-DIFFERENTIAL OPERATORS 7 −1 0 with d,b ∈ S and c ∈ S . (They may be singular at zero but our j l integrand is supported away from zero so this is irrelevant. ) The advantage of all this is that X t r α (M ) (a(x,θ)ψ(x)) = a (x,θ)D ψ(x) α x α≤r m−r with a ∈S and so if m−r −k , the integral (2) will converge α absolutely. So we take (2) to be the definition of Z iφ(x,θ) e a(x,θ)dθ and it is clear that we have a distribution of order r. Note that our formal computation is valid if m−r −k and so the answer will be independent of the choice of r. Since this is so important to everything we do and it may appear a little artificial, let’s consider another approach as well. Let χ be a smooth bump function i.e.   1 forθ 1 χ(θ) =  0 forθ 2. Then we define ZZ iφ(x,θ) u,ψ= lim e χ(ǫθ)a(x,θ)ψ(x)dθdx. ǫ→0 The integral is compactly supported for any fixed ǫ 0 so there is no problems with it. We want to see what happens as ǫ goes to m zero. If a∈S and m−k then the integral is uniformly absolutely convergent and so will converge to the convergent integral ZZ iφ(x,θ) e a(x,θ)ψ(x)dθdx. So as before the issue is what happens form≥−k. LettingM be as above, we have ZZ iφ(x,θ) t k u,ψ= lim e (M ) (χ(ǫθ)a(x,θ)ψ(x))dθdx. ǫ→0 Without loss of generality we can assume ǫ is within a compact set so we have that α −α D χ(ǫθ)≤C θ α θ8 M. S. JOSHI independentlyofǫ.Theimportantpointisthatthisestimateisuniform in ǫ all the way to ǫ = 0 - for fixed ǫ6= 0 the function χ(ǫθ) is, in fact, −∞ in S but not uniformly. Thus as before we can deduce X t r (M ) (χ(ǫθ)a(x,θ)ψ(x)) = a (ǫ,x,θ)∂ ψ α α α≤k m−r n k with a ∈S (R ×R ;R ) i.e. the behaviour of a at∞ is uniform α ǫ α x inǫ. And so taking r such thatm−r−k we see that the integral is uniformly convergent in ǫ and must converge to ZZ iφ(x,θ) t r e (M ) a(x,θ)ψ(x)dθdx, as the derivatives falling on χ will disappear when ǫ → 0. So the two approaches yield the same answer Note for the functional analysts, one can view this as the extension of a Frechet-space-valued function from a dense subset. In general, we will deal fairly freely with oscillatory integrals but it is important to realize that when we see such an expression that this is what we mean. As oscillatory integrals yield distributions rather than smooth func- tions, one thing we want to do is find their singularities. Proposition 3.5. If Z iφ(x,θ) u = e a(x,θ)dθ then n o ′ singsupp(u)⊂ x :∃θ (x,θ)∈ supp(a) and d φ(x,θ) = 0 θ Heuristically, this says that if the phase is oscillating nearx then the behaviour for large θ cancels and so there is no singularity. This is sometimes called stationary phase - or the WKB method. Proof. Suppose ′ d φ(x ,θ)6= 0,∀θ 0 θ then if we multiply by a bump function,ψ, supported sufficiently close to x we have 0 Z iφ(x,θ) ψu = e b(x,θ)dθ ′ with b a symbol such that d φ is non-zero on the support of b. θINTRODUCTION TO PSEUDO-DIFFERENTIAL OPERATORS 9 This means, similarly to above, we can put X ∂φ ∂ ′ −2 M =d φ θ ∂θ ∂θ j j j and M will be non-singular on the support of b and will, as before, have the property iφ iφ Me =e . ∞ Thus if f ∈C we have 0 Z iφ(x,θ) t r ψu,f = lim e (M ) (b(x,θ)χ(ǫθ))f(x)dθdx. ǫ→0 The important point here is that the derivatives do not fall on f as it is independent of θ. Any terms falling on χ, will have coefficients which are powers of ǫ and so will disappear in the limit. Thus, Z iφ(x,θ) t r ψu = e (M ) b(x,θ)dθ. t r m−r But (M ) b(x,θ) ∈ S so we conclude that ψu is smooth as this is true for any r. Example 3.6. Letφ(x,y,ξ)=hx−y,ξi thenφ is a phase function as ′ d φ =ξ is non-zero for any non-zero ξ and x ′ d φ =x−y ξ sotheassociated oscillatoryintegrals aresingular onlyonthediagonal. 4. Pseudo-differential Operators Definition 4.1. We define P to be a pseudo-differential operator on n R of order m, if it has a Schwartz kernel Z ihx−y,ξi K(x,y) = e a(x,y,ξ)dξ m n n n m n with a∈S (R ×R ;R ). We denote this class ΨDO (R ). x y ξ This is slightly different from what we defined before but we shall see that this definition is equivalent. We want our operators to act on distributions so first we must show that they preserve the class of smooth functions.10 M. S. JOSHI Theorem 4.2. If P is a pseudo-differential operator then ∞ n ∞ n P :C (R )→C (R ) 0 and is continuous and if P is properly supported ∞ n ∞ n P :C (R )→C (R ). 0 0 Proof. Let u be a smooth function then ZZ ihx−y,ξi Pu = e a(x,y,ξ)u(y)dξdy. For fixed x, this is a well-defined oscillatory integral in (y,ξ) and so, since the estimates are locally uniform in x, this gives a continuous function of x. The continuity is immediate from the fact that the oscillatory integral will be estimated by a finite number of derivatives in y of u. The formal x derivatives are of the same form and so will also be continuous. To check that the formal derivatives and the actual ones agree we evaluate the quotient - wlog we consider differentiation in x 1 Pu(x+he )−Pu(x) 1 = h ZZ ix+he ,ξ ix,ξ 1 e a(x+he ,y,ξ)−e a(x,y,ξ) j −iy,ξ e u(y)dξdy h = ZZ   ∂ ′′ ′′ i(s ,ξ +x ,ξ ′′ −iy,ξ 1 1 e a(s ,x ,y,ξ) e u(y)dξdy 1 s=x +ǫh 1 ∂s 1 with 0 ≤ ǫ ≤ 1. This is just the value of the formal derivative at x +ǫh and as h → 0 this must converge to the value of the formal 1 derivative at x and so the result follows. (just use induction for the higher terms). To define the action of pseudo-differential operators on distributions weneedtofindtheiradjoints. HoweverifK(x,y)istheSchwartzkernel t t of P then P has the kernel K(y,x). So P has a kernel Z iy−x,ξ e a(y,x,ξ)dξ and performing the change of variables η =−ξ we have Z ix−y,η e a(x,y,−η)dη.INTRODUCTION TO PSEUDO-DIFFERENTIAL OPERATORS 11 t ∞ n ThusP isapseudo-differentialoperatorandiscontinuousonC (R ). 0 Thus we have m n Theorem 4.3. If P ∈ ΨDO (R ) and is properly supported then P ′ n induces a map on D(R ) via t ′ n ∞ n Pu,φ=u,P φ,u∈D(R ),φ∈C (R ). 0 We have that properly supported pseudo-differential operators map distributions and preserve the space of compactly supported smooth functions. It is therefore not surprising that they preserve the class of smoothfunctions(whichisasubclassofdistributions.): ifP isproperly supported and u is smooth then Pu is a distribution. Now given any ∞ point x we can put u = u +u with u ∈ C and u such that 0 1 2 1 2 0 x 6∈ supp(P)◦supp(u ) which implies x 6∈ supp(Pu ). We thus have 0 2 0 2 that Pu equals Pu near x and thus is smooth near x . 1 0 0 Sincewehaveshownthatpseudo-differentialoperatorspreservesmooth functionsandaresingularonlythediagonal,itisnoweasytoshowthey have the additional property of pseudo-locality. Definition 4.4. The operator P is pseudo-local if singsupp(Pu)⊂ singsupp(u) ′ n for any u∈D(R ). Theorem 4.5. Properlysupportedpseudo-differentialoperatorsarepseudo- local. Proof. LetP be a pseudo-differential operator with Schwartz kernelK ′ n and u∈D(R ). Given ǫ 0, it is enough to show that the singularities of Pu lie within an ǫ neighbourhood of those of u. To see this we decompose both K and u into pieces supported near the singularities and pieces which are smooth. So suppose K =K +K 1 2 with the support of K contained in 1 x−yǫ/2 and K smooth and 2 u =u +u 1 2 with supp(u ) within an ǫ/2 neighbourhood of singsupp(u ) and u 1 1 2 smooth. Then Ku =K u +K u +Ku . 1 1 2 1 212 M. S. JOSHI Now from the relation of supports, we have supp(K u )⊂x:x−yǫ,y∈ singsuppu 1 1 and since the singular support is smaller than the support, the first term is OK. We also have Ku is smooth as u is. 2 2 This leavesK u . This turns out to be smooth also asK is and the 2 1 2 result will then follow. This is an important result so we will make it a theorem. ∞ n n Theorem 4.6. If K ∈ C (R ×R ) and is properly supported then ′ n ∞ n K induces a map from D(R ) to C (R ). Proof. Consider the map Ku :x7→K(x,y),u(y). This is well-defined for each x, as K(x,y) is a smooth function in y. We want to show the result is smooth in x. But by the definition of a distribution, we have for some k, X ′ α ′ kKu(x)−Ku(x)k≤ supD (K(x,y)−K(x,y)). y y α≤k The properness ofK will ensure that the supremum is over a compact set and thus the function will certainly be continuous. Note the same is true for the maps α α D :x7→D K(x,y),u(y) x x so the formal derivatives are continuous too. So, as in the proof that pseudo-differentialoperatorsarecontinuous,Kuissmoothasrequired. Since we have shown that smooth kernels kill all singularities, we will ignore them most of the time. So we shall regard two operators as equivalent if and only their Schwartz kernels differ by a smooth term and we write this as P ≡Q. We shall call an inverse up to a smoothing term a parametrix. Our theorem shows that such an operator will allow us to describe the singularities of u from those of Pu. Note that if K is smoothing then PK,KP are smoothing also for any pseudo-differential operator so theINTRODUCTION TO PSEUDO-DIFFERENTIAL OPERATORS 13 smoothing operators form an ideal in the class of pseudo-differential operators. The next thing we want to do is prove that the composition of two pseudo-differential operators is a pseudo-differential operator. In order to do this, we need a better understanding of the relationship between the symbol a and the associated operator P. Theorem 4.7. If P is a pseudo-differential operator of order m, then up to smooth terms the Schwartz kernel ofP can be written in the form Z ihx−y,ξi e c(x,ξ)dξ or in the form Z ihx−y,ξi e d(y,ξ)dξ m n n with c,d∈S (R ;R ). Note these are called the left and right quantizations. Proof. So we have initially that Z ihx−y,ξi K(x,y) = e a(x,y,ξ)dξ. We want to eliminate the dependence on one of the base variables. Recall from Taylor’s theorem that X X 1 α α α a(x,y,ξ)= ∂ a(x,x,ξ)(y−x) + (y−x) b (x,y,ξ) α y α α≤N−1 α=N 1 R N−1 α and b = (1−t) ∂ a(x,(1−t)x+ty,ξ)dt. It is immediate (and α y 0 α m m important) that ∂ a(x,x,ξ)∈S and b (x,y,ξ)∈S . α y α α Let’s consider the terms ∂ a(x,x,ξ)(y−x) , y Z Z ihx−y,ξi α α α ihx−y,ξi α e (x−y) ∂ a(x,x,ξ)dξ = D e ∂ a(x,x,ξ)dξ y ξ y and so integrating by parts we obtain Z Z ihx−y,ξi α α ihx−y,ξi α α α e (x−y) ∂ a(x,x,ξ)dξ = e (−1) D ∂ a(x,x,ξ)dξ. y ξ y (check this actually works)14 M. S. JOSHI We have applying the same argument to b that the result is true α up to a term of order m−N. To proceed further, we need to sum the series X 1 α α D ∂ a(x,x,ξ). ξ y α α This sum will not converge but we can make sense of it in an asymp- totic way. We will use the following result: m−j m Proposition 4.8. If a ∈S then there exists a∈S such that j N−1 X m−N a− a ∈S , ∀N m−j j=0 The proof of this is on the example sheet 1. Note an immediate corollary of this, which will come in handy later on, m−j n Corollary 4.9. If P ∈ ΨDO (R ) for j = 0,...,∞ then there j m−j n exists P ∈ ΨDO (R ) such that cl X m−N n P − P ∈ ΨDO (R ) j jN for all N. So with this proposition, we let c be an asymptotic sum of X 1 α α D ∂ a(x,x,ξ) ξ y α α and is then clear that Z Z ihx−y,ξi ihx−y,ξi m−N e c(x,ξ)dξ− e a(x,y,ξ)dξ∈ ΨDO , ∀N which establishes the first half of the result. The second half is identical, except that the Taylor expansion is in y not x. Notewehaveactuallydonebetterthanthestatementofourtheorem in that the proof gives us asymptotic formulaes forc,d α X X 1 (−1) α α α α c(x,ξ)∼ D ∂ a(x,x,ξ), d(y,ξ)∼ D ∂ a(y,y,ξ). ξ y ξ x α α α α Note that unlike our original symbol which depended on x,y and was definitely not unique, these symbols are essentially unique.INTRODUCTION TO PSEUDO-DIFFERENTIAL OPERATORS 15 If n Z 1 ihx−y,ξi K(x,y)= e a(x,ξ)dξ 2π then this is really just a Fourier transform across x =y. Putting w = ′ x−y,x =x, we have n Z 1 ′ ′ ′ iw,ξ ′ L(x,w)=K(x,x −w) = e a(x,ξ)dξ 2π and so Z ′ −iw.ξ ′ a(x,ξ) = e L(x,w)dw. i.e. a is determined by K. A smooth error on K, for K properly sup- ported, would translate into an error of order−∞ ona, as the Fourier transform of a compactly supported smooth function is Schwartz. Definition 4.10. If P has Schwartz kernel Z ihx−y,ξi e a(x,ξ)dξ m with a ∈ S , up to smooth terms, then the total left symbol of P, m −∞ σ (P), is a(x,ξ). We will regard σ (P) as an element of S /S . L L So the total left symbol determines P up to order −∞. The same arguments work on the other side and we make a similar definition. Definition 4.11. If P has Schwartz kernel Z ihx−y,ξi e b(y,ξ)dξ m with b ∈ S ,up to smooth terms, then the total right symbol of P, m −∞ σ (P), is b(y,ξ). We will regard σ (P) as an element of S /S . R R m n j Whyuseleftandrightsymbols? IfwehaveoperatorsP ∈ ΨDO (R ) j then giving P the left form and P the right one, they can be written 1 2 (up to smooth terms) nZ 1 ix.ξ P u = e p(x,ξ)uˆ(ξ)dξ 1 2π and n Z Z 1 ix.ξ P v = e q(y,ξ)v(y)dydξ. 2 2π So it is then immediate from the Fourier inversion theorem that16 M. S. JOSHI n Z 1 ihx−y,ξi P P v = e p(x,ξ)q(y,ξ)u(y)dydξ. 1 2 2π SoP P is a pseudo-differential operator of orderm +m with total 1 2 1 2 symbolp(x,ξ)q(y,ξ).Of course, we want a more symmetric expression in terms of outputs and inputs - we want to compute the left symbol of the composite in terms of the left symbols of the inputs. m j Theorem 4.12. IfP ∈ ΨDO are properly supported and have total j m +m 1 2 symbols p (x,ξ) then P P ∈ ΨDO and has total symbol j 1 2 α X i α α a(x,ξ)∼ D p (x,ξ)D p (x,ξ). 1 2 ξ x α α The only thing left to prove is the correctness of the asymptotic expansion. Thisisjustanexerciseinbinomialexpansions-firstconvert the left symbol of p to a right one and then convert the product to a 2 left one. 5. Classicality So far we have been working with general pseudo-differential op- erators - this class is in fact larger than we need and we can work with a smaller class which has some nice properties and will include all the operators we actually need. Recall that a homogeneous function smoothed off at the origin is a symbol so we can require a symbol to have an asymptotic expansion in homogeneous terms. Definition 5.1. A symbola(x,θ) of orderm is classical if there exists a sequence of functions a (x,θ), homogeneous of order m−j in θ m−j such that N−1 X m−N a− (1−φ)(θ)a (x,θ)∈S m−j j=0 with φ a compactly supported function equal to 1 near 0. Definition 5.2. A pseudo-differential operator is classical if its total m n left (or right) symbol is classical. We denote this class ΨDO (R ) cl Note that the asymptotic expansion is unique and so knowing the operator and knowing the expansion of its symbol are really the same thing up to smooth terms which we always ignore It is obvious that composition preserves classicality and that differ- ential operators are classical pseudo-differential operators. The mostINTRODUCTION TO PSEUDO-DIFFERENTIAL OPERATORS 17 important thing about classical operators is that they have a unique principal symbol. m n Definition 5.3. If P ∈ ΨDO (R ) then if P has Schwartz kernel cl Z ihx−y,ξi e p(x,ξ)dξ with asymptotic expansion X p (x,ξ) m−j then the principal symbol of P is p (x,ξ) and is denoted σ (P)(x,ξ). m m n We denote the space of smooth homogeneous functions on R × x n m n n (R −0) which are homogenousof degreem inθ byH (R ;R −0). θ We express the fact that σ (P)(x,ξ) determines P up to one order m lower by using a short exact sequence. Allthis means is that the image of one map is exactly equal to the kernel (null-space) of the next. Theorem 5.4. The following sequence is exact σ m−1 n m n m m n n 0− →ΨDO (R )− →ΨDO (R )−→H (R ;R −0)− →0. cl cl The reason the principal symbol is particularly useful is that the formula for products is nice. ′ m n m n Theorem 5.5. If P ∈ ΨDO (R ), Q ∈ ΨDO (R ) are properly cl cl supported and have classical expansions X X p (x,ξ), q ′ (x,ξ) m−j m−k P then PQ is also classical with asymptotic expansion r ′ (x,ξ) m+m−l where X 1 α α ′ r = D p .∂ q ′ m+m−l m−j m−k ξ x α j+k+α=l and thus σ ′(PQ) =σ (P)σ ′(Q). m+m m m Proof. We know PQ is a pseudo-differential operator with asymptotic expansion X 1 α α D p(x,ξ)∂ q(x,ξ). ξ x α α α Now D p(x,ξ) has asymptotic expansion ξ X α D p m−j ξ18 M. S. JOSHI α α and D p is homogeneous of degree m−j−α and ∂ q(x,ξ) has m−j ξ x asymptotic expansion X α ∂ q ′ m−k x α ′ with ∂ q ′ homogeneous of degree m − k. The result follows by m−k x collecting the terms at each level of homogeneity. Thestatement aboutprincipalsymbolsnowfollowsfromconsidering the top level of homogeneity. Now we have done all this, we can start constructing parametrices - ie inverses up to terms of order −∞. Definition 5.6. A classical pseudo-differential operator is said to be m elliptic of order m if P ∈ ΨDO and σ (P) is never zero. m cl Example 5.7. The Laplacian X 2 Δ = D x j n P 2 plus any first order perturbation has principal symbol, ξ , and is j j=1 therefore elliptic. Or more generally, if g (x) is a Riemannian metric ij n ij on R and g (x) is the inverse matrix then the variable coefficient Laplacian X ij Δ = g D D +E, x x i j P ij with E first order, has principal symbol g ξξ , and is therefore el- i j 2 liptic. OnR , we also have that the Neumann operator D +iD x y has principal symbol ξ+iη and is therefore elliptic. Theorem 5.8. If P is a properly supported, elliptic, classical pseudo- differential operator of order m then there exists −m Q∈ ΨDO cl such that ∞ PQ−Id,QP −Id∈C . Proof. First, note that the principal and total symbol of Id are both 1. So we know that for the result to be true σ (P)σ (Q) = 1. m −m −1 So we putσ (Q) =σ (P) . The assumption of ellipticity ensures −m m that σ (Q) is smooth and thus we can pick such a Q. −mINTRODUCTION TO PSEUDO-DIFFERENTIAL OPERATORS 19 th Now our formula for products of symbols tells us that the l term in the expansion for PQ will be X 1 α α D p (x,ξ)∂ q (x,ξ) (3) m−j −m−k ξ x α α+j+k=l and we need this to be equal to zero for each l≥ 1. So suppose that q ,...,q have been chosen so that (3) is −m −m−N satisified for l≤N then we can define X 1 −1 α α q =−p D p (x,ξ)∂ q −m−N−1 m−j −m−k m ξ x α α+j+k=N+1,kN+1 and (3) will be satisfied forl =N +1 also. Thus by induction, we can construct q so that (3) is true for all k. −m−k Let q have classical expansion q and then we have m−k −∞ ∞ PQ−Id∈ ΨDO =C . ′ A similar argument constructs Q such that ′ −∞ ∞ QP −Id∈ ΨDO =C . m −∞ But regarding ΨDO /ΨDO as a semi-group, left inverses and cl right inverses must be equal and the result follows. We can now prove elliptic regularity. m Theorem 5.9. (Weyl’s Lemma) If P ∈ ΨDO is proper, classical ′ n and P is elliptic of order m and u∈D(R ) and ∞ Pu∈C then ∞ u∈C . ′ n More generally, for any u∈D(R ), singsupp(Pu) = singsupp(u). ∞ Proof. Let Q be a parametrix forP then QP −Id =R andRu∈C . So by pseudo-locality, theorem 4.5, singsupp(u) = singsupp((Id+R)u) = singsupp(QPu) ⊂ singsupp(Pu) ⊂ singsupp(u).20 M. S. JOSHI Note this is actually a deep result and is quite hard to prove by other means It also relies heavily on the fact that P is elliptic and is definitely not true for a general pseudo-differential operator. 6. Continuity So, we have seen that elliptic operators have parametrices which provide inverses up to smooth terms. i.e. they provide an inverse at the singularity level. Sobolev Spaces can be used to measure the badness of a singularity and we will examine how a pseudo-differential operator maps between them. s n ′ n Definition 6.1. We shall say u∈H (R ) if u∈S (R ) and 2 s/2 2 n uˆ(ξ)(1+ξ ) ∈L (R ). Note that the lower s is, the more singular u becomes. It is trivial that s s−1 D :H →H . x j 2 There are two ways to fail to be inL - a distribution can be too big s at ∞ or too singular at some point. The same is true of H so, since what we really want to measure is how singular a distribution is, we will often work with two related spaces. Definition 6.2. s n s n H (R ) =u :u∈H (R ) and supp(u) compact.. c s n s n ∞ H (R ) =u :φu∈H (R ),∀φ∈C loc 0 These are the compact and local spaces. The theorem we want to prove is m n Theorem 6.3. If P ∈ ΨDO (R ) and is properly supported then cl s s−m P :H →H . c c Mostoftheworkinvolvedisinprovingthesimplestcase-m = 0,s = 2 0. ie the L continuity of zeroth order pseudo-differential operators. c To do this we will need to understand adjoints of pseudo-differential operators - we already understand transposes and these are similar.

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