Chapter 4 Elliptic Operators This chapter by Victor Guillemin 4.1 Differential operators on Rn Let U be an open subset of Rn and let Dk be the differential operator, 1 ∂ . ∂xk√−1 For every multi-index, α = α1, . . . , αn, we define DαnDα = Dα1 .1 n···A differential operator of order r: P : C∞(U) → C∞(U) , is an operator of the form P u = � aαDα u , .aα ∈ C∞(U)α ≤r| |Here α = α1 + αn.| | ···The symbol of P is roughly s peaking its “ rth order part”. More explicitly it is the function o n U × Rn defined by (x, ξ) → � aα(x)ξα =: p(x, ξ) . |α|=r The following prop e rty of symbols will be used to define the notion of “symbol” for differential operators on manifolds. Let f : U → R be a C∞ function. Theorem. The operator e−itf P e itf uu ∈ C∞(U) →is a sum r� tr−iPiu (4.1.1) i=0 Pi being a differential operator of order i which doesn’t depend on t. Moreover, P0 is mu ltiplication by the function p0(x) =: P (x, ξ) ∂f with ξi = ∂xi , i = 1, . . . n. Lecture 16� � �� �� � � � � �Proof. It suffices to check this for the o perators Dα . Consider first Dk : e−itf Dke itf ∂f u = Dk u + t . ∂xk Next consider Dα e−itf Dα itf e−itf (Dα1 itf e u = Dαn )e u1 n···= (e−itf D1e itf )α1 itf )αn ···(e−itf Dne u which is by the above ∂f �α1 ∂f �αn�D1 + t �Dn + t ∂x1 ··· 2xn and is clearly of the form (4.1.1). Moreover the tr term of this operator is just multiplication by ∂f�α1 ∂ �αn�� . (4.1.2) ∂x1 ··· ∂xn Corollary. If P and Q are differen tial operators and p(x, ξ) and q(x, ξ) their symbols, the symbol of P Q is p(x, ξ) q(x, s). Proof. Suppose P is of the order r and Q of the order s. Then e−itf P Qe itf �e−itf P e itf ��e−itf Qe itf �uu = r s = �p(x, df)t + ��q(x, df)t + �u··· ··· r+s = �p(x, df)q(x, df)t + �u . ··· Given a differential operator P = � aαDα α ≤r| |we define its transpose to be the operator u ∈ C∞(U) → � Dα aαu =: Pt u . α ≤r| |Theorem. For u, v ∈ C∞0 (U) �P u, v� =: P uv dx = u, Pt. Proof. By integration by parts � 1 � ∂ Dk u, v = Dkuv dx = uv dk � � √−1 ∂xk 1 � ∂ � = u v dx = uDkv dx −√−1 ∂xk = u, dk v . Thus �Dα u, v = �u, Dα vand �aαDα u, v = �Dα u, aαv = u, Dα aαv , .Exercises. If p(x, ξ) is the symbol of P , p(x, ξ) is the symbol of pt . Ellipticity. P is elliptic if p(x, ξ) /∈ 0 for all x ∈ U and ξ ∈ Rn − 0. 4.2 Differential operators on manifolds. Let U and V b e open subsets of Rn and ϕ : U V a diffeomorphism. →Claim. If P is a differentia l operator of order m on U the operator )∗P ϕ∗uu ∈ C∞(V ) → (ϕ−1is a differential operator of order m on V . Proof. (ϕ−1)∗Dαϕ∗ = �(ϕ−1)∗D1ϕ∗�α1 �(α−1)∗Dnϕ∗�αn so it suffices to check this for Dk and for Dk··· this follows from the chain rule Dkϕ∗f = � ∂ϕi ϕ∗Dif . ∂xk This invariance under coo rdinate changes means we can define differential operators on manifolds. Definition. Let X = Xn be a real C∞ manifold. An operator, P : C∞(X) → C∞(X), is an mth order differential operator if, for every co ordinate patch, (U, x1, . . . , xn) the restriction map P u1Uu ∈ C∞(X) →is given by an mth order differential operator, i.e., restricted to U, P u = � aαDα u , .aα ∈ C∞(U)α ≤m| |Remark. Note that this is a non-vacuous definition. More explicitly let (U, x1, . . . , xn) and (U′, x1′ , . . . , xn′ ) be coordinate patches. Then the map u → P u1U ∩ U′ is a differential operator of order m in the x-coordinates if and only if it’s a differential operator in the x′-coordinates. The symbol of a differential operator Theorem. Let f : X → R be C∞ function. Then the operator e−itf P e−itf uu ∈ C∞(X) →can be written as a sum m� tm−iPi i=0 Pi being a differential operator of order i which doesn’t depend on t. Proof. We have to check tha t for every coordinate patch (U, x1, . . . , xn) the operator e−itf P e itf 1Uu ∈ C∞(X) →has this prop e rty. This, however, follows from Theorem 4.1.� � � �� � � � In particular, the operator, P0, is a zeroth order operator, i.e., multiplication by a C∞ function, p0. Theorem. There exists C∞ function σ(P) : T∗X C→not depending on f such that p0(x) = σ(P )(x, ξ) (4.2.1) with ξ = dfx. Proof. It’s clear that the function, σ(P ), is uniquely determined at the points, ξ ∈ T∗by the property (4.2.1), x so it suffices to prove the local existence of such a function on a neighborhood of x. Let (U, x1, . . . , xn) be a coordinate patch centered a t x and let ξ1, . . . , ξn be the co tangent co ordinates on T∗U defined by ξ ξ1 dx1 + + ξn dkn .→ ···Then if P = � aαDα on U the function, σ(P ), is given in these coordinates by p(x, ξ) = � aα(x)ξα . (See (4.1.2).) Composition and transposes If P and Q are differential o perators of degree r and s, P Q is a differential operator of degree r + s, and σ(P Q) = σ(P )σ(Q). Let FX be the sigma field of B orel subsets of X. A measure, dx, on X is a measure on this sigma field. A measure, dx, is smooth if for every coordinate patch (U, x1, . . . , xn) . The restriction of dx to U is of the form ϕ dx1 . . . dxn (4.2.2) ϕ being a non-negative C∞ function and dx1 . . . dxn being Lebesgue measure on U . dx is non-vanishing if the ϕ in (4.2.2) is strictly positive. Assume dx is such a measure. Given u and v ∈ C∞0 (X) one defines the L2 inner pro duct u, vof u and v to be the integral u, v = uv dx . Theorem. If P is an mth order differential operator there is a unique mth order : C∞(X) → C∞(X) differential operator, Pt, having the property �P u, v = u, Pt vfor all u, v