(CRASH COURSE ON) PARA-DIFFERENTIAL OPERATORS

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1 (CRASH COURSE ON) PARA-DIFFERENTIAL OPERATORS BENOIT PAUSADER We refer to [1] for an excellent introduction to seudo-differential oerators and to [] for a comlete treatment. We also refer to htts://math.berkeley.edu/~evans/semiclassical.df for comrehensive lecture notes online. 1. Pseudo-differential oerators 1.1. The algebra of differential oerators. Our goal is to understand the Algebra of differential oerators A := {differential oerators} = { α a α (x) α x, α N d, a α C b (Rd )}. In fact, our rimary goal is to find the invertible elements and how to invert them. We can already isolate remarkable (commutative) subalgebras: the algebra of oerators of order 0: A 0 := {a(q); a C b (Rd }, (a(q)f)(x) = a(x)f(x) and the algebra of oerators with constant coefficients A P := {a(p ); a C[X 1,..., X d ]}; F(a(P )) = a(q)f. From this, we can make two remarks: In A 0, it is easy to find the invertible elements: they are the functions that never vanish. Their inverse is simle: a 1 = 1/a. it is fairly natural to extend A P to A c = F 1 A 0 F, the algebra of Fourier multiliers. In articular, this way, we can easily find the inverse of oerators in A c. A 0 is invariantly defined, but A c is not. Considering Pseudo-differential oerators on manifold requires some care. It is remarkable that A 0 and A c are so simle, seem to generate A, yet A is much more subtle. This is mainly due to the fact that A is noncommutative. 1.. Vector fields. There is a simle way to incrementally generate A from A 0. Indeed, we may consider A j the set of oerators of order j or less. This is a module over A 0. After oerators of degree 0, one can form the A 1 by adjoining the set of vector fields, namely of oerators of the form χ : (χf)(x) = d χ j (x) j f(x), χ j Cb (Rd ). j=1 Denoting X the set of vector fields, which is invariantly defined, we have that A 1 = A 0 + X can also be invariantly defined. One can then build A = A 1 + X A 1, e.g. choosing an orthonormal basis E i, one has f = d E j E j f j=1 and one can then construct A 3,... 1

2 BENOIT PAUSADER {Vector} {KNQuant} {weyl} We note however that this algebra is noncommutative since X is: [ a j j, b k k ] = d (a j j b k b j j a k ) k X j k k j=1 However, we note a fortunate fact: A is commutative to main order: the roduct of two vector fields is a differential oerator of order, yet their commutator is only of order 1: the term of order exactly does not deend of the order of the roduct Quantization rule. In fact, by considering the different coordinates (or by considering the actions of a vector on the coordinate functions x 1, x,..., x d ), one can write any vector field χ = (χ 1,..., χ d ) as d χ = χ j (x) j. (1.1) j=1 We want a way to assign a symbol to a vector. By the considerations of the revious subsection, one we secify a rule for vectors, this will automatically give a rule for A. A natural choice would be to associate to each vector χ as in (1.1) the symbol 1 χ(x, ξ) = d χ j (x)(iξ j ) j=1 and this is the Kohn-Nirenberg quantization. This leads to the quantization for elements in A 1 : L := a j (x) j + b(x) σ KN (L)(x, ζ) := ia j (x)ξ j + b(x) (1.) This gives a natural identification of differential oerators and symbols: a α (x) α a α (x)i α ξ α, α α and we could very well work with this rule. However, from the oerator viewoint, this has a severe limitation: it is very difficult to know from the symbol when an oerator is self-adjoint, whereas for a A 0, a(q) is self-adjoint if and only if a is real. Since self-adjoint oerators are so imortant (e.g. for energy-estimates), we will refer the Weyl quantization which retains the roerty that an oerator is self-adjoint if and only if its symbol is real. For differential oerators of order 1, it is easy to see what the rule should be: L := a j (x) j + b(x) = 1 (a j j + j a j ) + (b(x) ( j a j )) σ w (L)(x, ζ) := ia j (x)ξ j + (b(x) div( a)). (1.3) We can then verify that (i) real valued symbols corresond to self-adjoint oerators and (ii) the highest order term is the same for both quantizations (but lower order terms differ). In fact, there is a natural 1-arameter choice of quantizations all simly forced once we choose the corresondance to x ξ: x i ξ j = i(θx i j + (1 θ) j x i ) and we see that the Kohn-Nirenberg quantization corresonds to θ = 1, while the Weyl quantization corresonds to the case θ = 1/. We note that there are other natural roerties that one might want to reserve. Unfortunately, with neither of the quantization rules defined above can one have that an oerator is ositive if and only if its symbol is. When such a roerty is needed, one can consider the Bargmann transform. For other comarison, we refer to htt:// 1 The additional factor of i is suggested by looking at the Fourier transform. However, it clearly does not make a difference.

3 (CRASH COURSE ON) PARA-DIFFERENTIAL OPERATORS Beyond olynomial symbols. Considering classical differential oerators, we obtain olynomial symbols. If we are to find a framework where we can invert them, we need to consider rational symbols. We also want to be able to roject, thus we would like to consider comactly suorted symbols. We will consider general olynomial like symbols: Definition 1.1. A symbol of degree 0 is a smooth function a(x, ξ) such that for all α, β N d su (x,ξ) R d ξ β D α x D β ξ a(x, ξ) C α,β. We note this a S 0. More generally, b is a symbol of degree (noted b S ) if b S 0, b(x, ξ) := (1 + ξ ) b(x, ξ). Note that this does give a generalization of Rational functions and of comactly suorted functions. As an examle, a α ξ α S, and ϕ(ξ)a(x, ξ) S q α whenever ϕ Cc (R d ) and a Cc (R d R d ), q 0. From now on, whenever we talk about a symbol, it will be a symbol according to the above definition. Now that we have extended the notion of symbols, we need to find a way to transform them into oerators that would agree with the ma (1.3). This is easily done using the Fourier transform: if σ is a symbol, then A slightly simler version is {O w (σ)f} (x) := 1 (π) d F {O w (σ)f} (ξ) = 1 (π) d e i x y, σ( x + y R d where σ denotes the Fourier transform in the first variable only., )f(y)dyd. R σ(ξ η, ξ + η ) f(η)dη (1.4) {Pseudo} Definition 1.. A seudo-differential oerator of order is an oerator of the form O w (σ) for some σ S. Note that, using formula (1.4), one can also define symbols which are less regular and the corresonding seudo-differential oerators. This already allows to define arametrix for the inverse of ellitic oerators, see Subsection Commutation roerties. Let S denote the set of symbols. We now have a corresondance A S. This corresondance is obviously linear. But A is also equied with a roduct. We can ask how to generalize this to S. Besides S is also naturally equied with a notion of a roduct (the standard ointwise roduct of functions), but this is commutative so it cannot be the exact analog. However, we know that the roduct in A is commutative at to order, so we can hoe that (1) If A A j, B A k, σ(ab) σ(a)σ(b) () If a S j, b S k are olynomial, then is of degree j+k-1 O w (ab) O w (a)o w (b) A j+k 1

4 4 BENOIT PAUSADER {PoissonB} and this can be readily verified. The exact formula for a roduct # satisfying O w (ab) = O w (a)#o w (b) can then be comuted for a, b S. For general seudo-differential oerators, however, this is only true in the sense that the difference is an oerator with smooth kernel. We refer to the reference given at the beginning for more on this. Looking at vector fields, we can conjecture that where we introduce the Poisson bracket: σ([a, B]) = i{σ(a), σ(b)} {a, b} = x a ξ b ξ a x b. (1.5) 1.6. Ellitic oerators. Informally, a symbol is ellitic if it does not vanish outside of a neighborhood of 0. This is the natural generalization to the criterion for invertibility on A 0. A symbol σ is called ellitic if there exists C, δ > 0 such that σ(x, ζ) C 1 ξ δ, ξ C. The rototyical examle is the symbol of the Lalacian: σ w ( ) = ξ, which has an inverse of symbol 1/ ξ. We say that a seudo-differential oerator is ellitic if its symbol is ellitic 3. One can then find a first ansatz for an aroximate inverse by setting q(x, ξ) := 1 χ(ξ) σ(x, ξ) where χ 1 on the ball where we do not have control over σ. One can then verify that O w (σ)o w (q) = Id + S 1, O w (q)o w (σ) = Id + S where S 1 and S are smoothing oerators by one order.. Paradifferential oerators The urose of this section is to make the revious qualitative discussion more quantitative..1. Paradifferential oerators. A small twist is that we want to consider oerators and symbols which are not necessarily smooth. When considering roducts, in the case when a function is much smoother than the other, we want to kee track of this information (since the rougher one will force the regularity of the roduct). A simle but far-reaching remark by Bony is that we can always searate the smooth art from the rough art: fg = k 1,k P k1 f P k g adiffintro} = = k k 1 k 10 P k1 f P k g + (P k 10 f)p k g + k = T f g + T g f + R(f, g) k k 1 10 P k1 f P k g + (P k 10 g)p k f + k k 1 k <10 j <10 P k1 f P k g P k f P k+j g where (rovided f and g have a reasonable amount of smoothness, say f, g H d ) R(f, g) is smoother than f and g, T f g is as smooth as g and T g f is as smooth as f (see Lemma. for more (.1) Indeed, we want to allow ellitic oerators to have a finite dimensional kernel. Otherwise, would not be ellitic on T d! 3 Note that this is a roerty of its rincial symbol.

5 (CRASH COURSE ON) PARA-DIFFERENTIAL OPERATORS 5 recise estimates). This simle decomosition, called the araroduct decomosition allows to better kee track of the regularity and we want to extend it for seudo-differential oerators (it amounts to secifying the term where the derivative will fall ). In the following, for concreteness, we will secify to the case of dimension d =. Recall that a symbol σ if of degree if and only if (1 + ζ ) σ(x, ζ) is a symbol of degree 0. For this reason, we will in the next subsection concentrate on redefining a more quantified version of symbols of degree 0... Oerator bounds. A symbol denotes a function σ(x, ζ) C(R d R d : R). In this section, we will bound them using the following two norms 4 σ S a,b := su ζ β β ζ α x σ(x, ζ), σ S a,b x,ζ := su ζ α a, β b α a, β b ζ β β ζ α x σ(, ζ) L x. These are the most imortant norms. However, we may more generally define σ S a,b := su ζ β β ζ α x σ(, ζ) L x. ζ α a, β b Note that there is a large variation of norms and notation for symbols. A few useful roerties are 5 ab S min{r1,r},min{q1,q} {a, b} S min{r 1,r } 1,min{q 1,q } 1 P k a S r,q a S r 1,q 1 1 b S r,q, {, } = { 1, } a S r 1,q 1 1 b S r,q, {, } = { 1, } sk P k a S r+s,q, {, }. Given a symbol σ(x, ζ), we define the corresonding aradifferential oerator (comare with (1.4)): ξ η ξ + η F {T σ f} (ξ) = χ( ) σ(ξ η, R ξ + η ) f(η)dη, where σ denotes the Fourier transform in the first coordinate and χ Cc (R : R) is a function satisfying 1 [ 1/00,1/00] χ 1 [ 1/100,1/100]. It is clear that if σ is real, T σ is a self-adjoint oerator on L. But in fact T σ also acts on L saces. We first observe that aramultiliers define nice linear oerators. Proosition.1. Let a be a symbol and 1 q, let 6 γ > d/, γ N then and More generally, if (.) {SymNorm1} (.3) {SymNorm1Gen (.4) {ProSymb1} P k T a f L q a S 0,4γ P [k,k+]f L q (.5) {LqBdTa} P k T a f L a S 0,4γ P [k,k+] f L. (.6) {LBdTa} 1 r = 1 + 1, 1, q, r, (.7) {Homogeneity q 4 These norms measure symbols of degree 0. Thanks to (.5), the degree can then be trivially added. Another way to say this is that a symbol of degree can be written σ = (1 ) / σ 0, where σ 0 is an oerator of degree 0. 5 Recall the Poisson bracket introduced in (1.5). 6 as will be evident from the roof, we could take γ smaller when <, but we will not be concerned with the ζ-regularity which will always be automaitc.

6 6 BENOIT PAUSADER then omogeneity} Rescaling} SymEstimI} P k T a f L r a S 0,4γ P [k,k+] f L q. (.8) {LrBdTa} Proof. Fact 1: Insecting the Fourier transform, we directly see that P k T a f = P k T a P [k,k+] f, and therefore, the Fourier localization of f is guaranteed and, u to rescaling, we may assume that k = 0, i.e. Fact : It suffices to show (.8) for k = 0. Indeed, introducing the rescaling oerator (δ h f)(x) = f(hx), we recall that δ h f L = h d f L. (.9) Besides, letting h = k and writing down the Fourier transform, we find that and observing that δ h (P k T a f) = P 0 T a h(δ h f), a h (x, ζ) := a(hx, h 1 ζ) (.10) a h S 0,t = h d a S 0,t, t 0 and using (.7) and (.9), we see from (.10) that the claim at k = 0 imlies the claim for all k s. Fact 3: The claim holds when k = 0. We comute that g, P 0 T a h = g(x)h(y)i(x, y)dxdy, R d I(x, y) = a(z, ξ + η R 3d )ei ξ,x z e i η,z y ξ η χ( ξ + η )ϕ 0(ξ)dηdξdz = a(z, η + θ R 3d )ei θ,x z e i η,x y θ χ( η + θ )ϕ 0(η + θ)dηdθdz. We now observe that (1 + x y ) γ I(x, y) R3d a(z, η + θ = ) (1 + x z ) γ χ( θ η + θ )ϕ 0(η + θ) [(1 θ ) γ e i θ,x z (1 η ) γ e i η,x y ] dηdθdz (.11) so that and we conclude that g, P 0 T a h a S 0,4γ which is sufficient for our estimate. (1 + x y ) γ I(x, y) a S 0,4γ a S 0,4γ g(x) h(y) R 4 g L r h L q, 1 (1 + x y ) γ dxdy One of the main interest of T a is that it linearizes the roduct as observed in (.1): Lemma.. Let f, g L we have fg = T f g + T g f + HH(f, g) where HH is smoothing in the sense that for γ > 0 and r,, q as in (.7), there holds that P k HH(f, g) L r γk su γl P µ f L P ν g L q. l 0 µ,ν l+0

7 Proof. Starting from (CRASH COURSE ON) PARA-DIFFERENTIAL OPERATORS 7 HH(f, g) = fg T f g T g f and exanding on the frequency rojections, we see that unless j l 10 and k j Therefore P k HH(f, g) L r γk and the result follows by Hölder s inequality. P k HH(P j f, P l g) 0, j k 10 γ(k j) 0 j 10 rj P j fp j+j g L r Proosition.3. Let 1 q and γ > d/, γ N. Given a and b symbols, we may decomose where the error E obeys the following bounds T a T b = T ab + i T {a,b} + E(a, b) (.1) {ProComSym P k E(a, b)f L q (1 + k ) a S,6γ+ for 1 q and more generally, letting there holds that b S,6γ+ P [k 5,k+5] f L q (.13) {RemRemBound 1 r = , 1 r, 1,, 3 (.14) {TrilinearHo P k E(a, b)f L r (1 + k ) a S,6γ+ b 1 S,6γ+ P [k 5,k+5] f L 3. (.15) {RemRemBound Moreover E(a, b) = 0 if both a and b are indeendent of x. Proof. We rove (.13). The bounds in (.15) follow similarly. We begin by writing T a T b T ab = E s + E r, [ ] ξ θ η η F {E s f} (ξ) = χ( )χ( θ ) χ( ξ R 4 ξ + θ θ + η ξ + η ) ã(ξ θ, ξ + θ ) b(θ η, η + θ ) f(η)dηdθ, ξ η [ F {E r f} (ξ) = χ( R 4 ξ + η ) ã(ξ θ, ξ + θ ) b(θ η, η + θ ) ã(ξ θ, ξ + η ) b(θ η, ξ + η ) ] f(η)dηdθ, and observe directly that if ã(x, ζ) and b(y, ζ) are suorted on {x = 0} and {y = 0}, both E s and E r vanish. We also observe that, using Lemma.1 together with (.4), P k T Pj ap l bf L q + P k T Pj at Pl bf L q rj sl P j a S r,4γ P lb S s,4γ P [k 5,k+5]f L q. Thus in the following, we mays assume that a = P k 10 a and b = P k 10 b. In this case, E s = 0. Using this, it suffices to consider F ( Er k f ) ξ η [ (ξ) = ϕ k R d ξ + η ) ã(ξ θ, ξ + θ ) b(θ η, η + θ ) where we define ã(ξ θ, ξ + η ) b(θ η, ξ + η ) ] f(η)dηdθ, ϕ k (ξ, η, θ) = ϕ k (ξ)ϕ [k,k+] (η)ϕ [k,k+] (θ). (.16) {Underlineh

8 8 BENOIT PAUSADER We note that Er k = P k E r has kernel whose Fourier transform is m(ξ, η) = = e iφ ξ η [ ϕ k R 3d ξ + η ) a(y 1, ξ + θ )b(y, η + θ ) a(y 1, ξ + η )b(y, ξ + η ] ) dy 1 dy dθ Φ = y 1, θ ξ + y, η θ where in articular, {DerPh} y1 Φ = θ ξ, y Φ = η θ. (.17) Integrating by arts using (.17), we may rewrite this kernel as m(ξ, η) = i e iφ ξ η ϕ k R 3d ξ + η ) [ 1 a(y 1, ξ + η ) b(y, ξ + η ) a(y 1, ξ + η ) 1b(y, ξ + η ] ) dy 1 dy dθ + i 1 e iφ ξ η ϕ k s=0 R 3d ξ + η ) ( 1 a(y 1, ξ + η [ ) b(y, ξ + η + s θ ξ ) b(y, ξ + η ] ) [ a(y 1, ξ + θ + s η θ ) a(y 1, ξ + η ] ) 1 b(y, η + θ ) a(y 1, ξ + η [ ) 1 b(y, η + θ ) 1b(y, ξ + η ] ) ) dy 1 dy dθds The first integral gives the Poisson bracket. After another integration by arts using (.17), the second integral can be rewritten m r (ξ, η) = s=0 e iφ ξ η ( ϕ k R 3d ξ + η ) 11 a(y 1, ξ + η ) b(y, ξ + η 1 a(y 1, ξ + η ) 1b(y, ξ + η + s θ η ) + a(y 1, ξ + η + s θ η ) 11b(y, η + θ )(1 s) ) dy 1 dy dθds. + s θ ξ )(1 s) We may now use Lemma.6 and Lemma.4 to finish the roof. Note that we could go further in the exansion (.1). Indeed, a look at the above formula shows that the main contribution will come from the second Poisson bracket: {{a, b}} := 11 a : b 1 a : 1 b + a : 11 b. Further contributions are determined by higher order Poisson brackets.

9 (CRASH COURSE ON) PARA-DIFFERENTIAL OPERATORS 9 Lemma.4. For 0 s 1, define ξ η m k (ξ, η) = ϕ k R 3d ξ + η )eiφ {{a, b}}dy 1 dy dθ ( {{a, b}} = 11 a(y 1, ξ + η ) b(y, ξ + η + s θ ξ )(1 s) 1 a(y 1, ξ + η ) 1b(y, ξ + η + s θ η ) + a(y 1, ξ + η + s θ η ) 11b(y, η + θ ) )(1 s) Φ = y 1, θ ξ + y, η θ where ϕ k is defined in (.16). Then (F 1 m k )(x, z) L 1 x (1 + k ) a S,6γ+ b S,6γ+ [1 + x z ] γ. (.18) {EstEsFL1} Proof. Assume first that k 0. The Fourier transform is then I k,s (x, z) := J k,s (y 1, y )dy 1 dy R d J k,s (y 1, y ) = e iψ A k,s dηdξdθ R 3d The main information we use is that ξ η A k,s = ϕ k ){{a, b}}, ξ + η Ψ = y 1, θ ξ + y, η θ + ξ, x η, z ξ,η,θ α [A k,s] α k k 1 { ξ + η + θ k } a,+ α S ξ Ψ = x y 1, η Ψ = y z, θ Ψ = y 1 y, b S,+ α so that, integrating by arts, we find that, for 0 q, r, t γ, x y 1 q y z r y 1 y t J k,s (y 1, y ) = R ( 1)q+r+t e iψ q ξ r η t θ A k,sdηdξdθ 6 which is sufficient. k (3d (q+r+t))k a S,6γ+ b S,6γ+ If k 0, we simly observe that, using the Fourier transform in y 1, y in the integration, we may trade the derivatives in y 1, y by small factors of size O( k ), namely we may relace {{a, b}} by {{a, b}} = 1 ( a(y 1, ξ + η 4 ) ((θ ξ) ) b(y, ξ + η + s θ ξ )(1 s) + ((η θ) )a(y 1, ξ + η )((θ ξ) )b(y, ξ + η + s θ η ) ((η θ) ) a(y 1, ξ + η and since η + θ + ξ k, we obtain that which again, is sufficient. + s θ η )b(y, η + θ )(1 s) ), α ξ,η,θ [A k,s] α k 1 { ξ + η + θ k } a S 0,+ α b S 0,+ α

10 10 BENOIT PAUSADER imlemult1} Proosition.5. Assume that F (z) = z + h(z), where h is analytic for z 1/ and satisfies h(z) z 3, then there holds that E(u) H N F (u) = T F (u)u + E(u), u W 10, u H N 5 so long as u L 1/100. This also holds if z C, F : C C, where F (u) denotes the differential of F. Proof. Since F is analytic, it suffices to show this for h(x) = x n, n 0, but this as in the roof of Lemma.. Indeed, we may decomose P k (u n ) = n (P k u... P kn u) P k1 u k 1 k n, max{k,...,k n} k 3n + O(n ) k 1 k max{k 3,...,k n,k 3n} P k1 u P k u P k3 u... P kn u. The first term in this exansion leads to the araroduct u to smoothing terms, while the second leads to a smoothing terms. Recall the following simle Lemma for the oerator FM(f 1, f,..., f n )(ξ) = f(ξ1 ) f(ξ )... f(ξ d )m(ξ 1,..., ξ d )δ(ξ ξ 1 ξ d )dξ 1... dξ d. R d This oerator obeys simle bounds: Lemma.6. Assume that 1 1,,..., n, s and 1 s = n Then M(f 1, f,..., f n )(x) L s Fm L 1 References n f k L k. (.0) [1] S. Alinhac and P. Gérard, Pseudo-differential Oerators and the Nash-Moser Theorem, Graduate studies in mathematics, Vol 8, (007), ISBN-10: [] L. Hörmander, The analysis of linear differential oerators, I-IV. k=1 (.19) {Paracom} Brown University address: benoit.ausader@math.brown.edu