Introduction to pseudodifferential operators. 1. Kohn-Nirenberg calculus, simplest symbol classes S m

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1 February 8, 17) Introduction to pseudodifferential operators Paul Garrett garrett/ [This document is garrett/m/fun/notes 16-17/psidos.pdf] One goal is proof of local regularity of elliptic differential operators with variable coefficients, to understand regularity of Casimir operators on iemannian symmetric space G/K with G a semi-simple real Lie group and K maximal compact. egularity is most memorably discussed by enlarging the class of differential operators to include pseudodifferential operators. We give both the Kohn-Nirenberg and Weyl functional calculuses for pseudodifferential operators. Although the Kohn-Nirenberg calculus of pseudodifferential operators is more immediate and accessible, the Weyl calculus has some technical advantages. One is that in the Weyl case, operators are parametrized by functions on the Heisenberg group, with composition of operators given by convolution, and transpose by ϕ x) = ϕx 1 ). In contrast, expression of transposes in the Kohn-Nirenberg setting requires a computation, and composition requires an extension of the notion of pseudodifferential operator. Kohn-Nirenberg calculus, simplest symbol classes S m Schwartz kernels in the Kohn-Nirenberg setting The Schrödinger model Weyl calculus Examples of operators for Weyl calculus Schwartz kernels in the Weyl setting... Appendix: nuclearity of S and of D 1. Kohn-Nirenberg calculus, simplest symbol classes S m Fourier transform converts multiplication by x to differentiation /x, which rewrites constant-coefficient differential operators to integrals against polynomials. Enlarging the class of functions beyond polynomials, the multiplier operator attached to a function ϕ is f e πixξ ϕξ) fξ) dξ That is, with Fourier transform F, multiplier operator attached to ϕ = F 1 multiplication by ϕ F Extending this in another fashion gives variable-coefficient differential operators and sufficiently general operators to discuss solvability and regularity of partial differential equations. The fundamental complication is that the non-commutativity of variable-coefficient differential operators cannot easily be reflected in the commutative point-wise multiplication of functions, nor in convolution on n. Nevertheless, other aspects of this set-up are suggestive and encouraging. With x, ξ n, for px, ξ) = α k c αx) πiξ) α, the operator OPp) = OP KN p) is OPp)f)x) = px, D)fx) = e πiξ x px, ξ) fξ) dξ = c α x) α fx) n This is a variable-coefficient differential operator. We can attempt to extend p OPp) from polynomial functions of ξ to various broader classes, by the same formula in terms of Fourier transform: the pseudodifferential operator OPp) = OP KN p) = px, D) attached to the symbol px, ξ) in the Kohn-Nirenberg calculus is OPp)f)x) = px, D)fx) = e πiξ x px, ξ) fξ) dξ n 1 α k

2 The notation px, D) is suggested by the conversion of multiplication by ξ j to application of /x j by Fourier transform.. Schwartz kernels in the Kohn-Nirenberg setting Schwartz Kernel Theorem is that every continuous linear T : D m ) D n ) has a Schwartz kernel K D m n ), meaning that T ϕ)ψ) = Kϕ ψ) Similarly, every continuous linear T : S m ) S n ) has Schwartz kernel K S m n ), again meaning that T ϕ)ψ) = Kϕ ψ) Very nice Schwartz kernels K, for example in S n n ), unsurprisingly correspond to very nice if uninteresting) operators. [.1] Claim: For p S n n ), the Schwartz Kernel K of px, D) = OP KN p) is F 1 px, x t) where F is Fourier transform in the second argument. Thus, the pseudodifferential operators given by p S n n ) give exactly the operators S n ) S n ) with kernels in S n n ). Proof: By direct computation, for ϕ S m ) and ψ S n ), ) px, D)ϕ)ψ) = ψx) n e πiξ x px, ξ) ϕξ) dξ m dx = ψx) e πiξ x px, ξ) e πiξ t ϕt) = ψx) ϕt) e πiξ x t) px, ξ) = ψx) ϕt) F 1 px, x t) Partial) Fourier transform is an automorphism of S n n ), as is a non-singular change of variables, giving the result. /// 3. The simplest symbol classes Traditional terminology is that various classes of functions px, ξ) are symbol classes. The simplest symbol classes are S m = S m Ω) = {p C Ω n ) : compact; E Ω, α, β, α x β ξ px, ξ) E,α,β 1 + ξ ) m β } This is the simplest example within a larger family of symbol classes studied by Hörmander: S m = S m 1, where S m ρ,δ = {p C Ω n ) : compact E Ω, α, β, α x β ξ px, ξ) E,α,β 1 + ξ ) m δ α ρ β } [3.1] emark: In more abstract situations, the space Ω n would be the tangent bundle over Ω. The basic result is [3.] Theorem: The operators OPp) = px, D) for p S m are continuous maps DΩ) EΩ), which extend to continuous maps E Ω) D Ω).

3 Proof: [... iou...] /// [3.3] emark: The continuity of the extension requires knowing that the transpose px, D) : D E is a pseudodifferential operator, which is not obvious in the Kohn-Nirenberg context. [3.4] emark: Naturally, a parametrization of operators by functions is most useful when composition of operators is reflected directly in some sort of composition of the parametrizing functions. A fundamental complaint one might have about the Kohn-Nirenberg parametrization is that the non-commutativity of multiplication operators and derivative operators is not well reflected in the commutative convolution of functions on n. The Weyl parametrization below) has the virtue of overcoming this particular difficulty. The two operators 4. The Schrödinger model D = d X = multiplication by x dx on differentiable functions on do not commute, and this is not a pathology. Their commutator is [D, X] = D X X D = 1 and the commutator does commute with both of them. Further, neither of these two operators stabilizes L ), although multiplication by x has an obvious pointwise sense, and d dx has a distributional sense on L functions, since these are in L 1 loc. It is non-trivial to guess a good normalization, but we can make inferences. With Fourier transform F normalized by F fξ) = e πiξx fx) dx for f in the Schwartz space S D F f ) = πi)f Xf) That is, we find a normalization constant of ±πi in the relation between D and X. Further, we want these operators to occur in the complexified) image of a representation of a Lie algebra attached to a unitary Lie group representation. Thus, the image of the real) Lie algebra should be skew-hermitian operators. Integration by parts shows that x is already skew, but multiplication by x needs a factor of ±i. Thus, we might renormalize by taking and then the commutator is D = 1 π x [D, X] = DX XD = X = multiplication by ix i π The simplest Lie algebra whose representation theory can reflect this non-commutativity is the Heisenberg Lie algebra h = { } which has a Lie algebra representation σ alg σ alg : 1 D σ alg : 1 X σ alg : 1 i π 3

4 The fact that the Lie bracket is preserved is a direct computation. We want an associated unitary Lie group representation σ gp of the Heisenberg group H = { 1 1 } 1 and the standard idea would be to exponentiate to define σ gp on H by σ gp e πα ) = expσ alg α) for α h) where the first exponentiation α e πα is exponentiation of matrices, namely e A = 1 + A 1! + A! + A3 3! +... but where the second exponentiation, that of endomorphisms of L ), is not assured to make sense. The π in the exponent is in anticipation of compatibility with our normalization of Fourier transforms. Exponentiate σ alg to a group representation σ gp as follows. For a, b, the effect of exponentiated ad +bx on a function f L ) is characterized in a manner that should determine it uniquely, provided it exists. Let gx, t) = exp tad + bx) ) fx) for t real) The characterization of exponentiation is That is, gx, t) = πad + bx) gx, t) and gx, ) = fx) t t gx, t) = πa 1 + b ix) gx, t) and gx, ) = fx) π x This differential equation with boundary condition can be solved directly. earrange to t a ) gx, t) = πibx gx, t) x The left-hand side is a directional derivative: fix x and let ht) = gx at, t), so the equation becomes or Thus, The condition gx, ) = fx) gives d ht) = πibx at) ht) dt d ) log ht) dt = πibx at) at πibxt gx at, t) = e ) constant depending on x) at πibxt gx at, t) = e ) fx) eplace x by x + at At t = 1 at πibxt+ gx, t) = e ) fx + at) expad + bx) fx) = e πibx+ a ) fx + a) 4

5 That is, σ gp exp a b ) fx) = expad + bx) fx) = e πibx+ a ) fx + a) Including the center gives σ gp exp a c b ) fx) = exp ad + bx + c i π ) fx) = eπibx+ a )+ic fx + a) This is the Schrödinger model on L ) of the unique [1] ) infinite-dimensional unitary representation of the Heisenberg group with character ω : 1 c 1 e ic 1 on the center Z of H. For the moment, we do not need uniqueness of this representation, only existence. 5. Weyl s symbol calculus: operators from functions An association of operators to functions is a functional calculus. Except for the uninteresting case that the operators commute, the multiplication on functions cannot be the commutative) pointwise multiplication. Convolution on a form of the Heisenberg group is sufficiently non-commutative to reflection composition of pseudo-) differential operators. For any representation τ of of a unimodular group G on a topological vector space V, and for suitable functions η, ϕ on G, the integrated actions ϕ v = G ϕg) g v dg compose by η ϕ v) = η ϕ) v where is the standard convolution determined completely by the previous relation. Formulaically, that relation determines it as η ϕ)x) = ηxg 1 ) ϕg) dg G Thus, for a function ϕ on H, the associated action of ϕ in the Schrödinger model on f L ) is ϕ f)x) = ϕh) h fx) dh = ϕh) σ gp h) f ) x) dh = ϕh) e πibx+ a )+ci fx + a) da db dc H H H Here there is a complication: the center Z of H acts by the character ω, and the integral ωz) dz = Z e ic dc [1] The uniqueness is the Stone-vonNeumann theorem. 5

6 diverges, so the function ϕ on H cannot be constant along the center for the integral for ϕ f to behave properly. To remedy this minor flaw, note that c e ic factors through the quotient /πz, so the group representation above factors through the quotient group H red = H / 1 πz 1 1 called the reduced Heisenberg group. The reduced Heisenberg group has the same Lie algebra h, but smaller center / Z red = Z 1 πz 1 1 Pretending some foresight, we give the reduced center Z red total measure 1, rather than the more natural π, for later simplification. Since this Schödinger representation depends so trivially on the center, we can specify functions on H red or H by specifying functions ϕ on the linear complement to z in h, and then writing V = h ϕ exp πv + z) ) = ω 1 exp πz) ϕv) for v V and z z) where the ω 1 is chosen to cancel the ω in the Schrödinger representation. Then we may write ϕ exp π a b ) = ϕ a b = ϕa, b) Extend this notation to allow reference to the center by ϕa, b, c) = e ic ϕ exp π a c b ) = e ic ϕ a c b [5.1] emark: When the identification of H/Z and h/z with is exaggerated, the naturality of the convolution product on H red is thereby neglected. 6

7 6. Examples of operators for Weyl calculus In the Schrödinger representation of H red on L ), functions ϕ on the reduced Heisenberg group H red produce recognizable operators f ϕ f = OP W ϕ)f. [6.1] To get an idea, try ϕa, b) = 1. Keeping in mind that Z red has total measure 1, not π, OP W ϕ)f)x) = ϕ f)x) = ϕh) h f)x) dh = e ic 1 e πibx+ a )+ic fx+a) da db dc H red /πz = e πibx+ a ) fx + a) da db eplacing a by a x and integrating in a gives x+a πib ϕ f)x) = e ) fa) da db = e πib x eplacing b by b gives by Fourier inversion. [6.] Next, try ϕ f)x) = e πibx f b) db = f x) ϕa, b) = a f b/) db A similar computation, integrating first in b, gives ϕ f)x) = e ic a e πibx+ a )+ic fx + a) da db dc = a e πibx+ a ) fx + a) da db /πz = a x) e πibx+a)/ fa) da db = a x) e πibx+a) fa) da db = a x) δx + a) fa) da = x x) f x) = 4π xf x) That is, up to the change of sign in the argument to f, OPϕ) is a multiplication operator. [6.3] Now try ϕa, b) = b A similar computation: ϕ f)x) = e ic b e πibx+ a )+ic fx + a) da db dc = b e πibx+ a ) fx + a) da db /πz x+a πib = b e ) fa) da db = i x+a πib e ) fa) da db π x = i e πibx+a) fa) da db = i δx + a) fa) da = i π x π x π x f x) That is, we can obtain constant coefficient differential operators. Some of the details above are mere artifaces of coordinate choices. In fact, a significant advantage here is the intrinsic sense of the map from functions to operators. 7

8 7. Computing Schwartz kernels The possibilities and normalizations for operators appearing as σ gp ϕ) are best understood by determining the kernel K ϕ of the operator attached to ϕ, in the sense of Schwartz ϕ f)x) = K ϕ x, a) fa) da Direct computation shows that Schwartz functions S V ) on the complementary subspace V to z inside h biject to kernels in S ). We would expect to extend by continuity to a bijection of tempered distributions on V to kernels in S ). Schwartz kernel theorem says that every operator S ) S ) is given by an element of S ), thus, by σ gp ϕ) for some ϕ S V ). [7.1] Computing the kernel A direct computation gives the kernel: using the notational convention above for ϕ a function on V h, ϕ f)x) = = H red ϕh) h f ) x) dh = /πz e ic ϕa, b) e πibx+ a )+ic fx + a) da db dc ϕa, b) e πibx+ a ) x+a πib fx + a) da db = ϕa x, b) e ) fa) da db = F 1 ϕ a x, x + a ) fa) da where F is Fourier transform in the second argument. Thus, the kernel K ϕ for the operator σ gp ϕ) is K ϕ x, a) = F 1 ϕ a x, x + a ) Partial) Fourier transform and linear changes of coordinates are automorphisms of the Schwartz space. [7.] Examples As a trivial confirmation, consider the identity map 1 on S, which has kernel To find ϕ S V ) producing this kernel, solve K 1 x, a) = δx a) S ) F 1 ϕ a x, x + a ) x + a ) = δa x) = δa x) 1 Then ϕ a x, x + a ) x + a ) = δa x) δ 8. Appendix: nuclearity of S and of D For us, nuclear topological vectorspaces M, N have the property that there is a topological vectorspace M N behaving as a genuine tensor product in a suitable category of locally convex topological vectorspaces. That is, there is a continuous bilinear M N M N such that, for every continuous bilinear β : M N X to 8

9 Paul Garrett: Introduction to pseudodifferential operators February 8, 17) a topological vector space X in the category there is a unique linear B : M N X giving a commutative diagram M N! M N X To build up a class of such spaces, begin with nuclear Fréchet spaces: countable projective) limits of Hilbert spaces with Hilbert-Schmidt transition maps. That ET n ) is nuclear Fréchet follows from two things: the Sobolev result that H T n ) = ET n ), and the ellich result that H k T n ) H k l T n ) is Hilbert-Schmidt for l n 1. That Schwartz spaces S n ) are nuclear Fréchet is perhaps less obvious, since the XXXXXXXXXXXXXXXX seen as follows. Let S = + x be the Hermite operator, also known as the Hamiltonian for) the quantum harmonic oscillator. Define norms on D n ) by f B k = Sk f, f L for f D) and let B k be the completion of D with respect to the B k -norm. Then one shows two things: that B k B k l is Hilbert-Schmidt for l n 1, and that S = B = lim k B k. As in [Garrett 14], for example, this proves that S is nuclear Fréchet. [Folland 1983] G. Folland, Lectures on Partial Differential Equations, Tata Inst. Fund. esearch, Bombay/Mumbai, [Folland 1983] G. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, [Garrett 14] P. Garrett, Hilbert-Schmidt operators, nuclear spaces, kernel theorem I garrett/m/fun/notes 1-13/6d nuclear spaces I.pdf [Grothendieck 1955] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Memoirs AMS 16, [Kohn-Nirenberg 1965] J.J. Kohn and L. Nirenberg, On the algebra of pseudo-differential operators, Comm. Pure Appl. Math ), [Weyl 198] H. Weyl, Gruppentheorie und Quantenmechanik, Hirzel, Leipzig, 198. Quantum Mechanics, Methuen, London, 1931; reprinted by Dover Publications, 195. Group Theory and 9