Exact fundamental solutions

Journées Équations aux dérivées partielles Saint-Jean-de-Monts, -5 juin 998 GDR 5 (CNRS) Exact fundamental solutions Richard Beals Abstract Exact fundamental solutions are known for operators of various types. We indicate a general approach that gives various old and new fundamental solutions for operators with double characteristics. The solutions allow one to read off detailed behavior, such as the presence or absence of analytic hypoellipticity. Recent results for operators with multiple characteristics are also described.. Introduction. The left or right inverse L of a linear partial differential operator L = P (x, D), when it exists, is given by integration against a Schwartz kernel L u(x) = K(x, y) u(y) dy. We refer the kernel K as the Green s kernel. Green s kernels for the classical operators ), t,, t and for standard related boundary value problems, were known classically. In this century much of the emphasis in PDE has been on extending results to the variable coefficient and nonlinear versions of these and more general operators through the development of techniques of functional analysis, a priori estimates, exploitation of the Fourier transform, oscillatory integrals, pseudodifferential operators, Fourier integral operators,.... Typically we no longer expect an exact fundamental solution, but rather a parametrix. Even in the 0th century there are exceptions, however. Exact fundamental solutions for the kinetic operators () x x x ), x x ) + x x x x x ) + x. x x x x I-

were computed by Kolmogorov [6], Chandrasekhar [7], and Aarão [], respectively. Folland [8] computed the fundamental solution for the Kohn-Rossi sublaplacian on the Heisenberg group n { ) ) } () H = x n+j + + x j ; x j x 0 x n+j x 0 j= then Folland and Stein [9] found the fundamental solution for the related operator (3) H + α x 0, α IC. This last was the first example of a discrete phenomenon in hypoellipticity. It gave rise to an industry: hypoelliptic operators with double characteristics. The corresponding heat kernel, the fundamental solution for (4) t H was found by Gaveau [] and Hulanicki [4]; derivations continue to be published [5]. The operators () and (3) are associated to problems in several complex variables on the strictly pseudoconvex domain Ω n, = {(z, w) IC n IC : Im w = z }. Fundamental solutions for the analogous operator associated to the domains (5) Ω n,k = {(z, w) IC n IC : Im w = z k }. were found by Greiner [] for the case n =, k = and recently for n =, general k [4] and general n, general k [5]. (In a related direction, one might mention exact formulas for the Cauchy-Szegö projection associated to such domains [3], [7], [0].) There is a simple, general approach to finding fundamental solutions to a class of operators that includes ()-(4) as well as degenerate elliptic operators like (6) x ) + x x ), x ) + x ) + x x 3 ). The second is the well-known Baouendi-Goulaouic example []: hypoelliptic but not analytic hypoelliptic. This feature can be read off from the explicit fundamental solution. We describe this method in the next section. There is (as yet) no such sweeping approach to equations with characteristics of multiplicity greater than two, such as those associated with the domains (5), or the degenerate elliptic operators (7) x + x k y = n j= x j ) + x k m j= y j ), k =,, 3,.... In section 3 we describe joint work with Gaveau and Greiner treating the operators (7) along the lines of [4], [5]. We note in passing that when the the dimensions n and m are even, the Green s kernel of the operator (7) is an algebraic function that can be written in closed form. I-

. Generalities for double characteristics. We illustrate the general approach first with the harmonic oscillator in IR n (8) L = λ x. The idea is to find the associated heat kernel and integrate with respect to time. Now L is the sum of two negative operators and we are looking for the associated semigroup. Some computation based on the Trotter product formula exp [ t(a + B) ] = lim n [ exp ( t n A) exp ( t n B)] suggest as Ansatz that the heat kernel for (8) has the form (9) P (x, y, t) = ϕ(t) exp( Q t (x, y)), t > 0, where Q t is a quadratic form in the variables (x, y). Taking symmetries into account we expect (0) Q t (x, y) = α(t) x + β(t)x y + α(t) y. Applying the operator / t + L to (9) we obtain conditions () α = α λ = β ; β = αβ ; ϕ ϕ = α. Since P must act like an approximate identity as t 0, we need α to blow up and β α as t 0. Therefore the desired solution of () is () ϕ(t) = α(t) = λ cosh(λt) ; sinh(λt) λ β(t) = sinh(λt) ; λ n/ [π sinh(λt)] /. Integration with respect to t on IR + gives the fundamental solution for L. Consider next the first of the two degenerate elliptic operators in (6): (3) L = x ) + x x ). To compute the Green s kernel we look first at the corresponding heat kernel P (x, y, t). Because of translation invariance in the second variable we may take the Fourier transform with respect to x. The operator has the form (8), with λ as the Fourier transform variable. We take the inverse Fourier transform of the solution given by (9), () and obtain, after a change of scale in the integration, the heat kernel (4) P (x, y, t) = e f(x,y,τ)/t τ / dτ, (πt) 3/ IR (sinh τ) / I-3

f(x, y, τ) = i(x y )τ + τ coth τ(x + y ) τ(sinh τ) x y. We integrate with respect to t to obtain the Green s kernel for L with pole at y: (5) K (x, y) = (τ) / dτ. π f(x, y, τ) / (sinh τ) / IR It is interesting to note that although we used the Fourier transform as a means of obtaining (5), the integral itself is not oscillatory. It is absolutely convergent for x y, and can be made absolutely convergent for x = y, x y by a change in the contour of integration. One can play exactly the same game with the Baouendi-Goulaouic operator ) ) ( (6) L 3 = + + x ). x x x 3 The result is the Green s kernel (7) K 3 (x, y) = (π) IR f(x, y, τ) (τ) / dτ, (sinh τ) / f(x, y, τ) = i(x 3 y 3 )τ + τ coth τ(x + y ) τ(sinh τ) x y + 4 (x y ). It is not difficult to establish from the formula (7) that K 3 is Gevrey everywhere where (x, x ) (y, y ), and analytic where x y. A change of the contour of integration allows one to show that K 3 is also analytic where x 3 y 3. On the other hand, one can estimate derivatives of g(s ) = K 3 ((s, x, 0), (0, 0, 0)), x 0 from below and show that K 3 cannot be better than Gevrey at (0, x, 0), (0, 0, 0). Details of these arguments are in [6]. The kinetic operators () may be treated in the same way as L above: compute the heat kernel from the Ansatz (9) and integrate with respect to time. In fact Chandrasekhar [7] notes in passing that for operators like () one can expect a heat kernel in the form (9), although he uses probabilistic considerations in his calculations. The Heisenberg operators () and (3) can be treated in the same way as L above [3]: Take the Fourier transform in the invariant direction, determine the heat kernel by using the Ansatz (9), then compute the inverse Fourier transform. Here one can take advantage of group invariance to compute with the pole at the origin. The result is the Gaveau-Hulanicki formula for the heat kernel: (8) P (x 0, x; 0, 0; t) = (πt) n+ e f(x,τ)/t IR (τ) n (sinh τ) n dτ, n f(x, τ) = ix 0 τ + τ coth τ Again, integration with respect to time gives the Green s kernel for H. I-4 j= x j.

3. Specifics for characteristics of higher multiplicity. With some manipulation (left as an exercise) the formula (5) for the Green s kernel of the degenerate elliptic operator L may be put into the form (9) K (x, y) = F (p) R /4 where (0) R = 4 and () F (p) = π 0 [ (x + y ) + (x y ) ], p = xy R /, ds s / ( s) / ( sp) = F (, ; ; p ); / this last is the Gauss hypergeometric function. The formula (9) calls for some comment. First, it reflects the fact that one expects the (canonical) Green s kernel for L to be homogeneous of degree with respect to the dilations () (x, y) (λx, λ x, λy, λ y ). In fact the denominator carries this homogeneity, while the function p that appears in the numerator is homogeneous of degree 0. Second, L is most singular at x = 0, where the numerator is identically and the denominator carries all the information. Third, away from x = 0, L is elliptic. Here the denominator is positive, while p ; p = x = y. The hypergeometric function has a logarithmic singularity at, which gives precisely the necessary singularity of G. The preceding comments, and similar formulas in [4], [5], suggest that one might obtain formulas of this type for higher degeneracies and more variables, such as the operators (7). This is the case. We describe the method, briefly, and write the result for n =, m =. Here the use of (x, y) differs from our previous usage, so we use (x (0), y (0) ) to denote the position of the pole and look for the Green s function K(x, y; x (0), y (0) ). We look first for an appropriate choice of function to carry the homogeneity with the dilations analogous to (). The correct choice turns out to be (3) R(x, y; x (0), y (0) ) = ( x k + x (0) k + k y y (0) ). The correct degree of homogeneity for K is n mk, so we look for K is the form (4) K = c F R (n+mk )/k, I-5

where c = c(n, m, k) is a dimensional constant and F is a function of some degreezero functions of (x, y, x (0), y (0) ). One good initial choice for the latter is the pair of functions defined for x 0, y 0 by (5) ρ = x k x (0) k Then R, v = x x(0) x x (0). 0 ρ ; v ; (x, y) = (x (0), y (0) ) (ρ, v) = (, ). A tedious calculation shows that for K of the form (4) to be annihilated by the operator (7) off the diagonal, it is necessary and sufficient that the numerator F satisfy (6) ( v )F vv + ( n)f v + k ( ρ )ρ F ρρ + [ k(n + k ) k(n + mk + k )ρ] ρf ρ (n + mk )(n + mk + k )ρf = 0. 4 This fairly nasty looking equation splits into a nice combination of one-variable hypergeometric operators and permits reasonable solutions by separation of variables. In fact we can write the operator acting on F as M + 4k N where (7) 4N = D ρ (D ρ + n k ) ρ (D ρ + n +mk k Here D v and D ρ denote the Euler derivatives M = v D v(d v + n ); )(D ρ + n +mk + ). k D v = v v, D ρ = ρ ρ. It is convenient now to use the variables v, σ, where so that D ρ = D σ and σ = ρ, (8) N = D σ (D σ + n ) σ(d k σ + n +mk )(D 4k σ + n +mk + ). 4k At this point we make the simplifying assumption that n =. Then the eigenfunctions of M on the interval (, ) are the Tchebychev polynomials. We normalize them here as P ν (v) = ω ν + ω ν, ω + ω = v, ν = 0,,,..., with eigenvalues ν. We write (9) M + 4k N = (M + ν ) + 4k (N ν /4k ) = (M + ν ) + k [ (D σ ν k )(D σ + ν k ) σ(d σ + m 4 )(D σ + m 4 + )]. I-6

Conjugation by σ ν/k changes the operator in square brackets to (30) D σ (D σ + ν k ) σ(d σ + m 4 + ν k )(D σ + m 4 + ν k + ). Functions annihilated by (30) that are regular at the origin are multiples of the hypergeometric function (3). F ν (σ) = F ( m 4 + ν k, m 4 + ν k + ; + ν k ; σ) Combining the remarks so far, we expect that the numerator F for our Green s kernel K should have the form (3) c ν σ ν /k ω ν F ν (σ), c ν = c ν, for some choice of constants c ν. We now make our second simplifying assumption, that m =. In this case (3) is the algebraic function (33) F ν (σ) = ( σ + σ ) ν/k. Consideration of the desired behavior of the kernel at σ = leads us to choose After some calculation (3) becomes c ν = ν /k. (34) F = F (p, ρ) = ϕ (ρ ) ϕ + (ρ ) p, where the functions ϕ ± are ϕ + (σ) = ( + σ) /k + ( σ) /k, ϕ (σ) = ( + σ) /k ( σ) /k σ, and the new variable p is the analogue for this case of the variable in (0): Thus p = ρ /k v = x x(0) R /k p ; p = (x, y) = (x (0), y (0) ). Note that our final expression for the Green s function (4) is an algebraic function of (x, y, x (0), y (0) ). A similar expression occurs in [5], but as the argument for an integral transform. We obtained the specific formula (34) for the case n =, m =. For m = and general n, the formula is very similar: (35) F = F (p, ρ) = ϕ (ρ ) [ ϕ+ (ρ ) p ] n/. I-7

For general even m we consider the right side of (35) as a function of v and ρ and obtain F by multiplication and differentiation: { } ) (m )/ (36) F = ρ (n )/k ρ (m )/ ρ (n )/k ϕ (ρ ) [ ρ ϕ+ (ρ ) p ]. n/ If m is odd, the numerator F is a fractional integral of an expression like (36). I-8

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[7] A. Nagel, Vector fields and nonisotropic metrics. in Beijing Lectures in Harmonic Analysis, Annals of Math. Studies, Princeton Univ. Press, Princeton 986, pp. 4-306. Yale University beals@math.yale.edu I-0