Asymptotic analysis of fundamental solutions of hypoelliptic operators

2.1 Notation

The set of natural numbers { 1 , 2 , 3 , … } is denoted by ℕ .

For an open set U ⊂ ℝ n , let C 0 ∞ ⁢ ( U ) denote the space of infinitely differentiable functions with compact support in U. Let 𝒮 ⁢ ( ℝ n ) be the Schwartz space of all rapidly decaying smooth functions and let 𝒮 ′ ⁢ ( ℝ n ) be the space of tempered distributions.

Let the direct and the inverse Fourier transforms ℱ ± 1 : 𝒮 ⁢ ( ℝ n ) → 𝒮 ⁢ ( ℝ n ) be defined as follows:

They are extended to 𝒮 ′ ⁢ ( ℝ n ) in the standard way:

where 〈 ⋅ , ⋅ 〉 denotes the duality between the spaces 𝒮 ′ ⁢ ( ℝ n ) and 𝒮 ⁢ ( ℝ n ) .

We denote the unit sphere in ℝ n by 𝕊 n - 1 .

Throughout the paper, we use the symbol “ ∼ ” to denote asymptotic expansion that can be differentiated infinitely many times.

2.2 Fundamental solutions

An operator 𝐏 ⁢ ( i ⁢ ∂ x ) is said to be hypoelliptic if any distributional solution of the equation 𝐏 ⁢ ( i ⁢ ∂ x ) ⁢ u = 0 in ℝ n is an infinitely differentiable function (see, e.g., [2, Chapter 4, Section 4.1] or [3, Chapter 11, Section 11.1]). The operator 𝐏 ⁢ ( i ⁢ ∂ x ) is hypoelliptic if and only if the complex roots z = ξ + i ⁢ τ of the characteristic polynomial 𝐏 ⁢ ( z ) satisfy the inequality

with certain constants c 1 > 0 , c 2 > 0 , and 0 < γ ≤ 1 . We consider the following hypoelliptic differential equation:

where

P j ⁢ ( i ⁢ ∂ x ) , j = 1 , … , l , are hypoelliptic differential operators with constant coefficients, and m j ∈ ℕ .

The characteristic polynomial of the hypoelliptic differential operator 𝐏 ⁢ ( i ⁢ ∂ x ) has the following form:

where P j ⁢ ( ξ ) , j = 1 , … , l , are the characteristic polynomials of the differential operators P j ⁢ ( i ⁢ ∂ x ) , j = 1 , … , l .

We assume that the following conditions are satisfied:

1. ∇ ξ ⁢ ∏ j = 1 l P j ⁢ ( ξ ) ≠ 0 at the real zeros of P j ⁢ ( ξ ) , j = 1 , … , l .

2. The set of real zeros of P j ⁢ ( ξ ) consists of infinitely differentiable connected surfaces S j ( 1 ) , … , S j ( ϰ j ) , ϰ j ∈ ℕ , of dimension n - 1 .

3. The total curvature of the surfaces S j ( 1 ) , … , S j ( ϰ j ) , j = 1 , … , l , is not zero at any point.

It follows from (2.1) that the surfaces

are compact, and condition (i) implies that they have no boundary or common points. Condition (iii) implies that the surfaces S j ( 1 ) , … , S j ( ϰ j ) , j = 1 , … , l , are strictly convex (see [4, Chapter 7, Section 5, Theorem 5.6]). A class of such hypoelliptic differential operators with m j = 1 , j = 1 , … , l , was considered in [9, 10].

We will need the following result from [10, Chapter 7, Section 2].

Note that T j ⁢ ( ξ ) = 0 defines the surfaces S j , where

Below, “ p . v . ” will denote integrals in the sense of the Cauchy principal value. In particular,

Let α j ( ν ) ∈ C 0 ∞ ⁢ ( ℝ n ) be such that α j ( ν ) = 1 on S j ( ν ) and supp ⁡ α j ( ν ) ∩ supp ⁡ α k ( υ ) = ∅ if j ≠ k or ν ≠ υ , j , k = 1 , … , l , ν = 1 , … , ϰ j , υ = 1 , … , ϰ k .

Let

be a vector whose components are equal to 1 or -1, i.e., σ j ( ν ) = ± 1 , j = 1 , … , l , ν = 1 , … , ϰ j .

Consider the following distribution:

(2.6)

for all ψ ∈ 𝒮 ⁢ ( ℝ n ) .

It follows from the definition of α j ( ν ) that

Therefore, (2.6) can be rewritten in the form

Taking into account the definition of α j ( ν ) , we obtain

Let β j , 𝔦 ( ν ) , 𝔦 = 1 , … , N j ( ν ) , be a partition of unity in a neighborhood of the surface S j ( ν ) , i.e.,

where { U j , 𝔦 ( ν ) } 𝔦 = 1 N j ( ν ) is an open cover of the surface S j ( ν ) . Then we get

where φ j , 𝔦 ( ν ) ⁢ ( ξ ) := θ j , 𝔦 ( ν ) ⁢ ( ξ ) ⁢ ψ ⁢ ( ξ ) , φ j , 𝔦 ( ν ) ∈ C 0 ∞ ⁢ ( ℝ n ) , 𝔦 = 1 , … , N j ( ν ) , ν = 1 , … , ϰ j , j = 1 , … , l , and

We choose a partition of unity such that, if supp ⁡ β j , 𝔦 ( ν ) ∩ S j ( ν ) ≠ ∅ , then ∂ ξ n 𝔦 ⁡ T j ⁢ ( ξ ) ≠ 0 on supp ⁡ β j , 𝔦 ( ν ) for some n 𝔦 . Then one can introduce the following local coordinates in a neighborhood of supp ⁡ β j , 𝔦 ( ν ) :

where

It follows from the implicit function theorem that (2.8) has a C ∞ smooth inverse

Let