GAUSSIAN ESTIMATES FOR FUNDAMENTAL SOLUTIONS TO CERTAIN PARABOLIC SYSTEMS

Abstract Auscher proved Gaussian upper bound estimates for the fundamental solutions to parabolic equations with complex coefficients in the case when coefficients are time-independent and a small perturbation of real coefficients. We prove the equivalence between the local boundedness property of solutions to a parabolic system and a Gaussian upper bound for its fundamental matrix. As a consequence, we extend Auscher’s result to the time dependent case.

Here, we used the notation We also assume that there is a number M > 0 such that (1.5) In this article, we study fundamental solutions of the systems of equations (1.2).By a fundamental solution (or a fundamental matrix) Γ(x, t; y, s) to the system (1.2) we mean an N × N matrix of functions defined for t > s which, as a function of (x, t), satisfies (1.2) (i.e., each column is a solution of (1.2)), and is such that (1.6) lim t↓s R n Γ(x, t; y, s)f (y) dy = f (x) for any bounded continuous function f = (f 1 , . . ., f N ) T .
The adjoint system of (1.2) is given by where * A αβ = (A βα ) T , i.e., the transpose of A βα .Note that For X = (x, t) ∈ R n+1 and r > 0 we denote Definition 1.1.We say that the operators L and L * satisfy the local boundedness properties if there exists a constant B independent of r and Y such that for any u satisfying the system (1.2) in Q 2r (Y ) and any v satisfying the adjoint system (1.7) in Q * 2r (Y ) the following local boundedness properties hold: sup In the next theorem, we make the qualitative assumption that the coefficients A αβ ij (x, t) are smooth; however we emphasize that all qualitative estimates are only allowed to depend on the dimension n, parabolicity constants ν, M and the local boundedness constant B as appears in Definition 1.1.
Proof: We follow a technique as appears in [5], which is based on a method introduced by E. B. Davies in [3], [4].Let ψ be a Lipschitz function such that |∇ψ| ≤ γ with γ ≥ 0 to be determined later.Let S N (R n ) denote a function space whose elements are N × 1 column vectors of functions from Schwartz test function space.Define an operator P ψ s→t on S N (R n ) for t > s by setting Since lim t↓s P ψ s→t f (x) = f (x), the above differential inequality implies (1.15) The adjoint of P ψ s→t is given by (1.16) (P ψ s→t ) * g(y) = exp(−ψ(y)) R n Γ * (y, s; x, t) exp(ψ(x))g(x) dx, where (1.17) Γ * (y, s; x, t) = Γ(x, t; y, s) T .
A similar computation shows In particular, by setting ψ ≡ 0 so that γ = 0, we have . By setting u(x, t) := P 0 s→t f (x) and using property (1.11) with Y = (x, t) and r = √ t − s/2 we obtain Here, ω n denotes the volume of unit ball and similarly for ( In the case when γ > 0, i.e., ψ ≡ const., we set u(x, t) := e −ψ(x) P ψ s→t f (x) and carry out a similar estimate to get Hence Hence, the following L 2 → L ∞ estimate for P ψ s→t follows for γ > 0. (1.23) where N 1 = B 2 /2ω n κ.By a similar argument, we also have By duality, (1.22) and (1.24) imply L 1 → L 2 estimates On the other hand, (1.28) gives an upper bound for |Γ(x, t; y, s)| independent of x and y, Combining (1.32) and (1.33) together, Now set k 0 = 1/8κ = M 2 /8ν and and Then, we have the desired estimate (1.13).The proof is complete.
, then the local boundedness property (1.12) for L * follows from the local boundedness property (1.11) for L.
To see this, let v be a solution to ) and thus satisfies (1.11) by the assumption.Therefore v satisfies (1.12).
The following theorem is the converse of Theorem 1.1.Here, we also make the qualitative assumption that the coefficients A αβ ij (x, t) are smooth.
Note that we have localized energy inequality (see e.g.[11]) where N 0 = N 0 (ν, M ).Using this we estimate . (1.38) Note that the transpose of Γ(x, t; y, s) is Γ * (y, s; x, t), the fundamental solution to the adjoint system.
Note that lim s↑t η 2 |w| 2 (•, s) ≡ 0 by the choice of η and by (1.13).Therefore, as it is done in the proof of localized energy estimates we have Therefore, by the equivalence of norms in finite dimensional vector spaces, (1.41) and hence together with (1.39), we get . (1.43) We claim that the following estimate holds. (1.44) Then the estimate (1.35) follows from (1.43).
It remains to prove the claim (1.44).In the view of (1.13) and the definition Q = Q 5R (X) \ Q R/2 (X), it suffices to show the following estimates: Using Fubini's theorem and change of variable, A straightforward computation will show that the integral in (1.45) is bounded by A 0 R 2−n for some A 0 = A 0 (n, k 0 , K 0 ).For example, if n ≥ 3, we estimate Similarly, by a change of variable, we estimate Again, a straightforward calculation yields that For example, in the case when n ≥ 3, The proof is complete.
(2) For any p > 0, there is a constant c(p) such that for all ρ < R.

Applications
In this section, we will give examples of parabolic systems satisfying the local boundedness properties and thus satisfying the Gaussian upper bound estimates.For example, "almost diagonal" parabolic systems belong to such classes.This fact can be deduced by using a well-known technique due to Campanato.(see e.g.[7]).We will provide a proof for the sake of completeness.
Let us first recall some well-known facts.The following lemma is N. G. Meyers' integral characterization of Hölder continuous functions.See e.g.[13] and [12,Lemma 4.3,p. 50] for the proof.
Lemma 2.1.Let u ∈ L 2 (Q 2R ) and suppose there are positive constants µ ≤ 1 and H such that for any X ∈ Q R and any r ∈ (0, R).Here u X,r denotes the average of u over Q r (X).Then u is Hölder continuous with the exponent µ in Q R and [u] µ,µ/2;QR ≤ N (n, µ)H.
A proof of the following lemma can be found in [15], [9].
Lemma 2.3.There exists a constant P 0 = P 0 (n, ν, M ) such that for any solution u to the system Here, u R denotes the average of u over Q R .
Proposition 2.1.Let a(x, t) = [a αβ (x, t)] be a real n × n matrix satisfying uniform parabolicity condition There exists δ 0 = δ 0 (n, λ, Λ) such that if then local boundedness property holds for solutions to the system (1.2) associated with A αβ ij ; i.e., there is a number δ 0 such that if A αβ ij satisfies (2.3), then for any solution u to (1.1) , for some B = B(n, λ, Λ).
Proof: Note that the conditions (2.2) and (2.3) are invariant under the change of variables x → (x − x 0 )/R, t → (t − t 0 )/R 2 .Thus, we may and do assume that R = 1 and X 0 = 0. Let u be a solution to (1.2) in We apply Moser-Nash theory to each v i so that we have interior Hölder estimates, which, by a well-known argument, are equivalent to the following: (2.5) for some constants C 0 (n, λ, Λ) and γ > 0. The point is that this Dirichlet integral characterization of Hölder continuity is stable under perturbation, as we shall demonstrate.We note that Auscher also exploits the stability of (2.5) (or to be more precise, of its elliptic analogue) in his work on Gaussian bounds [1].Note that w := u − v satisfies (2.6) where Ãαβ := A αβ − a αβ I.
Since w = 0 on ∂ p Q r (X), we may use w itself as a test function to (2.6) and get (2.7) Therefore, we have (2.9)Let Γ be the fundamental matrix to the parabolic system (1.1) whose coefficients A αβ ij defined as in (2.19).By Proposition 2.1, Remark 2.1 and then by Theorem 1.1 we conclude that Γ has an upper bound (1.13) with constants K 0 , k 0 depending only on n, λ, Λ.
Observe that u is a solution to L a u = 0 if and only if u = ( u, u) is a solution to (1.1) with A αβ ij defined as in (2.19).Also, note that Γ is given by (2.23) Therefore, we have |Γ| = |Γ| op = |Γ| and thus |Γ| has an upper bound (1.13) with constants K 0 , k 0 depending only on n, λ, Λ.Note that in the proof of Proposition 2.1 we derived a uniform a-priori bound for Hölder norm for the solutions of L a u = 0. We also derived a uniform a-priori upper bound for the fundamental solution.Therefore, by a standard approximation argument (see e.g.[10, Chapter 2]), we are allowed to discard the extra smoothness assumptions on a(x, t).The proof is complete.