NONNEGATIVE SOLUTIONS OF PARABOLIC OPERATORS WITH LOWER-ORDER TERMS

We develop the harmonic analysis approach for parabolic operator with one order term in the parabolic Kato class on C 1;1 -cylindrical domain . We study the boundary behaviour of nonnegative solutions. Using these results, we prove the integral representation theorem and the existence of nontangential limits on the boundary of for nonnegative solutions. These results extend some rst ones proved for less general parabolic operators.


INTRODUCTION
In this paper we are interested in some aspects of the theory of the differential parabolic operator L = ∂ ∂t − div(A(x, t)∇ x ) + B(x, t).∇ x defined on Ω = D×]0, T [, where D is a bounded C 1,1 -domain of R n and 0 < T < ∞.The matrix A(x, t) is assumed to be real, symmetric, uniformly elliptic, i.e. 1 µ I ≤ A(x, t) ≤ µI for some µ ≥ 1, with Lipschitz coefficients.The vector B(x, t) is assumed to be in the parabolic Kato class as introduced by Zhang in [15], i.e.B ∈ L 1 loc and satisfies lim h→0 N α h (B) = 0, where for some constant α > 0.
In fact, the real starting points of this work are the famous papers [10] of Kemper, [5] of Fabes, Garofalo and Salsa, [9] of Heurteaux and [12] of Nyström.We recall here that, as was initially studied for the Laplace operator by Hunt and Wheeden in [7] and [8], the notion of kernel function, the integral representation theorem and the existence of nontangential limit at the boundary for nonnegative solutions (Fatou's theorem) for the heat equation have been developed by Kemper in [10] on Lipschitz domains.In his work an important role was played by the invariance of the heat equation under translations.The results of Hunt and Wheeden have been later extended to more general elliptic equations by Ancona in [1] and Gaffarelli, Fabes, Mortola and Salsa in [6].In [5], Fabes, Garofalo and Salsa are interested in the same problem for parabolic operators in divergence form with measurable coefficients on Lipschitz cylinders.When they attempted to adapt the techniques of [6] for their case, an interesting difficulty occurs, namely to prove the "doubling" property, which was essential for the proof of Fatou's theorem and which is equivalent to the existence of a "backward" Harnack inequality for nonnegative solutions (we refer the reader to [5] for more details).By proving some boundary Harnack principles for nonnegative solutions, they succeeded in resolving the problem for parabolic operators with time independent coefficients and they established all of Kemper's results in this case.In [9], Heurteaux took up the same problem for parabolic operators in divergence form with Lipschitz coefficients on more general Lipschitz domains, and by a straightforward adaptation of the idea of Ancona [1], he was able to extend the results of Fabes, Garofalo and Salsa to his situation.Recently, Nyström studied in [12] parabolic operators in divergence form with measurable coefficients on Lipschitz domains and he proved among other things, EJQTDE, 2003 No. 12, p. 2 the existence and uniqueness of a kernel function and established the integral representation theorem.
In this paper, our aim is to investigate the above mentioned results for our operator.The main difficulty is created by the lower order term where we cannot benefit from results proved for L having adjoint companions as is the case of operators in divergence form in [5], [9], [10] and [12].To overcome this difficulty our idea is based on the Green function estimates proved by the author in [13] and the Harnack inequality recently proved by Zhang in [15], under the above assumptions.Our method seems to be new and applies to similar parabolic operators and our results include their counterparts for the elliptic operator div(A(x)∇ x )+B(x).∇x with B in the elliptic Kato class, In Section 1, we give some notations and we state some known results that will be used throughout this paper.In Section 2, basing on the Green function estimates (Theorem 2.2, below), we prove a boundary Harnack principle and a comparison theorem for nonnegative L-solutions vanishing on a part of the parabolic boundary ∂ p Ω of Ω.In Section 3, using the previous results and the Harnack inequality (Theorem 2.1, below), we characterize the Martin boundary of the cylinder Ω with respect to the class of parabolic operators L that we deal with.More precisely, we prove that for every point Q ∈ ∂ p Ω there exists a unique (up to a multiplicative constant) minimal nonnegative L-solution, and then the Martin boundary of Ω with respect to L is homomorphic (or identical) to the parabolic boundary ∂ p Ω of Ω.In Section 4, we are able to define the kernel function and prove, basing on the previous results, the integral representation theorem for nonnegative L-solutions on Ω.In particular, we deduce a Fatou type theorem for our operator by proving that any nonnegative L-solution on Ω has a nontangential limit at the boundary except for a set of zero L-parabolic measure.

NOTATIONS AND KNOWN RESULTS
Let G be the L-Green function on Ω = D×]0, T [.We simply denote by For an open set Ω of R n+1 , let ∂ p Ω be the parabolic boundary of Ω, i.e. ∂ p Ω is the set of points on the boundary of Ω which can be connected to some interior point of Ω by a closed curve having a strictly increasing t-coordinate.For an arbitrary set Σ in Ω and a function u on Ω, we denote by R Σ u the nonnegative L-superparabolic envelope of u with respect to Σ which also called the "reduct" of u with respect to Σ, and defined by We next recall some known results that will be used in this work.
Theorem 2.1.(Harnackinequality [15]).Let 0 < α < β < α 1 < β 1 < 1 and δ ∈ (0, 1) be given.Then there are constants C > 0 and r 0 > 0 such that for all (x, s) ∈ R n × R, all positive r < r 0 and all nonnegative weak where All constants depend on B only in terms of the rate of convergence of N α h (B) to zero when h → 0. Theorem 2.2.(Green function estimates [13]).There exist positive constants k, c 1 and c 2 depending only on n, µ, T, D and on B only in terms of the rate of convergence of N α h (B) to zero when h → 0 such that for all x, y ∈ D and 0 < s < t ≤ T , where ϕ(x, y, u EJQTDE, 2003 No. 12, p. 4

BOUNDARY BEHAVIOUR
We prove in this section a boundary Harnack principle and a comparison theorem for nonnegative L-solutions vanishing on a part of the parabolic boundary, which will be used in the next section to characterize the Martin boundary of Ω = D×]0, T [.
By compactness of ∂D, the constants c 0 and r 0 can be chosen independent of z ∈ ∂D.
For Q ∈ R n+1 , r > 0 and h > 0, we denote by T Q (r, h) the cylinder We have the following result.Theorem 3.1(Boundary Harnack principle).Let Q ∈ ∂D×]0, T [, r ∈]0, r 0 ] and λ > 0. Then there exists a constant C > 0 depending only on n, µ, λ, D, T and on B in terms of the rate of convergence of N α h (B) to zero when h → 0 such that for all nonnegative L-solutions u on 2 ), we have for all M ∈ Ω \ T Q (r, λr), where M r = Q + (0, λr, r 2 ).
Proof.Without loss of generality we assume Q = (0, 0, S) ≡ (0, ψ(0), S), where ψ as defined above is the function which, after a suitable rotation, describes ∂D as a graph around (0, 0).In view of the minimum principle, it suffices to prove the theorem for M ∈ Ω ∩ ∂T Q (r, λr).
By Theorem 2.2, we have Using the fact that |λr − From the inequality e −α ≤ ( m αe ) m , for all m > 0, α > 0, it follows that Since M ∈ Ω ∩ ∂T Q (r, λr), we need to study the following three cases: If t = r 2 , 0 < x n ≤ λr, and |x | ≤ r, then If x n = λr, |t| ≤ r 2 , and |x | ≤ r, then Note that the same estimate holds when the pole A lies in Ω ∩ T Q (εr, λεr) with 0 < ε < 1.The constant C then depends also on ε.

EJQTDE, 2003 No. 12, p. 6
For the general case, by considering the set Σ = Ω \ T Q ( 2 3 r, 2 3 λr) we see that the function v = R Σ u is an L-potential on Ω with support in Ω ∩ ∂Σ and then there exists a positive measure µ supported in Ω ∩ ∂Σ such that R Σ u = Ω∩∂Σ G A dµ(A).For all M ∈ Ω \ T Q (r, λr), we have which completes the proof. 2 In the sequel, for λ > 0, we denote by C λ the set We next have the following result.
Proof.Without loss of generality we assume Q = (0, 0, S).We first prove the estimate for u EJQTDE, 2003 No. 12, p. 7 Put and By Theorem 2.2, we have Using the fact that For the general case, by considering the set Σ = Ω \ T Q (r, λr) we see that the functions R Σ u and R Σ v are two L-potentials on Ω with support in ∂T Q (r, λr) EJQTDE, 2003 No. 12, p. 8 and then there exist two positive measures σ and ν supported in ∂T Q (r, λr) From the last inequality we then deduce which means and so the required estimate follows from the equalities R Σ u = u on Σ and

MINIMAL NONNEGATIVE L-SOLUTIONS
In this section we exploit the results of Section 2 to characterize the Martin boundary of Ω.More precisely we show that for every point Q ∈ ∂ p Ω there exists a unique (up to a multiplicative constant) minimal nonnegative Lsolution, and then the Martin boundary is identical to ∂ p Ω. We first introduce the notion of minimal nonnegative L-solution.Definition 4.1.A nonnegative L-solution u on a given domain Ω of R n+1 is called minimal if every L-solution v on Ω satisfying the inequalities 0 ≤ v ≤ u is a constant multiple of u.
In view of a limiting argument given by Lemma 2.1 in [15], we may assume that |B| ∈ L ∞ .We denote by H the set of L-solutions on Ω.We recall that (Ω, H) is a P-Bauer space in the sense of [4] and any minimal nonnegative L-solution is the limit of a sequence of extreme potentials (see [11], Lemma 1.1).Note that an extreme potential is a potential with point support, and by Theorem III in [2] any two potentials in the whole space R n × R with the same point support are proportional.Since the hypothesis of proportionality is satisfied if and only if it is satisfied locally (see [11]  Proof.Case 1: y ∈ ∂D and s > 0.

Consider a sequence (A
where M r 0 = Q + (0, λr 0 , r 2 0 ).By Theorem 3.1, there exists a constant C = C(n, µ, λ, T, B) > 0 such that for all r ∈]0, r 0 ], n ≥ n(r) ∈ N, we have On the other hand by the Harnack inequality (Theorem 2.1), there exists a constant Therefore, for all r ∈]0, r 0 ], n ≥ n(r) ∈ N, we have ϕ An (M ) ≤ CC , for all M ∈ Ω \ T Q (r, λr).This means that (ϕ An ) n is locally uniformly bounded and then it has a subsequence converging to a nonnegative L-solution ϕ on Ω vanishing on ∂Ω\{Q}.To prove that ϕ is minimal, denote by C Q (Ω) the set of all nonnegative Lsolutions on Ω vanishing on ∂Ω \ {Q}.We will show that C Q (Ω) is a half-line generated by a minimal nonnegative L-solution.Using the Harnack inequality and Theorem 3.1 again we see that C Q (Ω) is a convex cone with compact base B = {u ∈ C Q (Ω) : u(M r 0 ) = 1}, and by the Krein-Milman theorem it is generated by the extremal elements of B which are the minimal nonnegative L-solutions.To complete the proof, it suffices to prove that two minimal nonnegative L-solutions in Ω are proportional. Recall ), and this also gives for all M ∈ Ω ∩ (Q + C λ ) ∩ T Q (r 0 , λr 0 ).Using the non-thinness of Ω ∩ (Q + C λ ) at h 1 and h 2 we see that the previous inequality holds on Ω which means h 1 ≤ αh 2 , α ≥ 0, and consequently h 1 , h 2 are proportional.Case 2: y ∈ ∂D and s = 0.By the first case, there exists a minimal nonnegative L-solution h on Ω = D×] − 1, T [ vanishing on ∂ Ω \ {Q}.In view of the minimum principle, h(x, t) = 0 for t < 0. Clearly, the function h ≡ h /Ω is a minimal nonnegative L-solution on Ω. Case 3: y ∈ D and s = 0. Let h(x, t) = G(x, t; y, s).We prove that h is a minimal nonnegative Lsolution on Ω.Let u be a nonnegative L-solution on Ω such that u ≤ h.We define u by u(x, t) = u(x, t) if 0 < t < T 0 if − 1 ≤ t ≤ 0.
Denote by u the lower semi-continuous regularization of u, then u is a nonnegative L-superparabolic function on Ω = D×] − 1, T [ with harmonic support {(y, 0)} and u(x, t) ≤ G(x, t; y, 0), where G is the L-Green function of Ω.It follows that u is an L-potential of Ω with support {(y, 0)} and so there exists C ≥ 0 such that u(x, t) = C G(x, t; y, 0).This gives u(x, t) = C G(x, t; y, 0) = C h(x, t), and then h is minimal.

INTEGRAL REPRESENTATION AND NONTANGENTIAL LIMITS
Following the characterization of the Martin boundary in Section 3, we are now able to define the kernel function associated to our operator and the cylinder Ω.Let Q 0 = (x 0 , t 0 ) be a given point in Ω.
will be also denoted by (x , x n ) with x ∈ R n−1 and x n ∈ R, when we need.EJQTDE, 2003 No. 12, p. 3 For x ∈ D, let d(x) denotes the distance from x to the boundary ∂D of D.

Theorem 3 . 2 ( 4 ]
Comparison theorem).Let Q ∈ ∂D×]0, T [, λ > 0, and for ρ > 0 denote M ρ = Q + (0, λρ, ρ 2 ).Then there exists a constant C > 0 depending only on n, µ, λ, D, T and on B in terms of the rate of convergence of N α h (B) to zero when h → 0 such that for all r ∈]0, r 0 and for any two nonnegative L-solutions u, v on Ω\T Q (r, λr) continuously vanishing on ∂ p Ω\ T Q (r, λr), we have u Lemma 1.3), this property holds in Ω.It follows that every minimal nonnegative L-solution is the limit of a sequence c k G(x, t; y k , s k ) for some sequence of poles (y k , s k ) ⊂ Ω and constantsc k ∈ R + .By compactness of Ω, it is clear that if h(x, t) = lim k→+∞ c k G(x, t; y k , s k ) is a minimal nonnegative L-solution,then there exists a subsequence of ((y k , s k )) k which converges to a point (y, s) ∈ ∂ p Ω.The reverse of this result constitutes the object of the following theorem.Theorem 4.2.For each point Q = (y, s) ∈ ∂ p Ω, there exist sequences ((y k , s k )) k convergent to Q and (c k ) k in R + such that the function h(x, t) = lim k→+∞ c k G(x, t; y k , s k ) is a minimal nonnegative L-solution.
[9]t if h is a minimal nonnegative L-solution in Ω and E ⊂ Ω, then RE h = h or RE h is an L-potential of Ω, where RE h is the lower semi-continuous regularization of R E h .We say E is thin at h if REh is an L-potential of Ω.Using Theorem 3.1 and Theorem 3.2 we prove as in[9](Proposition 4.2) that Ω ∩ (Q + C λ ) is not thin at h. EJQTDE, 2003 No. 12, p. 10 Let h 1 , h 2 two minimal nonnegative L-solutions of C Q (Ω).By Theorem 3.2, there exists C