Scott N Armstrong
email: armstrong@math.lsu.edu

AND Boyan Sirakov
email: sirakov@ehess.fr

Charles K Smart
email: smart@math.berkeley.edu

FUNDAMENTAL SOLUTIONS OF HOMOGENEOUS FULLY NONLINEAR ELLIPTIC EQUATIONS

Keywords: 2000 Mathematics Subject Classification. Primary 35A08, 35J60, 91A15, 49N70, 35P30 fully nonlinear elliptic equation, Bellman-Isaacs equation, fundamental solution, isolated singularity, Liouville theorem, stochastic differential game

We prove the existence of two fundamental solutions Φ and Φ of the PDE F (D 2 Φ) = 0 in R n \ {0} for any positively homogeneous, uniformly elliptic operator F . Corresponding to F are two unique scaling exponents α * , α * > -1 which describe the homogeneity of Φ and Φ. We give a sharp characterization of the isolated singularities and the behavior at infinity of a solution of the equation F (D 2 u) = 0, which is bounded on one side. A Liouville-type result demonstrates that the two fundamental solutions are the unique nontrivial solutions of F (D 2 u) = 0 in R n \{0} which are bounded on one side in both a neighborhood of the origin as well as at infinity. Finally, we show that the sign of each scaling exponent is related to the recurrence or transience of a stochastic process for a two-player differential game.

Introduction and main results

The following fundamental result was proved by M. Bôcher in 1903.

Theorem 1 (Bôcher, [START_REF] Bôcher | Singular points of functions which satisfy partial differential equations of the elliptic type[END_REF]). Denote B r = {x ∈ R n : |x| < r}, and assume n ≥ 2.

(i) Suppose u ∈ C(B 1 \ {0}) is harmonic and bounded below or above in B 1 \ {0}. Then either u can be extended to a harmonic function in B 1 , or there exist constants a = 0 and C > 0 such that

aΦ -C ≤ u ≤ aΦ + C in B 1/2 \ {0}, where Φ(x) = |x| 2-n if n > 2 -log |x| if n = 2.
Hence, by the linearity of the Laplacian, u -aΦ can be extended to a harmonic function in B 1 .

(ii) Suppose u is harmonic and bounded below or above in R n \ B 1 , n ≥ 3. Then u(x) → a 0 as |x| → ∞, for some a 0 ∈ R.

The function Φ which appears in this theorem is known as the fundamental solution for the Laplacian. Bôcher's result easily implies the following extended Liouville theorem.

Theorem 2 (Bôcher, [START_REF] Bôcher | Singular points of functions which satisfy partial differential equations of the elliptic type[END_REF]). The set of all harmonic in R n \ {0} functions that are bounded from above or from below in a neighbourhood of zero as well as in a neighbourhood of infinity is in the form {aΦ + b | a, b ∈ R}.

In this article we construct fundamental solutions of the fully nonlinear equation (1.1) F (D 2 u) = 0, and extend Theorem 1 to solutions of (1.1), under the assumption that F is a uniformly elliptic and positively homogeneous operator. Here equation (1.1) is understood in the viscosity sense (c.f. [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF][START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF]). We recall that, in general, the best regularity available for a solution u of equation (1.1) is u ∈ C 1,γ loc , for some constant γ > 0 which depends on the operator F (see [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF][START_REF] Trudinger | On regularity and existence of viscosity solutions of nonlinear second order, elliptic equations[END_REF][START_REF] Nadirashvili | Singular viscosity solutions to fully nonlinear elliptic equations[END_REF]).

Before proceeding to the precise statements of our results, let us give some additional context. An extension of Theorem 1 to some linear equations appeared already in [START_REF] Bôcher | Singular points of functions which satisfy partial differential equations of the elliptic type[END_REF], while a thorough study of fundamental solutions and isolated singularities of linear equations, in view of more modern theories, was performed by Gilbarg and Serrin [START_REF] Gilbarg | On isolated singularities of solutions of second order elliptic differential equations[END_REF]. Later, in a sequence of papers, Serrin [START_REF] Serrin | Local behavior of solutions of quasi-linear equations[END_REF][START_REF] Serrin | Isolated singularities of solutions of quasi-linear equations[END_REF][START_REF] Serrin | Removable singularities of solutions of elliptic equations[END_REF] produced a deep study of singular solutions of general quasilinear divergence-form equations -div A(x, u, Du) = B(x, u, Du), p-harmonic functions being the model case. We refer to [START_REF] Véron | Singularities of solutions of second order quasilinear equations[END_REF][START_REF] Nicolosi | Precise point-wise growth conditions for removable isolated singularities[END_REF] for more developments and references on solutions of quasilinear equations. We also refer to [START_REF] Miranda | Partial differential equations of elliptic type[END_REF] for more on the existence of fundamental solutions of linear and quasilinear equations.

In recent years, there have been a number of studies of singular solutions of the fully nonlinear equation (1.1), in the particular case when F is a rotationally invariant (Hessian) operator, that is, F (D 2 u) depends only on the eigenvalues of D 2 u. The work most closely related to ours is the one by Labutin [START_REF] Labutin | Isolated singularities for fully nonlinear elliptic equations[END_REF], who gave, among other things, a partial extension of Bocher's theorem to solutions of Pucci extremal equations. Below we discuss in more detail that paper and the additional hypotheses it involved. We also note that in the last several years there has been a great amount of interest of singular solutions of conformally invariant fully nonlinear equations (we refer to [START_REF] Li | Some Liouville theorems and applications[END_REF][START_REF] Li | Conformally invariant fully nonlinear elliptic equations and isolated singularities[END_REF][START_REF] Caffarelli | Some remarks on singular solutions of nonlinear elliptic equations[END_REF] and the references in these works).

The essence of all results on isolated singularities is that if a function fails to be a solution at an isolated point and is bounded on one side in a neighbourhood of this point, then u behaves like a fundamental solution of the elliptic operator near the isolated point. In the literature the term fundamental solution usually refers to a solution in R n (or in some domain of interest) except at zero, which goes to infinity at the origin and is bounded away from it. We are going to use this term also for solutions in R n \ {0} that have the inverse behavior, that is, are bounded on bounded sets and tend to infinity at infinity. For instance, the fundamental solution for the p-Laplace equation is Φ p (x) = |x| (p-n)/(p-1) if p = n. In [START_REF] Serrin | Local behavior of solutions of quasi-linear equations[END_REF][START_REF] Serrin | Isolated singularities of solutions of quasi-linear equations[END_REF] only the case p < n was considered, but as remarked for instance in [START_REF] Kichenassamy | Singular solutions of the p-Laplace equation[END_REF], similar asymptotics as in Theorem 1(i) hold if p > n.

Let us now state our main results. We consider an arbitrary Isaacs operator, that is, a nonlinear map F from the set S n of n-by-n real symmetric matrices into R, with the following two properties.

(H1) F is uniformly elliptic and Lipschitz: for some constants 0 < λ ≤ Λ and all real symmetric matrices M and N , with N nonnegative definite, we have

λ trace(N ) ≤ F (M -N ) -F (M ) ≤ Λ trace(N ).
(H2) F is positively homogeneous of degree 1:

F (tM ) = tF (M ) for each t ≥ 0 and each symmetric M.

We emphasize that (H1) and (H2) will be the only hypotheses on F . In particular, we assume neither that F is convex or concave, nor that F is rotationally invariant.

In can be shown that (H1)-(H2) are equivalent to

(1.2) F (D 2 u) = sup α∈A inf β∈B -a α,β ij ∂ ij u or F (D 2 u) = inf α∈A sup β∈B -a α,β ij ∂ ij u
for index sets A, B and symmetric matrices A α,β = (a α,β ij ), with λI ≤ A α,β ≤ ΛI. In the following theorem we establish the existence and the main properties of the fundamental solution.

Theorem 3. There exists a non-constant solution of (1.1) in R n \ {0} that is bounded below in B 1 and bounded above in R n \ B 1 . Moreover, the set of all such solutions is of the form {aΦ + b | a > 0, b ∈ R}, where Φ ∈ C 1,γ loc (R n \ {0}) can be chosen to satisfy one of the following homogeneity relations: for all t > 0

(1.3) Φ(x) = Φ(tx) + log t or Φ(x) = t α * Φ(tx) α * Φ(x) > 0 in R n \ {0},
for some number α * ∈ (-1, ∞) \ {0} which depends only on F and n.

Definition 1.1. We call the number α * = α * (F ) the scaling exponent of F , and we set α * (F ) = 0 in the case the first alternative in (1.3) occurs.

Definition 1.2. We call the function Φ whose existence is asserted in Theorem 3, and normalized so that (1.4) min

∂B1

(sign(α * )Φ) = 1 if α * = 0, and

∂B1 Φ = 0 if α * = 0,
the upward-pointing fundamental solution of (1.1).

Remark 1.3. By (1.3) and (1.4), the upward-pointing fundamental solution is strictly decreasing in the radial direction, and we have

lim |x|→0 Φ(x) = ∞ if α * (F ) ≥ 0, and lim |x|→∞ Φ(x) = 0 if α * (F ) > 0, lim |x|→0 Φ(x) = 0 if α * (F ) < 0, and lim |x|→∞ Φ(x) = -∞ if α * (F ) ≤ 0.
For any F satisfying (H1)-(H2) we denote the dual operator F of F by F (M ) := -F (-M ).

Notice that the two operators appearing in (1.2) are dual in this sense, as are the Pucci extremal operators. By Theorem 3, the operator F has an upward-pointing fundamental solution which we denote by Φ. It follows that the function -Φ is another solution of F (D 2 u) = 0 in R n \ {0}, and we call it the downward-pointing fundamental solution of F . The following result shows Φ and -Φ are the only fundamental solutions of (1.1), and extends Theorem 2 to fully nonlinear operators.

Theorem 4. Suppose that u ∈ C(R n \ {0}) is a solution of the equation F D 2 u = 0 in R n \ {0}.
Suppose further that u is bounded from above or below in B 1 \ {0}, and that u is bounded from above or below in R n \ B 1 . Then either u ≡ b, or u ≡ aΦ + b, or u ≡ -a Φ + b for some a > 0, b ∈ R.

Prior to this paper, the existence of fundamental solutions of fully nonlinear equations was known only for certain rotationally invariant operators, when a direct calculation verifies that the function ξ α defined by

(1.5) ξ α (x) :=    |x| -α if α > 0, -log |x| if α = 0, -|x| -α if α < 0,
satisfies the equation, for some α > -1 (see [START_REF] Cutrì | On the Liouville property for fully nonlinear equations[END_REF][START_REF] Labutin | Isolated singularities for fully nonlinear elliptic equations[END_REF][START_REF] Felmer | Fundamental solutions and two properties of elliptic maximal and minimal operators[END_REF]). For example, a direct computation shows that the upward-pointing fundamental solutions of the Pucci extremal operators P - λ,Λ and P + λ,Λ are ξ λ(n-1)/Λ-1 and ξ Λ(n-1)/λ-1 , respectively. It follows from the uniqueness result above that the upward-pointing fundamental solution of a rotationally invariant operator must be radial, that is, ξ α for some α > -1.

It should be noted that, by well-known results from the theory of linear elliptic PDE, if Φ is the fundamental solution of a linear operator L, then LΦ is interpreted as the Dirac mass at the origin. For fully nonlinear operators this is not true, as noticed by Labutin [START_REF] Labutin | Isolated singularities for fully nonlinear elliptic equations[END_REF] who showed that if λ = Λ, then P + λ,Λ ξ Λ(n-1)/λ-1 vanishes near the origin in a reasonable weak sense.

The Liouville-type Theorem 4 is, to our knowledge, the first result of its kind for fully nonlinear operators, and in particular is new even for the Pucci extremal operators. Of course, even in the case F (D 2 u) = -∆u we cannot relax the hypothesis that u be bounded on one side in both a neighborhood of the origin and a neighborhood of infinity. This we recall by considering the function u

(x) = x 2 1 -x 2 2 and its Kelvin transform v(x) = |x| -n-2 (x 2 1 -x 2 2
), both of which are harmonic in R n \ {0}. These functions also show that we may not relax the hypotheses on the solution in our theorems classifying isolated singularities below.

Remark 1.4. Informally, the scaling exponents α * (F ) and α * ( F ) characterize the intrinsic internal scalings of the operator F , and we think of each scaling exponent as a kind of principal eigenvalue of a certain elliptic equation on the unit sphere. Indeed, several of the ideas we employ in our proof of Theorem 3 are related to the principal eigenvalue theory for fully nonlinear operators developed in [START_REF] Lions | Bifurcation and optimal stochastic control[END_REF][START_REF] Birindelli | First eigenvalue and maximum principle for fully nonlinear singular operators[END_REF][START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF][START_REF] Scott | Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations[END_REF]. As we will see, α * (F ) is given by

α * (F ) = sup {α ∈ (-1, ∞) \ {0} : there exists an (-α)-homogeneous supersolution of F (D 2 v) ≥ 0 such that αv > 0 in R n \ {0} ,
and satisfies

-1 < λ Λ (n -1) -1 ≤ α * (F ) ≤ Λ λ (n -1) -1.
Of course, if F is a linear operator, then α * (F ) = n-2. In Section 4 we discuss more properties of the scaling exponents. Furthermore, in Section 6 we will see that α * (F ) has an interesting stochastic interpretation. Corresponding to the Isaacs operator is a diffusion process controlled by two competing players. The sign of α * (F ) indicates whether the first player can force the diffusion to return to the origin, or whether the second player can force the process out to infinity (almost surely). In the case α * (F ) = 0, the diffusion is recurrent (that is, it returns infinitely many times to every neighborhood of the origin almost surely) but returns to the origin with zero probability. These facts explain why operators whose scaling exponents have different signs behave differently, and generalizes the well-known fact that Brownian motion is recurrent in dimensions n = 1, 2, and transient in dimensions n ≥ 3.

In order to prove Theorem 3 we first define α * (F ) according to the formula in Remark 1.4, and show that F satisfies a maximum principle with respect to (-α)-homogeneous functions, for α < α * (F ). The rest of our argument is quite different for the cases α * (F ) > 0 and α * (F ) ≤ 0. In the former case, we use a construction based on comparison principle and the Perron method, which permit to us to solve an associated Dirichlet problem on spaces of homogeneous functions, see Lemma 3.8. On the other hand, when α * (F ) ≤ 0 we appeal to an abstract topological fixed point theorem, the Leray-Schauder alternative, which helps us to build approximate fundamental solutions -Lemma 3.12.

As an application of Theorem 3, we are able to completely characterize the isolated singularities of solutions of F (D 2 u) = 0 that are bounded on one side in a neighborhood of the singularity. For brevity we introduce the following notation:

u ∼ 0 v if u(x) v(x) → a as x → 0 ; u ≈ 0 v if av -C ≤ u ≤ av + C in B 1/2 \ {0},
for some a, C > 0, and similarly if in these formulas 0 is replaced by ∞ and

B 1/2 \{0} is replaced by R n \ B 2 . When we write u ∼ 0 v (resp. u ≈ 0 v; u ∼ ∞ v; u ≈ ∞ v), it is to be understood that u, v -→ x→0 0 (resp. u, v -→ x→0 ∞; u, v -→ |x|→∞ 0; u, v -→ |x|→∞ -∞). Theorem 5. Suppose u ∈ C(B 1 \ {0}
) is a viscosity solution of the equation

(1.6) F (D 2 u) = 0 in B 1 \ {0}.
If u is bounded above or below in a neighborhood of the origin, then precisely one of the following five alternatives holds : (i) the singularity is removable, that is, u can be defined at the origin so that u ∈ C 1,γ (B 1 ) and F (D 2 u) = 0 in B 1 ; (ii) α * (F ) ≥ 0, and u ≈ 0 Φ;

(iii) α * ( F ) ≥ 0, and u ≈ 0 -Φ;

(iv) α * (F ) < 0, u can be defined at the origin so that (u(x) -u(0)) ∼ 0 Φ(x);

(v) α * ( F ) < 0, u can be defined at the origin so that (u(x) -u(0))

∼ 0 -Φ(x).
This theorem generalizes a result of Labutin [START_REF] Labutin | Isolated singularities for fully nonlinear elliptic equations[END_REF], who proved (i)-(iii) above under the supplementary assumptions that F is rotationally invariant and there exist (fundamental) solutions u and v of F (D 2 u) = 0 in R n \ {0} such that u(x) → ∞ and v(x) → -∞ as |x| → 0. In light of our results, this latter assumption is equivalent to α * (F ) ≥ 0 and α * ( F ) ≥ 0. Alternatives (iv) and (v) in Theorem 5 are new even for the Pucci extremal operators.

Our next result is an analogue of Theorem 5 for solutions of F (D 2 u) = 0 near infinity. Since we do not have a Kelvin transform available, this is not simply a corollary of Theorem 5, although the arguments are very similar. Theorem 6. Suppose u ∈ C(R n \ B 1 ) is a viscosity solution of the equation

(1.7) F (D 2 u) = 0 in R n \ B1 .
If u is bounded above or below in R n \ B 1 , then precisely one of the following five alternatives holds :

(i) u ∞ = lim |x|→∞ u(x) exists, and min ∂Br u ≤ u ∞ ≤ max ∂Br u, for all r > 1;

(ii) α * (F ) > 0, u ∞ := lim |x|→∞ u(x) exists, and

(u(x) -u ∞ ) ∼ ∞ Φ(x); (iii) α * ( F ) > 0, u ∞ := lim |x|→∞ u(x) exists, and (u(x) -u ∞ ) ∼ ∞ -Φ(x);
(iv) α * (F ) ≤ 0, and u ≈ ∞ Φ;

(v) α * ( F ) ≤ 0, and u ≈ ∞ -Φ.

We expect the scaling exponents α * (F ) and α * ( F ) to govern many properties of equations which involve the operator F . In addition to the behavior of the fundamental solutions and isolated singularities of F (D 2 u) = 0, we have described another such property in Remark 1.4, and we do not doubt many others are to come. For instance, the results and techniques in this paper can be used to generalize and sharpen several theorems concerning the removability of singularities, critical exponents and Liouville type results for equations like F (D 2 u) ± u p = 0. We refer in particular to results obtained by Cutrì and Leoni [10, Theorems 3. This paper is organized as follows. In the next section, we give some preliminary definitions and recall some standard results for fully nonlinear equations which we use in our arguments. In Section 3 we study the scaling number α * (F ) and construct fundamental solutions of (1.1), establishing the existence part of Theorem 3. The uniqueness part of Theorem 3 is a consequence of Theorem 4, and is postponed to the end of Section 5. In Section 4 we discuss some examples. We study the singularities of solutions of (1.1) and prove our main results in Section 5. Finally, in Section 6 we show that the scaling exponent α * (F ) is related to the behavior of a certain controlled stochastic process.

Preliminaries

We begin by introducing some notation. The set of n-by-n real symmetric matrices is denoted by S n , and I n is the identity matrix. If M, N ∈ S n , then we write M ≥ N if M -N is nonnegative definite. If x, y ∈ R n , we denote by x ⊗ y the symmetric matrix with entries 1 2 (x i y j + x j y i ). If U is a matrix, then the transpose of U is written U t .

For 0 < λ ≤ Λ we define the operators

P + λ,Λ (M ) := sup A∈ λ,Λ
[-trace(AM )] and

P - λ,Λ (M ) := inf A∈ λ,Λ [-trace(AM )] ,
for M ∈ S n , and where λ, Λ ⊆ S n is the subset of S n consisting of A for which λI n ≤ A ≤ ΛI n . The nonlinear operators P + λ,Λ and P - λ,Λ are called the Pucci maximal and minimal operators, respectively. For ease of notation, we will often drop the subscripts and simply write P + and P -. A convenient way to write the Pucci extremal operators is

(2.1) P + (M ) = -λ µj >0 µ j -Λ µj <0 µ j and P -(M ) = -Λ µj >0 µ j -λ µj <0 µ j ,
where µ 1 , . . . , µ n are the eigenvalues of M .

In this article, we require our nonlinear operator F : S n → R to be uniformly elliptic in the sense that (H1) there exist 0 < λ ≤ Λ such that for every M, N ∈ S n ,

P - λ,Λ (M -N ) ≤ F (M ) -F (N ) ≤ P + λ,Λ (M -N )
, and positively homogeneous of order one: (H2) For all M ∈ S n and t ≥ 0, F (tM ) = tF (M ).

Notice that we have written (H1) in a different but equivalent way from how it appeared in the introduction. We remark that (H1) and (H2) are satisfied for both F = P -and F = P + , and that these hypotheses imply P -(M ) ≤ F (M ) ≤ P + (M ).

Every differential equation and differential inequality in this paper is assumed to be satisfied in the viscosity sense, which is the appropriate notion of weak solution for elliptic equations in nondivergence form. For basic definitions as well as a nice introduction to the theory of viscosity solutions of elliptic equations, we refer to [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF] and [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF]. The survey [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF] is a complete and deep account of the early theory of viscosity solutions, while the book [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF] describes the more recent regularity theory, made possible by the breakthrough article [START_REF] Caffarelli | Interior a priori estimates for solutions of fully nonlinear equations[END_REF].

To simplify the reader's task, we mention some standard results which will be used in this article. In what follows we suppose that Ω is an open subset of R n , the operator F satisfies (H1), f ∈ C(Ω), and u satisfies the differential inequalities Theorem 4.3]). Suppose in addition u ≥ 0. Then for each compact subsets

P -(D 2 u) ≤ |f | and P + (D 2 u) ≥ -|f | in Ω. • Strong maximum principle ([7, Proposition 4.9]). Suppose that u, v ∈ C( Ω) satisfy F (D 2 w) ≤ f ≤ F (D 2 v) in Ω and u ≤ v in Ω. If u(x 0 ) = v(x 0 ) at some point x 0 ∈ Ω, then u ≡ v in Ω. • Harnack inequality ([7,
K 1 ⊂ K 2 ⊆ Ω, there is a constant C depending on n, Λ, λ, K 1 , K 2 , Ω, such that sup K1 u ≤ C inf K1 u + f L n (K2) .
• Local C 1,γ estimates ([7, Theorem 8.3]). For each compact subsets K 1 ⊂ K 2 ⊆ Ω, and each p > n, there exist constants 0 < γ < 1 and C depending on n, p, Λ, λ, K 1 , K 2 , Ω, such that

u C 1,γ (K1) ≤ C v L ∞ (K2) + f L p (K2) .
• Stability under uniform convergence ([7, Proposition 2.9]). Suppose that

u k , f k ∈ C(Ω) are such that F (D 2 u k ) ≤ f k in Ω for each k ≥ 1. Also assume that u k → u and f k → f locally uniformly in Ω. Then u satisfies the inequality F (D 2 u) ≤ f in Ω. • Transitivity of inequalities in the viscosity sense ([7, Theorem 5.3]). Sup- pose that G, H are nonlinear operators satisfying (H1) such that F (M ) + G(N ) ≥ H(M + N ). Suppose also that F (D 2 u) ≤ f and v, g ∈ C(Ω) are such that G(D 2 v) ≤ g. Then the function w := u + v satisfies H(D 2 w) ≤ f + g. •
The supremum of a family of subsolutions is a subsolution ([7, Proposition 2.7]). Likewise, the infimum of a family of supersolutions is a supersolution.

For the rest of this article, we assume that the dimension n of our space is at least 2. For each α ∈ R we define the radial function ξ α ∈ C ∞ (R n \ {0}) by (1.5).

Notice that we have chosen the signs in the definition of ξ α to ensure continuity in the following sense

(2.2) ξ α -1 α -→ α 0 ξ 0 , ξ α + 1 -α -→ α 0 ξ 0 ,
meant to exploit the fact that if u is a solution of (1.1) then so is au + b, for each a > 0, b ∈ R.

For each α ∈ R and all σ > 0, we define the rescaling operator

T α σ : C(R n \{0}) → C(R n \ {0}) by (2.3) (T α σ u)(x) := σ α u(σx) if α = 0, u(σx) + log(σ) if α = 0.
Notice that the function ξ α is invariant under the rescaling operator T α σ , that is,

T α σ ξ α = ξ α for every σ > 0. For each α ∈ [-1, ∞) \ {0}
, we define the following spaces of homogeneous functions

H α := {v ∈ C(R n \ {0}) : αv ≥ 0, T α σ v = v for every σ > 0}, H + α := {v ∈ C(R n \ {0}) : αv > 0, T α σ v = v for every σ > 0},
and for α = 0 we set

H 0 := H + 0 := {v ∈ C(R n \ {0}) : T 0 σ v = v for every σ > 0}. We define the scaling exponent of F by (2.4) α * (F ) := sup α ∈ (-1, ∞) \ {0} : there exists v ∈ H + α such that F (D 2 v) ≥ 0 in R n \ {0} .
In order to estimate α * (F ), let us calculate (2.5)

D 2 ξ α = |α|(α + 2)|x| -α-4 x ⊗ x -|α||x| -α-2 I n if α = 0, 2|x| -4 x ⊗ x -|x| -2 I n if α = 0.
Observe that for α = 0, the eigenvalues of D 2 ξ α (x) are |α|(α + 1)|x| -α-2 with multiplicity one, and -|α||x| -α-2 with multiplicity n-1. Similarly, the eigenvalues of D 2 ξ 0 are |x| -2 with multiplicity one and -|x| -2 with multiplicity n -1. Thus inserting D 2 ξ α (x) into the Pucci extremal operators, we discover that (2.6)

P -D 2 ξ α = |α||x| -α-2 (λ(n -1) -Λ(α + 1)) if α = 0, |x| -2 (λ(n -1) -Λ) if α = 0,

and

(2.7)

P + D 2 ξ α = |α||x| -α-2 (Λ(n -1) -λ(α + 1)) if α = 0, |x| -2 (Λ(n -1) -λ) if α = 0.
In particular, we see that

(2.8) sign P -(D 2 ξ α ) = sign λ Λ (n -1) -1 -α , (2.9) sign P + (D 2 ξ α ) = sign Λ λ (n -1) -1 -α .
Since F ≥ P -, it immediately follows from (2.8) and the definition of α * (F ) that

α * (F ) ≥ λ Λ (n -1) -1 > -1.
We postpone demonstrating an upper bound for α * (F ), since for this we need a result in the next section (see Corollary 3.6).

Existence of fundamental solutions

In this section we construct fundamental solutions of the equation

(3.1) F (D 2 u) = 0 in R n \ {0}.
More precisely, we prove the following result, which represents the existence portion of Theorem 3.

Proposition 3.1. The number α * (F ) defined in (2.4) is finite and there exists a solution Φ ∈ H + α * (F ) of the equation

F (D 2 Φ) = 0 in R n \ {0}.
The function Φ is unique in the following sense: If α > -1 and u ∈ H + α is a solution of (3.1), then α = α * (F ) and either u ≡ tΦ for some t > 0, or u ≡ Φ + c for some c ∈ R.

To help the reader avoid misunderstandings, we recall that in this paper the spaces H + α contain positive functions if α > 0, and negative functions if α < 0. We begin by proving a comparison principle on spaces of homogeneous functions, suitable for our purposes.

Proposition 3.2. Suppose that α ≥ -1 and f ∈ C(R n \ {0}), u, v ∈ H α satisfy the differential inequalities (3.2) F (D 2 u) ≤ f ≤ F (D 2 v) in R n \ {0}.
Then the following statements hold :

(i) if α = 0, then u -v is constant; (ii) if α = 0 and f ≡ 0, then either u ≡ 0, or v ≡ 0, or F (D 2 u) = 0 = F (D 2 v) and u ≡ tv for some t > 0 ; (iii) if αf 0, then either u < v, or u ≡ v and F (D 2 u) = f = F (D 2 v).
Proof. Suppose α = 0. Then the function w := u -v is constant on the ray {tx : t > 0} for each x ∈ R n \ {0}. Moreover, by (H1) and the transitivity of the inequalities in the viscosity sense we have

P -(D 2 w) ≤ 0 in R n \ {0}.
Set M := max ∂B1 w = sup R n \{0} w. By the strong maximum principle, w ≡ M .

Next we consider the case α > 0. Since v ∈ H α we have v ≥ 0, so by the strong maximum principle and F (D 2 v) ≥ f ≥ 0 we conclude that either v ≡ 0 (and then

f ≡ 0) or v > 0 in R n \ {0}, that is, v ∈ H + α .
In the latter case the function w s := u -sv is strictly negative in R n \ {0} for large s, namely, for s > (max ∂B1 u)/(min ∂B1 v). Heuristically, we have

P -(D 2 w s ) ≤ F (D 2 u) -sF (D 2 v) ≤ (1 -s)f in R n \ {0},
for every s > 1. Using the transitivity of differential inequalities in the viscosity sense, we easily see that w s is a viscosity subsolution of P -(D 2 w s ) ≤ (1 -s)f . Define

s 0 := inf {s > 0 : w s < 0 in R n \ {0}} , s 1 := inf {s > 1 : w s < 0 in R n \ {0}} .
If u ≡ 0 then s 0 > 0. Hence, supposing that f ≡ 0 we see that P -(D 2 w s ) ≤ 0 implies that either u = s 0 v, which proves (ii), or w s0 < 0 in R n \ {0}. In the latter case we get w s0-δ < 0 in R n \ {0} for δ < (min ∂B1 |w s |)/(max ∂B1 v), which is a contradiction with the definition of s 0 .

Finally, if f 0 and s 1 > 1, we get P -(D 2 w s1 ) 0 and w s1 ≤ 0 which leads to a contradiction with the strong maximum principle and the definition of s 1 . Hence

s 1 = 1 and we conclude that u ≤ v. Since F (D 2 u) ≤ F (D 2 v), (iii) follows, for α > 0.
If α < 0 we reason in exactly the same way, noticing that either u ≡ 0 or u < 0, and in the latter case defining w s = v -su. This function is positive for large s and satisfies

P + (D 2 w s ) ≥ (1 -s)f in R n \ {0}.
The next lemma establishes that the set of α > -1 for which there exists a

supersolution u ∈ H + α of F (D 2 u) ≥ 0 in R n \ {0} is an interval. Lemma 3.3. Assume that -1 < α < α * (F ). Then there exists a supersolution u ∈ H + α of the inequality (3.3) F (D 2 u) ≥ ξ α+2 in R n \ {0}.
Proof. By the definition of α * we can select β > -1 satisfying α < β < α * and for which there exists a supersolution v ∈ H + β of the inequality (3.4)

F (D 2 v) ≥ 0 in R n \ {0}.
First we suppose that 0 < α < β. Define τ := β/α > 1 and w(x) := (v(x)) 1/τ . Notice that w ∈ H + α . Formally, we have

F (D 2 w) = 1 τ v 1/τ -1 F D 2 v -(1 -1/τ )v -1 Dv ⊗ Dv ≥ λ(τ -1)|Dv| 2 v 1/τ τ 2 v 2 ,
where we have used (3.4) and (H1), which implies

F (M -N ) ≥ F (M ) -P + (N ). As v ∈ H + β , by differentiating v(x) = t β v(tx) with respect to t we see that x • Dv = -βv, hence |Dv| ≥ β|x| -1 v. Thus we formally estimate (3.5) F (D 2 w) ≥ τ -2 λ(τ -1)β 2 v 1/τ |x| -2 = λα 2 (τ -1)w|x| -2 ≥ c(min ∂B1 w)ξ α+2 ,
since the homogeneity of w implies w ≥ (min ∂B1 w)ξ α . So dividing w by a constant we get (3.3).

To verify (3.5) in the viscosity sense, we select a smooth test function ϕ and

x 0 ∈ R n \ {0} such that x → w(x) -ϕ(x) has a local minimum at x = x 0 . We must demonstrate that (3.6) F (D 2 ϕ(x 0 )) ≥ λα 2 (τ -1)w(x 0 )|x 0 | -2 .
We may suppose without loss of generality that ϕ(x 0 ) = w(x 0 ) and ϕ ≥ w > 0. Let ψ(x) := (ϕ(x)) τ . Then v(x 0 ) = ψ(x 0 ) and x → v(x) -ψ(x) has a local minimum at x = x 0 . Recalling (3.4), we have F (D 2 ψ(x 0 )) ≥ 0. A routine calculation reveals that

D 2 ψ(x) = τ (ϕ(x)) τ -1 D 2 ϕ(x) + τ (τ -1) (ϕ(x)) τ -2 Dϕ(x) ⊗ Dϕ(x). Thus 0 ≤ F D 2 ϕ(x 0 ) + (τ -1) (ϕ(x 0 )) -1 Dϕ(x 0 ) ⊗ Dϕ(x 0 ) ≤ F D 2 ϕ(x 0 ) + τ -1 ϕ(x 0 ) P + (Dϕ(x 0 ) ⊗ Dϕ(x 0 )) = F D 2 ϕ(x 0 ) - λ(τ -1) w(x 0 ) |Dϕ(x 0 )| 2 . (3.7)
We will next derive a lower bound for |Dϕ(x 0 )|. Owing to the homogeneity of w, at any point x = 0 we have w(x

+ sx) = w(x)(1 + s) -α , so ∂ ∂s (w(x + sx))| s=0 = w(x) ∂ ∂s (1 + s) -α s=0 = -αw(x).
Since w -ϕ has a minimum at x 0 , we see that

x 0 • Dϕ(x 0 ) = ∂ ∂s (ϕ(x 0 + sx 0 ))| s=0 ≤ ∂ ∂s (w(x 0 + sx 0 ))| s=0 ≤ -αw(x 0 ). Hence |Dϕ(x 0 )| ≥ αw(x 0 ) |x 0 | .
Inserting into (3.7), we obtain (3.6). Recalling that w ∈ H + α , we see that a large multiple of w satisfies (3.3).

In the case α = 0 < β, we define w(x) := β -1 log v(x). Then w ∈ H + 0 , and formally we see that

F (D 2 w) = F D 2 v βv - Dv ⊗ Dv βv 2 ≥ λ|Dv| 2 βv 2 ≥ λβ|x| -2 = λβξ 2 in R n \ {0}.
This differential inequality is easily verified in the viscosity sense, as above.

Similarly, in the case α < 0 = β, we define w(x) := -exp(αv(x)). It is easily verified that w ∈ H + α , and formally we have

F (D 2 w) = F (wD 2 v -αwDv ⊗ Dv) ≥ -λα|w||Dv| 2 ≥ λ|α||w||x| -2 in R n \ {0}.
This inequality can also be routinely verified in the viscosity sense, so that, since |w| ≥ (min ∂B1 |w|)|ξ α |, some positive multiple of w satisfies (3.3).

If α < 0 < β we can combine the last two cases to obtain the desired supersolution.

Finally, we consider the case that -1

< α < β < 0. With τ := β/α < 1, we define w(x) := -(-v(x)) 1/τ . Formally we compute F (D 2 w) = τ -1 (-v) 1/τ -1 F D 2 v -(1/τ -1)(-v) -1 Dv ⊗ Dv ≥ λ(1 -τ )(-v) 1/τ |Dv| 2 τ 2 (-v) 2 ≥ λα 2 (1 -τ )|x| -2 (-w). Since λα 2 (1 -τ )|x| -2 |w| ≥ c|x| -α-2 in R n \ {0}
, we may again argue as above to conclude that a multiple of w satisfies (3.3).

From the previous two results we deduce a maximum principle.

Corollary 3.4. Assume that -1 ≤ α < α * (F ), α = 0. Suppose that u ∈ H α satisfies the inequality

(3.8) F (D 2 u) ≤ 0 in R n \ {0}.
Then u ≡ 0. If α * (F ) > 0, then there does not exist a function u ∈ H + 0 satisfying the inequality (3.8).

Proof. According to Lemma 3.3, there exists a function v ∈ H + α such that

F (D 2 u) ≤ 0 < ξ α+2 ≤ F (D 2 v) in R n \ {0},
and we conclude by Proposition 3.2.

Corollary 3.5. For any 0 < λ ≤ Λ, (3.9)

α * P - λ,Λ = λ Λ (n -1) -1 and α * P + λ,Λ = Λ λ (n -1) -1.
Proof. Recalling (2.8), from the definition of α * we see that

α * (P -) ≥ λ Λ (n -1) -1.
However, if α * (P -) > λ Λ (n -1) -1, then we see that Corollary 3.4 and (2.8) are incompatible. This verifies the first equality in (3.9). The second equality is proved with a similar argument.

Corollary 3.6. For any operator F which satisfies (H1) and (H2),

λ Λ (n -1) -1 ≤ α * (F ) ≤ Λ λ (n -1) -1.
Proof. Since P -≤ F ≤ P + , the result immediately follows from (3.9) and the definition of α * (F ).

We now split the proof of Proposition 3.1 into two parts, and consider separately the cases α * (F ) > 0 and α * (F ) ≤ 0. Lemma 3.7. Suppose α ≥ 0, f ∈ H + α+2 , and u ∈ H + α satisfy (3.10)

F (D 2 u) = f in R n \ {0}.
Then α < α * (F ).

Proof. Employing the local C 1,γ estimates for uniformly elliptic equations, we deduce that u ∈ C 1 (R n \{0}). By the homogeneity of u, we have u(x) = |x| -α u(x/|x|), so

(3.11) |Du(x)| ≤ C|x| -α-1 for every x ∈ R n \ {0}.
First we consider the case α > 0. Let 1 2 < τ < 1 be a number to be selected, and set w(x) := (u(x))

1/τ . Notice that w ∈ H + β for β := α/τ > α. From (3.11) we easily obtain the estimate

(3.12) |Dw(x)| ≤ C|x| -β-1 .
We claim that if τ is selected sufficiently close to 1, then w satisfies the inequality

(3.13) F (D 2 w) ≥ 0 in R n \ {0}.
Take a smooth test function ϕ and a point x 0 = 0 such that ϕ(x 0 ) = w(x 0 ), and x → w(x) -ϕ(x) has a local minimum at x = x 0 . Observe that Dw(x 0 ) = Dϕ(x 0 ). Set ψ := ϕ τ . The function x → u(x) -ψ(x) has a local minimum at x = x 0 . Thus

F D 2 ψ(x 0 ) ≥ f (x 0 ).
Following the calculations in the proof of Lemma 3.3, we obtain the estimate

f (x 0 ) τ (w(x 0 )) τ -1 ≤ F D 2 ϕ(x 0 ) + Λ(1 -τ ) ϕ(x 0 ) |Dϕ(x 0 )| 2 .
Rearranging, we have by the homogeneity of w

F D 2 ϕ(x 0 ) ≥ c|x 0 | -α-2 (w(x 0 )) 1-τ - C(1 -τ ) w(x 0 ) |x 0 | -2β-2 ≥ c|x 0 | -β-2 -C(1 -τ )|x 0 | -β-2 .
Taking 1 -τ > 0 to be sufficiently small, we obtain F D 2 ϕ(x 0 ) ≥ 0, which verifies that for such τ the function w satisfies (3.13). It now follows from the definition (2.4) of α * (F ) that α < β ≤ α * (F ).

Next we consider the case α = 0. Define the function v := exp(βu), where β > 0 will be selected.

Then v ∈ H + β ∩ C 1 (R n \ {0}
), and if u ∈ C 2 we check that

D 2 u = 1 β D 2 v v - 1 β Dv ⊗ Dv v 2 .
Formally, for some c > 0 we have

cβv|x| -2 ≤ F D 2 v - 1 v Dv ⊗ Dv ≤ F (D 2 v) + Λ v |Dv| 2 .
This calculation can be made rigorous by arguing with smooth test functions, so that in the viscosity sense we have

F (D 2 v) ≥ cβv|x| -2 - Λ v |Dv| 2 .
Using |Dv| = β|Du|v and the estimate (3.11), we obtain

F (D 2 v) ≥ cβv|x| -2 -Λk 2 β 2 v|x| -2 . Thus F (D 2 v) ≥ 0 in R n \ {0}, provided that we select β := c/Λk 2 > 0.
The next lemma is the key to the proof of Proposition 3.1 in the case α * (F ) > 0.

Lemma 3.8. Suppose that 0 < α < α * (F ) and f ∈ H α+2 . Then there exists a unique solution u ∈ H α of the equation

(3.14) F (D 2 u) = f in R n \ {0}. Moreover, if f ≡ 0, then u ∈ H + α . Proof. According to Lemma 3.3, there exists a supersolution w ∈ H + α of F (D 2 w) ≥ f in R n \ {0}. Let us define u(x) := sup {ũ(x) : ũ ∈ C(R n \ {0}
) is a subsolution of (3.14), and ũ ≤ w} .

Obviously the zero function is a subsolution of (3.14), so u is well-defined and u ≥ 0. If ũ ∈ C(R n \ {0}) is a subsolution of (3.14), then so is T α σ ũ for any σ > 0, by the scaling invariance of the equation. Thus u ∈ H α by construction. Standard arguments from viscosity solution theory (see [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF]) imply that u is a solution of (3.14). The uniqueness of u follows at once from Proposition 3.2. If f ≡ 0, then by the strong maximum principle u > 0 and hence u ∈ H + α .

We are now ready to prove Proposition 3.1 in the case that α * (F ) > 0.

Proposition 3.9. Suppose that α * (F ) > 0. Then there exists a function

Φ ∈ H + α * such that F D 2 Φ = 0 in R n \ {0}. Moreover, if β > -1 and u ∈ H + β satisfy F (D 2 u) = 0 in R n \ {0}, then β = α * (F ) and u ≡ tΦ for some t > 0. Proof. For each 0 < α < α * , let u α ∈ H + α denote the unique solution of F (D 2 u α ) = ξ α+2 in R n \ {0}.
We claim that (3.15) sup

|x|=1 u α (x) → +∞ as α → α * .
Suppose on the contrary that there exists a sequence α j → α * such that sup

j≥1 sup |x|=1 u αj (x) ≤ C.
By the homogeneity of the functions u α , it follows easily that

sup x∈K u αj ≤ C
for any compact subset K ⊆ R n \ {0}. Therefore we have the estimate

u αj C γ (K) ≤ C.
By taking a subsequence, if necessary, we may assume that u αj converges locally uniformly on R n \ {0} to a function u ∈ C(R n \ {0}). It is immediate that u ∈ H α * , and that u is a solution of the equation

F (D 2 u) = ξ α * +2 in R n \ {0}.
By the strong maximum principle u ∈ H + α * . This contradicts Lemma 3.7 and therefore (3.15) holds.

Define the functions v α by

v α (x) := c -1 α u α (x), where c α := max |x|=1 u α (x).
Then v α ∈ H + α . In fact, using homogeneity and the Harnack inequality we have

cξ α ≤ v α ≤ ξ α in R n \ {0}
for some c > 0. Using the homogeneity of F , we see that v α is a solution of

F (D 2 v α ) = c -1 α ξ α+2 in R n \ {0}. Hence for every compact subset K ⊆ R n \ {0}, we have the estimate v α C γ (K) ≤ C.
Thus there exists a function Φ ∈ C(R n \ {0}) such that, up to a subsequence, v α → Φ locally uniformly on R n \ {0}.

It immediately follows that

cξ α * ≤ Φ ≤ ξ α *
and F (D 2 Φ) = 0, since c -1 α ξ α+2 tends to zero locally uniformly on R n \ {0}. The uniqueness assertions in the last statement in the proposition are immediately obtained from Corollary 3.4, the definition of α * (F ), and Proposition 3.2.

The proof of the existence of the fundamental solution in the case α * (F ) < 0 is more subtle. Because Lemma 3.3 provides negative supersolutions for (3.3), resp. (3.14), we do not know how to extend Lemma 3.8 to α < 0. Instead, our proof of Proposition 3.1 in the case α * (F ) ≤ 0 relies on the following version of the Leray-Schauder alternative (c.f. Rabinowitz [START_REF] Rabinowitz | Théorie du degré topologique et applications à des problèmes aux limites nonlinéaires[END_REF] or Chang [START_REF] Chang | Methods in nonlinear analysis[END_REF]).

Proposition 3.10 (Leray-Schauder alternative). Let X be a real Banach space, K ⊆ X a convex cone, and A : R × K → K be a compact and continuous mapping such that A(0, u) = 0 for every u ∈ K. Then there exist unbounded, connected sets

C + ⊆ [0, ∞) × K and C -⊆ (-∞, 0] × K such that (0, 0) ∈ C + ∩ C -and A(λ, u) = u for every (λ, u) ∈ C + ∪ C -.
We use the Leray-Schauder alternative to control the norms of approximate fundamental solutions. We apply it to the Banach space X = C(∂B 1 ), and the convex cone K := {u ∈ C(∂B 1 ) : u ≤ 0}. Observe that for each α < 0 the convex cone H α is isomorphic to K via the map u(x) → ũ(x) := u(x/|x|).

The following lemma will provide the map A to which we are going to apply Proposition 3.10. Lemma 3.11. For every -1 ≤ α, β < 0 and v ∈ H α , there exists a unique function u ∈ H α that satisfies the equation

(3.16) F (D 2 u + α|x| -2 (u -v)I n ) = |x| -2 (βu -αv) + α|x| -α-2 in R n \ {0}.
Moreover, u ∈ H + α and we have the estimate (3.17) max

∂B1 |u| ≤ α β (1 + nΛ)(1 + max ∂B1 |v|).
Proof. We recall once more that the spaces H α contain negative functions when α < 0. Notice that the zero function is a smooth, strict supersolution of (3.16), since α < 0 implies

F -α|x| -2 v(x)I n ≥ 0 > -αv|x| -2 + α|x| -α-2 for every x ∈ R n \ {0}.
We are going to show that the function w(x) := -C|x| -α , with a properly selected C > 0, is a subsolution of (3.16). Notice we have

(3.18) D 2 (|x| -α ) + α|x| -α-2 = α(α + 2)|x| -α-4 x ⊗ x 0,
so, inserting w into the left-hand side of (3.16), we discover that

F D 2 w + α|x| -2 (w -v)I n = F -Cα(α + 2)|x| -α-4 (x ⊗ x) -(αv)|x| -2 I n ≤ P + -Cα(α + 2)|x| -α-4 (x ⊗ x) -(αv)|x| -2 I n ≤ P + (αv)|x| -2 I n = nΛ|x| -2 (αv).
Select

C := α β (nΛ + 1) 1 + max ∂B1 |v| , so that v ≥ (-max ∂B1 |v|)|x| -α easily implies nΛ|x| -2 (αv) ≤ (-Cβ + α)|x| -2-α -α|x| -2 v = |x| -2 (βw -αv) + α|x| -α-2 ,
allowing us to conclude that w is a subsolution of (3.16).

Let us define the function

u(x) := sup {w(x) : w ∈ C(R n \ {0}
) is a subsolution of (3.16) and w ≤ 0} .

It is clear that -C|x| -α ≤ u(x) ≤ 0 for every x ∈ R n \ {0}, giving us (3.17). Moreover, we have u ∈ H α due to the scaling invariance of (3.16) and the definition of u -since if w is a subsolution of (3.16), then so is T α σ w, for all σ > 0. Standard viscosity solution arguments (c.f. [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF]) imply that u is a solution of (3.16). Since the zero function is a smooth strict supersolution, it cannot touch u from above, as u is a viscosity subsolution. Thus u < 0, that is, u ∈ H + α . To establish the uniqueness of u, we notice that the equation is "proper" with respect to the space H + α . By this we mean that if u solves (3.17), then the function u + c|x| -α is a strict supersolution of (3.16) for any c > 0, a fact which is easy to check with the help of (H1) and (3.18). Let us suppose that u 1 , u 2 ∈ H α are two solutions of (3.16), such that c := max ∂B1 (u 2 -u 1 ) ≥ 0. Then by the strong maximum principle we must have u 2 ≡ u 1 + c|x| -α , which is impossible unless c = 0, since otherwise u 1 + c|x| -α is a strict supersolution. The uniqueness of u follows.

Using the Leray-Schauder alternative and the solution operator from the previous lemma, we build approximate fundamental solutions. Lemma 3.12. For every k ≥ 1 and -1 < β < 0, there exists a number α < 0 satisfying

(3.19) min{α * (F ), β} ≤ α ≤ cβ,
for some constant 0 < c = c(n, Λ) < 1/2, and a function u ∈ H + α satisfying the equation (3.16) with u = v, that is,

(3.20) F (D 2 u) = |x| -2 (β -α)u + α|x| -α-2 in R n \ {0},
and for which

(3.21) max ∂B1 |u| = k. Proof. Fix k ≥ 1, β ∈ (-1, 0), and recall that K := {w ∈ C(∂B 1 ) : u ≤ 0}. Given (α, w) ∈ [-1, 0) × K, let u ∈ H +
α be the unique solution of (3.16) for v given by v(x) := |x| -α w(x/|x|), and define an operator A : [-1, 0) × K → K by setting A(α, w) := ũ. Let us extend the domain by setting A(α, w) := 0 for all α ≥ 0, and A(α, w) := A(-1, ṽ) for α < -1.

By using (3.17) as well as Hölder estimates and stability properties of viscosity solutions under local uniform convergence, the map

A : R × K → K.
is easily seen to be continuous and compact.

We now apply Proposition 3.10 to deduce the existence of an unbounded and connected set C ⊆ (-∞, 0] × K such that (0, 0) ∈ C and A(α, ũ) = ũ for every (α, ũ) ∈ C, that is, u(x) = |x| -α u(x/|x|) satisfies (3.20). We claim that (3.22) C ⊆ [min{β, α * }, 0] × K.

Suppose that -1 < α < β and ũ ∈ K are such that (α, ũ) ∈ C. Then we see that the function u(x) := |x| -α ũ(x/|x|) belongs to H + α and satisfies

F (D 2 u) ≤ 0 in R n \ {0}. By Corollary 3.4, we see that α ≥ α * (F ). Thus C ∩ [-1, min{β, α * }) × K = ∅. Since C is connected, we deduce (3.22).
Since C is unbounded and connected, and (3.22) holds, we can find (α, ũ) ∈ C such that u(x) := |x| -α ũ(x/|x|) satisfies (3.21). We already know that u also satisfies equation (3.20). Finally, we notice that (3.17) with u = v and (3.21) give

1 2C 1 ≤ k C 1 (1 + k) ≤ α β ,
where C 1 = nΛ + 1 is the constant in (3.17). Thus if we select c := 1/2C 1 , then the second inequality in (3.19) must hold.

We now construct fundamental solutions in the case α * (F ) < 0, using the approximate fundamental solutions from Lemma 3.12. Proposition 3.13. If α * (F ) < 0, then there exists a solution

Φ ∈ H + α * of the equation F (D 2 Φ) = 0 in R n \ {0}.
Moreover, if β > -1 and u ∈ H + β satisfies F (D 2 u) = 0 in R n \ {0}, then β = α * (F ) and u ≡ tΦ for some t > 0.

Proof. Choose a sequence 1 < k j → ∞, and apply Lemma 3.12 with β = α * (F ) to find numbers α j such that α * ≤ α j ≤ cα * < 0, and u j ∈ H + αj which satisfy the equation

F (D 2 u j ) = |x| -2 (α * -α j )u j + α j |x| -αj -2 in R n \ {0},
as well as max

∂B1 |u j | = k j .
By taking a subsequence, we may assume that α j → α as j → ∞ for some number

α * ≤ α ≤ cα * < 0. Define w j (x) := u j (x)/k j . Observe that w j is (-α j )-homogeneous, (3.23) max ∂B1 |w j | = 1,
and that w j is a solution of the equation

F (D 2 w j ) = |x| -2 (α * -α j )w j + α j k -1 j |x| -αj -2 in R n \ {0}.
Since the right-hand side of the expression above is locally uniformly bounded, Hölder estimates imply that for any compact K ⊆ R n \ {0},

w j C γ (K) ≤ C,
for some constants C, γ > 0. By passing to a further subsequence we may assume that w j → Φ locally uniformly in R n \ {0}, for some function Φ which necessarily belongs to H α . It follows that Φ is a solution of the equation

F (D 2 Φ) = |x| -2 (α * -α )Φ in R n \ {0}.
Notice that (3.23) implies that max ∂B1 |Φ| = 1, so Φ ≡ 0. Since (α * -α )Φ ≥ 0, the definition of α * implies α ≤ α * , and thus we deduce α = α * . Therefore Φ is a solution of

F (D 2 Φ) = 0 in R n \ {0}.
By the strong maximum principle Φ ∈ H + α * . The uniqueness assertions follow from Corollary 3.4, the definition of α * , and Proposition 3.2.

Our construction of Φ in the case α * (F ) = 0 is a variation of the above argument. It is complicated somewhat by the need to bend the approximate solutions so that their limit lies in the set H + 0 . Proposition 3.14. If α * (F ) = 0, then there exists a solution Φ ∈ H + 0 of the equation

F (D 2 Φ) = 0 in R n \ {0}. Moreover, if β > -1 and u ∈ H + β satisfies F (D 2 u) = 0 in R n \ {0}, then β = 0 and u ≡ Φ + c for some c ∈ R.
Proof. Select a sequence ε j → 0, ε j > 0, and use Lemma 3.12 with β = -ε j to find numbers α j < 0 satisfying (3.24) -ε j ≤ α j ≤ -cε j , and functions u j ∈ H + αj which satisfy the equation

(3.25) F (D 2 u j ) = |x| -2 (-ε j -α j )u j + α j |x| -αj -2 in R n ,
and for which

(3.26) max ∂B1 |u j | = 1, so 0 < -u j ≤ |x| -αj .
We may improve (3.24) by observing that -ε j -α j ≤ α j , as otherwise we would have F (D 2 u j ) ≤ 0 in R n \ {0}, in violation of Corollary 3.4. Thus we have

(3.27) - 1 2 ε j ≤ α j ≤ -cε j ,
In particular, we have α j → 0 as j → ∞, and by taking a subsequence we may assume that the quantity

(3.28) -α -1 j (ε j + α j ) → b for some number 1 ≤ b ≤ (1 -c)/c.
The right-hand side of (3.25) is locally bounded by Cε j . Recalling (3.18), (3.24) and (3.26), it is simple to verify that the function |x| -αj + u j is nonnegative in R n \ {0} and F D 2 (|x| -αj + u j ) is locally bounded by Cε j . Therefore we may use the Harnack inequality to deduce that |x| -αj + u j is locally bounded by Cε j . That is, (3.29) max

K R (|x| -αj + u j ) ≤ min K R (|x| -αj + u j ) + C R ε j = C R ε j , in each set K R = {x : R -1 ≤ |x| ≤ R}.
Moreover, the regularity estimates imply

(3.30) |x| -αj + u j C 1,γ (K R ) ≤ C R ε j
Define w j := α -1 j log(-u j ). It is straightforward to check that w j ∈ H + 0 , since u j ∈ H + αj . Using (3.27) and (3.30), it is simple to verify that

w j C γ (K R ) = α -1 j log(|x| -αj -(|x| -αj + u j )) C γ (K R ) ≤ C R .
Hence, by taking a further subsequence, we may assume that w j → Φ locally uniformly in R n \ {0} as j → ∞, for a function Φ ∈ H + 0 . Formally differentiating, we find

D 2 u j = α j u j D 2 w j -α 2 j u 2 j Dw j ⊗ Dw j .
Since F is positively homogeneous and α j u j > 0, from (3.25) a standard viscosity solution argument yields that w j is a solution of the equation

(3.31) F (D 2 w j -α j u j Dw j ⊗ Dw j ) = -α -1 j (ε j + α j )|x| -2 + u -1 j |x| -αj -2
. It follows from (3.27) and (3.29) that α j u j and u -1 j |x| -αj tend locally uniformly to zero. Therefore we can use Theorem 4 in [START_REF] Sirakov | Solvability of uniformly elliptic fully nonlinear pde[END_REF], and pass to the limit j → ∞ in (3.31). Recalling (3.28), we discover that Φ is a solution of the equation

F (D 2 Φ) = (b -1)|x| -2 in R n \ {0}.
As b ≥ 1 and α * = 0, we may apply Lemma 3.7 with α = 0, to deduce that b = 1. The uniqueness assertions again follow from Corollary 3.4, the definition of α * (F ), and Proposition 3.2.

Proof of Proposition 3.1. Proposition 3.1 is immediately obtained from Propositions 3.9, 3.13, and 3.14.

Remark 3.15. In light of Proposition 3.1 and Corollary 3.4, the scaling exponent α * (F ) may also be expressed as

(3.32) α * (F ) = min α ∈ (-1, ∞) : there exists v ∈ H + α such that F (D 2 v) ≤ 0 in R n \ {0} .

Examples and discussion

In this section we discuss several examples. Example 4.1 (Operators with radial fundamental solutions). Felmer and Quaas [START_REF] Felmer | Fundamental solutions and two properties of elliptic maximal and minimal operators[END_REF] observed that a certain class of operators F have radial fundamental solutions. The key hypothesis is that F is invariant with respect to orthogonal changes of coordinates, that is, (4.1) F Q t M Q = F (M ) for every real orthogonal matrix Q and M ∈ S n .

In particular, they noticed that if F satisfies (4.1) and some additional hypotheses, then for some α > -1,

F (D 2 ξ α ) = 0 in R n \ {0}.
Of course, that (4.1) suffices for Φ = ξ α * is an immediate consequence of Theorem 4, since any rotation of a fundamental solution is a fundamental solution.

Let us generalize this observation. Notice that (4.1) is stronger than the condition

(4.2) F ≡ const on each set {M ∈ S n : spec(M ) = {-1, . . . , -1, k} }, k ≥ 0, which is equivalent to (4.3) F (ay ⊗ y -I n ) = F (az ⊗ z -I n ) for all a ≥ 1 and |y| = |z| = 1.
For fixed |y| = 1 and a ≥ 1, we see that

λ(n -1) -Λ(a -1) ≤ P -(ay ⊗ y -I n ) ≤ F (ay ⊗ y -I n ) ≤ P + (ay ⊗ y -I n ) = Λ(n -1) -λ(a -1).
Since F is continuous, there exists a constant 1

≤ ã ≤ Λ λ (n -1) + 1 such that F (ãy ⊗ y -I n ) = 0. If (4.
3) holds, then by (2.5)

F (ãz ⊗ z -I n ) = 0 for every |z| = 1. It follows that F (D 2 ξ α ) = 0 in R n \ {0}, for α = ã -2. Thus α * (F ) = ã -2 and Φ(F ) = ξ ã-2
, and so we see that (4.3) implies that the fundamental solution of F is radial.

Example 4.2 (Concave and Convex operators). Let us review some well-known, elementary facts regarding the fundamental solutions of linear elliptic operators. The Laplacian -∆ has scaling exponent α * (-∆) = n -2 and fundamental solution ξ n-2 . If L is any linear, uniformly elliptic operator with constant coefficients, given by Lu =i,j a ij u xixj , then there is a change of coordinates through which L is transformed into -∆. It follows that α * (L) = n -2 and the level sets of the fundamental solution Φ = Φ(L) are ellipsoids. Moreover, it is clear that the level sets of Φ(L) distinguish L among all linear operators, up to a positive constant multiple. In other words, as it is simple to check,

tr(A D 2 ξ n-2 ) = 0 on ∂B 1 ⇐⇒ A = cI.
From these facts, we will argue that if F is a convex operator which is not the maximum of multiples of the same linear operator, then

α * (F ) > n -2.
For such an operator F , there exist two linear operators L 1 and L 2 such that L 1 = cL 2 for every c > 0, and F ≥ max{L 1 , L 2 }. Let Φ 1 := Φ(L 1 ) and Φ 2 := Φ(L 2 ) be the fundamental solutions for L 1 and L 2 , respectively. Since L 1 and L 2 are not proportional, we see that Φ 1 and Φ 2 are not proportional, by the above. Since

F (D 2 Φ i ) ≥ 0 in R n \{0} for i = 1, 2, we have α * (F ) ≥ n-2, and by Proposition 3.2 α * (F ) = n -2 implies Φ 1 = Φ = Φ 2 .
A similar argument shows that if F is concave and not the minimum of multiples of one linear operator, then α * (F ) < n-2. The underlying idea behind this example was previously observed in different contexts in [START_REF] Lions | Bifurcation and optimal stochastic control[END_REF][START_REF] Scott | Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations[END_REF][START_REF] Armstrong | Long-time asymptotics for fully nonlinear homogeneous parabolic equations[END_REF].

It would be interesting to discover more about the relationship between the two scaling exponents α * (F ) and α * ( F ). In particular, we do not know whether there is an operator F satisfying (H1)-(H2) for which both α * (F ) and α * ( F ) are negative. We show now that there is no such operator F which also satisfies (4.3). Proposition 4.3. Suppose that Φ and Φ are radial functions. Then

(4.4) λ Λ (n -2) ≤ max{α * (F ), α * ( F )} and min{α * (F ), α * ( F )} ≤ Λ λ (n -2).
Proof. For ease of notation let us write α := α * (F ) and α := α * ( F ). Since Φ = ξ α and Φ = ξ α, we have that

F ((α + 2)x ⊗ x -I n ) = F (-(α + 2)x ⊗ x + I n ) = 0 for all x ∈ ∂B 1 .
Let us suppose that max{α, α} ≤ k := λ(n -2)/Λ ≥ 0. Select x, y ∈ ∂B 1 with x • y = 0 (so the matrices x ⊗ x and y ⊗ y commute and attain their diagonal form in the same basis of R n ) and observe that by (H1) we have

0 = -2Λk + 2λ(n -2) = P -((k + 2)x ⊗ x + (k + 2)y ⊗ y -2I n ) ≤ P -((α + 2)x ⊗ x + ( α + 2)y ⊗ y -2I n ) ≤ F ((α + 2)x ⊗ x -I n ) -F (-(α + 2)y ⊗ y + I n ) = 0.
It follows that k = α = α, and we obtain the first inequality in (4.4). The second inequality is obtained by a similar argument, using P + in place of P -in the above calculation.

Remark 4.4. The inequalities in (4.4) are sharp. To see this, consider the operators

F 1 (M ) := -Λ(µ 1 (M ) + µ n (M )) -λ n-1 i=2 µ i (M ), F 2 (M ) := -λ(µ 1 (M ) + µ n (M )) -Λ n-1 i=2 µ i (M ),
where

µ 1 (M ) ≤ µ 2 (M ) ≤ • • • ≤ µ n (M ) are the eigenvalues of M ∈ S n .
It is easy to check that F 1 = F1 and F 2 = F2 , that F 1 and F 2 satisfy (H1)-(H2) as well as (4.1), and their fundamental solutions are ξ λ Λ (n-2) and ξ Λ λ (n-2) , that is,

α * (F 1 ) = λ Λ (n -2) and α * (F 2 ) = Λ λ (n -2).

Characterization of singularities and a Liouville theorem

In this section we study the behavior near the origin of a solution u ∈ C (B 1 \ {0}) of the equation (5.1)

F D 2 u = 0 in B 1 \{0}
which is bounded on one side, and the behavior near infinity of a solution u ∈ C(R n \ B 1 ) of (5.1) in R n \ B 1 which is bounded on one side. Throughout this section, we take α * = α * (F ) and Φ to be the scaling exponent and fundamental solution, respectively, for the operator F obtained in Proposition 3.1, and α * = α * ( F ) and Φ be the scaling exponent and upward-pointing fundamental solution, respectively, for the dual operator F given by F (M ) := -F (-M ).

We make repeated use of the quantities Proof. This is a simple consequence of the Harnack inequality and the fact that if a function x → u(x) is a solution of F D 2 u = 0, then so is x → u(x/r).

5.1. Classification of isolated singularities. Our proof of Theorem 5, though considerably more complicated, borrows some ideas from Labutin [START_REF] Labutin | Isolated singularities for fully nonlinear elliptic equations[END_REF], who proved Theorem 5 for F = P - λ,Λ , in the case that α * (P - λ,Λ ) ≥ 0. The idea is to show that either the singularity at the origin of a solution u of (5.4)

F (D 2 u) = 0 in B 1 \ {0}
is removable, or else is bounded between two multiples of Φ near the origin. Then we use the Harnack inequality and the strong maximum principle to squeeze the gap as we blow up the function u at the origin. A similar idea will establish the corresponding conclusions in the case that α * (F ) < 0. We divide the proof of Theorem 5 into five lemmas. The first step is the following result, which states that a nonnegative solution u of (5.1) must either be bounded near the origin, or α * (F ) ≥ 0 and ρ(r) ≥ c > 0.

Lemma 5.2. Assume that u ∈ C (B 1 \ {0}) is a solution of (5.4) which is bounded below. Suppose that either (i) α * (F ) ≥ 0 and lim inf r→0 ρ(r) = 0, or (ii) α * (F ) < 0. Then u is bounded in B 1/2 \ {0}.

Proof. Note that adding a constant to u modifies neither the hypotheses nor the conclusion of the lemma. So we can assume u is positive in B 1 \ {0}.

We first consider the case (i). By adding a constant to Φ in the case that α * (F ) = 0, we may assume that Φ > 0 in B 1 . Let r k → 0 be such that ρ

(r k ) → 0 as k → ∞. Select a point x k ∈ B r k \ {0} such that u(x k ) ≤ (1/k)Φ(x k ).
According to the Harnack inequality, there exists a constant C > 0, depending only on n and the ellipticity constants Λ and λ, such that

u(x) ≤ C(1/k)Φ(x) for all |x| = |x k |.
According to the maximum principle,

u ≤ C(1/k)Φ + M (1/2) in B 1/2 \ B r k .
Passing to the limit k → ∞, we obtain

(5.5) u ≤ M (1/2) in B 1/2 \ {0}.
Thus u is bounded above in the punctured ball B 1/2 \ {0}.

We now consider the case α * (F ) < 0. By Lemma 5.1 we have M (r) ≤ Cm(r) for every 0 < r < 1/2. Define

a := -max ∂B 1/2
Φ > 0 and ψ(x) := a -1 (Φ(x) + a).

We may assume without loss of generality that ϕ(0) = 0 = u(0).

We may also assume without loss of generality that Dϕ(0) = 0. To see this, define ũ(x) := u(x) -x • Dϕ(0), and φ(x) := ϕ(x) -x • Dϕ(0), and notice that ũ -φ has a strict local maximum at the origin, D φ(0) = 0, φ(0) = 0 = ũ(0), and ũ is a solution of (5.4). Moreover, our hypotheses (i) or (ii) hold for ũ. To get the second condition in case (ii), notice that lim inf

|x|→0 ũ(x) Φ(x) = lim inf |x|→0 u(x) -x • Dϕ(0) Φ(x) ≤ lim inf |x|→0 u(x) Φ(x) + C lim |x|→0 |x| |x| -α *
The last expression on the right vanishes, since α * > -1. Finally, we remark that our conclusion holds for ũ if and only if it holds for u. Therefore, we may assume that u = ũ and Dϕ(0) = 0.

Owing to (a), there exists a unit vector z ∈ ∂B 1 and a sequence {y j } ⊆ B 1 \ {0} such that y j → 0 and u(y j ) ≥ 0 and z • y j > |y j | 2 for all j.

For ε > 0, we define ψ ε (x) := ϕ(x) -εz • x. Since u -ϕ has a strict local maximum at zero, we can fix r, δ > 0 such that u(x) -ϕ(x) ≤ -δ for every |x| = r and u < ϕ in B r \ {0}.

For each ε < δ/(2r), we have u(0) = ψ ε (0) = 0 and u(x) -ψ ε (x) ≤ -δ 2 for all |x| = r.

Notice that

ψ ε (y j ) = ϕ(y j ) -εz • y j ≤ - ε 2 |y j | + o (|y j |) as j → ∞.
Thus for j large enough, we have u(y j ) -ψ ε (y j ) ≥ -ψ ε (y j ) > 0, as well as

|y j | < r. Let x ε ∈ B r such that u(x ε ) -ψ ε (x ε ) = max Br (u -ψ ε ) > 0.
Since x ε = 0 and u is a viscosity solution away from the origin, we deduce that

F (D 2 ϕ(x ε )) = F (D 2 ψ ε (x ε )) ≤ 0.
It is clear that x ε → 0 as ε → 0, since ψ ε → ϕ uniformly. We now pass to the limit ε → 0, to obtain (5.6).

Lemma 5.5. Suppose that α * (F ) ≥ 0, and u ∈ C (B 1 \ {0}) is a solution of (5.4) which is bounded below. Then This verifies (5.13) in the case α * (F ) > 0. Now suppose that α * (F ) = 0. For each 0 < r < r 0 , define the function

v r (x) := u(rx) Φ(x r ) , x ∈ B 1/(2r) \ {0}.
It is clear that v r satisfies the equation F (D 2 v) = 0 in B 1/(2r) \ {0}. To get a lower bound for v r , we notice that

v r (x) ≥ a Φ(rx) Φ(x r ) = a Φ(x) -log r Φ(x r /r) -log r → a locally uniformly as r ↓ 0. Since v r x r r = ρ(r) → a as r → 0,
the Harnack inequality provides the bound v r L ∞ (K) ≤ C K for all 0 < r < r 1 ≤ r 0 and compact subsets K ⊆ B 1/(2r1) \ {0}. As before, using the Hölder estimates we can find a subsequence r j ↓ 0, a point y ∈ ∂B 1 , and a function v ∈ C(R n \ {0}) for which v rj → v locally uniformly in R n \ {0} and r -1 j x rj → y as j → ∞. We immediately deduce that

F (D 2 v) = 0 in R n \ {0},
as well as v(y) = a and v ≥ a in R n \ {0}. The strong maximum principle implies v ≡ a. Therefore, lim sup

x→0 u(x) Φ(x) = lim sup r→0 max x∈∂B1 u(rx) Φ(rx) = lim sup r→0 max x∈∂B1 v r (x)Φ(x r ) Φ(rx) = lim sup r→0 max x∈∂B1 v r (x) (Φ(x r /r) -log r) Φ(x) -log r = a.
This completes the proof of (5.13). In particular, ā = a.

We have shown above -see the argument leading to (5.12) -that by adding a constant to u so that ρ(1/2) ≥ 2ā = 2a, we can deduce that u ≥ aΦ in B 1/2 \ {0}. By a symmetric argument, we can show that by subtracting a constant from u so that ρ(1/2) ≤ a/2, we have u ≤ ρ(r)Φ in B 1/2 \ B r , so ρ is decreasing, and u ≤ āΦ = aΦ in B 1/2 \ {0}. Therefore (5.9) holds. Then a < ∞, and

(5.16) lim x→0 u(x) Φ(x) = a.
Proof. Our hypothesis (5.15) implies that there exists 0 < r 0 < 1 such that ρ(r) > 0 for 0 < r ≤ r 0 . For such r, since u ≤ ρ(r)Φ on ∂B r ∪ {0}, the maximum principle implies that u ≤ ρ(r)Φ on Br . In particular, ρ(r) = min Br u/Φ and u < 0 near the origin. It follows that the map r → ρ(r) is decreasing in r ∈ (0, r 0 ) and lim r↓0 ρ(r) = a. By a similar argument, we see that u ≥ ρ(r)Φ on ∂B r ∪ {0}, so by the maximum principle the map r → ρ(r) satisfies ρ(r) = max Br u/Φ for all 0 < r < r 0 , and is therefore increasing in r ∈ (0, r 0 ). In particular, a ≤ lim sup r→0 ρ(r) < ρ(r 0 ) < ∞. For every 0 < r ≤ r 0 , select x r ∈ ∂B r such that u(x r ) = ρ(r)Φ(x r ). Define the function v r (x) := r α * u(rx), 0 < r ≤ r 0 , x ∈ B1/(2r) . By the homogeneity of Φ, for all s ∈ (0, r 0 ] we have (5.17) ρ(s)Φ ≤ v r ≤ ρ(s)Φ in Bs/r , in particular ρ(r)Φ ≤ v r ≤ ρ(r)Φ on B1 , and v r (x r /r) = ρ(r)Φ(x r /r). As in the proof of the previous lemma, using the Hölder estimates, we can find a subsequence r j ↓ 0 for which v rj → v locally uniformly in R n \ {0}, and x rj /r j → y for some v ∈ C(R n \ {0}) and y ∈ ∂B 1 . Passing to limits we deduce that v is a solution of the equation

F (D 2 v) = 0 in R n \ {0}.
By letting r → 0 in (5.17) with s = r 0 , we obtain v ≤ aΦ in R n \ {0} and v(y) = aΦ(y). By the strong maximum principle, v ≡ aΦ. It follows that the full sequence {v r } r>0 converges to aΦ locally uniformly as r ↓ 0. Thus

lim sup x→0 u(x) Φ(x) = lim sup r→0 max x∈∂Br u(x) r -α * Φ(x/r) = lim sup r→0 max ∂B1 v r Φ = a.
The proof is complete.

We now combine the previous five lemmas into a proof of Theorem 5.

Proof of Theorem 5. Let us assume that u is bounded below in a neighborhood of the origin and first consider the case when α * (F ) < 0.

Then, according to Lemmas 5.2 and 5.3, we can define u at the origin so that u ∈ C(B 1 ). If lim inf x→0 (u(x)-u(0))/Φ(x) > 0 then Lemma 5.6 implies alternative (iv). If on the other hand lim inf x→0 (u(x) -u(0))/Φ(x) ≤ 0 then by Lemma 5.4 u is a subsolution of (5.1) in the whole ball B 1 .

Next, consider the case α * (F ) ≥ 0. Now, if lim inf x→0 u(x)/Φ(x) > 0 then Lemma 5.5 implies alternative (ii). On the other hand, if lim inf x→0 u(x)/Φ(x) ≤ 0 then by Lemmas 5.2-5.4 u can be defined at the origin so that it is again a subsolution of (5.1) in the whole ball B 1 .

So let us suppose now u is a subsolution of (5.1) in B 1 . If α * ( F ) ≥ 0, or if α * ( F ) < 0 and lim inf x→0 (-u(x) + u(0))/ Φ(x) ≤ 0, then by applying Lemma 5.4 Proof. Notice that ρ(r) is decreasing, since the maximum principle implies that for all ε > 0, u ≤ ρ(r)Φ + ε in R n \ B r , so by letting ε → 0 we get max R n \Br (u/Φ) = max ∂Br (u/Φ). Thus we have a ≤ ā := lim sup r→∞ ρ(r) ≤ ρ(1).

Set v r (x) = r α * u(rx). By a rescaling argument similar to that in Lemma 5.6, we find that v r → aΦ locally uniformly as r → ∞. It follows that a = ā.

Proof of Theorem 6. The proof is nearly identical to the proof of Theorem 5, with Lemmas 5.7, 5.8, 5.9, 5.10, and 5.11 taking the place of Lemmas 5.2, 5.3, 5.4, 5.5, and 5.6, respectively.

5.3. The extended Liouville theorem. In this subsection we use Theorems 5 and 6 to prove Theorem 4. We proceed by considering each of the alternatives provided by Theorem 5. Case (i): the singularity at the origin is removable. In this case, the function u is a solution of F (D 2 u) = 0 in the whole space R n , and u is bounded from above or below in R n . It now follows from the Liouville theorem for uniformly elliptic equations that u is constant. (The Liouville theorem is an immediate consequence of the Harnack inequality, see Remark 4 in Chapter 4 of [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF]).

Case (ii)(a): α * (F ) > 0 and u(x) = aΦ(x) + O(1) as x → 0. We may assume that a = 1. By Theorem 6 the limit u ∞ exists, so by adding a constant to u, we may assume that u → 0 as |x| → ∞. The maximum principle immediately yields that (1 -ε)Φ -ε ≤ u ≤ (1 + ε)Φ + ε for every ε > 0. Now we let ε → 0 to deduce that u ≡ Φ.

Case (ii)(b): α * (F ) = 0 and u(x) = Φ(x) + O(1) as x → 0. If (i) in Theorem 6 occurs, we apply it in R n \ B r for each r > 0, and get u ∞ = ∞, a contradiction. If (v) in Theorem 6 holds, then clearly u attains its minimum in R n \ {0}, so by the strong maximum principle u is a constant, a contradiction again.

Hence alternative (iv) in Theorem 6 remains. Now we have

Φ -C ≤ u ≤ Φ + C in B 1 \ {0}, aΦ -C ≤ u ≤ aΦ + C in R n \ {0},
for some a > 0. Hence a = 1. By adding a constant to u, we may suppose that max ∂B1 (u -Φ) = 0. By the maximum principle, for every 0 < c < 1 we have

u ≤ Φ + c(Φ -min ∂B1 Φ) in B 1 \ {0}, u ≤ Φ -c(Φ -max ∂B1 Φ) in R n \ B 1 .
Sending c → 0 we find u ≤ Φ in R n . Since max ∂B1 (u-Φ) = 0, the strong maximum principle implies that u ≡ Φ. Case (iii): α * ( F ) ≥ 0 and u(x) = -Φ(x) + O(1) as x → 0. We may repeat our arguments in case (ii) above, or simply apply them to -u and the dual operator F , to deduce that u ≡ -Φ.

Case (iv): α * (F ) < 0, u(0) = 0, and lim x→0 u(x)/Φ(x) = 1. By the maximum principle, for every r > 0 and ε > 0 we have u + ε ≥ ρ(r)Φ in B r \ {0}.

Thus u ≥ ρ(r)Φ in B r \ {0} for any r > 0, so ρ(r) = sup Br\{0} u/Φ, and r → ρ(r) is increasing. A similar argument ensures r → ρ(r) is decreasing.

If (i) in Theorem 6 occurs, by applying the inequality in its statement for each r > 0 and by using u(0) = 0 we get u ∞ = 0, so u attains its minimum in R n \ {0}, a contradiction. The same excludes (v) in Theorem 6.

Hence alternative (iv) in Theorem 6 remains, and we obtain ρ(r), ρ(r) → 1 both as r → 0 and as r → ∞. The monotonicity of ρ and ρ now obviously implies that ρ ≡ ρ ≡ 1, and thus u ≡ Φ.

Case (v): α * ( F ) < 0, u(0) = 0 and lim x→0 -u(x)/ Φ(x) = 1. We may apply the result we have proven in case (iv) to -u and F to deduce that u ≡ -Φ.

Proof of Theorem 3. The theorem is immediately obtained by appealing to Proposition 3.1 and Theorem 4.

Applications to stochastic differential games

In this section we give an interpretation of the scaling exponent α * (F ) in terms of two-player stochastic differential games. In particular, we generalize the wellknown fact that Brownian motion is recurrent in dimension n = 2 and transient in dimensions n ≥ 3. For a review of the connection between viscosity solutions of second-order elliptic and parabolic equations and stochastic differential games, we refer to Fleming and Souganidis [START_REF] Fleming | On the existence of value functions of two-player, zero-sum stochastic differential games[END_REF] and Kovats [START_REF] Kovats | Value functions and the Dirichlet problem for Isaacs equation in a smooth domain[END_REF].

Let us briefly describe the probabilistic setting (see [START_REF] Kovats | Value functions and the Dirichlet problem for Isaacs equation in a smooth domain[END_REF] for more details). We are given a probability space (Ω, F, P), a filtration of σ-algebras {F t } t≥0 which is complete with respect to (F, P), and a d-dimensional Weiner process {W t } t≥0 adapted to F t . Also given are compact metric spaces A and B, which are the control sets for Players I and II, respectively, and a function σ : A × B → M n×d .

We are interested in a random process {X t } t≥0 governed by the stochastic differential equation (6.1) dX t = σ(a t , b t )dW t ,

X 0 = x ∈ R n .
Here a t and b t are A and B-valued F t -progressively measurable stochastic processes, called the admissible control processes for Players I and II, respectively. The set of admissible control processes for Player I is denoted by M, and for Player II is denoted by N . Here we do not distinguish between controls {a t }, {ã t } ∈ M for which P [a t = ãt for almost every t ≥ 0] = 1. An admissible strategy for Player I is a mapping γ : N → M, and similarly an admissible strategy for Player II is a mapping θ : M → N . We denote the set of admissible strategies for Players I and II by Γ and Θ, respectively.

For each r > 0, we let the random variable τ r = τ x,a,b,r denote the first time the process X t hits the sphere ∂B r . Similarly, we define τ 0 = τ x,a,b,0 to be the first time the process X t touches the origin, and also denote τ ∞ = ∞.

We first consider a game played in the annulus B R \B r , for 0 < r < |x| < R ≤ ∞, and for which the payoff functional is the map J = J x,r,R : M × N → R given by

     F + (D 2 v) = 0 in B R \ Br , v = 1 on ∂B r , v = 0 on ∂B R .
Similarly, the lower value function v - r,R is the unique viscosity solution of the boundary-value problem

(6.3)      F -(D 2 v) = 0 in B R \ Br , v = 1 on ∂B r , v = 0 on ∂B R .
As F + ≤ F -, it is clear from the maximum principle that v - r,R ≤ v + r,R in B R \B r . We now show that we may use the fundamental solutions of F + and F -to estimate the value functions. We only treat v + r,R , since v - r,R can be estimated similarly. We let Φ and α = α * (F + ) > -1 denote the fundamental solution and scaling exponent of F + . Denote m(r) := min |x|=r Φ(x) and M (r) := max |x|=r Φ(x) for each r > 0. Fix two radii 0 < r < R, and notice that, by comparison,

  2 and 4.1], Labutin [16, Theorem 1], Felmer and Quaas [11, Theorem 1.3 and 1.4].

Lemma 5 . 1 .

 51 ρ(r) := min ∂Br u Φ and ρ(r) := max ∂Br u Φ , defined if Φ does not vanish on ∂B r . Suppose u and Φ have constant sign and u is a solution of (5.1) in B 1 \ {0} (resp. in R n \ B 1 ). Then there exists a constant C = C(n, λ, Λ) such that for each r ∈ (0, 1/2) (resp. r ∈ (2, ∞)) we have M (r) ≤ Cm(r) and ρ(r) ≤ C 2 ρ(r).

  then there is a constant C > 0 such that(5.9) aΦ -C ≤ u(x) ≤ aΦ + C in B 1/2 \ {0}.the strong maximum principle implies that v ≡ aΦ. From this we deduce that the full sequence {v r } r>0 converges to the function aΦ locally uniformly as r ↓ 0

Lemma 5 . 6 .

 56 Assume that α * (F ) < 0 and u ∈ C(B 1 ) is a solution of (5.4) such that u(0) = 0 and (5.[START_REF] Kovats | Value functions and the Dirichlet problem for Isaacs equation in a smooth domain[END_REF] a := lim inf r→0 ρ(r) > 0.

J

  x,r,R [a t , b t ] := P [τ x,a,b,r < τ x,a,b,R ] .Player I wishes to maximize the payoff and Player II wishes to minimize it. Thus, Player I wishes the process to exit the annulus B R \ Br on the inner boundary ∂B r while Player II tries to force the process to exit on the outer boundary ∂B R .We define the upper value of the game byv + r,R (x) := sup γ∈Γ inf b∈N J x,r,R [γ(b), b] ,and the lower value of the game byv - r,R (x) := inf θ∈Θ sup a∈M J x,r,R [a, Θ(a)] .For each M ∈ S n , we define the upper Isaacs operator byF + (M ) := -min a, b)σ(a, b) T M ) ,and the lower Isaacs operator byF -(M ) := -max b∈B a, b)σ(a, b) T M ) .It is clear thatF + (M ) ≤ F -(M )for all M ∈ S n . According to [15, Theorem 4.3], we have the following characterization of the upper and lower value functions. Proposition 6.1. Let 0 < r < R < ∞. The upper value function v + r,R is the unique viscosity solution of the boundary-value problem (6.2)

  ) -M (R) M (r) -M (R) ≤ v + r,R (x) ≤ Φ(x) -m(R) m(r) -m(R) , r < |x| < R.Let us suppose first that α = 0. From these inequalities we obtainm(1)|x| -α -M (1)R -α M (1)(r -α -R -α ) ≤ v + r,R (x) ≤ M (1)|x| -α -m(1)R -α m(1)(r -α -R -α ) , r < |x| < R.Suppose now that α > 0, and fix x ∈ R n \ {0}. Then we obtain limr→0 log v + r,R (x) log r = α * (F ),and this limit is uniform in R, for large R. That is, for any R > 0 we have (6.5) α * (F + ) = lim r→0 sup γ∈Γ inf b∈N log P τ x,γ(b),b,r < τ x,γ(b),b,R log r .

Acknowledgements. This research was conducted in part while the first author was a visitor at Le Centre d'analyse et de mathématique sociales (CAMS) in Paris.

Then ψ ≤ 0 on ∂B 1/2 , ψ ≤ 1 on B1/2 , and ψ > 1/2 in a neighborhood of the origin, say in B r0 \ {0}. It is clear that F (D 2 ψ) = 0 in B 1/2 \ {0}. According to the maximum principle, u ≥ m(r)ψ in B 1/2 \ Br .

Hence u ≥ C -1 M (r)ψ in B 1/2 \ Br . Since ψ > 1/2 near the origin, it follows that sup 0<r<r0 M (r) ≤ 2CM (r 0 ) < ∞.

Therefore, u is bounded in B 1/2 \ {0}.

The next auxiliary result says that under the conclusion of the previous lemma we can define u as a continuous function at the origin.

Lemma 5.3. Suppose that u ∈ C (B 1 \ {0}) is a bounded solution of (5.4). Then u can be defined at the origin so that u ∈ C(B 1 ).

Proof. We must show that lim x→0 u(x) exists. Define

Choose ε > 0. Define v(x) := u(x) -u 0 + ε, and fix 0 < r = r(ε) < 1/2 so small that v > 0 in B r \ {0}. By making r smaller, if necessary, we can find x 1 ∈ ∂B r such that v(x 1 ) ≤ 2ε. Let 0 < s < r. Choose x 2 ∈ B s \ {0} such that v(x 2 ) ≤ 2ε. By the Harnack inequality, there exists a constant

By the maximum principle, v ≤ Cε in B r \ B s . We send s → 0 to deduce that v ≤ Cε in B r \ {0}.

Thus sup

Br\{0}

u ≤ u 0 + Cε.

It follows that lim sup x→0 u(x) ≤ u 0 .

Lemma 5.4. Assume that u ∈ C(B 1 ) is a solution of (5.4) such that u(0) = 0. Suppose that either (i) α * (F ) ≥ 0, or (ii) α * (F ) < 0 and lim inf r→0 ρ(r) ≤ 0. Then (a) max ∂Br u ≥ 0 for every r ∈ (0, 1), and (b) u is a subsolution of the equation F (D 2 u) = 0 in the whole ball B 1 .

Proof. We first prove statement (a). In the case that (i) holds, we argue just as we did to obtain (5.5) in the proof of Lemma 5.2, by replacing 1/2 by r and u by u + C, where C is chosen so that u + C is positive in B 1 , and then showing that u(x) + C ≤ max ∂Br (u + C) in B r . Setting x = 0 in this equality gives (a). Next, consider the case that (ii) holds, and suppose on the contrary that max ∂Br u < 0 for some r > 0. By multiplying u by a positive constant, we may assume that u ≤ Φ on ∂B r .

Since u(0) = Φ(0), the maximum principle implies that u ≤ Φ, that is, u/Φ ≥ 1 in B r \ {0}. This contradicts the second hypothesis in (ii). We have established (a).

Let us now prove (b). Consider a smooth test function ϕ for which the function u -ϕ has a strict local maximum at the origin. We must show that (5.6) F (D 2 ϕ(0)) ≤ 0.

Proof. We may assume that u, Φ > 0 in B 1 \ {0} (if necessary, add constants to u and Φ). The maximum principle implies that for any 0 < r < 1/2,

Since max ∂B 1/4 Φ > max ∂B 1/2 Φ and max ∂B 1/4 u < ∞, we deduce that sup 0<r<1/4 ρ(r) < ∞.

In particular, Lemma 5.1 implies that (5.10) ā := lim sup r→0 ρ(r) < ∞.

Suppose now that a := lim inf r→0 ρ(r) > 0. By adding a positive constant to u, we may assume that ρ(1/2) ≥ 2ā -note a, ā do not change when we add a constant to u, since Φ → +∞ as x → 0. We claim that for sufficiently small r > 0, (5.11) ρ(r) = min

Then for 0 < r < r 0 , we have u ≥ ρ(r)Φ on ∂B 1/2 ∪ ∂B r . By the maximum principle, u ≥ ρ(r)Φ on B1/2 \ B r . Hence (5.11) holds for every r ∈ (0, r 0 ). We deduce that r → ρ(r) is increasing on the interval (0, r 0 ), and thus lim r↓0 ρ(r) = a. In particular, (5.12) u ≥ aΦ in B 1/2 \ {0}.

For each 0 < r < r 0 , select x r with |x r | = r such that u(x r ) = ρ(r)Φ(x r ). We now employ a rescaling argument to show that a = ā. That is, we claim that (5.13) lim

To prove (5.13) we consider the cases α * (F ) > 0 and α * (F ) = 0 separately. Suppose first that α * (F ) > 0. For each 0 < r < r 0 and x ∈ B 1/(2r) , we define

Recalling (5.10), (5.12), and the homogeneity of Φ > 0, for every compact set

and F (D 2 v r ) = 0 in B 1/(2r) . Thus using the Hölder estimates, we can find a function v ∈ C(R n \ {0}) and a sequence r j → 0 such that

By taking a further subsequence, we may also assume that r -1 j x rj → y as j → ∞ for some y ∈ ∂B 1 . According to (5.12), we have

to F and -u we see that u is supersolution of (5.1) in the whole ball, and therefore the singularity is removable, giving us alternative (i). In the case that α * ( F ) < 0 and lim inf x→0 (-u(x) + u(0))/ Φ(x) > 0, Lemma 5.6 implies that alternative (v) holds. This completes the proof in the case that u is bounded below. If u is bounded above, then we repeat our argument with -u in place of u, and F in place of F .

5.2.

Classification of singularities at infinity. In this subsection we study the behavior near infinity of a solution u ∈ C(R n \ B 1 ) of the equation

Our approach mirrors the proof of Theorem 5 given in the previous subsection.

) is a solution of (5.18) which is bounded above, and either (i) α * (F ) > 0, or (ii) α * (F ) ≤ 0 and lim inf r→∞ ρ(r

Proof. By subtracting a constant from u we can assume u < 0 in R n \B 1 . According to Lemma 5.1, there exists a constant 0 < C < 1 such that M (r) ≤ Cm(r) for all r ≥ 2. Thus it suffices to show that M (r) is bounded below. We first consider case (i). Recall we assume that min ∂B1 Φ = 1. By the maximum principle, for every r > 1 we have

Evaluating this expression at a point |x| = r 0 such that Φ(x) < 1/2 if |x| ≥ r 0 , we discover that m(r 0 ) ≤ u(x) ≤ (1/2)M (r) for all r > 2, verifying that M (r) is bounded below. We now consider case (ii). By subtracting a positive constant from Φ in the case α = 0, we may assume that Φ < 0 in R n \B 1 . Lemma 5.1 implies that ρ(r) ≤ Cρ(r) for all r ≥ 2. Thus for any ε > 0, there exists r > 2 such that ρ(r) < ε. By the maximum principle, we have

) is a bounded solution of (5.18). Then lim |x|→∞ u(x) exists.

Proof. Let u 0 := lim inf |x|→∞ u(x). Let ε > 0, and define v(x) := u(x) -u 0 + ε. If we take r > 1 very large, then v > 0 in R n \ B r . We can find a point

By the maximum principle,

Letting s → ∞, we deduce that v ≤ Cε in R n \B |x1| . Therefore, lim sup |x|→∞ v(x) ≤ Cε. This implies that lim sup |x|→∞ u(x) ≤ u 0 + Cε.

Lemma 5.9. Assume that u ∈ C(R n \ B 1 ) is a bounded solution of (5.18) and lim |x|→∞ u(x) = 0. Suppose that either (i) α * (F ) > 0 and lim inf r→∞ ρ(r) ≤ 0, or (ii) α * (F ) ≤ 0. Then m(r) ≤ 0 for all r > 1.

Proof. Suppose on the contrary that (i) holds but m(r) > 0 for some r > 1. Let c > 0 be so small that cΦ ≤ m(r) on ∂B r . By the maximum principle, for any ε > 0 we have

Thus u ≥ cΦ, a contradiction to our assumption that lim inf r→∞ ρ(r) ≤ 0. This completes the proof in case (i). In the case that (ii) holds, we argue as in the last paragraph in the proof of Lemma 5.7, to show that for each r 0 > 1 we have u(x) -C ≥ εΦ(x) + min ∂Br 0 (u -C) in B r \ Br0 , for sufficiently large r. Letting r → ∞, ε → 0, and recalling lim |x|→∞ u(x) = 0 yields the result.

Lemma 5.10. Suppose that α * (F ) ≤ 0 and u ∈ C(R n \ B 1 ) is a solution of (5.18) which is bounded above. Then lim sup r→∞ ρ(r) < ∞. Moreover, if a := lim inf r→∞ ρ(r) > 0, then there exists a constant C > 0 such that

Proof. We may assume that u, Φ < 0 in R n \ B 1 . The maximum principle implies that for any r > 2,

In particular, m(r 0 ) ≤ ρ(r) max ∂Br 0 Φ -min ∂B2 Φ for all r 0 > 2. Since we have min ∂Br 0 Φ < min B2 Φ if r 0 is sufficiently large, we deduce that sup r>r0 ρ(r) < ∞.

The Harnack inequality implies that ā := lim sup r→∞ ρ(r) < ∞.

Suppose now that a := lim inf r→∞ ρ(r) > 0. By subtracting a positive constant from u, we may assume that ρ(2) ≥ 2a. By the maximum principle, u ≤ ρ(r)Φ in Br \ B 2 for all r > 2 such that ρ(r) < 2a. Sending r → ∞, we find that u ≤ aΦ in R n \ {0}. A scaling argument very similar to the one in the proof of Lemma 5.5 confirms that

In particular, ā = a. By adding a constant to u so that ρ(2) ≤ 1 2 a, we find that u ≥ aΦ in R n \ B 2 , by the maximum principle. Thus we have (5.19).

Lemma 5.11. Assume that α * (F ) > 0 and u ∈ C(R n \ B 1 ) is a solution of (5.18) such that 0 = lim |x|→∞ u(x), and a := lim inf r→∞ ρ(r) > 0. Then a < ∞ and lim r→∞ ρ(r) = a, that is,

→ a as |x| → ∞.

We can let R → ∞ to obtain We conclude that if α > 0, then whichever strategy γ ∈ Γ Player I selects, the second player can find a control process b ∈ N so that the resulting diffusion process is transient. That is, with probability 1, the process {X t } converges to infinity in the sense that it eventually leaves every bounded set and never returns.

To see this, recall that the random process {X t } eventually must leave any ball since its variance is positive. Furthermore, we see from (6.6) that the process {X t } returns to any given ball infinity often with probability zero. Let us suppose instead that α = α * (F + ) < 0. Then (6.4) can be rewritten as That is, Player I can find a strategy which ensures the process {X t } returns to the origin almost surely. Therefore the process {X t } is recurrent in a very strong sense.

The case α = α * (F + ) = 0 is a compromise between the two cases discussed above. The estimate (6. From these inequalities, we see that v + r,R (x) → 1 as R → ∞, and v + r,R (x) → 0 as r → 0. Thus, Player I has a strategy γ which ensures that the diffusion {X t } will almost surely return to every neighborhood of the origin infinitely many times, but Player II may select a control process which ensures that the process {X t } never touches the origin (almost surely). Since α * (-∆) = 0 in dimension n = 2, we recover the well-known fact that this is how Brownian motion behaves in the plane.