Wiener-Landis criterion for Kolmogorov-type operators
aa r X i v : . [ m a t h . A P ] J a n WIENER-LANDIS CRITERIONFOR KOLMOGOROV-TYPE OPERATORS
ALESSIA KOGOJ, ERMANNO LANCONELLI, AND GIULIO TRALLI
Abstract.
We establish a necessary and sufficient condition for a boundary point to beregular for the Dirichlet problem related to a class of Kolmogorov-type equations. Ourcriterion is inspired by two classical criteria for the heat equation: the Evans–Gariepy’sWiener test, and a criterion by Landis expressed in terms of a series of caloric potentials. Introduction
Aim of this paper is to establish a necessary and sufficient condition for the regularityof a boundary point for the Dirichlet problem related to a class of hypoelliptic evolutionequations of Kolmogorov-type. Our criterion is inspired both to the Evans–Gariepy’s Wienertest for the heat equation, and to a criterion by Landis, for the heat equation too, expressedin terms of a series of caloric potentials.The partial differential operators we are dealing with are of the following type(1.1) L = div ( A ∇ ) + h Bx, ∇i − ∂ t , where A = ( a i,j ) i,j =1 ,...,N and B = ( b i,j ) i,j =1 ,...,N are N × N real and constant matrices, z = ( x, t ) = ( x , . . . , x N , t ) is the point of R N +1 , ∇ = ( ∂ x , . . . , ∂ x N ) , div and h , i stand forthe gradient, the divergence and the inner product in R N , respectively.The matrix A is supposed to be symmetric and positive semidefinite. Moreover, letting E ( s ) := exp ( − sB ) , s ∈ R , we assume that the following Kalman condition is satisfied: the matrix C ( t ) = Z t E ( s ) AE T ( s ) ds, is strictly positive definite for every t >
0. As it is quite well known, the condition C ( t ) > t > L in (1.1), i.e to the smoothness of u whenever L u is smooth (see, e.g., [9]). We also assume the operator L to be homogeneous of degreetwo with respect to a group of dilations in R N +1 . As we will recall in Section 2, this isequivalent to assume A and B taking the blocks form (2.1) and (2.2).Under the above assumptions, one can apply results and techniques from potential theoryin abstract Harmonic Spaces, as presented, e.g, in [2]. As a consequence, for every boundedopen set Ω ⊆ R N +1 and for every function f ∈ C ( ∂ Ω , R ), the Dirichlet problem(1.2) L u = 0 in Ω , u | ∂ Ω = f, Mathematics Subject Classification.
Key words and phrases.
Kolmogorov operators, Potential analysis, Wiener test. has a generalized solution H Ω f in the sense of Perron–Wiener–Brelot–Bauer. The function H Ω f is smooth and solves the equation in (1.2) in the classical sense. However, it may occurthat H Ω f does not assume the boundary datum. A point z ∈ ∂ Ω is called L -regular for Ω iflim z → z H Ω f ( z ) = f ( z ) ∀ f ∈ C ( ∂ Ω , R ) . Aim of this paper is to obtain a characterization of the L -regular boundary points in termsof a serie involving L -potentials of regions in Ω c , the complement of Ω, within different levelsets of Γ, the fundamental solution of L . More precisely, if z ∈ ∂ Ω and λ ∈ ]0 ,
1[ are fixed,we define for k ∈ N Ω ck ( z ) = ( z ∈ Ω c : (cid:18) λ (cid:19) k log k ≤ Γ( z , z ) ≤ (cid:18) λ (cid:19) ( k +1) log ( k +1) ) ∪ { z } . Then, our main result is the following
Theorem 1.1.
Let Ω be a bounded open subset of R N +1 and let z ∈ ∂ Ω . Then z is L -regular for ∂ Ω if and only if (1.3) ∞ X k =1 V Ω ck ( z ) ( z ) = + ∞ . Here and in what follows, if F is a compact subset of R N +1 , V F will denote the L -equilibrium potential of F , and cap ( F ) will denote its L -capacity . We refer to Section 3 forthe precise definitions.From Theorem 1.1, one easily obtains a corollary resembling the Wiener test for theclassical Laplace and Heat operators. Corollary 1.2.
Let Ω be a bounded open subset of R N +1 and z ∈ ∂ Ω . The followingstatements hold: ( i ) if ∞ X k =1 cap (Ω ck ( z )) λ k log k = + ∞ then z is L -regular; ( ii ) if z is L -regular then ∞ X k =1 cap (Ω ck ( z )) λ ( k +1) log ( k +1) = + ∞ . We can make the sufficient condition for the L -regularity more concrete and more geo-metrical with the following corollary. Corollary 1.3.
Let Ω be a bounded open subset of R N +1 and z ∈ ∂ Ω . If ∞ X k =1 | Ω ck ( z ) | λ Q +2 Q k log k = + ∞ then z is L -regular. In particular, the L -regularity of z is ensured if Ω has the exterior L -cone property at z . IENER-LANDIS CRITERION FOR KOLMOGOROV-TYPE OPERATORS 3 If E is a subset of either R N or R N +1 , | E | stands for the relative Lebesgue measure.Moreover, Q is the homogeneous dimension recalled in Section 2, and the L -cone propertywill be defined precisely in Section 7. We just mention here that it is a natural adaptationof the parabolic cone condition to the homogeneities of the operator L .Before proceeding, we would like to comment on Theorem 1.1 and Corollary 1.2.A boundary point regularity test for the heat equation involving infinite sum of (caloric)potentials was showed by Landis in [12]. A similar test for a Kolmogorov equation in R was obtained by Scornazzani in [14]. Our Theorem 1.1 contains, extends, and improvesthe criterion in [14]. The Wiener test for the heat equation was proved by Evans andGariepy in [3]. The extension of such a criterion to the Kolmogorov operators (1.1) isan open, and seemingly difficult, problem. Our Corollary 1.2, which is a straightforwardconsequence of Theorem 1.1, is a Wiener-type test giving necessary and sufficient conditionswhich look “almost the same”. As a matter of fact, in Theorem 1.1 we have considered the L -potentials of the compact sets Ω ck ( z ) which are built by the difference of two consecutivesuper-level sets of Γ( z , · ). These level sets correspond with the sequence of values λ − k log k .The exact analogue of the Evans-Gariepy criterion would have required the sequence withinteger exponents λ − k . The presence of the logarithmic term, which makes the growth ofthe exponents slightly superlinear, is crucial for our proof of Theorem 1.1. Moreover, suchpresence is also the responsible for the non-equivalence of the necessary and the sufficientcondition in Corollary 1.2. To complete our historical comments, we mention that a potentialanalysis for Kolmogorov operators of the kind (1.1) first appeared in [14], in [4], and in [9].We also mention that the cone criterion contained in Corollary 1.3 has been recently provedin [6], where such a boundary regularity test has been showed for classes of operators moregeneral than (1.1). For further bibliographical notes concerning Wiener-type tests for bothclassical and degenerate operators, we refer the reader to [10].The paper is organized as follows. In Section 2 we show some structural properties of L and fix some notations. Section 3 is devoted to the potential theory for L , while in Section4 a crucial estimate of the ratio between the fundamental solution Γ at two different poles isproved. In Section 5 the only if part of Theorem 1.1 is proved. The if part, the core of ourpaper, is proved in Section 6, where the estimates of Section 4 play a crucial rˆole. Section7 is devoted to the proof of Corollary 1.2 and Corollary 1.3.2. Structural properties of L In [9, Section 1] it is proved that the operator L is left-translation invariant with respectto the Lie group K whose underlying manifold is R N +1 , endowed with the composition law( x, t ) ◦ ( ξ, s ) = ( ξ + E ( s ) x, t + s ) . Furthermore, a fundamental solution for L is given byΓ ( z, ζ ) = Γ (cid:0) ζ − ◦ z (cid:1) for z, ζ ∈ R N +1 , where, Γ ( z ) = Γ ( x, t ) = t ≤ , (4 π ) − N/ √ det C ( t ) exp (cid:0) − (cid:10) C − ( t ) x, x (cid:11) − t tr B (cid:1) for t > . A.E. KOGOJ, E. LANCONELLI, AND G. TRALLI
We assume the operator L to be homogeneous of degree two with respect to a groupof dilations. This last assumption, together with the hypoellipticity of L , implies that thematrices A and B take the following form with respect to some basis of R N (see again [9,Section 1]):(2.1) A = (cid:20) A
00 0 (cid:21) for some p × p symmetric and positive definite matrix A ( p ≤ N ), and(2.2) B = . . . B B . . . . . . B n , where B j is a p j − × p j block with rank p j ( j = 1 , , ..., n ), p ≥ p ≥ ... ≥ p n ≥ p + p + ... + p n = N . For such a choice we have tr B = 0, and the family of automorphismsof K making L homogeneous of degree two can be taken as δ r : R N +1 −→ R N +1 , δ r ( x, t ) = δ r ( x ( p ) , x ( p ) , . . . , x ( p n ) , t ):= (cid:16) rx ( p ) , r x ( p ) , . . . , r n +1 x ( p n ) , r t (cid:17) ,x ( p i ) ∈ R p i , i = 0 , . . . , n, r > . We denote by Q + 2 (= p + 3 p + ... + (2 n + 1) p n + 2) the homogeneous dimension of K with respect to ( δ r ) r> . We explicitly remark that Q is the homogenous dimension of R N with respect to the dilations D r : R N −→ R N , D r ( x ) = (cid:16) rx ( p ) , r x ( p ) , . . . , r n +1 x ( p n ) (cid:17) . Under these notations, the matrix C ( t ) and the fundamental solution of L with pole atthe origin can be written as follows ([9, Proposition 2.3], see also [7]): C ( t ) = D √ t C (1) D √ t and Γ ( x, t ) = t ≤ , c N t Q exp (cid:16) − h C − (1) D √ t x, D √ t x i (cid:17) for t > . We observe that Γ is δ r -homogeneous of degree − Q .Throughout the paper we denote by |·| the Euclidean norms in R N , R p k or R . We alsodenote, for x ∈ R N , | x | C := 14 (cid:10) C − (1) x, x (cid:11) . For all x ∈ R N , we have(2.3) | E (1) x | C ≥ σ C | x | where 4 σ C is the smallest eigenvalue of the positive definite matrix E T (1) C − (1) E (1). Werecall that the homogeneous norm k·k : R N −→ R + is a D λ -homogeneous function of degree1 defined as follows k x k = n X i =0 (cid:12)(cid:12)(cid:12) x ( p i ) (cid:12)(cid:12)(cid:12) i +1 , for x = (cid:16) x ( p ) , . . . , x ( p n ) (cid:17) ∈ R p × . . . R p n = R N . We call homogeneous cylinder of radius r > C r := (cid:8) x ∈ R N : k x k ≤ r (cid:9) × (cid:8) t ∈ R : | t | ≤ r (cid:9) = δ r ( C ) , and define C r ( z ) := z ◦ C r . Remark 2.1.
The norms k·k and |·| can be compared as follows (2.4) σ min n | x | , | x | n +1 o ≤ k x k ≤ ( n + 1) max n | x | , | x | n +1 o ∀ x ∈ R N , where σ = min | x | =1 k x k .Indeed, on one side we simply have k x k ≤ n X i =0 | x | i +1 ≤ ( n + 1) max n | x | , | x | n +1 o ∀ x ∈ R N . On the other hand, for any x = 0 , we get k x k min n | x | , | x | n +1 o ≥ n X i =0 (cid:12)(cid:12) x ( p i ) (cid:12)(cid:12) i +1 | x | i +1 = n X i =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) x | x | (cid:19) ( p i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i +1 = (cid:13)(cid:13)(cid:13)(cid:13) x | x | (cid:13)(cid:13)(cid:13)(cid:13) ≥ σ. Some recalls from Potential Theory for L : L -potentials and L -capacity We briefly collect here some notions and results from Potential Theory applied to theoperator L .For every open set Ω ⊆ R N +1 we denote L (Ω) := { u ∈ C ∞ (Ω) | L u = 0 } . and we call L - harmonic in Ω the functions in L (Ω) . We say that a bounded open set V ⊆ Ω is L -regular if for every continuous function ϕ : ∂V −→ R , there exists a unique function, h Vϕ in L ( V ), continuous in V , such that h Vϕ | ∂V = ϕ. Moreover, if ϕ ≥ h Vϕ ≥ u : Ω −→ ] − ∞ , ∞ ] is called L -superharmonic in Ω if( i ) u is lower semi-continuous and u < ∞ in a dense subset of Ω;( ii ) for every regular set V , V ⊆ Ω, and for every ϕ ∈ C ( ∂V, R ), ϕ ≤ u | ∂V , it follows u ≥ h Vϕ in V. We will denote by L (Ω) the family of the L -superharmonic functions in Ω. Since the operator L endows R N +1 with a structure of β -harmonic space satisfying the Doob convergence A.E. KOGOJ, E. LANCONELLI, AND G. TRALLI property (see [13, 2, 6]), by the Wiener resolutivity theorem, for every f ∈ C ( ∂ Ω), theDirichlet problem ( L u = 0 in Ω u | ∂ Ω = f has a generalized solution in the sense of Perron–Wiener–Bauer–Brelot given by H Ω f := inf { u ∈ L (Ω) | lim inf Ω ∋ z → ζ u ( z ) ≥ f ( ζ ) ∀ ζ ∈ ∂ Ω } . The function H Ω f is C ∞ (Ω) and satisfies L u = 0 in Ω in the classical sense. However, itis not true, in general, that H Ω f continuously takes the boundary values prescribed by f . Apoint z ∈ ∂ Ω such that lim Ω ∋ z → z H Ω f ( z ) = f ( z ) for every f ∈ C ( ∂ Ω)is called L -regular for Ω.For our regularity criteria we still need a few more definitions. We denote by M ( R N +1 )the collection of all nonnegative Radon measure on R N +1 and we callΓ µ ( z ) := Z R N +1 Γ( z, ζ ) dµ ( ζ ) , z ∈ R N +1 , the L -potential of µ .If F is a compact set of R N +1 and M ( F ) is the collection of all nonnegative Radonmeasure on R N +1 with support in F , the L -capacity of F is defined ascap ( F ) := sup { µ ( R N +1 ) | µ ∈ M ( F ) , Γ µ ≤ R N +1 } . We list some properties of the L -capacities cap. For every F , F and F compact subsets of R N +1 , we have:( i ) cap ( F ) < ∞ ;( ii ) if F ⊆ F , then cap ( F ) ≤ cap ( F );( iii ) cap ( F ∪ F ) ≤ cap ( F ) + cap ( F );( iv ) cap ( z ◦ F ) = cap ( F ) for every z ∈ R N +1 ;( v ) cap ( δ r ( F )) = r Q cap ( F ) for every r > vi ) if F = A × { τ } for some compact set A ⊂ R N , then cap ( F ) = | A | ;( vii ) if F ⊂ R N × [ a, b ], then we have(3.1) cap ( F ) ≥ | F | b − a . The properties ( i ) − ( v ) are quite standard, and they follow from the features of Γ. We wantto spend few words on the last two properties. Property ( vi ) was proved in [8, Proposizione5.1] in the case of the heat operator, namely with the capacity build on the Gauss-Weierstrasskernel. It can be proved verbatim proceeding in our situation: the main tools are the factsthat Γ has integral 1 over R N , and it reproduces the solutions of the Cauchy problems.Property ( vii ) appears to be new even in the classical parabolic case (at least to the best ofour knowledge), and it can be deduced readily from ( vi ). As a matter of fact, if a compactset F lies in a strip R N × [ a, b ], we have( b − a ) cap ( F ) = Z ba cap ( F ) d τ ≥ Z ba cap ( F ∩ { t = τ } ) d τ = Z ba | F ∩ { t = τ }| d τ = | F | . IENER-LANDIS CRITERION FOR KOLMOGOROV-TYPE OPERATORS 7
The last notions we need are the ones of reduced function and of balayage of 1 on F . Theyare respectively defined by W F := inf { v | v ∈ L ( R N +1 ) , v ≥ R N +1 , v ≥ F } , and V F ( z ) = lim inf ζ −→ z W F ( ζ ) , z ∈ R N +1 . From general balayage theory we have that V F is less or equal than 1 everywhere, identically1 in the interior of F , it vanishes at infinity, is a superharmonic function on R N +1 andharmonic on R N +1 \ ∂F . Furthermore, the following properties will be useful for us. Let F, F , F be compact subsets of R N +1 , and let ( F n ) n ∈ N be a sequence of compact subsets of R N +1 , we have:( i ) if F ⊆ F ⊆ R N +1 , then V F ≤ V F ;( ii ) V F ∪ F ≤ V F + V F ;( iii ) if F ⊆ S n ∈ N F n , then V F ≤ P ∞ n =1 V F n . The first property is a consequence of the definition of balayage; for the second and thethird one we refer respectively to [1, Proposition 5.3.1] and [1, Theorem 4.2.2 and Corollary4.2.2].Now, following the same lines of the proof of [8, Teorema 1.1], we have the existence of aunique measure µ F ∈ M ( F ) such that(3.2) V F ( z ) = Γ µ F ( z ) = Z R N +1 Γ( z, ζ ) d µ F ( ζ ) ∀ z ∈ R N +1 , and µ F ( R N +1 ) = cap ( F ) .V F is also called the L - equilibrium potential of F and µ F the L - equilibrium measure of F .The proof of this fact relies on the good behavior of Γ, a representation formula of Riesz-typefor L -superharmonic functions proved in [2, Theorem 5.1], and a Maximum Principle for L (see [2, Proposition 2.3]).Fix now a bounded open set Ω compactly contained in R N +1 , and z = ( x , t ) ∈ ∂ Ω.Let us denote by G r = { ( x, t ) ∈ C r ( z ) r Ω : t ≤ t } . From general balayage theory and proceeding, e.g., as in [11, Theorem 4.6], we can charac-terize the regularity of the boundary point of Ω by the following condition:the point z ∈ ∂ Ω is L -regular if and only if(3.3) lim r → V G r ( z ) > . A crucial estimate
We start by recalling the following identity, whose proof can be found in [9, Remark 2.1](see also [7]),(4.1) E ( λ s ) D λ = D λ E ( s ) ∀ λ > , ∀ s ∈ R . In what follows we will need the following lemma.
Lemma 4.1.
For > t > τ we have the following matrix inequality E T ( t ) C − ( t − τ ) E ( t ) ≥ C − ( − τ ) . A.E. KOGOJ, E. LANCONELLI, AND G. TRALLI
Proof.
Since for symmetric positive definite matrices we have M ≤ M ⇒ M − ≥ M − (see [5, Corollary 7.7.4]) and recalling that E − ( t ) = E ( − t ), it is enough to prove that(4.2) E ( − t ) C ( t − τ ) E T ( − t ) ≤ C ( − τ ) . From the very definition of the matrix C we get E ( − t ) C ( t − τ ) E T ( − t ) = e tB (cid:18)Z t − τ e − sB Ae − sB T ds (cid:19) e tB T = Z t − τ e ( t − s ) B Ae ( t − s ) B T ds= Z − τ − t e − σB Ae − σB T d σ. Since − τ > − t > A is nonnegative definite, we have Z − τ − t e − σB Ae − σB T d σ ≤ Z − τ e − σ B Ae − σ B T d σ = C( − τ )which proves (4.2) and the lemma. (cid:3) A crucial role in the proof of our main theorem will be played by the ratio Γ( z,ζ )Γ(0 ,ζ ) , for z = ( x, t ) and ζ = ( ξ, τ ) with 0 > t > τ . We use the following notations µ = − t − τ ∈ (0 , , M ( z ) = (cid:12)(cid:12)(cid:12) D √− t x (cid:12)(cid:12)(cid:12) , M ( ζ ) = (cid:12)(cid:12)(cid:12) D √− τ ξ (cid:12)(cid:12)(cid:12) . Lemma 4.2.
There exists a positive constant C such that, for any z = ( x, t ) , ζ = ( ξ, τ ) with > t > τ and µ ≤ min { , σ ( n +1) } , we have Γ( z, ζ )Γ(0 , ζ ) ≤ (cid:18) − µ (cid:19) Q e C √ µM ( z ) M ( ζ ) . Proof.
In our notations we can writeΓ( z, ζ )Γ(0 , ζ ) = ( t − τ ) − Q e − (cid:12)(cid:12)(cid:12)(cid:12) D √ t − τ ( x − E ( t − τ ) ξ ) (cid:12)(cid:12)(cid:12)(cid:12) C ( − τ ) − Q e − (cid:12)(cid:12)(cid:12)(cid:12) D √− τ ( E ( − τ ) ξ ) (cid:12)(cid:12)(cid:12)(cid:12) C = (cid:18) − µ (cid:19) Q e (cid:12)(cid:12)(cid:12)(cid:12) D √− τ ( E ( − τ ) ξ ) (cid:12)(cid:12)(cid:12)(cid:12) C − (cid:12)(cid:12)(cid:12)(cid:12) D √ t − τ ( x − E ( t − τ ) ξ ) (cid:12)(cid:12)(cid:12)(cid:12) C . Let us deal with the exponential term (cid:12)(cid:12)(cid:12) D √− τ ( E ( − τ ) ξ ) (cid:12)(cid:12)(cid:12) C − (cid:12)(cid:12)(cid:12) D √ t − τ ( x − E ( t − τ ) ξ ) (cid:12)(cid:12)(cid:12) C =(4.3)= 14 (cid:10) C − ( − τ ) E ( − τ ) ξ, E ( − τ ) ξ (cid:11) − (cid:10) C − ( t − τ ) ( x − E ( t − τ ) ξ ) , ( x − E ( t − τ ) ξ ) (cid:11) . Lemma 4.1 says in particular that we have (cid:10) C − ( − τ ) E ( − τ ) ξ, E ( − τ ) ξ (cid:11) − (cid:10) C − ( t − τ ) E ( t − τ ) ξ, E ( t − τ ) ξ (cid:11) ≤ . IENER-LANDIS CRITERION FOR KOLMOGOROV-TYPE OPERATORS 9
Using this in (4.3) we get (cid:12)(cid:12)(cid:12) D √− τ ( E ( − τ ) ξ ) (cid:12)(cid:12)(cid:12) C − (cid:12)(cid:12)(cid:12) D √ t − τ ( x − E ( t − τ ) ξ ) (cid:12)(cid:12)(cid:12) C ≤ (4.4) ≤ − (cid:10) C − ( t − τ ) x, x (cid:11) + 12 (cid:10) C − ( t − τ ) x, E ( t − τ ) ξ (cid:11) ≤ (cid:10) C − ( t − τ ) x, E ( t − τ ) ξ (cid:11) ≤ (cid:0)(cid:10) C − ( t − τ ) x, x (cid:11) (cid:10) C − ( t − τ ) E ( t − τ ) ξ, E ( t − τ ) ξ (cid:11)(cid:1) . We are going to bound (cid:10) C − ( t − τ ) x, x (cid:11) and (cid:10) C − ( t − τ ) E ( t − τ ) ξ, E ( t − τ ) ξ (cid:11) separately.We have (cid:10) C − ( t − τ ) x, x (cid:11) = (cid:28) C − (cid:18) µ − (cid:19) D √− t x, D √− t x (cid:29) ≤ (cid:13)(cid:13)(cid:13)(cid:13) C − (cid:18) µ − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) M ( z ) , where k A k stands for the operator norm of a matrix A (i.e. its biggest eigenvalue forsymmetric matrices). By (2.4), for any vector v with | v | = 1 we getmin n(cid:12)(cid:12) D √ µ v (cid:12)(cid:12) , (cid:12)(cid:12) D √ µ v (cid:12)(cid:12) n +1 o ≤ σ √ µ k v k ≤ n + 1 σ √ µ max n | v | , | v | n +1 o = n + 1 σ √ µ. From µ ≤ σ ( n +1) we then deduce (cid:12)(cid:12) D √ µ v (cid:12)(cid:12) ≤ n +1 σ √ µ . Hence, since µ is also less than , (cid:28) C − (cid:18) µ − (cid:19) v, v (cid:29) = (cid:10) C − (1 − µ ) D √ µ v, D √ µ v (cid:11) ≤ (cid:13)(cid:13) C − (1 − µ ) (cid:13)(cid:13) (cid:12)(cid:12) D √ µ v (cid:12)(cid:12) ≤ ( n + 1) σ (cid:13)(cid:13) C − (1 − µ ) (cid:13)(cid:13) µ ≤ ( n + 1) σ (cid:13)(cid:13)(cid:13)(cid:13) C − (cid:18) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) µ ∀ | v | = 1 . This gives(4.5) (cid:10) C − ( t − τ ) x, x (cid:11) ≤ ( n + 1) σ (cid:13)(cid:13)(cid:13)(cid:13) C − (cid:18) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) µM ( z ) . On the other hand, by the commutation property (4.1), we get (cid:10) C − ( t − τ ) E ( t − τ ) ξ, E ( t − τ ) ξ (cid:11) = D C − (1 − µ ) D √− τ E ( t − τ ) ξ, D √− τ E ( t − τ ) ξ E ≤ (cid:13)(cid:13) C − (1 − µ ) (cid:13)(cid:13) (cid:12)(cid:12)(cid:12) D √− τ E ( t − τ ) ξ (cid:12)(cid:12)(cid:12) = (cid:13)(cid:13) C − (1 − µ ) (cid:13)(cid:13) (cid:12)(cid:12)(cid:12) E (1 − µ ) D √− τ ξ (cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13) C − (1 − µ ) (cid:13)(cid:13) (cid:13)(cid:13) E T (1 − µ ) E (1 − µ ) (cid:13)(cid:13) M ( ζ ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) C − (cid:18) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13) E T (1 − µ ) E (1 − µ ) (cid:13)(cid:13) M ( ζ ) . Since 0 < µ ≤ , the term (cid:13)(cid:13) E T (1 − µ ) E (1 − µ ) (cid:13)(cid:13) is bounded from above by a universalconstant C . Thus we have(4.6) (cid:10) C − ( t − τ ) E ( t − τ ) ξ, E ( t − τ ) ξ (cid:11) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) C − (cid:18) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) M ( ζ ) . Plugging (4.5) and (4.6) in (4.4), we get (cid:12)(cid:12)(cid:12) D √− τ ( E ( − τ ) ξ ) (cid:12)(cid:12)(cid:12) C − (cid:12)(cid:12)(cid:12) D √ t − τ ( x − E ( t − τ ) ξ ) (cid:12)(cid:12)(cid:12) C ≤ C n + 1 σ (cid:13)(cid:13)(cid:13)(cid:13) C − (cid:18) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) √ µM ( z ) M ( ζ ) . Therefore Γ( z, ζ )Γ(0 , ζ ) ≤ (cid:18) − µ (cid:19) Q e C √ µM ( z ) M ( ζ ) with C = C n +1 σ (cid:13)(cid:13) C − (cid:0) (cid:1)(cid:13)(cid:13) . (cid:3) Necessary condition for regularity
The characterization in (3.3), together with the following lemma, will give the necessityof (1.3) in Theorem 1.1.
Lemma 5.1.
For every fixed p ∈ N , let us split the set G r as follows G r = G pr ∪ G ∗ pr , where G pr = ( z ∈ G r | Γ( z , z ) ≥ (cid:18) λ (cid:19) p log p ) ∪ { z } , and G ∗ pr = ( z ∈ G r | Γ( z , z ) ≤ (cid:18) λ (cid:19) p log p ) . Then, lim r −→ V G r ( z ) = lim r −→ V G pr ( z ) . Proof.
From the monotonicity and subadditivity properties of the balayage, we have V G pr ( z ) ≤ V G r ( z ) ≤ V G pr ( z ) + V G ∗ pr ( z ) . Furthermore, by (3.2), V G ∗ pr ( z ) ≤ (cid:18) λ (cid:19) p log p cap ( G ∗ pr ) . On the other hand, from the monotonicity and homogeneity properties of the capacities, itfollows cap ( G ∗ pr ) ≤ cap ( C r ( z )) = cap ( z ◦ δ r ( C )) = r Q cap ( C ( z )) . Hence cap ( G ∗ pr ) goes to zero as r goes to zero. This proves the lemma. (cid:3) Proof of necessary condition in Theorem 1.1.
Assume ∞ X k =1 V Ω ck ( z ) ( z ) < + ∞ . We are going to prove the non regularity of the boundary point z . The assumption impliesthat for every ε >
0, there exists p ∈ N such that ∞ X k = p V Ω ck ( z ) ( z ) < ε. On the other hand, with the notations of the previous lemma, for any positive rG pr ⊆ ∞ [ k = p Ω ck ( z ) , IENER-LANDIS CRITERION FOR KOLMOGOROV-TYPE OPERATORS 11 so that, V G pr ( z ) ≤ ∞ X k = p V Ω ck ( z ) ( z ) < ε. Then, from Lemma 5.1, we get lim r → V G r ( z ) ≤ ε for every ε >
0, which implieslim r −→ V G r ( z ) = 0 . Hence, by (3.3), the boundary point z is not L -regular. (cid:3) Sufficient condition for regularity
In this section we prove the if part of Theorem 1.1. This is the core of our main resultand requires three lemmas. Lemma 6.1.
Suppose we have a sequence of compact sets { F k } k ∈ N in R N +1 such that ( F k ∩ F h = ∅ if k = h, ∀ r > ∃ ¯ k such that F k ⊆ G r for k ≥ ¯ k. Suppose also that the following two conditions hold true: ( i ) + ∞ X k =1 V F k ( z ) = + ∞ ;( ii ) sup h = k sup (cid:26) Γ( z, ζ )Γ( z , ζ ) : z ∈ F h , ζ ∈ F k (cid:27) ≤ M . Then we have V G r ( z ) ≥ M for every positive r. Proof.
Let
A > M , and fix any r >
0. Let us pick m, n ∈ N with m < n such that n [ k = m F k ⊆ G r and n X k = m V F k ( z ) ≥ A. We are going to denote by G m,n = S nk = m F k and by W m,n ( z ) = P nk = m V F k ( z ). We want toestimate W m,n on G m,n .Take z ∈ G m,n . We have then z ∈ F h for some h ∈ { m, . . . , n } . Of course we have V F h ( z ) ≤
1. On the other hand, if k = h we get V F k ( z ) = Z F k Γ( z, ζ ) d µ k ( ζ ) = Z F k Γ( z, ζ )Γ( z , ζ ) Γ( z , ζ ) d µ k ( ζ ) ≤ M V F k ( z ) . Hence V F k ≤ M V F k ( z ) in F h . By the continuity of the equilibrium potentials (outside oftheir relative compact sets) there exists an open neighborhood O h of F h such that V F k ≤ M V F k ( z ) + 12 k ∀ k ∈ { m, . . . , n } , k = h. We put O = S h O h . In O we get W m,n ≤ M n X k = m V F k ( z ) + n X k = m k ≤ M n X k = m V F k ( z ) . If we consider the function v m,n = M P nk = m V Fk ( z ) W m,n , we thus get v m,n ≤ O .Moreover, the function v m,n is a nonnegative H -superharmonic in R N +1 , it is H -harmonicin R N +1 r G m,n , and it vanishes at the infinity. If we take any function u ∈ Φ G m,n we have u − v m,n ∈ H ( R N +1 r G m,n ) , lim inf z →∞ u ( z ) − v m,n ( z ) ≥ , lim inf z → ζ ∈ ∂G m,n u ( z ) − v m,n ( z ) ≥ u ( ζ ) − ≥ . The maximum principle infers that u − v m,n has to be nonnegative in R N +1 r G m,n . Onthe other hand, u ≥ ≥ v m,n in G m,n . Therefore u ≥ v m,n in R N +1 , for every u ∈ Φ G m,n .This implies that V G m,n ( z ) ≥ v m,n ( z ) = W m,n ( z )2 + M P nk = m V F k ( z ) for all z ∈ R N +1 . In particular this has to be true at z = z , i.e. V G m,n ( z ) ≥ P nk = m V F k ( z )2 + M P nk = m V F k ( z ) . Since the function s s M s is increasing, we deduce V G m,n ( z ) ≥ A M A > M . This concludes the proof since V G r ≥ V G m,n . (cid:3) In order to simplify the notations, from now on we assume z = 0 ∈ ∂ Ω. This is notrestrictive because of the left-invariance property. We want to choose suitably the compactsets F k of the previous lemma. For any fixed λ ∈ (0 , ck (0) = ( z ∈ Ω c : (cid:18) λ (cid:19) k log k ≤ Γ(0 , z ) ≤ (cid:18) λ (cid:19) ( k +1) log ( k +1) ) . Now, we set α ( k ) = k log k and denote T k = max ( x,t ) ∈ Ω ck (0) − t = (cid:16) c N λ α ( k ) (cid:17) Q . We fix q ∈ N such that(6.1) q ≥ q := 4 + m log (cid:0) λ (cid:1) , where m = max (cid:26) , Q log 6 , σ C log 6 , Q log 2log 8 , Q log ( n +1 σ )log 8 (cid:27) , and σ C , σ are the constants in (2.3) and (2.4). We also denote by p = 1 + h q i = 1 + the integer part of q . IENER-LANDIS CRITERION FOR KOLMOGOROV-TYPE OPERATORS 13 So q ≤ p ≤ q < q −
1. For any k ∈ N we want to consider the setsΩ ckq (0) = ( z ∈ Ω c : (cid:18) λ (cid:19) α ( kq ) ≤ Γ(0 , z ) ≤ (cid:18) λ (cid:19) α ( kq +1) ) . Moreover, we put(6.2) Ω ckq (0) = (cid:0) Ω ckq (0) ∩ { t ≥ − T ∗ kq } (cid:1) ∪ (cid:0) Ω ckq (0) ∩ { t ≤ − T ∗ kq } (cid:1) := F (0) k ∪ F k where T ∗ kq = T kq + p = (cid:16) c N λ α ( kq + p ) (cid:17) Q . First we notice that, since kq + p < q ( k + 1), F k lies strictly below F k +1 , namely(6.3) min ( x,t ) ∈ F h t = − T hq > − T ∗ kq = max ( ξ,τ ) ∈ F k τ ∀ h, k ∈ N , h > k. Lemma 6.2.
We have + ∞ X k =1 V F (0) k (0) < + ∞ . Proof.
We are going to prove that F (0) k is contained in a homogeneous cylinder C r k so that(6.4) + ∞ X k =1 (cid:18) λ (cid:19) α ( kq +1) r Qk < + ∞ . This is enough to prove the statement since V F (0) k (0) = Z F (0) k Γ(0 , ζ ) d µ F (0)k ( ζ ) ≤ (cid:18) λ (cid:19) α ( kq +1) cap ( F (0) k ) , and by monotonicity and homogeneity we have cap ( F (0) k ) ≤ cap ( C r k ) = cap ( C ) r Qk . Inorder to prove (6.4), we have to find a good bound for r k .Fix z = ( x, t ) ∈ F (0) k . Since in particular z ∈ Ω ckq (0), we have (cid:12)(cid:12)(cid:12) D √− t ( E ( − t ) x ) (cid:12)(cid:12)(cid:12) C ≤ log c N λ α ( kq ) ( − t ) Q ! . On the other hand, by (4.1) and (2.3), we get (cid:12)(cid:12)(cid:12) D √− t ( E ( − t ) x ) (cid:12)(cid:12)(cid:12) C = (cid:12)(cid:12)(cid:12) E (1) D √− t x (cid:12)(cid:12)(cid:12) C ≥ σ C (cid:12)(cid:12)(cid:12) D √− t x (cid:12)(cid:12)(cid:12) and then(6.5) (cid:12)(cid:12)(cid:12) D √− t x (cid:12)(cid:12)(cid:12) ≤ σ C log c N λ α ( kq ) ( − t ) Q ! . Therefore, from (2.4), we deduce1 √− t k x k = (cid:13)(cid:13)(cid:13) D √− t x (cid:13)(cid:13)(cid:13) ≤ ( n + 1) max (cid:26)(cid:12)(cid:12)(cid:12) D √− t x (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) D √− t x (cid:12)(cid:12)(cid:12) n +1 (cid:27) ≤ ( n + 1) max σ C log c N λ α ( kq ) ( − t ) Q ! , σ n +1 C log n +1) c N λ α ( kq ) ( − t ) Q ! . Let us remark that from our choice α ( k ) = k log k we can check that the sequence α ( kq + p ) − α ( kq ) is monotone increasing . In particular α ( kq + p ) − α ( kq ) ≥ α ( q ) − α ( q ) ≥ q log ( q ) ≥ q log 6. By our choice of q (6.1), we have then α ( kq + p ) − α ( kq ) ≥ Q λ ) and so( T ∗ kq ) Q = c N λ α ( kq + p ) ≤ c N λ α ( kq ) e − Q ∀ k. This fact and the fact that the functions s s log β αs Q are increasing in the interval(0 , e − β α Q ] allow to bound the term k x k further. Indeed, having 0 < − t ≤ T ∗ kq , we get k x k ≤ ( n + 1) max √− tσ C log c N λ α ( kq ) ( − t ) Q ! , √− tσ n +1 C log n +1) c N λ α ( kq ) ( − t ) Q ! ≤ ( n + 1) max q T ∗ kq σ C log c N λ α ( kq ) ( T ∗ kq ) Q ! , q T ∗ kq σ n +1 C log n +1) c N λ α ( kq ) ( T ∗ kq ) Q ! . Since q log 6 ≥ σ C log ( λ ) we have also ( T ∗ kq ) Q ≤ c N λ α ( kq ) e − σ C , which says log c N λ α ( kq ) ( T ∗ kq ) Q ! ≥ σ C and implies k x k ≤ ( n + 1) σ C q T ∗ kq log c N λ α ( kq ) ( T ∗ kq ) Q ! . Summing up, we have just proved that( x, t ) ∈ F (0) k = ⇒ k x k ≤ n +1 σ C q T ∗ kq log c N λ α ( kq ) ( T ∗ kq ) Q ! =: r k and0 < − t ≤ T ∗ kq ≤ ( n + 1) T ∗ kq ≤ r k , namely F (0) k ⊆ C r k . We are left with verifying (6.4) with this definition of r k . We have thus to prove that + ∞ X k =1 (cid:18) λ (cid:19) α ( kq +1) − α ( kq + p ) ( α ( kq + p ) − α ( kq )) Q < + ∞ . The sequences α ( kq + 1) − α ( kq + p ) and α ( kq + p ) − α ( kq ) are asymptotically equivalentrespectively to (1 − p ) log( kq + p ) and p log( kq + p ). Hence, the series is equivalent to + ∞ X k =1 kq + p ) ( p −
1) log λ log Q ( kq + p ) , which is convergent since p ≥ q > λ ) . This proves (6.4), and therefore the lemma. (cid:3) Lemma 6.3.
There exists a positive constant M such that Γ( z, ζ )Γ(0 , ζ ) ≤ M ∀ z ∈ F h , ∀ ζ ∈ F k , ∀ h, k ∈ N , h = k. IENER-LANDIS CRITERION FOR KOLMOGOROV-TYPE OPERATORS 15
Proof.
Fix any h, k ∈ N with h = k . If h ≤ k −
1, then Γ( z, ζ ) = 0 and the statement istrivial. Thus, suppose h ≥ k + 1. For every z = ( x, t ) ∈ F h and ζ = ( ξ, τ ) ∈ F k we have µ = − t − τ ≤ T hq T ∗ kq = (cid:18) λ α ( hq ) λ α ( kq + p ) (cid:19) Q = (cid:18) λ (cid:19) Q ( α ( kq + p ) − α ( hq )) . By monotonicity we have α ( hq ) − α ( kq + p ) ≥ α ( kq + q ) − α ( kq + p ) ≥ α (2 q ) − α ( q + p ) ≥ α (2 q ) − α ( q + 1) ≥ ( q −
1) log (2 q ). By our choice of q (6.1) we have then α ( hq ) − α ( kq + p ) ≥ (cid:16) q − (cid:17) log (8) ≥ Q { log 2 , log ( n +1 σ ) } log ( λ )which implies µ ≤ min { , σ ( n +1) } . This fact allows us to exploit Lemma 4.2 and getΓ( z, ζ )Γ(0 , ζ ) ≤ (cid:18) − µ (cid:19) Q e C √ µM ( z ) M ( ζ ) ≤ Q e C √ µM ( z ) M ( ζ ) , for some structural positive constant C . To prove the statement we need to show that theterm µM ( z ) M ( ζ )is uniformly bounded for z ∈ F h and ζ ∈ F k . By estimating as in (6.5) we have M ( z ) = (cid:12)(cid:12)(cid:12) D √− t x (cid:12)(cid:12)(cid:12) ≤ σ C log c N λ α ( hq ) ( − t ) Q ! ≤ σ C log c N λ α ( hq ) ( T ∗ hq ) Q ! = 1 σ C log (cid:18) λ (cid:19) ( α ( hq + p ) − α ( hq )) , and analogously M ( ζ ) ≤ σ C log (cid:18) λ (cid:19) ( α ( kq + p ) − α ( kq )) . In order to bound µM ( z ) M ( ζ ) we are thus going to estimate the term( α ( kq + p ) − α ( kq ))( α ( hq + p ) − α ( hq )) (cid:18) λ Q (cid:19) ( α ( kq + p ) − α ( hq )) ≤ ( α ( kq + p ) − α ( kq ))( α ( hq + p ) − α ( hq )) (cid:18) λ Q (cid:19) ( α ( kq + p ) − α ( kq + q − α ( hq − − α ( hq )) = ( α ( kq + p ) − α ( kq )) (cid:18) λ Q (cid:19) ( α ( kq + p ) − α ( kq + q − ! ( α ( hq + p ) − α ( hq )) (cid:18) λ Q (cid:19) ( α ( hq − − α ( hq )) ! =: A k · B h . Since p < q − α ( n + s ) − α ( n ) is asymptotically equivalent to s log( n + s ) as n goesto ∞ , it is easy to check that the sequences A k and B h are convergent to 0. Therefore theyare a fortiori bounded. This proves the lemma. (cid:3) Proof of sufficient condition in Theorem 1.1.
As we noticed, it is not restrictive to assume z = 0. Then, our assumption implies ∞ X k =1 V Ω ck (0) (0) = + ∞ for some fixed λ ∈ (0 , q ∈ N as in (6.1). There exists at least one i ∈ { , . . . , q − } such that ∞ X k =1 V Ω ckq + i (0) (0) = + ∞ . We can assume without loss of generality that i = 0, i.e. ∞ X k =1 V Ω ckq (0) (0) = + ∞ . Let us split the sets Ω ckq (0) as in (6.2). In this way we have defined the sequence of compactsets F k . We want to check that such a sequence satisfies the hypotheses of Lemma 6.1.First of all, from (6.3), we have that the F k ’s are disjoint. Moreover, since F k ⊂ Ω ckq (0),it is easy to see that the sets converge from below to the point 0 (e.g., using that Γ(0 , · ) is δ r -homogeneous of degree − Q ). Lemma 6.3 provide the existence of a positive constant M for which condition ( ii ) in Lemma 6.1 holds true. The last assumption we have to verify isthe condition ( i ). To do this, we recall that the subadditivity of the equilibrium potentialsimplies that V Ω ckq (0) ≤ V F k + V F (0) k . Lemma 6.2 says that P k V F (0) k (0) is convergent. We then deduce + ∞ X k =1 V F k (0) = + ∞ , which is condition ( i ).Then, we can apply Lemma 6.1 and infer that V G r (0) ≥ M for all positive r . The regularityof the point 0 is thus ensured by the characterization in (3.3). (cid:3) The Wiener-type test, and the cone condition
In this section we want prove Corollary 1.2, and Corollary 1.3.First, we want to show how one can deduce the Wiener-type test of Corollary 1.2 fromTheorem 1.1: it follows easily from the representation of the potentials (3.2).
Proof of Corollary 1.2.
For every k ∈ N , we denote by µ k the L -equilibrium measure ofΩ ck ( z ) . Then, keeping in mind the very definition of Ω ck ( z ), we have: V Ω ck ( z ) ( z ) = Z Ω ck ( z ) Γ( z , ζ ) dµ k ( ζ ) dζ ≤ (cid:18) λ (cid:19) ( k +1) log ( k +1) µ k (Ω ck ( z )) = cap (Ω ck ( z )) λ ( k +1) log ( k +1) . IENER-LANDIS CRITERION FOR KOLMOGOROV-TYPE OPERATORS 17
Analogously, V Ω ck ( z ) ( z ) ≥ cap (Ω ck ( z )) λ k log k . Hence, ∞ X k =1 cap (Ω ck ( z )) λ k log k ≤ ∞ X k =1 V Ω ck ( z ) ( z ) ≤ ∞ X k =1 cap (Ω ck ( z )) λ ( k +1) log ( k +1) . The assertions ( i ) and ( ii ) directly follow from these inequalities, and from Theorem 1.1. (cid:3) The main statement in Corollary 1.3 follows from the sufficient condition ( i ) we have justproved, and from (3.1). In fact, we have(7.1) cap (Ω ck ( z )) ≥ | Ω ck ( z ) | ( c N λ k log k ) Q since Ω ck ( z ) ⊂ R N × [ t − T k , t ] where we recall that T Q k = c N λ k log k .Finally, we have to deal with the proof of the cone condition. To this aim, we need somedefinitions. We call L -cone of vertex 0 ∈ R N +1 a set of the form K R ( B ) := { ( D r ( ξ ) , − r ) : ξ ∈ B, ≤ r ≤ R } for some bounded set B ⊂ R N with non-empty interior, and for some positive R . We call L -cone of vertex z the left-translated cone z ◦ K R ( B ) . Definition 7.1.
Let Ω be a bounded open subset of R N +1 and z ∈ ∂ Ω . We say that Ω hasthe exterior L -cone property at z if there exists an L -cone of vertex z which is completelycontained in Ω c . We can now complete the proof.
Proof of Corollary 1.3.
As we said, from (7.1) we get ∞ X k =1 cap (Ω ck ( z )) λ k log k ≥ c − Q N ∞ X k =1 | Ω ck ( z ) | λ Q +2 Q k log k and the first part of the proof follows. If we suppose that Ω has the exterior L -cone propertyat z , we want to prove that the series on the r.h.s. is divergent. In particular, we are goingto prove that the terms of that series are uniformly bigger than a positive constant, for k big enough.Without loss of generality, we can assume z = 0. Denote F θr := (cid:26) z ∈ R N +1 : 1 r ≤ Γ(0 , z ) ≤ θr (cid:27) , for r > , and for θ > , and let r k = λ k log k . For any θ > k such that we haveΩ ck ( z ) ⊇ F θr k ∩ K R ( B ) ∀ k ≥ ¯ k. On the other hand F θr k ∩ K R ( B ) = δ r Qk (cid:18) F θ ∩ K R r − Qk ( B ) (cid:19) . We claim that there exist ¯ k ≥ ¯ k and a non-empty open set A ⊂ R N +1 such that(7.2) A ⊆ F θ ∩ K R r − Qk ( B ) ∀ k ≥ ¯ k . If this is the case, we get | Ω ck ( z ) | ≥ (cid:12)(cid:12)(cid:12)(cid:12) δ r Qk (cid:18) F θ ∩ K R r − Qk ( B ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = r Q +2 Q k | F θ ∩ K R r − Qk ( B ) | ≥ r Q +2 Q k | A | for all k ≥ ¯ k , which is exactly the desired relation | Ω ck ( z ) | λ Q +2 Q k log k ≥ | A | > ∀ k ≥ ¯ k . Hence, we are left with the proof of the claim (7.2). Take ¯ k ≥ ¯ k such thatsup ξ ∈ int( B ) Γ(0 , ( ξ, − < R Q r k ∀ k ≥ ¯ k . Consider A := (cid:26) ( D ρ ( ξ ) , − ρ ) : ξ ∈ int( B ) , and 1 θ Γ(0 , ( ξ, − < ρ Q < Γ(0 , ( ξ, − (cid:27) , which is open, and non-empty since int( B ) = ∅ and θ >
1. Moreover A ⊂ F θ by construction,and A ⊂ K R r − Qk ( B ) for k ≥ ¯ k because of the inequality ρ Q < R Q r k . The proof is thuscomplete. (cid:3) Acknowledgments
A.E.K. and G.T. have been partially supported by the Gruppo Nazionale per l’AnalisiMatematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale diAlta Matematica (INdAM).
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