Heat kernels and Green functions on metric measure spaces Jiaxin Hu - PowerPoint PPT Presentation

Heat kernels and Green functions on metric measure spaces Jiaxin Hu Tsinghua University, Beijing, China (Joint with Alexander Grigor’yan) December 10-14, 2012 (Hong Kong) 1 / 23

Programme Background Conditions Theorems . 2 / 23

Metric measure space ( M , d ): a metric space (locally compact, separable). µ : a Radon measure (locally finite, inner regular) ( µ (Ω) > 0 for any open Ω � = ∅ ). ( M , d , µ ): a metric measure space. A metric space: Hata’s tree. 3 / 23

Dirichlet form ( E , F ): a Dirichlet form in L 2 ( M , µ ) that is regular, strongly local. ց • DF: a closed Markovian symmetric form. • regular: C 0 ( M ) ∩ F is dense in both F and C 0 ( M ). • strongly local: E ( f , g ) = 0 for any f , g ∈ F where f is constant in some neighborhood of supp( g ). 4 / 23

Heat semigroup { P t } t ≥ 0 : a heat semigroup in L 2 ( M , µ ): (a) strongly cts, contractive, symmetric in L 2 ; (b) Markovian in L ∞ : P t f ≥ 0 if f ≥ 0, and P t f ≤ 1 if f ≤ 1. ( E , F ) ⇔ { P t } t ≥ 0 : E ( f , g ) = lim t → 0 E t ( f , g ) t → 0 t − 1 ( f − P t f , g ) . := lim 5 / 23

Restricted Dirichlet form Restricted DF : ( E , F (Ω)), where F (Ω) := C 0 (Ω) ∩ F in F -norm , for a non-empty open Ω ⊂ M . ( E , F (Ω)) ⇔ { P Ω t } . Generator: L Ω P Ω t f − f L Ω f := lim in L 2 -norm . t t → 0 6 / 23

Heat kernel { p t } t > 0 : a heat kernel . ւ • symmetric: p t ( x , y ) = p t ( y , x ); • Markovian: p t ( x , y ) ≥ 0, and � M p t ( x , y ) d µ ( y ) ≤ 1; • semigroup property; • identity approximation. 7 / 23

Heat kernel: examples Sierpinski gaskets (’88) and carpets (’92, ’99) � � β/ ( β − 1) � � | x − y | p t ( x , y ) ≍ t − α/β exp − c , t 1 /β Gasket Carpet 8 / 23

Purpose To find equivalence conditions for the following estimate: ( UE ) Upper estimate: the heat kernel p t ( x , y ) exists, has a H¨ older continuous in x , y ∈ M version, and satisfies � − 1 � c d ( x , y ) �� C p t ( x , y ) ≤ V ( x , R ( t )) exp 2 t Φ t for all t > 0 and all x , y ∈ M , where R : = F − 1 and � s 1 � Φ ( s ) := sup r − . F ( r ) r > 0 Interesting case: F ( r ) = r β ( β > 1) , V ( x , r ) ∼ r α , then Φ ( s ) = cs β/ ( β − 1) , and � � β/ ( β − 1) � � d ( x , y ) C p t ( x , y ) ≤ t α/β exp − c . t 1 /β 9 / 23

Conditions How ? Volume doubling condition : for all x ∈ M , r > 0, V ( x , 2 r ) ≤ C D V ( x , r ) , ( VD ) where V ( x , r ) := µ ( B ( x , r )). Then, for all 0 < r 1 ≤ r 2 , � α � r 2 V ( x , r 2 ) V ( x , r 1 ) ≤ c . r 1 Reverse volume doubling condition : for all x ∈ M and 0 < r 1 ≤ r 2 , � α ′ V ( x , r 2 ) � r 2 V ( x , r 1 ) ≥ c − 1 . ( RVD ) r 1 If M is connected and unbounded, then ( VD ) ⇒ ( RVD ). 10 / 23

Conditions The (uniform elliptic) Harnack inequality: for any function u ∈ F that is harmonic and non-negative in B ( x 0 , r ), u ≤ C H esup B ( x 0 ,δ r ) u , einf ( H ) B ( x 0 ,δ r ) where the constants C H and δ are independent of the ball B ( x 0 , r ) and the function u . A function u ∈ F is harmonic in Ω if E ( u , ϕ ) = 0 for any ϕ ∈ F (Ω) . 11 / 23

Harnack inequality Harnack inequality: Harmonic function u is nearly constant in B ( x 0 , δ r ). 12 / 23

Conditions The resistance condition ( R F ): res ( B , KB ) ≃ F ( r ) µ ( B ) , ( R F ) where K > 1, r is the radius of B , and F is continuous increasing such that for all 0 < r 1 ≤ r 2 , � β � β ′ � r 2 � r 2 ≤ F ( r 2 ) C − 1 F ( r 1 ) ≤ C ( β > 1) . r 1 r 1 The resistance and capacity are defined by 1 res ( A , Ω) := cap( A , Ω) , cap( A , Ω) := inf {E ( ϕ ) : ϕ is a cutoff function of ( A , Ω) } for any A ⋐ Ω. 13 / 23

Conditions Interesting case: F ( r ) = r β ( β > 1) , V ( x , r ) ∼ r α , then condition ( R F ) becomes res ( B , KB ) ≃ F ( r ) µ ( B ) ≃ r β − α . 14 / 23

Conditions Condition ( G F ) : the Green function g B exists and is jointly continuous off the diagonal, and � R F ( s ) ds g B ( x 0 , y ) ≤ C sV ( x , s ) ( y ∈ B \ { x 0 } ) , ( G F ≤ ) d ( x 0 , y ) K � R F ( s ) ds g B ( x 0 , y ) ≥ C − 1 sV ( x , s ) ( y ∈ K − 1 B \ { x 0 } ) , d ( x 0 , y ) K ( G F ≥ ) where K > 1 and C > 0, and B := B ( x 0 , R ). The Green function g Ω is defined by � G Ω f ( x ) = g Ω ( x , y ) f ( y ) d µ ( y ) , Ω and the Green operator G Ω : E ( G Ω f , ϕ ) = ( f , ϕ ) , ∀ ϕ ∈ F (Ω) . 15 / 23

Conditions Condition ( E F ) : for any ball B of radius r , E B ≤ CF ( r ) , esup ( E F ≤ ) B δ 1 B E B ≥ C − 1 F ( r ) . ( E F ≥ ) einf where C > 1 and δ 1 ∈ (0 , 1). The function E B is defined by E B ( x ) = G B 1 ( x ) = E x ( τ B ) , where τ B is the first exit time from B . 16 / 23

Conditions Namely, function E B satisfies the Poisson-type equation: −L B E B = 1 weakly , � that is, E ( E B , ϕ ) = B ϕ d µ for any ϕ ∈ F ( B ). Note: if the Green function g B exists, then � E B ( x ) = g B ( x , y ) d µ ( y ) . B Note: Condition ( E F ) can be written E B ≤ C einf C − 1 F ( r ) ≤ esup δ 1 B E B . B 17 / 23

Theorem 1 Theorem 1(Grigor’yan, Hu, 2012): Assume that ( M , d , µ ): a metric measure space. ( E , F ): a regular, strongly local DF in L 2 ( M , µ ). ( VD ) and ( RVD ) hold. Then we have the following equivalences: ( H ) + ( R F ) ⇔ ( G F ) ⇔ ( H ) + ( E F ) . Remark: Condition ( RVD ) is needed only for ( H ) + ( E F ) ⇒ ( R F ≥ ) . 18 / 23

Theorem 1 Ideas of the proof: Maximum principles for subharmonic functions. ( Subharmonic : E ( u , ϕ ) ≤ 0 for any ϕ ∈ F (Ω)) If u is continuous on Ω, then esup u = sup u . ∂ Ω Ω 19 / 23

Theorem 1 Ideas of the proof: The Riesz measure associated with a superharmonic function: if 0 ≤ f ∈ dom( L Ω ) is superharmonic in Ω, then −L Ω f d µ ( x ) = d ν f ( x ) , a non-negative Borel measure on Ω, namely, � E ( u , ϕ ) = ϕ ( x ) d ν f ( x ) for any ϕ ∈ C 0 (Ω) ∩ F . Ω Consequently, if f is harmonic in Ω \ S for a compact set S , � g Ω ( x , y ) d ν f ( y ) ( x ∈ Ω) . f ( x ) = S 20 / 23

Theorem 1 The hardest part of the proof: The annulus Harnack for the Green function from ( H ): g Ω ( x 0 , · ) = sup g Ω ( x 0 , · ) sup ∂ B Ω \ B B g Ω ( x 0 , · ) = C inf ∂ B g Ω ( x 0 , · ) , ≤ C inf where C > 0 is independent of the ball B = B ( x 0 , R ) and Ω. 21 / 23

One more condition Near-diagonal lower estimate: The heat kernel p t ( x , y ) exists, has a H¨ older continuous in x , y ∈ M version, and satisfies c p t ( x , y ) ≥ ( NLE ) V ( x , R ( t )) , for all t > 0 and all x , y ∈ M such that d ( x , y ) ≤ η R ( t ), where η > 0 is a sufficiently small constant. Recall that R = F − 1 , for example, R ( t ) = t 1 /β ( β > 1) . 22 / 23

Theorem 2 Theorem 2 (Grigor’yan, Hu, 2012): Assume that ( M , d , µ ): a metric measure space. ( E , F ): a regular, strongly local DF in L 2 ( M , µ ). ( VD ) and ( RVD ) hold. Then we have the following three equivalences: ( H ) + ( R F ) ⇔ ( G F ) ⇔ ( H ) + ( E F ) ⇔ ( UE ) + ( NLE ) . The End of Talk 23 / 23