ON PARABOLIC HARNACK INEQUALITY

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1 ON PARABOLIC HARNACK INEQUALITY JIAXIN HU Abstract. We show that the parabolic Harnack inequality is equivalent to the near-diagonal lower bound of the Dirichlet heat kernel on any ball in a metric measure-energy space where the Dirichlet form is strongly local and regular. This gives a new characterization of the parabolic Harnack inequality. 1. Framework and main theorem Let M, d, µ, E, F)) be a metric measure-energy space, namely, the M, d) is a locally compact separable metric space, and µ is a Radon measure supported on M, and E, F) is a Dirichlet form on L M) := L M, µ). Recall that F is a Hilbert space with respect to the inner product E 1, ) defined by E 1 f, g) := E f, g) + f, g), where, ) is the usual inner product in L M). Assume that M is connected, that is, the set Bx 0, r)\bx 0, r) is non-empty for any x 0 M and any 0 < r < r 0, where r 0 := diam M) 0, ] is the diameter of M, and Bx, r) := {y M : dy, x) < r} is a ball in M. For the measure µ, we assume that there exist c 1 0, 1] and α 1 > 0 such that, for all x, y M and all 0 < r 1 < r < r 0, 1.1) V x, r 1 ) V y, r ) c 1 r1 r ) α1, where V x, r) = µ Bx, r)) is the volume of the ball. Let C 0 M) be the space of all continuous functions on M with compact support. A Dirichlet form E, F) is said to be regular if F C 0 M) is dense both in F in the E 1 -norm) and in C 0 M) in the sup-norm). A Dirichlet form E, F) is said to be strongly local if Ef, g) = 0 for any f, g F such that suppf) and suppg) are compact, and f is constant in a neighborhood of suppg). The reader may consult the details in [3, Chapter 1]. Date: February, Mathematics Subject Classification. Primary:58J35; Secondary: 31B05; 31C5. Key words and phrases. Dirichlet form, heat kernel, parabolic Harnack inequality. 1

2 HU Let {T t } be the unique strongly continuous semigroup on L M) L M) that corresponds to the Dirichlet form E, F). Recall that Ef, g) = lim t 1 fx) T t fx)) gx) dµx) for any f, g F. t 0 M Denote by the infinitesimal generator of the semigroup {T t } in L M), that is 1.) lim t 0 t 1 T t f f) f = 0 for f dom ), the space of all functions in L M) such that the above limit exists under L -norm. If there exists a measurable function p on 0, ) M M such that, for any t > 0 and µ-almost all x M, T t fx) = pt, x, y)fy) dµy) f L M) ), then p is called the heat kernel of E, F). Let be a non-empty open subset of M. Define M F 0 ) = {f F CM) : f M\ = 0}, where CM) denotes the space of all continuous functions on M. Let F) be the closure of F 0 ) under the E 1 -norm. If E, F) is regular, the space F) is dense in L ), and so E, F)) is a regular Dirichlet form on L ) [3, Theorem 4.4.3, p.154]. Denote by {Tt } and the semigroup and the generator of E, F)) in L ), respectively. Let p t, x, y) be the heat kernel of E, F)) if it exists. We extend p to the whole domain 0, ) M M such that for any t > 0, if x / or if y /. p t, x, y) = 0 In the sequel, we will assume that the heat kernel p exists for any non-empty open M. This assumption is mild. In fact, if the global heat kernel p = p M exists, and satisfies an on-diagonal upper bound 1.3) pt, x, y) mt) for all t > 0 and a.a. x, y M and for some continuous m : 0, ) 0, ), then the Dirichlet heat kernel p t, x, y) for any open M also exists and satisfies the same upper bound. In order to state the parabolic Harnack inequality, we need to introduce a function ρ : [0, ) [0, ) that is assumed to be continuous and strictly increasing. Assume that there exist c, c 0, 1] and β 1, β > 0 such that ) β 1.4) c r1 ρr ) β1 1) ρr ) c r1 r for all 0 < r 1 < r < r 0. A typical example for such function is r ρr) = r β for some β > 0.

3 PARABOLIC HARNACK INEQUALITY 3 For an open M and an open I 0, ), and for f L ), a function u : I F) is said to satisfy weakly 1.5) u f in I, if for Lebesgue almost every t I, the function ut, ) is Fréchet differentiable in L ), and its Fréchet derivative, denoted by, satisfies that 1.6) t, x)ϕx) dµx) + E ut, ), ϕ) fx)ϕx) dµx) for any non-negative ϕ F). A function u : I F) is called a weak solution of the heat equation in I if 1.7) u = 0 weakly in I, that is, for any ϕ F) and for Lebesgue almost every t I, 1.8) t, x)ϕx) dµx) + E ut, ), ϕ) = 0. Note that given initial data u t=0 = f L ), the function u = T t f is a weak solution of 1.7) in 0, ), by using the spectral family. A function u : I F) is said to be a weak supersolution resp. subsolution) of the heat equation in I if 1.9) u 0 resp. 0) weakly in I. We say that the ρ-parabolic Harnack inequality, denoted by PHIρ), holds on a metric measure-energy space M, d, µ, E, F)) if PHIρ): For δ 0, 1), there exists a constant C H > 0 such that, for all balls Bx 0, r) in M and all s ρδr), and for all non-negative bounded weak solution u of 1.7) in 0, s] Bx 0, r), we have that 1.10) sup u C H inf u, Q Q + δ δ where Q δ and Q+ δ are defined by Q δ Q + δ = [s 3ρδr)/4, s ρδr)/] Bx 0, δr), = [s ρδr)/4, s] Bx 0, δr). We emphasize here that the constant C H above is independent of r, x 0, s and u, but may depend on δ. The following figure Figure 1) illustrates the PHI ρ). The purpose of this paper is to give an equivalent characterization for the PHIρ). To do this, we introduce a condition, called the local lower estimate of the Dirichlet heat kernel on any ball. More precisely, we say that condition LLEρ) holds on a metric measure-energy space M, d, µ, E, F)) if

4 4 HU Figure 1. LLEρ): For δ 0, 1) and 0 < λ 1 < λ <, there exists a constant C D 0, 1) such that, for all balls Bx 0, r) in M and all t [λ 1 ρδr), λ ρδr)], and for µ-almost all x, y B x 0, δr), 1.11) p Bx0,r)t, x, y) C D V x 0, δr). Condition LLEρ) says that the Dirichlet heat kernel p B t, x, y) of E, F B ) satisfies the near-diagonal lower bound for x, y close to the center of the ball, see Figure. This condition was stated in [5] for ρt) = t β with β > 1. See also [] for the classical case on R n. Note that the constant C D in 1.11) is independent of r, x 0 and t, x, y, but may depend on δ and λ 1, λ. Figure. We state the main theorem of this paper. Theorem 1.1. Assume that E, F) is a strongly local and regular Dirichlet form on L M), and that the heat kernel p of E, F) exists on 0, ) M M. Then, we

5 have that PARABOLIC HARNACK INEQUALITY 5 1.1) PHI ρ) LLE ρ). Remark 1.. Let ρr) = r β for β > 1. If M, d) satisfies the chain condition, it was proved in [5] that LLE ρ) + locality of E, F) + V α) are equivalent to the two-sided exponential decay of the heat kernel p ) ) β/β 1) dx, y) 1.13) pt, x, y) t α/β exp c for all 0 < t < r β 0 and µ-almost all x, y M, where condition V α) means that for all x M and 0 < r < r 0. t 1/β V x, r) r α, α > 0, Remark 1.3. By Theorem 1.1, we see that PHI ρ) + locality of E, F) + V α) are equivalent to 1.13), if ρr) = r β, β > 1. This result is not new, see [6, 7, 1]. Here we provides an alternative approach for proving it. Question 1.4. Would Theorem 1.1 still hold without the strong locality of E, F)?. Proof of P HIρ) LLEρ) From now on, we will drop the letter ρ in PHI ρ) and LLE ρ) for simplicity. In this section, we show the implication P HI LLE. Proof. The proof of P HI LLE. Fix δ 0, 1) and a ball B := Bx 0, r) M. Set t 0 := ρδr)/4. Let φ F such that 0 φ 1 in M, and φ = 1 in a neighborhood of B. For x M, we define φx), 0 < t t 0, ut, x) = p B t t 0, x, z) dµz), t > t 0. B Then u is the bounded weak solution of = u in 0, ) B. In fact, for any 0 < t < t 0, the Fréchet derivative t, ) = 0, and E ut, ), ϕ) = Eφ, ϕ) = 0 for any ϕ FB), by using the strong locality of E, F). Thus, for 0 < t < t 0,.1) t, )ϕx) dµx) = E ut, ), ϕ) = 0 for any ϕ FB). B A Dirichlet form E, F) is said to be local if Ef, g) = 0 for any f, g F with disjoint compact supports. The symbol f g means that there are positive constants c i, c i > 0 i = 1, ) such that c 1gc 1 x) fx) c gc x) for any x in the common domain of f and g.

6 6 HU It is easy to see that.1) also holds for t > t 0. We apply PHI to the above solution u. Then, for a.e. x, y Bx 0, δr), which gives that 1 = u t 0, y) C H u t 0, x) = C H p B t 0, x, z) dµz), B B p B t 0, x, z) dµz) C 1 H. Hence, there must be a point z 0 Bx 0, r) such that.) p B t 0, x, z 0 ) C 1 H V x 0, r) c V x 0, δr), by using 1.1), where c = C 1 H c 1δ α 1. On the other hand, applying PHI to the solution ut, ) = p B t, x, ), it follows from.) that c V x 0, δr) p B t 0, x, z 0 ) C H p B s, x, y) for a.a. y Bx 0, δr) and all s [3t 0, 4t 0 ]. Therefore, we conclude that p B s, x, y) C 1 H c V x 0, δr) for µ-almost all x, y Bx 0, δr) and all s [3t 0, 4t 0 ]. This proves that LLE) holds for λ 1 = 3/4 and λ = 1. The proof given here is motivated by the argument in [7, p.153]. 3. Proof of LLEρ) P HIρ) In this section we show that LLE implies PHI. The proof is more involved. Recall that the following comparison principle holds for any regular Dirichlet form on the metric measure space [4, Lemma 4.16]. Proposition 3.1 comparison principle [4]). Let E, F) be a regular Dirichlet form on L M). Let a, b) 0, ) and M be open, and let 0 f L ). If u : a, b) F is a non-negative weak supersolution of the heat equation in a, b), and ut, ) f L ) 0 as t a, then ut, x) Tt fx) for Lebesgue-a.e. t a, b) and µ-a.e. x. The above comparison principle will give rise to the following key estimate. Lemma 3.. Assume that E, F) is a strongly local and regular Dirichlet form on L M). Let U be an open subset of M, and let u be a weak supersolution of the heat equation in a, b) for an interval a, b) 0, ). Then 3.1) ut, x) m p U t a, x, z) ua, z) m) dµz) U for Lebesgue-a.e. t a, b) and µ-a.e. x U, where m := inf a,b) U u.

7 PARABOLIC HARNACK INEQUALITY 7 Proof. Let K U be open. Let φ F be a cut-off function in K, U), that is, the function 0 φ 1 on M, and φ = 1 in a small neighborhood of K, and φ = 0 on M\U. Such a test function exists because E, F) is regular. Set vt, x) = ut, x) m) φx) for t, x) a, b) M. Clearly, the function v 0 in a, b) M. We claim that v is a weak supersolution of the heat equation in a, b) K. In fact, we see that for any 0 ϕ FK), E vt, ), ϕ) = E ut, ) m)φ, ϕ) = E ut, ), ϕ) + E ut, )φ 1), ϕ) me φ, ϕ) = E ut, ), ϕ) by using the strong locality of E, F). Therefore, v t, x)ϕx) dµx) + E vt, ), ϕ) K = t, x)φx)ϕx) dµx) + E ut, ), ϕ) K = t, x)ϕx) dµx) + E ut, ), ϕ) 0 K for any 0 ϕ FK), since u is the weak supersolution of the heat equation in a, b). This proves our claim. On the other hand, we have that vt, ) f ) := ua, ) m 0 in the L K)-norm, as t a. By the above comparison principle, we conclude that vt, x) = ut, x) m Tt afx) K for Lebesgue-a.e. t a, b) and µ-a.e. x K. Now letting K approach U and noting that Tt af K Tt af, U we obtain 3.1). Remark 3.3. Under the assumptions of Lemma 3., we have that, for any weak subsolution of the heat equation in a, b) U, 3.) m ut, x) p U t a, x, z) m ua, z)) dµz) U for Lebesgue-a.e. t a, b) and µ-a.e. x U, where m := sup a,b) U u. For a function u on, the oscillation of u on M is defined by osc u = esssup u essinf u. We will see that condition LLE will yield an estimate of the oscillation of a weak solution of the heat equation. Proposition 3.4. Let E, F) be a strongly local and regular Dirichlet form on L M), and let δ 0, 1). If condition LLE holds, then for any ball Bx 0, r) in M and any s > ρr), and for any weak solution u of 1.7) in 0, ) Bx 0, r), 3.3) osc u ε [s ρδr),s] Bx 0,δr) osc [s ρr),s] Bx 0,r) u,

8 8 HU where ε = 1 C D 4 the constant C D is the same as in 1.11)). Proof. Let U = Bx 0, r) and = Bx 0, r). Fix s > ρr), and let Mr) := sup u and mr) := inf u. [s ρr),s] U [s ρr),s] U Define the set S by { S = x Bx 0, δr) : u s ρr), x) Consider the function u mr) if 3.4) µs) V x 0, δr) 1. } mr) + Mr). Otherwise, consider the function Mr) u and run the same argument as below but using 3.) instead of using 3.1), and the constant ε can be taken the same. Let t, x) [s ρδr), s] Bx 0, δr). Set t = t s ρr)). By 1.4), we see that λ ρδr) ρr) t ρr) ρδr) λ 1 ρδr), where λ = 1/ ) c δ β and λ1 = c 1 δ β 1 1. Thus, by 1.11), p U t, x, z) C D V x 0, δr) for a.e. z Bx 0, δr). Therefore, using 3.1) with a = s ρr) and b = s, it follows from 3.4) that ut, x) mr) p U t, x, z) u s ρr), z) mr)) dµz) U Mr) mr) p U t, x, z) dµz) which gives that Therefore, C D 4 S Mr) mr)), m δr) C D 4 Mr) + 1 C ) D mr). 4 M δr) m δr) M r) m δr) 1 C ) D Mr) mr)), 4 proving 3.3) with ε = 1 C D 4. The proof given here is motivated by that in [, Lemma 5.), p. 335]. By Proposition 3.4, we will see that the weak solution is Hölder continuous.

9 PARABOLIC HARNACK INEQUALITY 9 Proposition 3.5. Assume that all the hypotheses in Proposition 3.4 hold. Then there are positive constants c and θ such that, for any ball Bx 0, r) in M and any s > ρr), and for any weak solution u of 1.7) in 0, s] Bx 0, r), 3.5) ut, x) ut, x ) c sup u [s ρr),s] Bx 0,r) ρ 1 t t ) + dx, x ) for any t, x), t, x ) [s ρr) + ρδr), s] Bx 0, 1 δ)r), where ρ 1 is the inverse of the function ρ. Proof. Let t > t, and let t, x), t, x ) [s ρr) + ρδr), s] Bx 0, 1 δ)r). Set l = ρ 1 t t ) + dx, x ). Assume that l < δr otherwise, the 3.5) is clear). Let k 1 be the integer such that δ k+1 l r < δk. Note that [ t ρlδ k+1 ), t ] Bx, lδ k+1 ) [s ρr), s] Bx 0, r), and t, x ) [t ρl), t] Bx, l). Therefore, it follows from 3.3) that ut, x) ut, x ) Let θ be such that ε = δ θ. Thus, osc u [t ρl),t] Bx,l) ε k 1 ε k 1 osc [t ρlδ k+1 ),t] Bx,lδ k+1 ) sup [s ρr),s] Bx 0,r) u. r u ) θ which yields the desired. ε k 1 = δ k 1) θ δ θ ) θ l, r Proposition 3.5 implies that any Dirichlet heat kernel p B t, x, y) is jointly continuous in 0, ) U U for any U B, if 1.3) holds for some continuous function m : 0, ) 0, ). We are now in a position to show the implication LLE P HI. Proof. LLE P HI. Let δ 0, 1) and t 0 = ρδr)/4. We shall prove 1.10) by contradiction. For s 4t 0, assume that there would exist points and s 0, z 0 ) Q + δ t 0, y 0 ) Q δ = [s 3t 0, s t 0 ] Bx 0, δr) = [s t 0, s] Bx 0, δr) such that 3.6) ut 0, y 0 ) C H us 0, z 0 ) := C H a. By choosing a suitable C H, we will show that 3.6) implies that u is unbounded in [s 4t 0, s t 0 ] Bx 0, δr). But this is a contradiction.

10 10 HU To do this, let t [s 4t 0, s t 0 ] and δ 1 := δ. Then, by 1.4), we see that s 0 t t 0 ρδ 1 r). By condition LLE, it follows that, for a.e. z Bx 0, δ 1 r), p s 0 t, z 0, z) C D V x 0, δ 1 r) where = Bx 0, δ r) for some δ δ 1, 1). By Proposition 3.1, it follows that a = us 0, z 0 ) p s 0 t, z 0, z)ut, z) dµz) p s 0 t, z 0, z)ut, z) dµz) 3.7) Bx 0,δ 1 r) C Dλa µst, λ)), V x 0, δ 1 r) where St, λ) = {z Bx 0, δ 1 r) : ut, z) λa} for any λ > 0. Let σ = 1 ε where ε is the same as in 3.3). Define ) 1/α1 3.8) bλ) = δ 1. C D c 1 σλ Note that the function b is strictly decreasing on 0, ). We will choose the constant C H > 0 below so that bc H ) δ/, that is, ) 1/α1 3.9) 1 C D c 1 σλ 4. Hence, by 1.1), 3.10) C D σλ = c 1 ) α1 bλ) for any y Bx 0, δ 1 r) and any λ [C H, ). δ 1 V y, bλ)r) V x 0, δ 1 r) We claim that if there is a point t, y) [s 4t 0, s t 0 ] Bx 0, δ 1 r) such that ut, y) λa, and [t ρbλ)r), t] By, bλ)r) [s 4t 0, s t 0 ] Bx 0, δ 1 r), then there must be a point t, y ) [t ρbλ)r), t] By, bλ)r) such that ut, y ) Kλa, where K := 1 σ > 1. In fact, by 3.7) and 3.10), we have that ε µ St, σλ)) V x 0, δ 1 r) C D σλ 1 V y, bλ)r), and so the set By, bλ)r)\st, σλ) is non-empty. Hence, there exists a point y 0 By, bλ)r) such that ut, y 0 ) < σλa. Therefore, osc u ut, y) ut, y 0 ) 1 σ)λa, [t ρbλ)r),t] By,bλ)r) which combines with 3.3) to yield that osc u [t ρbλ)r),t] By,bλ)r) 1 σ)λa ε = Kλa.

11 It follows that there is a point PARABOLIC HARNACK INEQUALITY 11 t, y ) [t ρbλ)r), t] By, bλ)r) such that ut, y ) Kλa, proving our claim. We now show that 3.6) is impossible by suitably choosing the value of C H. In fact, if 3.6) were true, we would have from the above claim that there were a point t 1, y 1 ) [t 0 ρbc H )r), t 0 ] By 0, bc H )r) such that ut 1, y 1 ) KC H a. Repeating this procedure, we obtain a sequence of points t m, y m ) such that ut m, y m ) K m C H a, and t m+1 [t m ρbk m C H )r), t m ], y m+1 B y m, bk m C H )r). Here we have assumed that all the points {t m, y m )} lie in [s 4t 0, s t 0 ] Bx 0, δ 1 r), see Figure 3. In order to make this assumption valid, we need to choose the constant C H. To do this, we observe from 3.8) that, for i 0, which implies that Figure 3. dy i, y i+1 ) bk i C H )r = rδ 1 dx 0, y m ) dx 0, y 0 ) + m 1 i=0 dy i, y i+1 ) C D c 1 σk i C H ) 1/α1 ) 1/α1, δr + rδ 1 C i=0 D c 1 σk i C H ) 1/α1 = δr + rδ 1 1 K 1/α 1 ) 1 C D c 1 σc H δr = δ 1 r for each m 1,

12 1 HU provided that 3.11) C D c 1 σc H ) 1/α1 1 K 1/α 1 ) Note that 3.11) implies that bc H ) δ/ as required.) Therefore, we see that each y m Bx 0, δ 1 r) if 3.11) is satisfied. On the other hand, we have from 1.4) and 3.8) that ρ b ) ) K i C H r c ρδr) [ δ 1 b )] K i β1 C H [ ) ] β1 1/α1 = c ρδr) 4, C D c 1 σk i C H and hence, ρ b [ ) ) ) ] β1 1/α1 K i C H r c ρδr) 4 C i=0 i=0 D c 1 σk i C H ) β1 /α = ρδr) 4 β 1 ) 1 c 1 K β 1 1 α 1 C D c 1 σc H provided that 3.1) 4 β 1 c Therefore, we obtain that 1 4 ρδr), C D c 1 σc H s t m = s t 0 ) + ) β1 /α 1 1 K β 1 α 1 ) m 1 i=0 t i t i+1 ) 3 4 ρδr) + ρ b ) ) K i C H r i=0 ρδr) = 4t 0, and so each t m lies in the interval [s 4t 0, s t 0 ], if 3.1) is satisfied. Now we choose C H such that both 3.11) and 3.1) are satisfied. Finally, note that ut m, y m ) tends to infinity as m, and so u is unbounded in [s 4t 0, s t 0 ] Bx 0, δ 1 r). Acknowledgement. The author would like to thank Richard Bass for reading through the paper and for his valuable comments. This work was supported by NSFC Grant No ) and the Humboldt Foundation. References [1] Barlow, M.; Bass, R.; Kumagai, T. Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Japan ), [] Fabes, E.; Stroock, D. A new proof of Moser s parabolic Harnack inequality using the old ideas of Nash. Arch. Rational Mech. Anal ),

13 PARABOLIC HARNACK INEQUALITY 13 [3] Fukushima, M.; Oshima, Y.; Takeda, M. Dirichlet forms and symmetric Markov processes. Walter de Gruyter, Berlin, [4] Grigor yan, A.; Hu, J. Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces preprint). [5] Grigor yan, A.; Hu, J.; Lau, K.-S. Obtaining upper bounds of heat kernels from lower bounds. Comm. Pure Appl. Math. to appear). [6] Hebisch, W.; Saloff-Coste, L. On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier Grenoble) ), [7] Saloff-Coste, L. Aspects of Sobolev-type inequalities. Cambridge Univ. Press, Cambridge, 00. Department of Mathematical Sciences, Tsinghua University, Beijing , China. address: