The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-090 Wien, Austria An Estimate on the Heat Kernel of Magnetic Schrodinger Operators and Unformly Elliptic Operators with Non{negative Potentials Kazuhiro Kurata Vienna, Preprint ESI 564 (998) June 6, 998 Supported by Federal Ministry of Science and Transport, Austria Available via

2 An Estimate on the Heat Kernel of Magnetic Schrodinger Operators and Unformly Elliptic Operators with Non-negative Potentials Kazuhiro Kurata July 2, 998 Keywords: uniformly elliptic operator, magnetic Schrodinger operator, non-negative potential, heat kernel, weighted smoothing estimate, subsolution estimate Introduction and Main Results We consider the uniformly elliptic operator L E =?r(a(x)r) + V (x) with certain non-negative potential V and the Schrodinger operator L M = (i? r? a(x)) 2 + V (x) with a magnetic eld a(x) = (a (x); ; a n (x)); n 2. We use the notation L J for J = E or J = M. The purpose of this paper is to give an estimate of the fundamental solution ( or heat kernel)? J (x; t : y; s) to namely? J (x; t; y; s) satises (@ t + L J )u(x; t) = 0; (x; t) 2 R n (0; ); () (@ t + L J )? J (x; t; y; s) = 0; x 2 R n ; t > s; (2) lim? J (x; t; y; s) = (x? y): (3) t!s Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa -, Hachioji-shi, Tokyo, Japan, kurata@math.metro-u.ac.jp,

3 For the elliptic operator L E, we assume the following conditions for A(x) = (a ij (x)). Assumption (A.): a ij (x) is a real-valued measurable function and satises a ij (x) = a ji (x) for every i; j = ; ; n and x 2 R n. Assumption (A.2): There exists a constant > 0 such that jj 2 nx i;j= a ij (x) i j? jj 2 ; = ( ; ; n ) 2 R n : (4) To state our assumptions on V and a, we prepare some notations. We say U 2 (RH) if U 2 L loc(r n ) and satises sup y2b(x;r) ju(y)j jb(x; r)j B(x;r) and say U 2 (RH) q if U 2 L q loc (Rn ) and satises =q ju(y)j q dy jb(x; r)j B(x;r) jb(x; r)j ju(y)j dy; (5) B(x;r) ju(y)j dy; (6) for some constant and for every x 2 R n and r > 0, respectively. We can dene the function m(x; U) for U 2 (RH) q with q > n=2 as follows: m(x; U) = supfr > 0; r 2 U(y) dy g: (7) jb(x; r)j B(x;r) We note that if there exist positive constants K and K 2 such that K U (x) U 2 (x) K 2 U (x), then it is easy to see that there exist positive constants K 0 and K 0 2 such that K 0 m(x; U ) m(x; U 2 ) K 0 2m(x; U ): When n 3, since it is known U 2 (RH) n=2 actually belongs to (RH) n=2+ for some > 0, m(x; U) can be dened for U 2 (RH) n=2 ([Sh]). For other properties of the class (RH) q, see, e.g., [KS]. We denote by B(x) = (B jk (x)) the magnetic eld dened by B jk (x) j a k (x)?@ k a j (x). We use the notation m J (x): m E (x) = m(x; V ); m M (x) = m(x; jbj + V ) 2

4 for the operator L J, J = E or M, respectively. We assume the following conditions for V (x) and a(x) = (a (x); ; a n (x)). Assumption(V; a; B): For each j = ; ; n, a j (x) is a real-valued (R n )-function, V is non-negative. (i) For n 3, we assume V (x) and a(x) satisfy V + jbj 2 (RH) n=2 ; jrb(x)j m(x; V + jbj) 3 : (ii) For n = 2, we assume V (x) and a(x) satisfy for some q >. V + jbj 2 (RH) q ; jrb(x)j m(x; V + jbj) 3 Remark For n = 2, we may assume the condition (ii') instead of (ii) by employing Lemma (b). (ii') V 2 L loc(r 2 ); B(x) 0 and that m J (x) satises m J (x) ( + jx? yjm J (x)) m k 0=(k 0 +) J(y) 2 ( + jx? yjm J (x)) k 0 m J (x) (8) for some positive constants ; 2 ; k 0 and for every x; y 2 R 2, where m E (x) = q V (x) and m M (x) = q V (x) + B(x). We remark that it is known that m J (x) satisies (8) under the assumption (V; a; B) for n 3([Sh]) and even for n = 2 in the same way. We also note that if jbj + V 2 (RH), then it is easy to see that jb(x)j + V (x) m(x; jbj + V ) 2 holds. For example, the condition jbj + V 2 (RH) is satised for any a j (x) = Q j (x); V (x) = jp (x)j, where P (x) and Q j (x); j = ; ; n, are polymonials and is a positive constant. In this case, under the assumption (V; a; B) (i) or (ii), we see that there exists a positive constant m 0 such that m J (x) m 0, although in general we cannot say jbj + V is strictly positive for imhomogeneous polynomials. To state our main result, we introduce the notation:? 0 (x; t; y; s) = for some positive constant 0. (t? s) exp(? jx? yj 2 n=2 0 t? s ) 3

5 Theorem (a) Suppose A(x) and V (x) satisfy the assumptions (A:); (A:2) and (V; 0; 0). Then, there exist positive constants 0 and j (j = 0; ; 2) such that (0 )? E (x; t; y; s) exp? 2 (+m E (x)(t?s) =2 ) 0=2? 0 (x; t; y; s) (9) for x; y 2 R n and t > s > 0. (b) Suppose V (x) and a(x) satisfy the assumption (V; a; B). Then, there exist positive constants 0 and j (j = 0; ; 2) such that j? M (x; t; y; s)j exp for x; y 2 R n and t > s > 0.? 2 ( + m M (x)(t? s) =2 ) 0=2? 0 (x; t; y; s) (0) The number 0 is actually dened by 0 = 2=(k 0 + ), where k 0 is the constant in (8). The exponent 0 =2 would not be sharp. If we restrict for the case B 0 jb(x)j B 0 > 0, the following sharp estimate is known ([Ma], [Er,2] for n 3 and [LT] for n = 2): j? M (x; t; y; s)j D exp(?d 2 B 0 t)? D0 (x; t; y; s): More detail informations on the constants D j (j = 0; ; 2) can be seen in those papers. By using the parabolic distance: d P ((x; t); (y; s)) = max(jx? yj; jt? sj =2 ); we have the following decay estimate. orollary (a) Under the same assumptions as in Theorem, there exist positive constants j (j = ; 2) and 0 such that j? J (x; t; y; s)j exp? 2 (+m J (x)d P ((x; t); (y; s))) 2 0=( 0 +4)? 0 (x; t; y; s) for J = E and M, for every x; y 2 R n and t > s > 0. (b) Under the same assumptions as in Theorem, for eack k > 0 there exist positive constants k and 0 such that j? J (x; t; y; s)j for J = E and M. k ( + m J (x)d P ((x; t); (y; s))) k? 0 (x; t; y; s) 4

6 Remark 2 Actually we can show the estimate in Theorem for the operators L E =?r(a(x; t)r) + V (x; t) with time-dependent coecients, if we assume the uniform ellipticity (4) of A(x; t) and the existence of constants j ; j = ; 2, such that U(x) V (x; t) 2 U(x) and U satis- es the condition (U; 0; 0). For the magnetic Schrodinger operator L M = (i? r?a(x; t)) 2 +V (x; t), the estimate in Theorem still holds, if there exists positive constants j ; j = ; ; 5; such that U(x) V (x; t) 2 U(x), 3 jb 0 (x)j jb(x; t)j 4 jb 0 (x)j, and jrb(x; t)j 5 m(x; jb 0 j + U) 3, where a(x; t) is and B jk (x; t) j a(x; k a j (x; t) and U(x) and B 0 (x) satisfy the Assumption (U; a; B 0 ) ( except jrb 0 (x)j m J (x) 3 (= m(x; jb 0 j + Uj) 3 )), and if the upper bound : holds for some constants and 0. j? M (x; t; y; s)j? 0 (x; t; y; s) Remark 3 In particular, orollary (b) yields j? J (x; t; y; s)j k ( + m J (x)jx? yj) k ( + m J (x)jt? sj)? k 0 (x; t; y; s) k ( + m J (x)jx? yj)? k 0 (x; t; y; s) () for J = E or M. Let n 3. Then this implies j? J (x; y) + s? J (x; t; y; s) dtj k ( + m J (x)jx? yj) k jx? yj n?2 where? J (x; y) is the fundamental solution to L J u = 0. This estimate for the elliptic operator was proved by Shen [Sh,2]. Thus, orollary (b) is a generalization of his estimate. Remark 4 Recently we are informed by.shen that he obtained the following shape estimate[sh3] for the elliptic operators: under the assumption V 2 (RH) n=2 for n 3 and V 2 (RH) q with q > for n = 2, exp(? 2 d(x; y))jx? yj 2?n? E (x; y) 3 exp(? 4 d(x; y))jx? yj 2?n holds for some positive constants j (j = ; 2; 3; 4), where d(x; y) is dened by d(x; y) = inf m((t); V )j d(t) j dt: 0 dt 5

7 Here the inmum is taken over all curves such that (0) = x and () = y. Moreover, he gave the following estimate: ( + m(x)jx? yj) 0=2 d(x; y) 2 ( + m(x)jx? yj) 0 for some positive constants j (j = ; 2) and 0. In particular, it follows? E (x; y) 5 exp(? 6 ( + m E (x)jx? yj) 0=2 )jx? yj 2?n for some positive constants 5 and 6. We remark that this decay estimate also can be shown for the fundamental solution? M (x; y) to L M in a similar way. On the other hand, from orollary (a) it follows a somewhat weaker decay estimate: j? J (x; y)j exp(?( + m J (x)jx? yj) 2 0=( 0 +4) )jx? yj 2?n for J = E or M. We do not know whether his sharp estimate can be generalized to heat kernel estimates or not. We denote by e?tl J the semigroup generated by L J. Here we also denote by L J the self-adjoint operator determined from the form associated with L J ( see, e.g., [Si], [LS]). We obtain the following weighted smoothing estimate by using orollary (b). Theorem 2 Assume the same assumptions as in Theorem. Let J = E or M. Suppose < p q + and =p?=q < and put = n(=p?=q)(2 [0; n)). Then for each l 2 [0; (n? )=2] there exists a constant l such that km J (x) 2l e?tl J fk L q (R n ) l t l+(=2)kfk L p (R n ); t > 0: (2) orollary 2 Suppose the additional condition jbj + V 2 (RH). Then we have the following estimates: k(jbj + V ) l e?tl J fk L p (R n ) t l kfk L p (R n ); t > 0 (3) holds for < p < + and l 2 [0; n=2], and k(jbj + V ) l e?tl J fk L (R n ) t l+(n=2p)kfk L p (R n ); t > 0 (4) holds for p < + and l 2 [0; n=(2p 0 )]. Here =p 0 =? =p and l is a constant depending on l and p. 6

8 orollary 2 is an easy consequence of Theorem 2 by (jbj+v )(x) m J (x) 2. Note that (4) for the case l = 0 is a classical result. Theorem yields a weighted smoothing estimate with an exponential decay in time. Theorem 3 Assume the same assumptions as in Theorem and the additional assumption m J (x) m 0 > 0. (a) Let p < + and l 2 [0; n=(2p 0 )]. Then we have km J (x) 2l e?tl J fk L (R n ) exp(?(+m 0 t =2 ) 0 2 ) t l+(n=2p)kfk L p (R n ); t > 0: (b) Let p 2 and l 2 [0; n=(2p 0 )]. Then we have km J (x) 2l e?tl J fk L (R n ) exp(?( + m 2 0t)) t l+(n=2p)kfk L p (R n ); t > 0 for some positive constant and for every p < + and l 2 (0; n=(2p 0 )]. Especially, for the case B 0 jb(x)j B 0 exponential decay estimate in time: > 0, Theorem 3 (b) yileds an ke?tl M fk L (R n ) t n=2 exp(? 2B 0 t)kfk L (R n ); t > 0 (5) for some positive constant and 2, which is known (see, e.g., [Ma], [Er,2], [Ue], [LT]). Indeed, in this case m M (x) p B 0 holds. Note that Theorem 3 (a) gives weaker decay rate e?p B 0 t, since k 0 = 0 and 0 = 2. We also emphasize that Theorem 3 can be applied to any polynomial like magnetic eld B(x) which may be zero somewhere. Denition We say u(x; t) is a complex-valued weak solution to (@ t + L M )u = 0 in Q r (x 0 ; t 0 ), if u 2 L ((t 0? r 2 ; t 0 ); L 2 (B(x 0 ; r)); ) \ L 2 ((t 0? r 2 ; t 0 ); H (B(x 0 ; r); ) and satises + + B(x 0 ;r) t u(x; t)(x; t)dx? nx t 0?r B(x 2 0 ;r) j= t t 0?r 2 B(x 0 ;r) t t 0?r 2 B(x 0 ;r) D a j u(x; s)da j (x; s) dxds u(x; s)@ s (x; s) dxds V (x)u(x; s)(x; s) dxds = 0 (6) 7

9 for everu 2 f 2 L 2 ((t 0? r 2 ; t 0 ); H (B(x 0 ; r); s 2 L 2 ((t 0? r 2 ; t 0 ); L 2 (B(x 0 ; r); ); (x; t 0? r 2 ) = 0g, where is the complex conjugate of. Here, we used the notation D a j = xj? a j (x) and Q r (x 0 ; t 0 ) = f(x; t) 2 R n (0; +); jx? x 0 j < r; t 0? r 2 < t < t 0 g: A real-valued weak solution u to (@ t +L E )u = 0 in Q r (x 0 ; t 0 ) can be dened in a similar way. Our proof of Theorem is based on the following subsolution estimate. Theorem 4 Let u(x; t) be a weak solution t u + L J u = 0 in Q 2r (x 0 ; t 0 ). Then there exits positive constants j ; j = ; 2; such that sup ju(x; t)j exp? 2 (+rm J (x 0 )) 0=2 (x;t)2q r=2 (x 0 ;t 0 ) r Q n+2 r(x0 ;t 0 ) (7) Throughout this paper, we use the following notation: D a j = xj? a j (x); D = i? r? a; B(x 0 ; r) = fy 2 R n ; jy? x 0 j < rg; haru; rui = nx j;k= a xj u@ xk u; Q r (x 0 ; t 0 ) = f(x; t) 2 R n (0; +); jx? x 0 j < r; t 0? r 2 < t < t 0 g: 2 Proof of Theorem 4 We use the following inequalities. Lemma (a) ([Sh2]) Suppose n 2 and V (x) and a(x) satisfy the condition (V; a; B). Then there exists a constant 0 such that m(x; jbj + V ) 2 juj 2 dx 0 j(i? r? a(x))uj 2 + V (x)juj 2 dx juj 2 dxdt =2 : for u 2 0 (Rn ; ). 8

10 (b) ([AHS]) Suppose n = 2; V 0; V 2 L loc (R2 ), a 2 (R 2 ), and B(x) 0. Then the inequality (B(x) + V (x))juj 2 dx holds for u 2 0 (R n ; ). j(i? r? a(x))uj 2 + V (x)juj 2 dx We also prepare the following accioppoli-type inequality. Lemma 2 Let 0 < <. Let u be a weak solution to (@ s + L J )u = 0 in Q 2r (x 0 ; t 0 ) for J = E or J = M. Then there exists a constant such that sup ju(x; t)j t 0?(r) 2 tt 0 B(x 2 dx + 0 ;r) j(i? r? a)uj 2 + V juj 2 dxds juj (? ) 2 r Q 2 dxdt: 2 r(x0 ;t 0 ) Proof: Although the proof is standard, we give it here for the sake of completeness. We show the estimate for a weak solution u to (@ t +L E )u = 0 in Q 2r (x 0 ; t 0 ). Since we can show the esimate for a weak solution to (@ t +L M )u = 0 in the similar way, we just mention some modications we need at the end of this proof. Take functions (x) 2 (B(x 0 0; r)) and (t) 2 (R ) satisfying 0 (x), (x) on B(x 0 ; r) and jr(x)j =(? )r, and 0 (t), (t) on t t 0?(r) 2, (t) 0 on t t 0?r 2, j@ t (t)j =r 2 (? 2 ). For the sake of simplicity, we also t u 2 L 2 (Q 2r (x 0 ; t 0 )). Actually, we can remove this additional assumption by using the argument as in [AS]. Fix t 2 [t 0? (r) 2 ; t 0 ]. Multiplying 2 (t) 2 (x)u(x; t) to the equation and integrating over B(x 0 ; r) [t 0? r 2 ; t], we have + + =? 2 B(x 0 ;r) t t 0?r B(x 2 0 ;r) t t 0?r B(x 2 0 ;r) t t 0?r B(x 2 0 ;r) t t 0?r 2 B(x 0 ;r) u(x; t) 2 (x) 2 dx ha(x)ru(x; s); ru(x; s)i(s) 2 (x) 2 dxds V (x)u(x; s) 2 (s) 2 (x) 2 dxds u(x; s) 2 (x) 2 (s)@ s (s) dxds ha(x)ru(x; s); r( 2 (x))i(s) 2 u(x; s) dxds: (8) 9

11 Because of the ellipticity of A(x) and the positivity of V, we obtain by (8) + sup t 0?(r) 2 tt 0 B(x 0 ;r) (? ) u 2 j@ s j dxds u(x; t) 2 (x) 2 dx (9) jrujjuj 2 jrj dxds r 2 u 2 dxds jruj 2 dxds : By using (8) again, we have jruj dxds + haru; rui@ k u 2 2 dxds + u (? )r Q 2 dxds + 2 r(x0 ;t 0 ) u (? )r Q 2 dxds + 2 r(x0 ;t 0 ) 2 V u dxds Q r (x0 ;t 0 ) V u dxds jrujjrj 2 juj dxds jruj dxds: (20) It follows jruj dxds + V u dxds 2 u (? ) 2 r Q 2 dxds: (2) 2 r(x0 ;t 0 ) (9) and (2) yield the desired result. For L M, we can prove in a similar way by noting the following identities: D a j (u) = (D a j u) + u(i? r); D a j uv dx = ud a j v dx: 2 Proof of Theorem 3: Let k 2 N and dene p j (j = ; 2; ; k + ) by p j = 2=3 + ((j? )=k)(? (2=3)). Let j (x) 2 0 (B(x 0 ; p j r)) and j (t) 2 (R) be the functions satisfying 0 j ; j (x) on B(x 0 ; p j? r), 0

12 jr j (x)j k=r, and 0 j ; j (t) on t t 0? (p j? r) 2, j (t) 0 on t t 0? (p j r) 2, jr j (t)j k=r 2. By Lemma 2 (see also (2)), we have Q pj+ r(x 0 ;t 0 ) j(i? r? a)uj 2 2 j+ 2 j+ + V juj2 2 j+ 2 j+ dxds k2 juj r Q 2 dxds: 2 pj+ r (x 0 ;t 0 ) We write just = j+ and = j+, for simplicity. Since j(i? r? a)(u)j 2 2j(i? r? a)uj u 2 jrj 2 2, it follows that j(i? r? a)(u)j V juj dxds Q pj+ r(x 0 ;t 0 ) k2 juj r Q 2 dxds 2 pj+ r(x 0 ;t 0 ) for j = ; ; k. By using Lemma, we obtain t0 t 0?(p j+ r) 2 B(x 0 ;p j+ r) m J (x) 2 juj 2 dx dt k2 juj r Q 2 dxds: 2 pj+ r(x 0 ;t 0 ) By using m J (x) ( + p j+ rm J (x 0 ))?k 0=(+k 0) m J (x 0 ) on jx? x 0 j < p j+ r and noting 2=3 p j+ (see (8) and the remark after that), we have Q pj r(x 0 ;t 0 ) juj 2 dxdt t0 k 2 r 2 m J (x 0 ) 2( + rm J(x 0 )) 2k 0=(k 0+) k 2 ( + rm J (x 0 )) 2=(k 0+) t 0?(p j+ r) B(x 2 0 ;p j+ r) Q pj+ r(x 0 ;t 0 ) Q pj+ r(x 0 ;t 0 ) juj 2 dxdt juj 2 dxdt: juj 2 dxdt (22) for each j = ; 2; ; k. Here we used a trival inequality R R Q pj r (x 0 ;t 0 )( ) dxdt R R Q pj+ r(x 0 ;t ( ) dxdt for the case rm 0 ) J(x 0 ). By this proceedure, we can obtain the following: there exists a constant such that for every k 2 N Q 2r=3 (x 0 ;t 0 ) juj 2 dxdt k (k 2 ) k ( + rm J (x 0 )) k 0 Q r (x0 ;t 0 ) juj 2 dxdt; (23)

13 where 0 = 2=(k 0 + ). Since V (x) 0, the well-known subsolution estimate (see, e.g., [AS]) yields =2 sup juj juj dxdt Q r=2 (x 0 ;t 0 ) r Q 2 (24) n+2 2r=3 (x 0 ;t 0 ) for some constant. For the magnetic Schrodinger operator case, we have used Kato's inequality. ombining (23) and (24), we arrive at k=2 k k =2 sup juj juj dxdt Q r=2 (x 0 ;t 0 ) ( + rm J (x 0 )) k 0=2 r Q 2 (25) n+2 r (x0 ;t 0 ) for every k 2 N. Note that, by Stirling's formula k k e k k!(= p 2k) as k!, there exists a constant 0 such that k k 0 e k k! for k. Multiplying k =k! and taking the summation, we obtain X (( + rm J (x 0 )) 0=2 ) k ( sup juj) Q r=2 (x 0 ;t 0 ) k! k= X 0 (e p =2: ) k juj dxdt r Q 2 n+2 r(x0 ;t 0 ) k= Take > 0 so that e p <. Then we have sup juj exp(?( + rm J (x 0 )) 0=2 =2 ) juj dxdt Q r=2 (x 0 ;t 0 ) r Q 2 : n+2 r(x0 ;t 0 ) This complete the proof. 2 3 Proof of Theorem To show Theorem we prove the following proposition. Proposition Under the assumptions as in Theorem, there exist positive constants and 2 such that j? J (x; t; y; s)j exp(? 2 ( + m J (x)jt? sj =2 ) 0=2 ) (26) (t? s) n=2 for x; y 2 R n and t > s > 0. 2

14 Proof: Assume t? s 2jy? xj 2. Take r 2 = jt? sj=8. Then u(z; u) =? J (z; u; y; s) satises (@ t + L J )u(z; u) = 0 in Q 2r (x; t). Hence, by applying Theorem 4 to u(z; u), we obtain j? J (x; t; y; s)j sup juj Q r=2 (x;t) exp(?( + m J (x)jt? sj =2 ) 0=2 ) r n+2 =2 j?(z; u; y; s)j 2 dzdu : Q r (x;t) By using the maximum principle for L E and the diamagnetic inequality (see, e.g., [AS], [LS], [AHS]) for L M, we have j? J (z; u; y; s)j (u? s) exp jz? yj2? (27) n=2 (u? s) for some constant = (n; ). Since t? s u? s 7r 2 (7=8)(t? s) on (z; u) 2 Q r (x; t), it is easy to see =2 j? J (z; u; y; s)j 2 dzdu r n+2 Q r (x;t) (t? s) n=2: This yields the desired estimate. 2 Proof of Theorem : The positivity of? E (x; t; y; s) is a consequence of V 0 and the maximum principle. Hence Proposition and (27) imply j? J (x; t; y; s)j 2 exp(?(+jt?sj =2 m J (x)) 0=2 ) (t? s) exp? n jy? xj2 (t? s) for some constant. This concludes the desired estimate. 2 Proof of orollary : Let f(t) = (m J (x)t =2 ) 0=2 + jx? yj 2 =t for t > 0. The, an easy computation shows that inf f(t) (m J (x)jx? yj) 2 0=( 0 +4) t>0 for some positive constant. Thus, we obtain? yj2 j? J (x; t; y; s)j exp(?f(t? s)) exp(?jx ) n=2 (t? s) t exp(?(m J (x)(t? s) =2 ) 0=2 )? 0 (x; t; y; s) exp(?(m J (x)jx? yj) 2 0=( 0 +4) ) exp(?(m J (x)t =2 ) 0=2 ): This proves the part (a) since 2 0 =( 0 + 4) 0 =2. The part (b) is an easy consequence of the part (a). 2 3

15 4 Proof of Theorem 2, 3 To show Theorem 2, we prove the following inequality. Theorem 5 Let 2 [0; n). Then there exists a constant such that jm J (x) 2l (e?tl J f)(x)j t l+(=2)(m jfj)(x) (28) holds for every 0 < l (n? )=2. Here M f is the fractional maximal function dened by (M f)(x) = sup jfj dy; x2b jbj B?=n where the supremum is taken all balls B containing x. Theorem 2 is a consequence of Theorem 5 and the following lemma (see, e.g., [St]). Lemma 3 Let 0 < n. There exists a constant such that km fk q kfk p for < p q + and =q = =p? =n. Proof of Theorem 5: Let r = =m J (x). By orollary (b) we have jm J (x) 2l (e?tl J f)(x)j m J (x) 2l jf(y)j ( + m J (x)jx? yj) k t exp? n=2 r 2l t n=2 +X j=? f2 j? r<jx?yj2 j rg jf(y)j ( + 2 j? ) exp k jx? yj2 dy t? (2j r) 2 t dy: (29) By the assumption on l, we take 0 such that 2 = n?? 2l. Put = sup s>0 s e?s < + for 0. Then the right hand side of (29) is dominated by +X t n=2 j=? t n=2? t n=2? +X j=? +X j=? f2 j? r<jx?yj2 j rg (2 j ) n? ( + 2 j? ) k (2 j? ) 2 (2 j? r) 2?jf(y)j dy r 2l ( + 2 j? ) k t (2 j r) fjx?yj2 n? j rg jf(y)j dy : (2 j ) n? ( + 2 j? ) k (2 j? ) 2(M jfj)(x): (30) 4

16 Now, since n?? 2 = 2l > 0, by taking k > 2l we have and +X j= 0X j=? (2 j ) n? +X ( + 2 j? ) k (2 j? ) 2 j= (2 j ) n? 0X ( + 2 j? ) k (2 j? ) 2 j=? < +; j(k?2l) 2 (2 j ) 2l < +: Thus, we obtain the desired result. 2 Proof of Theorem 3: First, the estimate for the case l = 0 and p = is classical except the exponential factor in time. Under the assumption, by orollary (a) we have j? J (x; t; y; s)j? 0 (x; t; y; s) exp(?( + m J (x)jx? yj) 2 0=( 0 +4) ) exp(?( + m 0 t =2 ) 0=2 ) (3) for some positive constants and 0. Then by using this estimate we can prove the part (a) of Theorem 3 in a similar way as in the proof of Theorem 2. To show the part (b), we use the semigroup property and Theorem 2 and get km J (x) 2l e?tl J fk L (R n ) t l+(n=4)ke?(2=3)tl J fk L 2 (R n ) for some constant. Note that under the assumption m J (x) m 0, Lemma yields inf (L J ) m 2 for some positive constant. Here (L 0 J) is the spectrum of the operator L J. So, we have Using this estimate, we obtain km J (x) 2l e?tl J fk L (R n ) ke?(=3)tl J gk L 2 (R n ) e?m2 0 t kgk L 2 (R n ): t t l+(n=4) e?m2 0 ke?(=3)tl J fk L 2 (R n ) e?m2 0 t l+(n=4) t t n=2(=p?=2)kfk L p (R n ): In the last inequality, we used p 2 and Theorem 2. 2 REFERENES 5

17 [AHS] Avron J.E., Herbst I., Simon B., Schrodinger operators with magnetic elds, I. General interactions, Duke Math. J., 45(978), 847{883. [AS] Aronson D.G., Serrin J., Local behaviour of solutions of quasilinear parabolic equations, Arch. Rat. Mech. Anal., 25(967), 8{22. [Er] Erdos L., Estimates on stochastic oscillatory integrals and on the heat kernel of the magnetic Schrodinger operator, Duke Math. J., 76(994), 54{566. [Er2] Erdos L., Dia- and paramagnetism for nonhomogeneous magnetic elds, J.Math.Phys., 38(3)(997), 289{37. [LS] Leinfelder H., Simader., Schrodinger operators with singular magnetic vector potentials, Math.., 76(98), {9. [LT] Loss M., Thaller B., Optimal heat kernel estimates for Schrodinger operator with magnetic elds in two dimensions, omm. Math. Phys., 86(997), 95{07. [Ma] Malliavin P., Minoration de l'etat fondamental de l'equation de Schrodinger du magnetisme et calcul des variations,.r. Acad. Sci. Ser. I. Math. 302(986), 48{486. [KS] Kurata K., Sugano S., Fundamental Solution, eigenvalue asymptotics and eigenfunctions of degenerate elliptic operators with positive potentials, Preprint. [Sh] Shen., L p estimates for Schrodinger operators with certain potentials, Ann. Inst. Fourier, 45(995), 53{546. [Sh2] Shen., Estimates in L p for magnetic Schrodinger operators, Indiana Univ. Math. J., 45(996), [Sh3] Shen., by personal communication. [Si] Simon B., Maximal and minimal Schrodinger forms, J.Opt. Theo., (979), 37{47. [St] Stein, E.M., Harmonic Analysis: Real variable methods, Orthogonality, and Oscillatry integrals, Prinston Univ. Press, 993. [Ue] Ueki N., Lower bounds on the spectra of Schrodinger operators with magnetic elds, J.Fun.Ana., 20(994), 344{379. AKNOWLEDGEMENT This paper was writen while the author was visiting the Erwin Scgrodinger International Institute for Mathematical Physics in Vienna. The author wishes to thank Professor T. Homann-Ostenhof for his invitation and the 6

18 members of the Schrodinger institute for their hospitality. This work is partially supported by Tokyo Metropolitan University maintenance costs and by the Erwin Schrodinger institute for Mathematical Physics. In the early version of this paper, the author only obtained a polynomial decay estimate for heat kernels. The author wishes to thank Professor hongwei Shen for informing his sharp estimate for elliptic operators and his kindness to allow the author to use his argument here in order to improve and obtain exponetial decay for the heat kernel. He also pointed out that even in 2-dimensional case we can obtain the same results under the condition (V; a; B)(ii). The author also wishes to thank Professor Laszlo Erdos for his valuable comments. Especially, the author learned from him the observation which improves the exponent in the estimate of Theorem and also simplies the proof. The author also thank Professor B. Heler for his valuable advice which leads to the part (b) of Theorem 3. 7

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