 Applied Mathematics, 2010, 1, 425-430 doi:10.4236/am.2010.15056 Published Online November 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM Note on Gradient Estimate of Heat Kernel for Schrödinger Operators Shijun Zheng Department of Mathematical Sciences, Georgia Southern University, Statesboro, USA E-mail: szheng@georgiasouthern.edu Received August 27, 2010; revised September 24, 2010; accepted September 27, 2010 Abstract Let =HV  be a Schrödinger operator on n. We show that gradient estimates for the heat kernel of H with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for pL and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists V in the Schwartz class such that the long time gradient heat kernel estimate fails. Keywords: Heat Kernel, Schrödinger Operator, Functional Calculus 1. Introduction Consider a Schrödinger operator =HV  on n, where V is a real-valued potential in )(1nlocL. It is noted in [1,2] that for positive V, if H admits the following gradient estimates for its heat kernel ),(=),(yxeyxp tHt: for all nyx , and 0>t, ,|),(| /2||/2 tyxcnnt etcyxp  (1) ,|),(| /2||1)/2( tyxcnntxetcyxp  (2) then the kernel of )(Hj and its derivatives satisfy a polynomial decay as in (4), where j is a function in certain Sobolev space with support in ],22[ jj. As is well-known, the decay estimate in (4) implies the Littlewood-Paley inequality for )( npL [3-6]. For positive V, based on heat kernel estimates one can show (4) by a scaling argument . In this paper we will prove the general case, namely Theorem 1, by modifying the proofs in [7,8] and . However, in general the gradient estimates (1), (2) do not hold for all t. This situation may occur when H is a Schrödinger operator with negative potential, or the sub-Laplacian on a Lie group of polynomial growth, cf. [9-12]. A second part of this paper is to show such a counterexample, based on Theorem 1. Recall that for a Borel measurable function  :, one can define the spectral operator )(H by functional calculus ()= ()HdE, where dE is the spectral measure of H. The kernel of )(H is denoted ),)(( yxH in the following sense. Let A be an operator on a measure space ),(dM , d being a Borel measure on M. If there exists a locally integrable function AK: MM such that ,=()=(, ) ()()()()MAMMAf gAf gdKxy fygxdxdy for all gf , in )(0MC with fsupp and gsupp being disjoint, where )(0MC is the set of continuous functions on M with compact supports, then A is said to have the kernel ),(:=),( yxKyxA A. Throughout this paper, c or C will denote an absolute positive constant. The main result is the following theorem for )(1nlocLV . Theorem 1 Suppose that the kernel of tHe satisfies the upper Gaussian bound for 0,1= 0.>,|),(| /2||)/2(tetcyxe tyxcnntHx  (3) Let  be supported in 1,1][ and belong to )(21NnH for some fixed 0N and 0>. Then for each 0N, there exists a constant Nc independent of  such that for all j 1( )/2/22|()(,)|2(12||) ,xjnjnjN NNHHxycxy (4) S. J. ZHENG Copyright © 2010 SciRes. AM 426 where )(2=)( xx jj and )(:=ss HH denotes the usual Sobolev space with norm 22/22=(1/ )ssHLfddx f. When V is positive, a result of the above type was proved and applied to the cases for the Hermite and Laguerre operators . The observation was that if 0V, then the constants corresponding to Lemma 2 do not change for VH = with )(=)( 2xVxV (called scaling-invariance in what follows), according to the Feynman-Kac path integral formula  (())0()=(()) ,tVsdstHxefx Efte here xE is the integral over the path space  with respect to the Wiener measure x, nx and )(t stands for a brownian motion (generic path). For general V the technical difficulty is that we do not have such a scaling-invariance. We are able to overcome this difficulty by establishing Lemma 6, a scaling version of the weighted 1L inequality for ),)(( yxHj with sH , for which we directly use the scaling information indicated by the time variable appearing in Lemma 2. Thus this leads to the proof of the main theorem by combining methods of Hebisch and the author’s in [7,2]. In Section 3 we give a counterexample to show that for 2)cosh1)((=)(  xxV, , the estimates in (1) and (2) fail for t. Note that under the condition in Theorem 1, (4) is valid for all )(0Cj, j satisfying (i) }2|:|{supp jjxx  and (ii) ,2|)(| )( kjkkjcx  0=0k. A corollary is that (3) implies the Littlewood-Paley inequality 21/2()()(|()()|),1<|||=|)|(1|),)((| yxyxaikLdxyxyxLe || (,)/2 2(1)(, )(, )kLxynaLxLyLxye e  12,, ,,(1)(1||())(),nasasaCkCscCsc  where we note that by (6) S. J. ZHENG Copyright © 2010 SciRes. AM 427=0()(,)(, )!nikLnikLx yexy n ||(, )=0|| (,)=.!nkLxynnkLxy en It is easy to calculate that ,,10=>. asa asascaeass Hence taking /2= s and fixing 0>= 0ss give that .|)|)(1,,(|)|(1|),)((|20anaikLkanscdxyxyxLe  (b) Similarly we show the case for all j. If HjeL2=, then Lemma 2 tells that with jt2= /222 2|),(|jnHjcdxyxe  .|),(|22||2 sjcyxsHjcedxeyxe  For j let ajyxjjyxeyx |)|2(1=),( /2||/22, s<<0. Then similar to (a) we obtain /22| |/2(, )=|(, )|(12||)jxy jajjjLxyLxy exydx .)(=|)|(1sup)( ,,||)(asaxsxcsCxesC  It follows that, with /2= s and 0>= 0ss fixed,    |>|||/2 =|)|2(1|),)((| yxyxajjjikL dxyxyxLe /2 /22|| (,)/22(12)(,) (,)njanjLxkL xyjjjjjcLyLxy ee  ,|)|)(1,( /2 ankanc  where we set /2 1=2||( ,)jjjkLxy. We also need a basic property on the weighted 2 norm of Fourier coefficients of a compactly supported function in Sobolev space, which can be proved by elementary Fourier expansions. Lemma 5 Let 0s, 0>T and ])([0,=])([0,00 TCTH s denote the subspace of Sobolev space )( sH. Then we have for all ])([0,0THg s, 20ˆ() ,sHsTgn cg (8) where 1/2222)/|)(|(=}{ snsnTnn  and )(ˆng are the Fourier coefficents of g over the interval ][0,T. The inequality in (8) can be replaced by equality (however we will not use this improvement), which is a special case of the general norm characterization for periodic functions in ])([0,TH s, see e.g. . It follows from Lemma 4 and Lemma 5 the following weighted 1L estimates for ),)(( yxHj, which is an improved version of [2, Lemma 3.1], where the restriction 0V is removed. Lemma 6 Suppose )(1nlocLV and the kernel of tHe satisfies for all 0>t .|),(| /2||/2 tyxcnntH etcyxe   (9) If Nns1)/2(> , 0N and 10,10][supp, then /21()(),(2)( ,)2(),sup jjNsnnHLnjyHy yc here ||1:=xx. Proof. Let 1,1][. If ][0,2:=suppIg , then g has the Fourier series expansion on I ,)(ˆ=)( ikxkekgxg where dxexgkg ikx)(21=)(ˆ20. Let eeg )(=)( and ikkef =)( . Then yyyg )/log(=)( with ],[supp 1eeg , and so ).()(ˆ=)(ˆ=)(2 222HjkkHjHjikekjefkgeekgH  (10) It follows from Lemma 4, (10) and Lemma 5 that for each y NnkNjjkkgcdxyxyxH /2/2 |)|(1|)(ˆ|)(2|),)((| 1)/2()/2(1/2 |)|(1|)|(1|)(ˆ|=  kkkgc Nnk 1/211/2122)|)|(1(|)|(1|)(ˆ|  kkkgckNnk 1/2/2(1 )/2([0,2])0nNHcg 1/2([ 1,1])0,sHc where 1)/2(=nNs and the last inequality follows from a change of variable and interpolation. Remark 7 Let VVV =, 0V on n, 3n. Then the heat kernel estimate in (9) holds if V is in Kato class and KV, the global Kato norm of V, is less than 1)2(/:= /2 nnn, see . Also (9) holds S. J. ZHENG Copyright © 2010 SciRes. AM 428 whenever 0V is locally integrable on n, 1n. 2.1. Proof of Theorem 1 With (3) and Lemma 6 we are in a position to prove (4). The proof is similar to that of Proposition 3.3 in  in the case of positive V. For completeness, we present the details here. .),))(()(,(=),)(( dzyzHezxeyxH jtHtHxzjx   By (3) we have |),)((| yxHjx  2()/2 ||/()/()/ ()/ncxzt NNnNctex ztx ztzy t  dzyzHetyz jtHN|),))(((|)/(  ()/2()/ ()/ |((2))(,)|.nNNntH jctxy tzy teHzydz  Applying Lemma 6 with jt2=, we obtain |),)((| yxHjx   /2 12/NnnNnHctx yte  1/22()/ ,>0.nnNNnHctx yt  Remark 8 In the following section we will show that there exists V, the Schwartz class, such that (4) does not hold for j. By Theorem 1, this means that for such V the gradient upper Gaussian bound (3) does not hold for all t. 3. A Counterexample to the Gradient Heat Kernel Estimate Consider the solvable model VdxdH 22 /= , , where .sech1)(=)( 2xxV  We know from  that solving the Helmholtz equation for {0}\k ),,(=),( 2kxekkxeH yields the following formula for the continuum eigen- functions: ,),(||1))(sign(=),(1=ikxjekxPkijkkxe where ),tanh(=),( ikxpkxP is defined by the recursion formula 11(tanh ,)=((tanh ,))(tanh)(tanh ,),pxikdpx ikikxpx ikdx with 10p. Note that ),(=),( kxekxe  and the function ()22=1(, ,)(,)(,)1=(,)(,)ik xyjxyk exkeykPxkPy kejk  (11) is real analytic on 3. Moreover, H has only absolutely continuous spectrum )[0,= ac and point spectrum .},4,1,{=2 pp The corresponding eigenfunctions 1=}{ nne in 2L are Schwartz functions that are linear combinations of xxmtanhsech , m, 0. Let acac EHH= denote the absolutely continuous part of H and )[0,=EEac the corresponding orthogonal projection. If )(0C, then we have for all 21 LLf  , ,),)(()(),(=)()( 21=nnneefndyyfyxKxfH    where dxxexfef nn)()(=),(  and dkkyekxekyxK ),(),()()(2=),( 21 (12) is the kernel of acacEHH )(=)(, cf. . Since H has eigenfunctions in )( and pp is finite, from now on it is essential to check the kernel ),(=),)(( yxKyxHac instead of the kernel of )(H. 3.1. Decay for the Kernel of acjEH )( Let )(}{ 0=  Cjj satisfy }2||2:{supp)i( 2jjjxx  and jcx kjkkj,2|)(|)i(i )(, 0k. Let ),)((=),( yxHyx acjj. In  we showed that for each N ,,|)|2(12|),(| /2/2  jyxcyx NjjNj (13) but (with 1=) NjjNjxyxcyx   |)|2(12|),(| /2|)|/2(1 (14) only holds for 0j and does not hold for all 0t (or more precisely t), according to Theorem 1. Now consider the system jj}{ which satisfy (i), (ii) as in Section 1. We may assume )(2=)( xx jj for a fixed  in 1,1])([C with 1=)(x on ]21,21[. Let f and f be the Fourier transform and its inverse of f on . The following lemma S. J. ZHENG Copyright © 2010 SciRes. AM 429shows that (13) does not hold for ),)(( yxHacj when j. Lemma 9 Let ),( yxK j be the kernel of acj EH)(. a) For each 0N there exists Nc such that for all 0j, .|)|2(12|),(| /2/2 NjjNj yxcyxK  (15) b) For each 0N there exists Nc such that for all 0>j, precisely .|)|2(12|),(| 1/2/2  yxcyxK jjNj (16) In particular, the decay in (15) does not hold for all 0>j with 1>N. c) There exist positive constants C and c such that for all j, ,|)(||)(||),(| || dueuCyxCyxK uyxcjjj where )(=)( 2kkjj and it is easily to see that for each N, there exists Nc such that for all j .|)|2(12|)(| /2/2 NjjNj yxcyx  Proof. (a) Let /22=j. By (12), (11) and integration by parts we have ()2221=12,=(1)[()() (,)(,)],ik xyNNjNkjjix yKxyekjkPxkPykdk  which can be written as a finite sum of 2()221()()2=1(tanh) (tanh)[(())(())(())] ()mirsjxykkqkxy  (17)  m,0 , Nsri = , )(2kq are polynomials of degree 2. We obtain for each N and all 0j 11/2(1)|()(,)|=() =()=(2),NiNjNjxyKxyOOO using ).(=)(=))(()(=))((2)(22)(1221=)(2ssrriijkOqkOkOk This proves (15) for 0j. In order to show part (c) for j, using partial fractions we write ),( yxK j as a finite sum of 22212=1(tanh) (tanh)()()()(), mjxyk kqkxy(18) which is bounded by (up to a constant multiple) 22=1|[()]( )||[()]()|,jjabkkxyk xyk  where ba,. The general term in the sum is estimated by ,|)(||)(])([| ||22 dueuCyxkkbakuyxcjj  in terms of the identities 2||12=)()( kke x (19) .12=)())(sign( 2||kikkex x (20) (b) Finally we prove the sharp estimate in (16). For 0>j, (15) does not hold for 2N, instead we have only, with 0,1=N, ,|)|2(12|),(| /2/2NjjjyxcyxK  by using similar argument and noting (17), (19), (20). Indeed, let 0>J, 2N. It is easy to find }{ j satisfying (i’) and (ii’) such that ).(1=)( xxjJJ We have by (13) .22|),)((|)( 1)/2(1)/2(    NJNjJNacjNJcyxHyx  On the other hand, from (18) and the relevant steps in part (c) we observe that if 1>N, [0, )|| ||() ()(,)=()(, )(, )=finite sum of()sign( ),NNNcx ycx yxyH xyxy exkeykdkxyexye 1 (21) where 0, are of the form tanhtanh mcx y. This shows that the term ),)(()( yxHyxacJN cannot admit a decay of 1)/2(2 NJ for all 0>J, otherwise one would have ,2|),)(()(| 1)/2()[0,NJN yxHyx 1 which leads to a contradiction that the sum of those functions in (21) must vanish, by letting J. Remark 10 The argument in the proof of part (b) can be made rigorous by replacing )(][0,H1 with )(HL, and then let L to get the same contradiction. 3.2. The Derivative of the Kernel of jacΦHE Similar argument shows that S. J. ZHENG Copyright © 2010 SciRes. AM 430 /2(2)|(,)|2 /||jNNxj NKxycx y  holds for all 0>j but does not hold for all 0