L p -CONVERGENCE OF THE LAPLACE BELTRAMI EIGENFUNCTION EXPANSIONS

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1 L -CONVERGENCE OF THE LAPLACE BELTRAI EIGENFUNCTION EXPANSIONS ATSUSHI KANAZAWA Abstract. We rovide a simle sufficient condition for the L - convergence of the Lalace Beltrami eigenfunction exansions of functions on a closed Riemannian manifold. Nous fournissons une condition suffisante simle our la convergence L de Lalace Beltrami exansions de fonctions sur une variété riemannienne fermée. 1. Introduction Let be a closed smooth Riemannian manifold and the Lalace Beltrami oerator associated to the Riemannian metric [6]. We define the Hilbert sace L 2 ( as the comletion of C ( with resect to the inner roduct f, g := f(xg(xdx, where we simly write dx := dvol(x. The eigenfunctions {ω i } i of the oerator form an orthonormal basis of L 2 (. Then for a function f L 2 ( we have the eigenfunction exansion f(x = f i ω i (x, f i := f, ω i. In light of the Fourier analysis, we would like to roose the following uestion: when does the right-hand side L -converges to the function f(x?. It is a classical fact that the Fourier series exansion of a C 1 -function on the circle S 1 absolutely and uniformly converges [8, Theorem ]. oreover, a C 1 -function on the 2-shere S 2 admits a uniformly convergent series exansion by the sherical harmonics [4, Cha.10.4]. We refer the reader to the articles [5, 2] for results on the uniform convergence of the exansion of a function in terms of sherical harmonics in higher dimensions. The objective of this short article is to record the following simle theorem that works for a general closed smooth Riemannian manifold of dimension m athematics Subject Classification. 58J05, 35P10. Key words and hrases. Lalace Beltrami oerator, eigenfunction exansion, L -convergence, Riemannian geometry, Green s function. 1

2 2 ATSUSHI KANAZAWA Theorem 1.1. Let k be the minimal integer greater than m. Then the 4 Lalace Beltrami eigenfunction exansion of a function f L 2 ( L 2k k l -converges 1 if f C 2l ( for an integer 1 l k. At this oint, we do not know whether or not the above estimate rovides a shar bound for the Riemannian manifolds of dimension m 2 (see Section 3. The only result in this direction known to the author is the fact that if a function f is in the Sobolev sace H k (, then the convergence of the Lalace Beltrami eigenfunction exansion is also in H k ( [3, Theorem 3.1]. It is also worth mentioning that the Riesz mean version of the L -convergence roblem is discussed in [7]. We hoe that our simle result is useful in geometric analysis of Riemannian manifolds. 2. Proof of Theorem We denote by L ( the sace of L -integrable functions on where the norm is defined by f := ( f(x dx 1 for 1 < and f := max x ( f(x for =. The Hölder conjugate of is denoted by. The essential ingredients of the roof of Theorem 1.1 are the invertible ellitic oerator I, where I denotes the identity oerator, and its Green s function G(x, y = G x (y. We define an oerator Gr by Gr[f](x := G x (yf(ydy, where the domain of Gr will be secified in Lemma 2.1 below. We assume that m 3 until otherwise stated. Then we observe that G x (y L ( for < m because for sufficiently close x and m 2 y in the Green s function behaves as G x (y Cd(x, y 2 m for some constant C > 0. Here d(, denotes the distance defined by the Riemannian metric on. The following lemma is an analogue of Young s ineuality for convolutions [1]. Lemma 2.1. For 1 < m, 1, r 1 and = 1. m 2 r Then for f L ( we have Gr[f] L r and Gr[f] r K f, where K := su x ( Gx <. 1 Throughout the article, a fraction with 0 in the denominator is always understood as.

3 L -CONVERGENCE OF EIGENFUNCTION EXPANSIONS 3 Proof. The roof is almost identical to that of Young s ineuality, but we include it here for the sake of comleteness. First the generalized Hölder ineuality shows that Gr[f](x ( ( = G x (y r Gx (y 1 r f(y r f(y 1 r dy 1 ( G x (y f(y r dy G x (y (1 1 ( r dy f(y (1 1 r dy 1 G x (y f(y r dy Gx 1 r f 1 r, where we used relations (1 r = and (1 r =. Therefore it follows that Gr[f](x r dx f r ( Gx r G x (y f(y dy dx ( K r f r G y (x dx f(y dy K r f r K r f r. G y f(y dy oreover K < because is comact and 1 < m. m 2 Lemma 2.2. For an integer k > m and a function f 4 L2 (, we have Gr l 1 [f] L 2k k l+1 ( and Gr l [f] 2k K 2k Gr l 1 [f] 2k. k l 2k 1 k l+1 for 1 l k. Here Gr l denotes l-th iteration of Gr. Proof. We can rove the assertion by induction on l, by setting = 2k 2k 1, = 2k k l + 1, r = 2k k l in Lemma 2.1. Here the assumtion k > m m guarantees that <. 4 m 2 We leave it to the reader to check the above, and r satisfy the assumtions of Lemma 2.1 for 1 l k. We are now in a osition to rove Theorem 1.1. Proof of Theorem 1.1. Let k and l be integers such that k > m and 4 1 l k. Let f(x = f iω i (x be the eigenfunction exansion of a function f C 2l (. Since has no boundary, we have Gr l [( I l f] = ( 1 l f

4 4 ATSUSHI KANAZAWA by integration by arts. Let λ i be the eigenvalue of the eigenfunction ω i (x. Then it follows that for an arbitrary n N n f f i ω i 2k = Gr [ n ] l (I l f (1 λ i l f i ω i 2k k l k l K l n 2k (I l f (1 λ i l 2 (1 f i ω i 2k 1 where we reeatedly used Lemma 2.2 in the second line. On the other hand, the eigenfunction exansion of (I l f(x L 2 ( is given by (I l f(x = (1 λ i l f i ω i (x by the self-adjointness of I. Therefore we conclude that the righthand side of E.(1 converges to zero as n increases. This roves the L 2k k l -convergence of the eigenfunction exansion of f C 2l ( for m 3. We finally deal with the case where( m = 2. With the same notation as before, we have K 2 = su Gx2 x < because for sufficiently close x and y in the Green s function behaves as G x (y C log(d(x, y for some constant C > 0. Then, for = 2, k = 1 and l = 1, an almost identical argument works for Lemma 2.1 and Lemma 2.2 to obtain the estimate Gr[f] K 2f2. The rest follows from the same argument as the last aragrah. 3. Otimality Lastly we make an observation about the otimality of Theorem 1.1. The best result known for the shere is the following: Theorem 3.1 (Ragozin [5] and Kalf [2]. Assume that m 2. Then a function f C [ m 2 ] (S m admits a uniform convergent exansion in terms of sherical harmonics. oreover, the regularity assumtion in Theorem 3.1 is known to be otimal [2, Remark 2]. Although our general result does not recover the above otimal estimate, the defect is at worst twice differentiability in the sheres case. As was mentioned earlier, we do not know at this oint whether or not our estimate rovides a shar bound for m 2 and 1, and we hoe to come back to this otimality roblem in future work.

5 L -CONVERGENCE OF EIGENFUNCTION EXPANSIONS 5 Acknowledgement. The author is very grateful to C. Behan and J. a for useful discussion on the subject. References [1] V. I. Bogachev, easure Theory I, Berlin, Heidelberg, New York: Sringer- Verlag, [2] H. Kalf, On the exansion of a function in terms of sherical harmonics in arbitrary dimensions, Bull. Belg. ath. Soc. Simon Stevin, Volume 2, Number 4 (1995, [3] J. L. Kazdan, Alications of Partial Differential Euations to Some Problems in Differential Geometry, Lecture note, htt:// [4] O. D. Kellogg, Foundations of otential theory, Berlin: Sringer [5] D. L. Ragozin, Uniform convergence of sherical harmonic exansions. ath. Ann. 195 ( [6] S. Rosenberg, The Lalacian on a Riemannian anifold: An Introduction to Analysis on anifolds, ser. London athematical Society Student Texts. Cambridge Univ. Press, [7] C. D. Sogge, On the convergence of Riesz means on comact manifolds, Ann. of ath. (2 126 (1987, no. 2, [8] G. P. Tolstov, Fourier Series (Dover Books on athematics, [9] I. Vekua, On metaharmonic functions (Russ., English summary. Trudy Tbilisskogo at. Inst. 12 ( Deartment of athematics University of British Columbia 1984 athematics Rd Vancouver, BC, V6T 1Z2, CANADA. kanazawa@10.alumni.u-tokyo.ac.j