On the Weak Localization Principle of the Eigenfunction Expansions of the Laplace-Beltrami Operator by Riesz Method ABSTRACT 1.

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1 Malaysia Joural of Mathematical Scieces 9(): (05) MALAYSIA JOURAL OF MATHEMATICAL SCIECES Joural homepage: O the Weak Localizatio Priciple of the Eigefuctio Expasios of the Laplace-Beltrami Operator by Riesz Method, Avarjo Ahmedov ad * Ahmad Fadly urullah Rasedee Departmet of Process ad Food Egieerig, Faculty of Egieerig, Uiversiti Putra Malaysia, UPM Serdag, Selagor, Malaysia Istitute for Mathematical Research, Uiversiti Putra Malaysia, UPM Serdag, Selagor, Malaysia ahmadfadlyurullah@yahoocom *Correspodig author ABSTRACT I this paper we deal with the problems of the weak localizatio of the eigefuctio expasios related to Laplace-Beltrami operator o uit sphere The coditios for weak localizatio of Fourier-Laplace series are ivestigated by comparig the Riesz ad Cesaro methods of summatio for eigefuctio expasios of the Laplace- Beltrami operator It is show that the weak localizatio priciple for the itegrable fuctios f ( x ) at the poit x depeds ot oly o behavior of the fuctio aroud x but o the behavior of the fuctio aroud diametrically opposite poit x Keywords: Distributios, Fourier-Laplace series, localizatio, Riesz method, Sphere, Laplace-Beltrami operator ITRODUCTIO I this work, the problems of the weak localizatio of the eigefuctio expasios of Laplace-Beltrami operator o uit sphere are cosidered It is stated i Riema's localizatio theorem that the covergece or divergece for a oe-dimesioal Fourier series at a give poit depeds oly o the behavior of the fuctio f L i a arbitrary small eighborhood of the poit The localizatio theorem may also formulated as follows: if two fuctios coicide i the eighborhood of a poit, the the partial sums of their Fourier series have the same behavior

2 Avarjo Ahmedov & Ahmad Fadly urullah there The problems of localizatio of the eigefuctio expasios of the itegrable fuctios are ivestigated i the papers by Ili, 968, Boami ad Clerc, 973, Alimov, 974, Pulatov, 983, Topuria, 987, Rakhimov, 003 ad Ashurov, 0 For more referece o the problems of the covergece of the Fourier-Laplace series o uit sphere we refer the readers to Rakhimov, 003 We ote here that the ivestigatio of the localizatio problems of the Fourier-Laplace series related to distributios o uit sphere was started by Rakhimov, 003 He foud the sufficiet coditios for the localizatio of the Fourier-Laplace series of the distributios by Cesaro meas i the classes of ikolskii, Liouville ad Sobolev I the work of Ahmedov, 00, he proved the geeralized priciple of localizatio for the Fourier-Laplace series by Riesz Meas of the order s ( ) p, p I the curret work we ivestigate the weak localizatio of the eigefuctio expasios for the case of the Riesz meas of the spectral expasios of the Laplace-Beltrami operator o uit sphere We give positive aswer to the followig cojucture: The sufficiet coditios for localizatio ca be weakeed for the eigefuctio expasios of the Laplace-Beltrami operator o uit sphere? EIGEFUCTIO EXPASIOS OF THE LAPLACE- BELTRAMI OPERATOR Let S be a - dimesioal uit sphere i R We cosider a symmetric ad oegative elliptic operator, kow as the Laplace- Beltrami operator The Laplace-Beltrami operator complete orthoormal system of eigefuctios Y ( x), Y ( x),, Y ( x), k = 0,,, ( k ) ( k ) ( k ) ak correspodig to the eigevalues ak is the multiplicity of the eigevalue k 338 Malaysia Joural of Mathematical Scieces s has i s L S a k k k 0,,, where k The Riesz meas of the spectral expasios of the order 0 related to the Laplace-Beltrami operator is defied by k E f ( x) Y f, x, k k 0

3 O the Weak Localizatio Priciple of the Eigefuctio Expasios of the Laplace-Beltrami Operator by Riesz Method Here ak k k Y f, x Y ( x) f ( y) Y ( y) d ( y), k j j j S d ( y) si si si d d d d For ay x S the diametrically opposite of x is deoted by x It is obvious that spherical distace xx, betwee x ad x is equal to : xx, MAI RESULTS We proceed with the formulatio of the mai results of the paper Theorem Let f L S x S If f vaishes i a eighborhood of poit ad x S the the Riesz meas E f ( x) coverges to zero at x : lim E f ( x) 0 of order The order of the Riesz meas E f ( x) of the spectral expasios of the fuctio f is called critical order It is proved that (see Stei, 958) the coditio caot guaratee covergece of the Riesz meas E f ( x) for itegrable fuctios I the case of classes L S, p we have the followig: Theorem Let f L S, eighborhood of x, coverges to zero at x : p p p If f vaishes i some S the the Riesz meas E f ( x) lim E f ( x) 0 of order Malaysia Joural of Mathematical Scieces 339

4 3 Proof of Theorem Avarjo Ahmedov & Ahmad Fadly urullah 3 PROOF OF MAI RESULTS The Riesz meas of the spectral expasios of the Laplace-Beltrami operator ca be writte as the followig itegral operator with the kerel E f ( x) x, y, f ( y) d ( y) S ( ) ( ),, a k k k k ( ) ( ) j j k0 j x y Y x Y y Usig the Gegebauer polyomials Pk () t, by P k () t k k we ca represet x, y, k k k d k t t, k dt i the followig form k k k x, y, P k cos k 0 From this represetatio we ca coclude that x, y, depeds oly o the distace betwee x ad y Due to this fact, we ca deote the kerel as x, y, cos Theorem 3 Let x, y, be the kerel of Riesz meas of the spectral expasios If ( xy, ), as, the si 4 ( x, y, ) = ( ) si si / 340 Malaysia Joural of Mathematical Scieces

5 O the Weak Localizatio Priciple of the Eigefuctio Expasios of the Laplace-Beltrami Operator by Riesz Method 3 ( ) ( ) si si si where ( ) C, ( ) C; If, 0, 0 >, the ( x, y, ) C, 3 If 0, >, the For the proof, see Rasedee, 05 ( x, y, ) = C The itegral over uit sphere of the fuctio g ( xy), where ( xy) scalar product, ca be writte as follows 4 5 ; 0 0 S x, ycos x, ycos g( x y) d( y) g( ) dt( y) d g( ) d f ( y) dt( y) dt( y) Usig this represetatio we have here which yields 0 xy, cos E f ( x) cos f ( y) dt( y) xy, cos f ( y) dt( y) 0 E f ( x) cos d ad f( y) 0, if Let f L S y B ( x) B ( x), for some, 0 We the separate itegratio from 0 to ito the followig itegral E f ( x) cos d 0 Malaysia Joural of Mathematical Scieces 34

6 Usig the fact that f( y) 0 whe Avarjo Ahmedov & Ahmad Fadly urullah 0 xy, y B ( x) B ( x) cos d x, y, f ( y) d ( y) ad x, y, f ( y) d ( y) 0 B ( x ) x, y cos d x, y, f ( y) d ( y) x, y, f ( y) d ( y) 0 B ( x ) hece, allows us to rewrite the Riesz meas of spectral expasios of f as follows, E f ( x) cos d We estimate usig Theorem 3, the have si 4 ( ) E f ( x) ( ) d si si / 3 ( ) ( ) ( ) ( ) d d I I I si si si Estimatio of I 3 Let, ( ) we see that si 3 is itegrable o, for 34 Malaysia Joural of Mathematical Scieces

7 O the Weak Localizatio Priciple of the Eigefuctio Expasios of the Laplace-Beltrami Operator by Riesz Method, 0, the it ca be clearly see that ( ) ( ) C ( ) I 0, 3 d d si si Estimatio of I Let It is ot difficult to see that I C ( ) d 0, si si ( ) whe is itegrable o, whe, 0 si si Estimatio of I We have I 3 si ( ) ( ) d, si si / si si / which ca be writte as, 3 ( ) cos si d 3 ( ) si cos d si si / Malaysia Joural of Mathematical Scieces 343

8 The fuctios ad Avarjo Ahmedov & Ahmad Fadly urullah 3 ( ) cos si si / si si / 3 ( ) si are itegrable o, for ay, 0 The the Riema-Lebesgue Theorem implies that as Fourier coefficiets 3 Proof of Theorem From 3 ( ) cos si d 0, si si / si si / 3 ( ) si cos d 0, ( ) E f ( x) C d, si we obtai the E f ( x) 0 for If Lebesgue Theorem it ca be show that the itegral the usig the ( ) si d 344 Malaysia Joural of Mathematical Scieces

9 O the Weak Localizatio Priciple of the Eigefuctio Expasios of the Laplace-Beltrami Operator by Riesz Method ca be made less tha ay 0 If we have ( ) E f ( x) C d si si si We will show that the itegral is fiite C ( ) si ( ) si d d Let p satisfy Usig Holder iequality we obtai p q ( ) ( ) d q si si q si d p p ( ) d C d p q q si si p p p q f ( y) q dt y xy, cos d C p q si q 0 si p p d C f ( y) ds y q S 0 si q q Malaysia Joural of Mathematical Scieces 345

10 Avarjo Ahmedov & Ahmad Fadly urullah The first itegral is fiite because f Lp S, ad if we choose p, the q satisfies the followig coditio: q By similar method we ca show that the secod itegral is also fiite COCLUSIO I the curret work, sufficiet coditios for weak localizatio are established i the classes of itegrable fuctios It is show that the coverget of the Fourier Laplace series of the itegrable fuctios at oe poit requires ivestigatios of the behavior of the fuctio ot oly i the give poit, but also its diametrical opposite ACKOWLEDGEMET This paper has bee supported by Miistry of Higher Educatio (Mohe) for its MyPhd sposorship ad Uiversiti Putra Malaysia uder Research Uiversity Grat (RUGS), Project umber is RU REFERECES Ahmedov, A A, Rasedee, A F ad Rakhimov, A (03) O the sufficiet coditios of the localizatio of the Fourier-Laplace series of distributios from Liouville classes J Phys: Cof Ser 435: Article ID: 006 Ahmedov, A A, Rasedee, A F ad Rakhimov, A (03) Localizatio of Fourier-Laplace series of distributios Malaysia Joural of Mathematical Scieces 7(): Ahmedov, A A (009) The geeralized priciple of the localizatio o uit sphere Joural of Mathematical Aalysis ad Applicatios 56: 30-3 Ahmedov, A A (997) About spectral expasios of Laplace - Beltrami operator o sphere Uzbek Mathematical Joural 4: 0-7 Ahmedov, A A (0) The Estimatios for Maximal Operators Applied Mathematics Letter 4(5): Malaysia Joural of Mathematical Scieces

11 O the Weak Localizatio Priciple of the Eigefuctio Expasios of the Laplace-Beltrami Operator by Riesz Method Ahmedov, A A (00) The geeralized priciple of the localizatio for the -Fold Fourier Itegrals Malaysia Joural of Mathematical Scieces 4(): 09-5 Alimov, Sh A (974) O the localizatio of spectral decompositios Differ Urav 0(4): Alimov, Sh A, Ashurov, R R ad Pulatov, A K (99) Multiple Fourier series ad Fourier itegrals Commutative Harmoic Aalysis IV Spriger-Verlag: -97 Ashurov, R R (983) Summability almost everywhere of Fourier series i Lp with respect to eigefuctios Mathematical otes of the Academy of Scieces of the USSR 34(6): Ashurov, R, Butaev, A ad Pradha, B (0) O Geeralized Localizatio of Fourier Iversio Associated with a Elliptic Operator for Distributios Abstract ad Applied Aalysis: Article ID: , 3 pages doi:055/0/ Bastis, A I (983) Almost-everywhere covergece of expasios i eigefuctios of the Laplace operator o the sphere Mathematical otes of the Academy of Scieces of the USSR 33(6): Boami, A ad Clerc, J L (973) Sommes de Cesa'ro et multiplicateues de de'veloppemets et harmoique sphe'riques Tras Amer Math Soc 83: 83-3 Carbery, A ad Soria, F (988) Almost-everywhere covergece of Fourier itegrals for fuctios i Sobolev spaces ad a localizatio priciple Rev Mat Iberoamericaa 4(): Cazzaiga, F ad Travaglii, G (993) O poitwise covergece ad localizatio for Fourier series o compact Lie groups Arch Math 60: Hormader, L ( ) O the Riesz meas of spectral fuctios ad eigefuctio expasios for elliptic differetial operators, Recet advaces i the Basic Scieces Proc Aual Sci Cof Belfer Grad School Sci, Yeshiva Uiv ew York: 55-0 Malaysia Joural of Mathematical Scieces 347

12 Avarjo Ahmedov & Ahmad Fadly urullah Ili, V A (968) Problems of localizatio ad covergece for Fourier series i fudametal systems of the Laplace operator Russia Math Surveys 3: 59-6 Ili, V A ad Alimov, Sh A (97) Covergece coditios for spectral decompositios coected with the self-adjoit extesio of elliptic operators, I, II Differ Urav 7(4): , 7(5): Meaey, C (984) Localizatio of spherical harmoics expasio Moatsh Math: Peetre, T (964) Remark o eigefuctio expasios for elliptic operators with costat coefficiets Math Scad 5: 83-9 Pulatov, A K (98) O uiform covergece ad localizatio for arithmetic meas of Fourier-Laplace series J Izvestiya Vuzov, Phys Math Sci 3(4): Rakhimov, A A (003) O the localizatio coditios of the Riesz meas of distributio expasios of the Fourier series by eigefuctios of the Laplace operator o sphere Soviet Math Dokl 3(3): Rasedee, A F (05) Spectral expasios of the Laplace-Beltrami operator o the sphere Uiversity Putra Malaysia, Phd Thesis (Preprit) Stei, E M (958) Localizatio ad summability of multiple Fourier series Acta Mathematica 00(-): Topuria, S B (987) The Fourier-Laplace series o a sphere Izdat Tbilisi TGU -356 (i Russia) 348 Malaysia Joural of Mathematical Scieces