THE RIESZ-KOLMOGOROV THEOREM ON METRIC SPACES

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1 Miskolc Mathematical Notes HU e-issn Vol. 5 (24), No. 2, THE RIES-KOLMOGOROV THEOREM ON METRIC SPCES PREMYSŁW GÓRK ND NN MCIOS Received 24 Setember, 23 bstract. We study recomact sets in L.;/, where.;/ is a metric measure sace. Using Vitali and maximal function theorems we establish a full characterization of such sets. 2 Mathematics Subject Classification: 28C99; 46B5; 46E3 Keywords: metric measure saces, recomact sets, Riesz-Kolmogorov theorem. INTRODUCTION Comact sets are imortant object of analysis research because of their imortant roerties and huge alications. Therefore, question about full characterization of comact or recomact sets (sets such that their closure is comact) in different saces is very imortant. Riesz-Kolmogorov theorem gives necessary and sufficient conditions for recomactness of subset of L.R n / (see [6,]). This theorem lays a fundamental role in analysis. Esecially, it is frequently used in the theory of function saces, e.g. Sobolev and Besov saces. The goal of the aer is to give a characterization of recomact sets in L.;/, where.; / is a metric sace with doubling measure, i.e ositive Borel measure satisfying condition <.B.x;2r// C.B.x;r// < for all x 2, r > and some constant C > called doubling constant. We shall denote the average of locally integrable function f over the measurable set in the following manner.f / D f d:./ In this note we would like to resent the following observation. The first author enjoyed the suort by the Euroean Union in the framework of Euroean Social Fund through the Warsaw University of Technology Develoment Programme. c 24 Miskolc University Press

2 46 PREMYSŁW GÓRK ND NN MCIOS Theorem. Let be a doubling measure such that h.r/ WD inff.b.x;r// W x 2 g > for each r > and assume that < <. Let x 2, then the subset F of L.;/ is relatively comact in L.;/ if and only if the following conditions are satisfied: F is bounded; (.) lim jf.x/j d.x/ D ; uniformly forf 2 F ; (.2) nb.x ;R/ jf.x/.f / B.x;r/ j d.x/ D uniformly forf 2 F : (.3) R! lim r! We can find a bit similar theorem in [5], i.e. sufficient condition for recomactness of subset of L.;/ for <, where.;/ is a metric sace with finite measure. The roof of our theorem is obtained by aroriate modifications in the roof of theorem mentioned above [5]. We aly Vitali convergence theorem. Moreover, Lebesgue differentiation theorem as well as Hardy-Littlewood maximal function theorem will be used. Finally, let us mention about some generalizations of the Riesz-Kolmogorov theorem. For instance, Weil [] showed the comactness theorem in L.G/, where G is a locally comact grou. Pego [8] (see also [2]) formulated Kolmogorov theorem for D 2 in terms of the Fourier transform. There also exists a characterization of relatively comact subsets of general Banach saces [9]. 2. PROOF OF THEOREM First of all we assume that conditions (.)-(.3) hold. Let f n be a sequence of elements from F. We shall rove that f n has a subsequence converging in L.;/. Since the set F is bounded and the sace L.;/ is reflexive, we may assume that f n converges weakly to some f 2 L.;/. We can reresent the sace in the following manner [ D B.x ;n/: Now, we shall show that for each n nd f k B.x ;n/! f B.x ;n/ in L.;/: k! We aly Vitali convergence theorem (see e.g. [7]). We first rove that the sequence f k B.x ;n/ is -equi-integrable. For this urose we fix >. By (.3), there exists r such that for each k jf k.x/.f k / B.x;r/ j d.x/ 2 :

3 THE RIES-KOLMOGOROV THEOREM ON METRIC SPCES 46 Moreover, we fix a measurable set such that B 2 su jjf k jj L./ k2n h.r/: Then, using Minkowski and Hölder inequality we jf k B.x ;n/.x/j jf k.x/j jf k.x/.f k / B.x;r/ j d.x/ j.f k / B.x;r/ j d.x/ 2 C 2 C 2 C./ h.r/..b.x;r///.b.x;r// B.x;r/ B.x;r/ su jjf k jj L./ < : k2n f k.y/d.y/ C d.x/ jf k.y/j C d.y/d.x/ This ends the roof of -equi-integrability. Our next claim is that f k B.x ;n/! f B.x ;n/ in measure. Let us take >. k! We note x 2 W jfk B.x ;n/.x/ f B.x ;n/.x/j > D fx 2 B.x ;n/ W jf k.x/ f.x/j > g n x 2 B.x ;n/ W jf k.x/.f k / B.x;r/ j > n 3 [ x 2 B.x ;n/ W j.f k / B.x;r/.f / B.x;r/ j > o n 3 [ x 2 B.x ;n/ W jf.x/.f / B.x;r/ j > o 3 D [ 2 [ 3 : It suffices to rove that for each ı >, there exists K such that for each k K, the inequality. / C. 2 / C. 3 / < ı o

4 462 PREMYSŁW GÓRK ND NN MCIOS holds. Using Markov s inequality we get 3. / jf k.x/ 3 B.x ;n/ jf k.x/.f k / B.x;r/ j d.x/.f k / B.x;r/ j d.x/: Therefore, by assumtion (.3), there exists r such that for each r r we get. / < ı 3 : Next, we consider the set 2. We conclude from f k w! B.x ;n/ and r >.f k / B.x;r/!.f / B.x;r/ : k! f that for each x 2 This gives.f k / B.x;r/!.f /B.x;r/ in measure on B.x ;n/. By the definition, there exists K.r/ such that for each k K.r/ the inequality. 2 / < ı 3 holds. It remains to consider the set 3. Using Markov s inequality we have 3. 3 / D r WD jf.x/.f / B.x;r/ j d.x/: B.x ;n/ Thanks to Lebesgue differentiation theorem (see [4]) we get that lim r! f B.x;r/ D f.x/ a.e. on. Direct calculations leads us to the following estimates jf.x/ f B.x;r/ j jf.x/j C j.f / B.x;r/j 2 jf.x/j C j.f / B.x;r/ j 2 jf.x/j C jm.f /.x/j ; where M.f / is a maximal function of f. By virtue of the Hardy-Littlewood maximal function theorem (see [4]) we have km.f /k L.;/ C./kf k L.;/, where >. Therefore, since jf j is integrable, the Lebesgue dominated convergence theorem gives lim D r D r! 3 B.x ;n/ lim jf.x/.f / B.x;r/j d.x/ D r! and hence there exists r, such that for r r we have 3 jf.x/.f / B.x;r/ j d.x/ < ı 3 : B.x ;n/ Finally, taking Qr D minfr ;r g, we obtain the inequality.fx 2 W jf k B.x ;n/.x/ f B.x ;n/.x/j > g/ < ı

5 THE RIES-KOLMOGOROV THEOREM ON METRIC SPCES 463 for each k K.Qr/. Now, we show that f k! f in L.;/: k! Let us fix >. Since f 2 L.;/ and by (.2), there exists N such that jf k.x/ f.x/j d.x/ 2 : Hence, kf k nb.x ;N / f k L.;/ D jf k B.x ;N /.x/ f B.x ;N /.x/j d.x/ C jf k.x/ f.x/j d.x/ : nb.x ;N / This ends the roof of relatively comactness of F. Now, we shall show the converse. We assume that the family F is relatively comact. Hence, the boundedness is straightforward. To establish condition (.2), let us fix > and let U D fu ;U 2 ;:::;U n g be an -cover of F. For each k D ;:::;n we can select g k 2 U k and R k > such that jg k.x/j d.x/ < : nb.x ;R k / Let us fix f 2 F. Since U is an -cover of F, there exists k n such that f 2 U k. Thus, jf.x/ g k.x/j d.x/ < and for R D maxfr i W i ng we get jf.x/j d.x/ 2 nb.x ;R/ C 2 So condition (.2) holds. Using g k choosen as above, we obtain: jf.x/.f / B.x;r/ j d.x/ 2 C 2 2. C 2 2. nb.x ;R/ nb.x ;R/ jf.x/ / / jf.x/ jg k.x/ g k.x/j d.x/ jg k.x/j d.x/ < 2 : g k.x/j d.x/ j.g k / B.x;r/.g k / B.x;r/ j d.x/ D 2 I C 2 2. / I 2 C 2 2. / I 3 :.f / B.x;r/ j d.x/

6 464 PREMYSŁW GÓRK ND NN MCIOS We easily get I <. Now, using the Hardy-Littlewood maximal function theorem we obtain I 3 D.g k.y/ f.y//d.y/.b.x;r// d.x/ B.x;r/ jm.g k f /.x/j d.x/ C./ jg k.x/ f.x/j d.x/ C./ : Now it remains to rove that I 2 is arbitrary small. By Lebesgue differentiation theorem we obtain that lim gk.x/.g k / B.x;r/ D ; a.e. on Moreover, r! jg k.x/.g k / B.x;r/ j 2.jg k.x/j C jm.g k /.x/j /: Since jg k j and jm.g k /j are integrable, we can aly Lebesgue dominated convergence theorem. Thus, I 2 D jg k.x/.g k / B.x;r/ j d.x/ < and the roof is comlete. Let us remark that if is doubling and diam < or is continuous with resect to the metric (see [, 3]) and is comact, then for each r >, h.r/ is ositive. CKNOWLEDGEMENT The authors would like to thank the anonymous referee for careful reading of the aer, correcting errors and ointing out laces where the exosition is not entirely clear. REFERENCES [] M. Gaczkowski and P. Górka, Harmonic functions on metric measure saces: Convergence and comactness, Potential nal., vol. 3, , 29. [2] P. Górka, Pego theorem on locally comact abelian grous, J. lgebra l., vol. 3, 24. [3] P. Górka, Camanato theorem on metric measure saces, nn. cad. Sci. Fenn., Math., vol. 34, no. 2, , 29. [4] J. Heinonen, Lectures on nalysis on Metric Saces. Universitext, 2. [5]. Kałamajska, On comactness of embedding for sobolev saces defined on metric saces, nn. cad. Sci. Fenn., vol. 24, , 999. [6]. N. Kolmogorov, Über komaktheit der funktionenmengen bei der konvergenz im mittel, Machr. Ges. Wiss. Göttingen, vol. 9,. 6 63, 93. [7] G. Leoni, first course in Sobolev saces. Graduate Studies in Mathematics, MS, 29. [8] R. L. Pego, Comactness in l 2 and the fourier transform, Proc. m. Math. Soc., vol. 95, , 985. [9] R. S. Phillis, On linear transforms, Trans. m. Math. Soc., vol. 48, , 94.

7 THE RIES-KOLMOGOROV THEOREM ON METRIC SPCES 465 [] M. Riesz, Sur les ensembles comacts de foncions sommables, cta Szeged Sect. Math., vol. 6, , 933. []. Weil, L intégration dans les groues toologiques et ses alications. Hermann et Cie., Paris, 94. uthors addresses Przemysław Górka Warsaw University of Technology, Deartment of Mathematics and Information Sciences, Ul. Koszykowa 75, -662 Warsaw, Poland address: P.Gorka@mini.w.edu.l nna Macios Warsaw University of Technology, Deartment of Mathematics and Information Sciences, Ul. Koszykowa 75, -662 Warsaw, Poland