The Multiplication Operator in Sobolev Spaces with Respect to Measures

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1 Joural of Approximatio Theory 109, (2001) doi: jath , available olie at o The Multiplicatio Operator i Sobolev Spaces with Respect to Measures Jose M. Rodr@ guez 1 Departmet of Mathematics, Uiversidad Carlos III de Madrid, Aveida de la Uiversidad, 30, Legae s, Madrid, Spai jomaromath.uc3m.es Commuicated by Walter Va Assche Received Jauary 6, 1997; accepted i revised form November 6, 2000; published olie February 5, 2001 We cosider the multiplicatio operator, M, i Sobolev spaces with respect to geeral measures ad give a characterizatio for M to be bouded, i terms of sequetially domiated measures. This has importat cosequeces for the asymptotic behaviour of Sobolev orthogoal polyomials. Also, we study properties of Sobolev spaces with respect to measures Academic Press Key Words: Sobolev spaces; weights; orthogoal polyomials. 1. INTRODUCTION Weighted Sobolev spaces are a iterestig topic i may fields of mathematics (see, e.g., [HKM, K, Ku, KO, KS, T]. I [ELW1, EL, ELW2] the authors study some examples of Sobolev spaces with respect to geeral measures istead of weights, i relatio with ordiary differetial equatios ad Sobolev orthogoal polyomials. The papers [RARP1, RARP2] are the begiig of a theory of Sobolev spaces with respect to geeral measures. We are iterested i the relatioship betwee this topic ad Sobolev orthogoal polyomials. Let us cosider 1p< ad +=(+ 0,..., + k ) a vectorial Borel measure i R with 2 := k j=0 supp + j. The Sobolev orm of a fuctio f of class C k (R) iw k, p (2, +) is defied by k & f & p W k, p (2, +) := : f ( j) p d+ j. j=0 We talk about Sobolev orm although it ca be a semiorm; i this case we will take equivalece classes, as usual. 1 Research partially supported by a grat from DGES (MEC), Spai Copyright 2001 by Academic Press All rights of reproductio i ay form reserved.

2 158 JOSE M. RODRI GUEZ We say that + # M if every polyomial belogs to L 1 (+ 0 ) & L 1 (+ 1 ) & }}} & L 1 (+ k ). Therefore if + # M, every polyomial belogs to L p (+ 0 ) & L p (+ 1 ) & }}} & L p (+ k ) for ay 1p<. Obviously, every + # M is fiite. If 2 is a compact set, we have + # M if ad oly if + is fiite. If + # M, we deote by P k, p (2, +) the completio of polyomials P with the orm of W k, p (2, +). By a theorem i [LP] we kow that the zeros of the Sobolev orthogoal polyomials with respect to the scalar product i W k,2 (2, +) are cotaied i the disk [z: z 2 &M&], where the multiplicatio operator (Mf )(x)=x f(x) is cosidered i the space P k,2 (2, +). Cosequetly, the set of the zeros of the Sobolev orthogoal polyomials is bouded if the multiplicatio operator is bouded. The locatio of these zeros allows to prove results o the asymptotic behaviour of Sobolev orthogoal polyomials (see [LP]). I [LP] also appears the followig result: If 2 is a compact set ad + is a fiite measure i 2 sequetially domiated, the M is a bouded operator i P k,2 (2, +), where the vectorial measure + is sequetially domiated if *supp + 0 = ad d+ j = f j d+ j&1 with f j bouded for 1 jk. I that paper the authors ask for other coditios o M to be bouded. It is ot difficult to see that the multiplicatio operator ca be bouded whe the vectorial measure is ot sequetially domiated. I Sectio 4 below ad i [RARP2] other coditios are give i order to have the boudedess of M. Now, let us state the mai results here. We refer to the defiitios i Sectios 2 ad 4. I the paper, the results are umbered accordig to the sectio where they are proved. Here we obtai the followig characterizatio for the boudedess of the multiplicatio operator i terms of comparable orms. Observe that this characterizatio is closely related to sequetially domiated measures, sice we say that a vectorial measure + belogs to the class ESD (exteded sequetially domiated) if ad oly if d+ j = f j d+ j&1 with f j bouded for 1 jk. Theorem 4.1. Let us cosider 1p< ad +=(+ 0,..., + k ) a fiite vectorial measure with 2 a compact set. The, the multiplicatio operator is bouded i P k, p (2, +) if ad oly if there exists a vectorial measure +$#ESD such that the Sobolev orms i W k, p (2, +) ad W k, p (2, +$) are comparable o P. Furthermore, we ca choose +$=(+$ 0,..., +$ k ) with +$ j :=+ j ++ j+1 +}}}++ k. We have also ecessary coditios ad sufficiet coditios for M to be bouded. The followig are the most importat.

3 MULTIPLICATION OPERATOR IN SOBOLEV SPACES 159 Theorem 4.3. Let us cosider 1p< ad a fiite vectorial measure + with 2 a compact set. Assume that (2 ad, + ad )#C 0 ad that for each 1 jk we have + j (2"(J j&1 _ K j&1 ))=0, where K j&1 is a fiite uio of compact itervals cotaied i 0 ( j&1), ad J j&1 is a measurable set with d+ j = f j d+ j&1 i J j&1 ad f j bouded. The the multiplicatio operator is bouded i P k, p (2, +). Theorem 4.4. Let us cosider 1<p< ad +=(+ 0,..., + k ) a fiite vectorial measure such that 2 is a compact set ad there exist ' 0 >0, x 0 # R ad 0<k 0 k with + j ([x 0 &' 0, x 0 +' 0 ])=0 for k 0 < jk. Let us assume that x 0 is either right or left (k 0 &1)-regular. If + k0 ([x 0 ])>0 ad + k0 &1([x 0 ])=0, the the multiplicatio operator is ot bouded i P k, p (2, +). Theorem 4.5. Let us cosider 1p< ad +=(+ 0,..., + k ) a fiite vectorial measure with 2 a compact set. Assume that there exist x 0 # R, c>0, 0k 0 <k ad a ope eighbourhood U of x 0 such that d+ j+1 (x)c x&x 0 p d+ j (x), for x # U"[x 0 ] ad k 0 j<k. If there exists i>k 0 with + i ([x 0 ])>0 ad + i&1 ([x 0 ])=0, the the multiplicatio operator is ot bouded i P k, p (2, +). I order to prove these results we also obtai some results o Sobolev spaces with respect to measures, which are iterestig by themselves. Theorem 3.1. Let us cosider 1p< ad +=(+ 0,..., + k ) a fiite vectorial measure with 2 a compact set. Assume that there exist x 0 # R ad 0k 0 k with + k0 ([x 0 ])=0 ad satisfyig the followig property if k 0 <k: there exist a ope eighbourhood U of x 0 ad c>0 such that d+ j+1 (x)c x&x 0 p d+ j (x), for x # U ad k 0 j<k. Let us defie & :=(0,..., 0, : k0 $ x0, : k0 +1$ x0,..., : k $ x0 ) ad N :=*[k 0 jk : : j >0]. Give a Cauchy sequece [q ]/P i W k, p (2, +) ad u k0,..., u k # R there exists a Cauchy sequece [r ]/P i W k, p (2, +) with lim &q &r & W k, p (2, +)=0 ad r ( j) (x 0)=u j for k 0 jk. Cosequetly P k, p (2, ++&) is isomorphic to P k, p (2, +)_R N. Theorem 3.3. Let us cosider 1<p< ad +=(+ 0,..., + k ) a vectorial measure such that there exist ' 0 >0, x 0 # R ad 0<k 0 k with + j ([x 0 &' 0, x 0 +' 0 ])< for 0 jk 0 ad + j ([x 0 &' 0, x 0 +' 0 ])=0 for

4 160 JOSE M. RODRI GUEZ k 0 < jk (if k 0 <k). Let us assume that x 0 is either right or left (k 0 &1)-regular ad that + k0 &1([x 0 ])=0. The, for ay 0<'' 0, there is o positive costat c 1 with c 1 f (k 0 &1) (x 0 ) & f & W k, p (2, +), for every f # C c ([x 0 &', x 0 +']). If we have also that + is fiite ad 2 is a compact set, the there is o positive costat c 2 with for every q # P. c 2 q (k 0 &1) (x 0 ) &q& W k, p (2, +), We also obtai results which allow to decide i may cases whe two orms are comparable. We have also localizatio results o the multiplicatio operator. Now we preset the otatio we use. Notatio. I the paper k1 deotes a fixed atural umber; obviously W 0, p (2, +)=L p (2, +). All the measures we cosider are Borel ad positive o R; if a measure is defied o a proper subset E/R, we defie it o R"E as the zero measure. Also, all the weights are o-egative Borel measurable fuctios defied o R. If the measure does ot appear explicitly, we mea that we are usig Lebesgue measure. We always work with measures which satisfy the decompositio d+ j =d(+ j ) s +d(+ j ) ac = d(+ j ) s +hdx, where (+ j ) s is sigular with respect to Lebesgue measure, (+ j ) ac is absolutely cotiuous with respect to Lebesgue measure ad h is a Lebesgue measurable fuctio (which ca be ifiite i a set of positive Lebesgue measure); obviously the RadoNikodym Theorem gives that every _-fiite measure belogs to this class. Give a vectorial measure + ad a closed set E, we deote by W k, p (E, +) the space W k, p (2 & E, + E ). We deote by supp & the support of the measure &. IfA is a Borel set, A, / A, A, it(a) ad *A deote, respectively, the Lebesgue measure, the characteristic fuctio, the closure, the iterior ad the cardiality of A. By f ( j) we mea the j th distributioal derivative of f. P ad P deote respectively the set of polyomials ad the set of polyomials with degree less tha or equal to. We say that a -dimesioal vector satisfies a oedimesioal property if each coordiate satisfies this property. Fially, the costats i the formulae ca vary from lie to lie ad eve i the same lie. The outlie of the paper is as follows. Sectio 2 presets most of the defiitios we eed to state our results. We prove some useful results o Sobolev spaces i Sectio 3. Sectio 4 is dedicated to the proof of the results for the multiplicatio operator.

5 MULTIPLICATION OPERATOR IN SOBOLEV SPACES DEFINITIONS AND RESULTS Obviously oe of our mai problems is to defie correctly the space W k, p (2, +). There are two atural defiitios: (1) W k, p (2, +) is the biggest space of (classes of) fuctios f which are regular eough to have & f & W k, p (2, +)<. (2) W k, p (2, +) is the closure of a good set of fuctios (e.g., C (R) or P) with the orm &}& W k, p (2, +). However, both approaches have serious difficulties: We cosider first the approach (1). It is clear that the derivatives f ( j) must be distributioal derivatives i order to have a complete Sobolev space. Therefore we eed to restrict the measures + to a class of p-admissible measures (see Defiitio 8). Roughly speakig + is p-admissible if (+ j ) s, for 0< jk, is cocetrated o the set of poits where f ( j) is cotiuous, for every fuctio f of the space, because otherwise f ( j) is determied, up to zero-lebesgue measure sets (see Defiitios 4 ad 9 below). This will force (+ k ) s to be idetically zero. However, there will be o restrictio o the support of (+ 0 ) s. This reasoable approach excludes orms appearig i the theory of Sobolev orthogoal polyomials. Eve if we work with the simpler case of the weighted Sobolev spaces W k, p (2, w) (measures without sigular part) we must impose that w j belogs to the class B p (see Defiitio 2 below) i order to have a complete weighted Sobolev space (see [KO, RARP1]). The approach (2) is simpler: we kow that the completio of every ormed space exists (e.g., (C (R), &}& W k, p (2, +)) or(p, &}& W k, p (2, +))). We have two difficulties. The first oe is evidet: we do ot have a explicit descriptio of the Sobolev fuctios as i (1) (i Sectio 4 of [RARP2] there are several theorems which show that both defiitios of Sobolev space are the same for p-admissible measures). The secod problem is worse: The completio of a ormed space is by defiitio a set of equivalece classes of Cauchy sequeces. I may cases this completio is ot a fuctio space (see Theorem 3.1 below ad its Remark). However, sice we eed to work with the multiplicatio operator i P k, p (2, +), we have to choose this secod approach if + is ot p-admissible. First of all, we explai the defiitio of geeralized Sobolev space i [RARP1]. We start with some prelimiary defiitios. Defiitio 1. We say that two positive fuctios u, v are comparable o the set A if there are positive costats c 1, c 2 such that c 1 v(x)u(x) c 2 v(x) for almost every x # A. Sice measures ad orms are fuctios o measurable sets ad vectors, respectively, we ca talk about comparable

6 162 JOSE M. RODRI GUEZ measures ad comparable orms. We say that two vectorial weights or vectorial measures are comparable if each compoet is comparable. I what follows, the symbol a b meas that a ad b are comparable for a ad b fuctios, measures or orms. Obviously, the spaces L p (A, +) ad L p (A, &) are the same ad have comparable orms if + ad & are comparable o A. Therefore, i order to obtai our results we ca chage a measure + to ay comparable measure &. Next, we shall defie a class of weights which plays a importat role i our results. Defiitio 2. If 1p<, we say that a weight w belogs to B p ([a, b]) if ad oly if w &1 # L 1( p&1) ([a, b]). Also, if J is ay iterval we say that w # B p (J) if w # B p (I) for every compact iterval IJ. We say that a weight belogs to B p (J), where J is a uio of disjoit itervals i # A J i, if it belogs to B p (J i ), for i # A. Observe that if vw i J ad w # B p (J), the v # B p (J). The class B p (R) cotais the classical A p (R) weights appearig i harmoic aalysis (see [Mu1, GR]). The classes B p (0), with 0R, ad A p (R ) (1<p<) have bee used i other defiitios of weighted Sobolev spaces i [KO, K], respectively. Defiitio 3. We deote by AC([a, b]) the set of fuctios absolutely cotiuous o [a, b], i.e., the fuctios f # C([a, b]) such that f(x)&f(a) = x a f $(t) dt for all x #[a, b]. If J is ay iterval, AC loc (J) deotes the set of fuctios absolutely cotiuous o every compact subiterval of J. Defiitio 4. Let us cosider 1p< ad a vectorial measure += (+ 0,..., + k ) with absolutely cotiuous part w=(w 0,..., w k ). For 0 jk we defie the ope set 0 j := [x # R : _ a ope eighbourhood V of x with w j # B p (V)]. Observe that we always have w j # B p (0 j ) for ay 1p< ad 0 jk. I fact, 0 j is the greatest ope set U with w j # B p (U). Obviously, 0 j depeds o w ad p, although p ad + do ot appear explicitly i the symbol 0 j. Applyig Ho lder's iequality it is easy to check that if f ( j) # L p (0 j, w j ) with 1 jk, the f ( j) # L 1 loc(0 j ) ad f ( j&1) # AC loc (0 j ). From ow o we assume that w j is idetically 0 o the com- Hypothesis. plemet of 0 j.

7 MULTIPLICATION OPERATOR IN SOBOLEV SPACES 163 Remark. We eed this hypothesis i order to have complete Sobolev spaces (see [KO, RARP1]). This hypothesis is satisfied, for example, if we ca modify w j i a set of zero Lebesgue measure i such a way that there exists a sequece : z0 with w &1 j [(:, ]] ope for every. Ifw j is lower semicotiuous, the this coditio is satisfied. The followig defiitios also deped o w ad p, although w ad p do ot appear explicitly. Let us cosider 1p<, +=(+ 0,..., + k ) a vectorial measure ad y # 2. To obtai a greater regularity of the fuctios i a Sobolev space we costruct a modificatio of the measure + i a eighbourhood of y, usig the followig Muckehoupt weighted versio of Hardy's iequality (see [Mu2; M, p. 44]). This modified measure is equivalet i some sese to the origial oe (see Theorem A below). Muckehoupt iequality. Let us cosider 1p< ad + 0, + 1 measures i (a, b ] with w 1 :=d+ 1 dx. The there exists a positive costat c such that " b x g(t) dt "L p ((a, b], + 0 )c &g& L p ((a, b], + 1 ) for ay measurable fuctio g i (a, b], if ad oly if sup + 0 ((a, r]) &w &1 1 & L 1(p&1) ([r, b])<. a<r<b Defiitio 5. A vectorial measure + =(+ 0,..., + k) is a right completio of a vectorial measure +=(+ 0,..., + k ) with respect to y, if + k :=+ k ad there is a =>0 such that + j :=+ j o the complemet of ( y, y+=] ad + j :=+ j ++~ j, o (y, y+=] for 0 j<k, where +~ j is ay measure satisfyig: (i) (ii) +~ j ((y, y+=])<, 4 p (+~ j, + j+1)<, with &1 4 p (&, _) := sup &((y, r]) "\d_. y<r< y+= dx+ "L 1 ( p&1) ([r, y+=]) The Muckehoupt iequality guaratees that if f ( j) # L p (+ j ) ad f ( j+1) # L p (+ j+1), the f ( j) # L p (+ j). If we work with absolutely cotiuous measures, we also say that a vectorial weight w is a completio of + (or of w).

8 164 JOSE M. RODRI GUEZ Example. It ca be show that the followig costructio is always a completio: we choose w~ j :=0 if w j+1 B p ((y, y+=]); if w j+1 # B p ([y, y+=]) we set w~ j (x) :=1 i [ y, y+=]; ad if w j+1 # B p ((y, y+=])"b p ([y, y+=]) we take w~ j (x) :=1 for x #[y+=2, y+=], ad w~ j (x) := d dx {\ y+= w~ j (x) :=&w for x #(y, y+=2). x w &p+1 &1( p&1) j+1 + = = ( p&1) w &1( p&1) j+1(x) &1( p&1) w ( y+= x j+1 ) p if 1<p<, &1 j+1 &&1 + d L ([x, y+=]) dx (&w &1 j+1 &&1 L ([x, y+=]) ) if p=1, Remarks. (1) We ca defie a left completio of + with respect to y i a similar way. (2) If w j+1 # B p ([y, y+=]), the 4 p (+~ j, w j+1)< for ay measure +~ j with +~ j ((y, y+=])<. I particular, 4 p (1, w j+1)<. (3) If +, & are comparable measures, & is a right completio of & if ad oly if it is comparable to a right completio + of +. (4) If +, & are two vectorial measures with the same absolutely cotiuous part, the + is a right completio of + if ad oly if it is a right completio of &. Defiitio 6. For 1p< ad a vectorial measure +, we say that a poit y # R is right j-regular (respectively, left j-regular), if there exist =>0, a right completio w (respectively, left completio) of + ad j<ik such that w i # B p ([y, y+=]) (respectively, B p ([y&=, y])). Also, we say that a poit y # R is j-regular, if it is right ad left j-regular. Remarks. (1) A poit y # R is right j-regular (respectively, left j-regular), if at least oe of the followig properties is satisfied: (a) There exist =>0 ad j<ik such that w i # B p ([y, y+=]) (respectively, B p ([y&=, y])). Here we have chose w~ j =0. (b) There exist =>0, j<ik, :>0, ad $<(i& j) p&1, such that w i (x): x& y $, for almost every x #[y, y+=] (respectively, [ y&=, y]). See Lemma 3.4 i [RARP1]. (2) If y is right j-regular (respectively, left), the it is also right i-regular (respectively, left) for each 0i j.

9 MULTIPLICATION OPERATOR IN SOBOLEV SPACES 165 (3) We ca take i= j+1 i this defiitio sice by the secod remark after Defiitio 5 we ca choose w l=w l +1 # B p ([y, y+=]) for j<l<i, ifj+1<i. (4) If +, & are two vectorial measures with the same absolutely cotiuous part, the y is right j-regular (respectively, left) with respect to + if ad oly if it is right j-regular (respectively, left) with respect to &. Whe we use this defiitio we thik of a poit [b] as the uio of two half-poits [b + ] ad [b & ]. With this covetio, each oe of the followig sets (a, b) _ (b, c) _ [b + ]=(a, b) _ [b +, c){(a, c), (a, b) _ (b, c) _ [b & ]=(a, b & ] _ (b, c){(a, c), has two coected compoets, ad the set (a, b) _ (b, c) _ [b & ] _ [b + ]=(a, b) _ (b, c) _ [b]=(a, c) is coected. We oly use this covetio i order to study the sets of cotiuity of fuctios: we wat that if f # C(A) ad f # C(B), where A ad B are uio of itervals, the f # C(A _ B). With the usual defiitio of cotiuity i a iterval, if f # C([a, b))& C([b, c]) the we do ot have f # C([a, c]). Of course, we have f # C([a, c]) if ad oly if f # C([a, b & ])& C([b +, c]), where, by defiitio, C([b +, c])=c([b, c]) ad C([a, b & ])=C([a, b]). This idea ca be formalized with a suitable topological space. Let us itroduce some otatio. We deote by 0 ( j) the set of j-regular poits or half-poits, i.e., y # 0 ( j) if ad oly if y is j-regular, we say that y + # 0 ( j) if ad oly if y is right j-regular, ad we say that y & # 0 ( j) if ad oly if y is left j-regular. Obviously, 0 (k) =< ad 0 j+1 _ }}} _ 0 k 0 ( j). Observe that 0 ( j) depeds o p (see Defiitio 6). Remark. If 0j<k ad I is a iterval, I0 ( j), the the set I"(0 j+1 _ }}} _ 0 k ) is discrete (see the Remark before Defiitio 7 i [RARP1]). Defiitio 7. We say that a fuctio h belogs to the class AC loc (0 ( j) ) if h # AC loc (I) for every coected compoet I of 0 ( j). Defiitio 8. We say that the vectorial measure +=(+ 0,..., + k ) is p-admissible if (+ j ) s (R"0 ( j) )=0 for 1 jk. We use the letter p i p-admissible i order to emphasize the depedece o p (recall that 0 ( j) depeds o p).

10 166 JOSE M. RODRI GUEZ Remarks. (1) There is o coditio o supp (+ 0 ) s. (2) We have (+ k ) s #0, sice 0 (k) =<. (3) Every absolutely cotiuous measure is p-admissible. Defiitio 9 (Sobolev Space). Let us cosider 1p< ad += (+ 0,..., + k )ap-admissible vectorial measure. We defie the Sobolev space W k, p (2, +) as the space of equivalece classes of V k, p (2, +) :=[f: 2 Cf ( j ) # AC loc (0 ( j ) ) for 0 j<k ad with respect to the semiorm & f ( j ) & L p (2, + j )< for 0 jk], & f & W k, p (2, +) := \ : k j=0 1p & f ( j) & p L p (2, + j. )+ Remarks. This defiitio is atural sice whe the (+ j ) s -measure of the set where f ( j) is ot cotiuous is positive, the itegral f ( j) p d(+ j ) s does ot make sese. If we cosider Sobolev spaces with real valued fuctios every result i this paper also holds. At this momet we ca cosider also orms like the followig: & f & p = 1 &1 f p + 0 &1 x p&1 f $ p f $ p + f(0 + ) p, & f & p = 1 0 f p f $ p + f(0 + ) p. I the secod example, we ca write f(0) p istead of f(0 + ) p, sice f is ot defied at the left of 0, ad the this causes o cofusio. Obviously we always write (a+b) $ x0 istead of a$ x & 0 +b$ x + 0. Defiitio 10. Let us cosider 1p< ad + a p-admissible vectorial measure. Let us defie the space K(2, +) as K(2, +) :=[g: 0 (0) Cg # V k, p (0 (0), + 0 (0)), &g& W k, p (0 (0), + 0 (0)) =0]. K(2, +) is the equivalece class of 0 i W k, p (0 (0), + 0 (0) ). It plays a importat role i the geeral theory of Sobolev spaces ad i the study of the multiplicatio operator i Sobolev spaces i particular (see [RARP1, RARP2], Theorem A below, ad Theorem C i Sectio 4).

11 MULTIPLICATION OPERATOR IN SOBOLEV SPACES 167 Defiitio 11. Let us cosider 1 p < ad + a p-admissible vectorial measure. We say that (2, +) belogs to the class C 0 if there exist compact sets M, which are a fiite uio of compact itervals, such that (i) M itersects at most a fiite umber of coected compoets of 0 1 _ }}} _ 0 k, (ii) K(M, +)=[0], (iii) M M +1, (iv) M =0 (0). We say that (2, +) belogs to the class C if there exists a measure +$ 0 =+ 0 + m # D c m $ xm with c m >0, [x m ]/0 (0), DN ad (2, +$) # C 0, where +$=(+$ 0, + 1,..., + k ) is miimal i the followig sese: there exists [M ] correspodig to (2, +$) # C 0 such that if + 0 "=+$ 0 &c m0 $ xm with 0 m 0 # D ad +"=(+ 0 ", + 1,..., + k ), the K(M, +"){[0] if x m0 # M. Remarks. (1) The coditio (2, +)# C is ot very restrictive. I fact, the proof of Theorem A below (see [RARP1, Theorem 4.3]) gives that if 0 (0) "(0 1 _ }}} _ 0 k ) has oly a fiite umber of poits i each coected compoet of 0 (0), the (0, +)#C. If furthermore K(2, +)=[0], we have (2, +)#C 0. (2) Sice the restrictio of a fuctio of K(2, +) to M is i K(M, +) for every, the(2, +)#C 0 implies K(2, +)=[0]. (3) If (2, +)#C 0, the (2, +)#C, with +$=+. (4) The proof of Theorem A below gives that if for every coected compoet 4 of 0 1 _ }}} _ 0 k we have K(4, +)=[0], the (2, +)#C 0. Coditio *supp & 0 (0)k implies K(4, +)=[0]. The ext results, proved i [RARP1], play a cetral role i the theory of Sobolev spaces with respect to measures (see the proofs i [RARP1, Theorems 4.3 ad 5.1]). Theorem A. Let us suppose that 1p< ad +=(+ 0,..., + k ) is a p-admissible vectorial measure. Let K j be a fiite uio of compact itervals cotaied i 0 ( j), for 0 j<k ad + a right (or left ) completio of +. The: (a) If (2, +)#C 0 there exist positive costats c 1 =c 1 (K 0,..., K k&1 ) ad c 2 =c 2 (+, K 0,..., K k&1 ) such that k&1 c 1 : j=0 &g ( j) & L (K j )&g& W k, p (2, +), c 2 &g& W k, p (2, + )&g& W k, p (2, +), \g # V k, p (2, +).

12 168 JOSE M. RODRI GUEZ (b) If (2, +)#C there exist positive costats c 3 =c 3 (K 0,..., K k&1 ) ad c 4 =c 4 (+, K 0,..., K k&1 ) such that for every g # V k, p (2, +), there exists g 0 # V k, p (2, +), idepedet of K 0,..., K k&1, c 3, c 4 ad +, with &g 0 & g& W k, p (2, +)=0, k&1 c 3 : j=0 &g ( j) 0 & L (K j )&g 0 & W k, p (2, +)=&g& W k, p (2, +), c 4 &g 0 & W k, p (2, + )&g& W k, p (2, +). Furthermore, if g 0, f 0 are these represetatives of g, f respectively, we have for the same costats c 3, c 4 k&1 c 3 : j=0 &g ( j) 0 & f ( j) 0 & L (K j )&g& f & W k, p (2, +), c 4 &g 0 & f 0 & W k, p (2, + )&g& f & W k, p (2, +). Remark. Theorem A is proved i [RARP1] with the additioal hypothesis that +~ :=+ &+ is absolutely cotiuous, sice [RARP1] oly uses absolutely cotiuous completios, but the same proof also works i the geeral case. Theorem B. Let us cosider 1p< ad +=(+ 0,..., + k ) a p-admissible vectorial measure with (2, +)#C. The the Sobolev space W k, p (2, +) is complete. 3. RESULTS ON SOBOLEV SPACES We start this sectio with a techical result which shows how to modify a measure i order to have (2, +)#C 0. We use this propositio i the proof of Corollary 3.2 below. Propositio 3.1. Let us cosider 1p< ad +=(+ 0,..., + k ) a p-admissible vectorial measure. The there exists a measure + *+ 0 0 with + 0 *&+ 0 discrete ad fiite, (+*&+ 0 0 )(R"0 (0) )=0, ad such that +*:=(+ *, 0 + 1,..., + k ) is p-admissible ad (2, +*) # C 0. We have also V k, p (2, +) & L (0 (0) ) V k, p (2, +*) ad for every f # V k, p (2, +). & f & W k, p (2, +*)& f & W k, p (2, +)+& f & L (0 (0) ),

13 MULTIPLICATION OPERATOR IN SOBOLEV SPACES 169 Proof. Let us cosider the coected compoets [A m ] M m=1 (M # N _ []) of 0 1 _ }}} _ 0 k. For each m, choose k poits x 1 m,..., xk m # A m, ad ow defie the measure + 0 *:= k : k i=1 M : m=1 2 &m $ x i m. Obviously + 0 *&+ 0 is discrete ad fiite, ad (+ 0 *&+ 0 )(R"0 (0) )=0. Obviously +* is p-admissible sice + is p-admissible. We see ow that K(A m, +*)=[0]. Let us cosider q # K(A m, +*). For each y # A m, there is a 1 jk with y # 0 j.leti be the coected compoet of 0 j which cotais the poit y. Ifw j deotes the absolutely cotiuous part of + j, we have that I q ( j) (x) p w j (x) dx=0, sice q # K(A m, +*). Ho lder's iequality gives &q ( j) & L 1 (I$)&q ( j) & L p (I$, w j )&w &1 j & L 1(p&1) (I$)=0, for every compact iterval I$/I, sice w j # B p (0 j ). The &q ( j) & L 1 (I)=0 ad sice q ( j&1) is locally absolutely cotiuous i I, it has to be costat i I, ad cosequetly q ( j) #0iI. We have that q I # P j&1 P k&1. The we obtai q Am # P k&1, sice A m is a coected set. We coclude q=0 i A m sice q(x 1 m )=}}}=q(xk m )=0. The same argumet gives K(J, +)=[0] for every closed iterval J/A m with x 1,..., m xk # J. m For each m ad, let us cosider a compact iterval J, m with x 1,..., m xk # J m, m, J, m J +1, m ad J, m =A m & 0 (0). We defie ow M := m # D J, m, where D :=[m : A m 1 ad A m & (&, ){<]. Sice *D ad K(J, m, +*)=[0], this choice of [M ] gives (2, +*) # C 0. Assume ow that f # V k, p (2, +) & L (0 (0) ). We have that f(x i m ) & f & L (0 (0) ), for every m ad i, sice f is cotiuous at x i m. We have also f p d+ 0 *= f p d+ 0 + f p d(+ 0 *&+ 0 )& f & p L p (+ 0 ) +& f & p L (0 (0) ), & f & L p (+* 0 )& f & L p (+ 0 )+& f & L (0 (0) ), & f & W k, p (2, +*)& f & W k, p (2, +)+& f & L (0 (0) ),

14 170 JOSE M. RODRI GUEZ sice (+*&+ 0 0 )(R)=(+ 0 *&+ 0 )(0 (0) )1. The we have V k, p (2, +) & L (0 (0) )V k, p (2, +*). K A immediate computatio gives the followig techical result. Lemma 3.1. measure with Let us cosider 1p< ad +=(+ 0,..., + k ) a vectorial d+ j+1 (x)c p 1 x&x 0 p d+ j (x), for 0 j<k, x 0 # R ad x i a iterval I. Let. # C k (R) be such that supp.$[* 1, * 1 +t] _ [* 2, * 2 +t]i, with * 1 +t<* 2, max[ * 1 &x 0, * 1 +t&x 0, * 2 &x 0, * 2 +t&x 0 ]c 2 t ad &. ( j) & L (I)c 3 t &j for 0 jk. The, there is a positive costat c 0 which is idepedet of I, x 0, * 1, * 2, t, +,., ad g such that &.g& W k, p (2, +)c 0 &g& W k, p (I, +), for every g # C k (R) with supp (.g)i. Remarks. (1) The costat c 0 ca deped o c 1, c 2, c 3, p, ad k. (2) I the proof we oly use the hypothesis g # C k (R) to assure that g ( j) p d+ j has sese (although it ca be ifiite). Therefore, if + is p-admissible, the result is also true for every g # V k, p (2, +) with supp(.g)i. (3) Coditio d+ j+1 (x)c p x&x 1 0 p d+ j (x) meas that + j+1 is absolutely cotiuous with respect to + j, ad that the RadoNikodym derivative satisfies d+ j+1 d+ j c p x&x 1 0 p. Propositio 3.2 below shows that this coditio is ot as restrictive as it seems, sice may weights with aalytic sigularities ca be modified i order to satisfy it. We defie ow the fuctios log 1 x :=&log x, log 2 x :=log(log 1 x),..., log x :=log(log &1 x). With this defiitio we have the followig result, which is a cosequece of Muckehoupt iequality. Propositio 3.2. Let us cosider 1p< ad w=(w 0,..., w k ) a fiite vectorial weight i (a, b). Assume also that there exist 0k 0 <k, x 0 # R, a eighbourhood U of x 0, # N, c i >0, = i 0 ad : i, # i,..., 1 #i # R for k 0 ik such that (i) w i (x) e &c i x&x 0 &= i x # U ad k 0 ik, x&x 0 : i log # i 1 1 x&x 0 } } } log # i x&x 0 for

15 MULTIPLICATION OPERATOR IN SOBOLEV SPACES 171 (ii) (iii) (1+: i )p N if = i =0 ad k 0 <ik, w k B p (U). The there exists a weight w* i (a, b) such that the Sobolev orms W k, p ([a, b], w) ad W k, p ([a, b], w*) are comparable for every fuctio i W k, p ([a, b], w) ad satisfyig w* j+1 (x)c x&x 0 p w j *(x), for k$ 0 j<k ad x # U, for some k 0 k$ 0 <k. Furthermore, if k 0 {k$ 0 the we have w* k$0 # B p (U). The followig result reveals a big problem whe dealig with the completio of P. Furthermore, it allows to prove Theorem 4.5 about the multiplicatio operator. Theorem 3.1. Let us cosider 1p< ad +=(+ 0,..., + k ) a fiite vectorial measure with 2 a compact set. Assume that there exist x 0 # R ad 0k 0 k with + k0 ([x 0 ])=0 ad satisfyig the followig property if k 0 <k: there exist a ope eighbourhood U of x 0 ad c>0 such that d+ j+1 (x)c x&x 0 p d+ j (x), for x # U ad k 0 j<k. Let us defie & :=(0,..., 0, : k0 $ x0, : k0 +1$ x0,..., : k $ x0 ) ad N :=*[k 0 jk : : j >0]. Give a Cauchy sequece [q ]/P i W k, p (2, +) ad u k0,..., u k # R there exists a Cauchy sequece [r ]/P i W k, p (2, +) with lim &q &r & W k, p (2, +)=0 ad r ( j) (x 0)=u j for k 0 jk. Cosequetly P k, p (2, ++&) is isomorphic to P k, p (2, +)_R N. Remark. Observe that P k, p (2, ++&) is ot a space of fuctios eve whe P k, p (2, +) is a space of fuctios. I fact, if q # P is a elemet of P k, p (2, +), the it represets R N elemets of P k, p (2, ++&), ad therefore there are ifiitely may equivalece classes i P k, p (2, ++&) whose restrictio to P k, p (2, +) coicides with q. Hece, the values f ( j) (x 0 ) for k 0 jk do ot represet aythig related with the derivatives of f # P k, p (2, ++&). Proof. It is eough to see that, give sequeces [v k 0 ],..., [v k]/r, there exists a sequece [s ]/P covergig to 0 i the orm of W k, p (2, +) with s ( j) (x 0 )=v j for k 0 jk, sice the we ca take r :=q &s with v j) j :=q( (x 0)&u j.

16 172 JOSE M. RODRI GUEZ Let us cosider the polyomial h # P k&k0 with h ( j&k 0 ) (x 0 )=v j for k 0 jk, a fuctio. # C c (R) with 0.1 ad ad the fuctios.(x) := {1, if x #[&1,1], 0, if x [&2, 2],. t (x) :=. \x&x 0 t +, for each 0<tt 0, where t 0 is ay positive umber with supp. t0 /U. For each # N, defie the fuctio g :=h. t, where [t ] is a sequece covergig to 0, with 0<t <t 0, which will be chose later. Let us defie f :=g if k 0 =0 ad otherwise. Sice we have f (x) := x g (t) (x&t)k 0 &1 dt, x 0 +2t (k 0 &1)! f ( j) (x)= x g (t) (x&t)k0& j&1 x 0 +2t (k 0 & j&1)! dt, for 0 j<k 0, + is fiite ad 2 is compact, we obtai that & f ( j) & L p (2, + j )c & f ( j) & L (2)c &g & L 1 (R)c &h & L 1 ([x 0 &2t, x 0 +2t ]), (3.1) for 0 j<k 0.Ifk 0 <k, Lemma 3.1 gives that k k : & f ( j) & L p (2, + j )= : j=k 0 We ca apply Lemma 3.1 sice &. ( j) t &g ( j&k0) & L p (2, + j ) j=k 0 c &h. t & W k&k0, p ([x 0 &2t, x 0 +2t ], (+ k0,..., + k )) c &h & W k&k0, p ([x 0 &2t, x 0 +2t ], (+ k 0,..., + k )). (3.2) supp.$ t [x 0 &2t, x 0 &t ] _ [x 0 +t, x 0 +2t ]/[x 0 &2t, x 0 +2t ], max[ &2t, &t,t,2t ]=2t, & L (R)=t &j &. ( j) & L (R)ct &j for 0 jk&k 0, supp g =supp(h. t )[x 0 &2t, x 0 +2t ].

17 MULTIPLICATION OPERATOR IN SOBOLEV SPACES 173 If k 0 =k, iequality (3.2) is also true sice & f (k) & L p (2, + k )=&g & L p (2, + k )&h & L p ([x 0 &2t, x 0 +2t ], + k ). Iequalities (3.1) ad (3.2) ad the fact + k0 ([x 0 ])=}}}=+ k ([x 0 ])=0 allow us to choose t small eough i order that & f & W k, p (2, +)< 1. (3.3) If 2* is the covex hull of 2 _ [x 0 ], we ca choose p # P such that & f ( j) j) & p( & L (2*)< 1, (3.4) for 0 jk, sice f # C (R). This is deduced from the compactess of 2 ad Berstei's proof of the Weierstrass Theorem, where the Berstei polyomials approximate ay fuctio i C k ([a, b]) uiformly up to the k-th derivative (see, e.g., [D, p. 113]). I particular, we have that f ( j) (x 0)&p ( j) (x 0) < 1, for 0 jk. If we cosider the polyomial = # P k with = ( j) (x 0)= f ( j) (x 0)&p ( j) (x 0), for 0 jk, the there exists a positive costat c, which oly depeds o 2*, with &= ( j) & L (2*)< c, (3.5) for 0 jk. Therefore, the polyomial s :=p += satisfies s ( j) (x 0)=p ( j) (x 0)+= ( j) (x 0)= f ( j) (x 0)=g ( j&k 0 ) (x 0 )=h ( j&k 0 ) (x 0 )=v j, for k 0 jk, ad (3.3), (3.4), ad (3.5) show that there is a positive costat c, which does ot deped o, with &s & W k, p (2, +)< c. This fiishes the proof of Theorem 3.1. K The proof of Theorem 3.1 gives the followig result.

18 174 JOSE M. RODRI GUEZ Corollary 3.1. Let us cosider 1p< ad +=(+ 0,..., + k ) a fiite vectorial measure with 2 a compact set. Assume that there exist x 0 # R ad 0k 0 k with + k0 ([x 0 ])=0 ad satisfyig the followig property if k 0 <k: there exist a ope eighbourhood U of x 0 ad c>0 such that d+ j+1 (x)c x&x 0 p d+ j (x), for x # U ad k 0 j<k. Give sequeces [v k 0 ],..., [v k ]/R there exists a sequece [s ]/P covergig to 0 i the orm of W k, p (2, +) with s ( j) (x 0)=v j for k 0 jk. We have also the followig cosequeces of Theorem 3.1. Corollary 3.2. Let us cosider 1p< ad +=(+ 0,..., + k ) a fiite vectorial measure with 2 a compact set. Assume that there exist x 0 # R, 0k 0 <k, a ope eighbourhood U of x 0 ad c>0 such that d+ j+1 (x)c x&x 0 p d+ j (x), for x # U ad k 0 j<k. The x 0 is either right or left k 0 -regular. Proof. Without loss of geerality we ca assume that + is absolutely cotiuous, sice the j-regularity just depeds o the absolutely cotiuous part of the measure. Cosequetly + is p-admissible. Assume that x 0 is right or left k 0 -regular ad cosider the measure +* as i Propositio 3.1 with the additioal coditio x i {x m 0 for every m ad i. The (2, +*) # C 0 ad we have by Theorem A ad cosequetly g (k 0 ) (x 0 ) c &g& W k, p (2, +*), \g # V k, p (2, +), q (k 0 ) (x 0 ) c &q& W k, p (2, +*), \q # P, sice +* fiite ad 2 compact imply +*#M. The measure +* satisfies the hypotheses i Theorem 3.1 ad therefore there exists a sequece of polyomials [r ] with r (k 0 ) (x 0 )=1 ad lim &r & W k, p (2, +*)=0, which cotradicts the last iequality. Corollary 3.3. Let us cosider 1p< ad +=(+ 0,..., + k ) a fiite vectorial measure with 2 a compact set. Assume that there exist x 0 # R ad 0k 0 k with + k0 ([x 0 ])=0 ad satisfyig the followig property if k 0 <k: there exists a ope eighbourhood U of x 0 such that + j (U)=0 for k 0 < jk. Let us defie & :=(0,..., 0, : k0 $ x0, : k0 +1$ x0,..., : k $ x0 )

19 MULTIPLICATION OPERATOR IN SOBOLEV SPACES 175 ad N :=*[k 0 jk : : j >0]. Give a Cauchy sequece [q ]/P i W k, p (2, +) ad u k0,..., u k # R there exists a Cauchy sequece [r ]/P i W k, p (2, +) with lim &q &r & W k, p (2, +)=0 ad r ( j) (x 0 )=u j for k 0 jk. Cosequetly P k, p (2, ++&) is isomorphic to P k, p (2, +)_R N. The followig result (which will be used i the proof of Theorem 3.3) is a improvemet of Theorem 3.1 i [RARP2]. The same argumets used i the proof of Theorem 3.1 i [RARP2] give this result. Theorem 3.2. Let us cosider 1p<, +=(+ 0,..., + k ) a vectorial measure ad a closed set I2 with + I p-admissible ad (I, +)#C 0. Assume that KI is a fiite uio of compact itervals J 1,..., J ad that for every J m there is a iteger 0k m k satisfyig J m 0 (k m &1), if k m >0, ad + j (J m )=0 for k m < jk, if k m <k. If + j (K)< for 0< jk, the there exists a positive costat c 0 such that c 0 & fg& W k, p (2, +)& f & W k, p (I, +) (sup x # I g(x) +&g& W k, p (I, +) ), for every f, g # V k, p (I, +) ad defied o 2 with supp( fg)i ad g$=g"=}}}=g (k) =0 i I"K. Remark. The sets 0 ( j) are costructed with respect to (I, +). Theorem 3.2 gives the followig result correspodig to the case =1 ad k 1 =k. Corollary 3.4. Let us cosider 1p<, +=(+ 0,..., + k ) a vectorial measure ad a closed set I2 with + I p-admissible ad (I, +)#C 0. Assume that K is a compact iterval cotaied i I & 0 (k&1). If + j (K)< for 0< jk, the there exists a positive costat c 0 such that c 0 & fg& W k, p (2, +)& f & W k, p (I, +) (sup x # I g(x) +&g& W k, p (I, +) ), for every f, g # V k, p (I, +) ad defied o 2 with supp( fg)i ad g$= g"=}}}=g (k) =0 i I"K. We eed some techical result. Lemma 3.2. Let us cosider 1<p<, c 1, c 2 >0 ad w a (oe-dimesioal ) weight. (1) If w satisfies &w& L 1 ([:, ;])<c 1 ad c 2 <&w &1 & L 1(p&1) ([:, ;]),

20 176 JOSE M. RODRI GUEZ the there exists a weight vw such that &v& L 1 ([:, ;])<c 1 ad c 2 <&v &1 & L 1(p&1) ([:, ;])<. (2) If w # L 1 ([a, b]) ad satisfies &w &1 & L 1(p&1) ([a, a+=])=, for every =>0, the there exists a weight vw such that v # L 1 ([a, b]), &v &1 & L 1(p&1) ([a+=, b])<, for every =>0, ad &v &1 & L 1(p&1) ([a, b])=. Proof. We first prove (1). For each t>0, let us cosider the fuctio w t :=max(t, w), which obviously satisfies w t w. Recall that if + is a _-fiite measure i X, every measurable fuctio g0 satisfies Therefore we have that X gd+= +([x # X : g(x)*]) d*. 0 a(t) :=&w t & L 1 ([:, ;])= 0 [x #[:, ;] :max(t, w(x))*] d* = b p (t) 1( p&1) := ; : t = 0 [x #[:, ;] :w(x)*] d*+(;&:) t, &1( p&1) w t [x #[:, ;] :mi(t &1( p&1), w(x) &1( p&1) )*] d* = t&1( p&1) 0 [x #[:, ;] :w(x) &1( p&1) *] d* t &1( p&1) (;&:)<. Sice a(t) ad b p (t) are cotiuous fuctios for t>0 ad lim a(t)=&w& L 1 t 0 + ([:, ;]), we ca take v :=w t for small eough t>0. lim b p (t)=&w &1 & L 1( p&1) t 0 + ([:, ;]),

21 MULTIPLICATION OPERATOR IN SOBOLEV SPACES 177 I order to prove (2), let us choose x 0 :=b ad x +1 #(a, mi[a+2 &, x ]] such that &w &1 & L 1(p&1) ([x +1, x ])>1. By part (1) we ca take a weight v w i [x +1, x ] with ad 1<&v &1 & L 1(p&1) ([x +1, x ])<, &v & L 1 ([x +1, x ])&w& L 1 ([x +1, x ])+x &x +1. If we defie v i (a, b] byv :=v i (x +1, x ], we have vw, &v &1 & L 1(p&1) ([a, b])=, ad this fiishes the proof. &v &1 & L 1(p&1) ([x, b])<, &v& L 1 ([a, b])&w& L 1 ([a, b])+b&a, K Theorem A gives that if + is p-admissible, (2, +)#C 0 ad x 0 is (k&1)- regular, the we have c 1 f (k&1) (x 0 ) & f & W k, p (2, +), for every f # W k, p (2, +). The followig result, which will be used to prove Theorem 4.4, says that this iequality is always false if x 0 is ot (k&1)- regular. Theorem 3.3. Let us cosider 1<p< ad +=(+ 0,..., + k ) a vectorial measure such that there exist ' 0 >0, x 0 # R ad 0<k 0 k with + j ([x 0 &' 0, x 0 +' 0 ])< for 0 jk 0 ad + j ([x 0 &' 0, x 0 +' 0 ])=0 for k 0 < jk (if k 0 <k). Let us assume that x 0 is either right or left (k 0 &1)-regular ad that + k0 &1([x 0 ])=0. The, for ay 0<'' 0, there is o positive costat c 1 with c 1 f (k 0 &1) (x 0 ) & f & W k, p (2, +), for every f # C c ([x 0&', x 0 +']). If we have also that + is fiite ad 2 is a compact set, the there is o positive costat c 2 with for every q # P. c 2 q (k 0 &1) (x 0 ) &q& W k, p (2, +),

22 178 JOSE M. RODRI GUEZ Remark. If E is a closed set, we deote by C c (E) the set of fuctios f # C c (R) with supp fe. Proof. I order to prove the first part of the theorem, without loss of geerality we ca assume that k 0 =k, sice otherwise we ca chage 2 to 2 & [x 0 &' 0, x 0 +' 0 ]. Let us deote by w the absolutely cotiuous part of +. Observe that the fact that x 0 is either right or left (k&1)-regular is equivalet to w k B p ([x 0, x 0 +'])_ B p ([x 0 &', x 0 ]), for every '>0. We ca assume that w j (x)1 for 0 j<k if x #[x 0 &' 0, x 0 +' 0 ] ad w k (x)#b p ([x 0 &', x 0 +']"[x 0 ]), sice otherwise we ca chage w j (x) to max (w j (x), 1) ad w k (x) accordig to Lemma 3.2 i [x 0 &' 0, x 0 +' 0 ]. This icreases the right had side of the first iequality ad does ot chage the fact that w k B p ([x 0, x 0 +'])_ B p ([x 0 &', x 0 ]) for every '>0. Observe that it is eough to prove the first part of Theorem 3.3 for almost every ' #(0,' 0 ] (with respect to Lebesgue measure). Let us fix 0<'' 0 with + k ([x 0 &'])=+ k ([x 0 +'])=0 (the set of ''s i (0, ' 0 ] which do ot satisfy this is at most deumerable sice + k ([x 0 &' 0, x 0 +' 0 ])<). Sice w k # B p ((x 0, x 0 +'])"B p ([x 0, x 0 +']), the fuctio U(t) := x 0 +' x 0 +t &1( p&1) w k is positive ad cotiuous o (0, ') ad lim t 0 + U(t)=; sice for ay sequece [y ] with y z0as lim x0& y x 0 &' w &1( p&1) k =, for large eough there exists a poit x #(0,') such that x0+' w &1( p&1) k x 0 +x = x 0 & y x 0 &' &1( p&1) w k. (3.6) We have also x z0 as. Therefore, we ca choose decreasig sequeces [y ] ad [x ] satisfyig (3.6) ad + k ([x 0 & y ])=+ k ([x 0 + x ])=0 for every. Let us defie S :=supp(+ k ) s ad &1( p&1) h :=w k (/ [x0 &', x 0 & y ]"S&/ [x0 +x, x 0 +']"S ). Observe that h # L 1 (R), sice w k # B p ([x 0 &' 0, x 0 +' 0 ]"[x 0 ]).

23 MULTIPLICATION OPERATOR IN SOBOLEV SPACES 179 If we defie g (x) := x x 0 +' h (t) (x&t)k&1 (k&1)! the g (k&1) = x h x 0 +' # AC loc (R). We have also dt, g (x)= x x 0 +' g (k&1) (t) (x&t)k&2 dt, (k&2)! g ( j) (x)= x x 0 +' g (k&1) (t) (x&t)k& j&2 dt, (k& j&2)! for 0 jk&2. Therefore there exists a positive costat c such that &g ( j) & L p ([x 0 &', x 0 +'], + j )= \ x 0 +' x 0 &' } x x 0 +' g (k&1) (t) (x&t)k& j&2 dt } p d+ j (x) + 1p (k& j&2)! c &g (k&1) & L 1 ([x 0 &', x 0 +']), (3.7) for 0 jk&2, sice + j ([x 0 &', x 0 +'])< for 0 jk. Sice + k ([x 0 &'])=+ k ([x 0 +'])=+ k ([x 0 & y ])=+ k ([x 0 +x ])=0 ad + k ([x 0 &', x 0 +'])<, give ay =>0 we ca choose a fuctio I # C c ((x 0 &', x 0 & y ) _ (x 0 +x, x 0 +')) such that &I &h & L p ([x 0 &', x 0 +'], + k )= ad &I &h & L 1 ([x 0 &', x 0 +'])=, by Lemma 3.1 i [R] (recall that h # L p ([x 0 & ', x 0 + '], + k ) & L 1 ([x 0 &', x 0 +']) ad h =0 o (x 0 & y, x 0 +x )). (This Lemma is just a versio of the classical approximatio result.) Sice + k ([x 0 &', x 0 +']) < ad I # C c ((x 0 &', x 0 & y ) _ (x 0 +x, x 0 +')), by a covolutio of I with a approximatio of idetity, we ca fid a fuctio H # C c ((x 0&', x 0 & y ) _ (x 0 +x, x 0 +')) such that &I &H & L p ([x 0 &', x 0 +'], + k )= ad &I &H & L 1 ([x 0 &', x 0 +'])=. The we have &H &h & L p ([x 0 &', x 0 +'], + k )2= ad &H &h & L 1 ([x 0 &', x 0 +'])2=. (3.8) We ow defie G (x) := x x 0 +' H (t) (x&t)k&1 (k&1)! dt.

24 180 JOSE M. RODRI GUEZ Let us fix a fuctio. # C (R) satisfyig 0.1 i R,.=1 i [x 0 &'2, ) ad.=0 i (&, x 0 &'], ad defie F :=G.. Assume that there is a positive costat c 1 with c 1 f (k&1) (x 0 ) & f & W k, p (2, +), for every f # C c ([x 0 &', x 0 +']). By Remark 1 after Defiitio 11 we have ([x 0 &', x 0 +'], +)#C 0, sice w k # B p ([x 0 &', x 0 +']"[x 0 ]) ad w j (x)1 for 0 j<k, if x #[x 0 &', x 0 +'], ad this implies that 0 (0) "(0 1 _ }}} _ 0 k ) has at most three poits ([x 0 &', x 0, x 0 +']) ad K([x 0 &', x 0 +'], +)=[0]. By Corollary 3.4, with K :=[x 0 &', x 0 &'2], we have c 1 G (k&1) (x 0 ) =c 1 F (k&1) (x 0 ) &F & W k, p (2, +) c &G & W k, p ([x 0 &', x 0 +'], +), ad cosequetly G (k&1) (x 0 ) c &G & W k, p ([x 0 &', x 0 +'], +), (3.9) for every. I order to apply Corollary must be p-admissible; otherwise, applyig Corollary 3.4, we ca obtai (3.9) for + ad istead of + (see Defiitio 15 i Sectio 4), ad we have + ad +. By (3.8), we have that there exists a positive costat c, idepedet of ad =, such that &g ( j) j) &G( & L p ([x 0 &', x 0 +'], + j ) \ x 0 +' x 0 &' \ x0+' x h (t)&h (t) c &h &H & L 1 ([x 0 &', x 0 +'])2c=, k& j&1 p x&t (k& j&1)! dt d+ + j (x) + 1p for 0 j<k, sice + j ([x 0 &', x 0 +'])< for 0 jk. This iequality ad (3.8) show that there exists a positive costat c such that if we choose h as g (k) &g &G & W k, p ([x 0 &', x 0 +'], +)c=, (3.10) (observe that if we chage g (k) Lebesgue measure, this would chage &g (k) We have also by (3.8) g (k&1) i a set B of zero & L p ([x 0 &', x 0 +'], + k ) if + k (B)>0). (x 0 )&G (k&1) (x 0 ) x0+' h (t)&h (t) dt x 0 &h &H & L 1 ([x 0 &', x 0 +'])2=.

25 MULTIPLICATION OPERATOR IN SOBOLEV SPACES 181 Therefore, by (3.9) ad (3.10), we obtai for some positive costat c g (k&1) (x 0 ) &2= G (k&1) (x 0 ) c &G & W k, p ([x 0 &', x 0 +'], +) for every ad =>0. Cosequetly c(&g & W k, p ([x 0 &', x 0 +'], +)+c=), g (k&1) (x 0 ) c &g & W k, p ([x 0 &', x 0 +'], +), for every. Therefore by (3.7) we have that there exists a positive costat c such that c g (k&1) (x 0 ) &g (k&1) & L 1 ([x 0 &', x 0 +'])+&g (k&1) & L p ([x 0 &', x 0 +'], + k&1 ) +&g (k) & L p ([x 0 &', x 0 +'], + k ), for every. Sice w k&1 1 i [x 0 &', x 0 +'], there exists a positive costat c such that c g (k&1) (x 0 ) &g (k&1) & L p ([x 0 &', x 0 +'], + k&1 )+&h & L p ([x 0 &', x 0 +'], + k ) =&g (k&1) & L p ([x 0 &', x 0 +'], + k&1 )+&h & L p ([x 0 &', x 0 +'], w k ), for every, sice g (k) =h =0 i S=supp (+ k ) s. For each =>0 there exists $>0 with + k&1 ([x 0 &$, x 0 +$])<=, sice + k&1 ([x 0 ])=0 ad + k&1 ([x 0 &', x 0 +']) is fiite. Recall that g (k&1) # AC([x 0 &', x 0 +']). Therefore, we have that c g (k&1) (x 0 ) = 1p &g (k&1) & L ([x 0 &$, x 0 +$]) +&g (k&1) & L p ([x 0 &', x 0 &$] _ [x 0 +$, x 0 +'], + k&1 ) +&h & L p ([x 0 &', x 0 +'], w k ) == 1p g (k&1) (x 0 ) +&g (k&1) & L p ([x 0 &', x 0 &$] _ [x 0 +$, x 0 +'], + k&1 ) +(2 g (k&1) (x 0 ) ) 1p, (3.11) sice g (k&1) (x)= x h x 0 +', &g (k&1) & L ([x 0 &$, x 0 +$])= g (k&1) (x 0 ), ad (3.6) shows Sice g (k&1) lim g (k&1) &h & p L p ([x 0 &', x 0 +'], w k )= x 0 & y x 0 &' w =2 g (k&1) (x 0 ). &1( p&1) k + x 0 +' &1( p&1) w k x 0 +x (x)= x h x 0 +' ad g (k&1) (x 0 )= x 0 +' &1( p&1) x 0 +x w k, we have that (x 0 ) = x 0 +' &1( p&1) x 0 w k =, sice w k B p ([x 0, x 0 +']).

26 182 JOSE M. RODRI GUEZ Claim. We have that &g (k&1) & L p ([x 0 &', x 0 &$] _ [x 0 +$, x 0 +'], + k&1 ) is bouded. If we have the claim, the as i (3.11), we obtai c= 1p (recall that lim g (k&1) (x 0 ) =), ad sice =>0 is arbitrary we coclude that c=0, which is a cotradictio. This fiishes the proof of the first part of Theorem 3.3, except for the claim. We ow prove the claim. We have for x #[x 0 +$, x 0 +'] 0g (k&1) (x) x 0 +' x 0 +$ &1( p&1) w k. The fact (3.6) gives g (k&1) (x 0 &')=0, ad therefore g (k&1) (x)= x h x 0 &'. The we have for x #[x 0 &', x 0 &$] 0g (k&1) (x) x0&$ x 0 &' &1( p&1) w k. This fiishes the proof of the claim. If we have also that + is fiite ad 2 is a compact set, the we obtai the result for polyomials, sice we ca approximate the k th derivative of each fuctio i C k (R) uiformly i 2 by polyomials. K Theorems 3.3 ad A give the followig result. Corollary 3.5. Let us cosider 1<p< ad +=(+ 0,..., + k ) a p-admissible vectorial measure with (2, +)#C 0 ad such that there exist ' 0 >0, x 0 # R ad 0<k 0 k with + j ([x 0 &' 0, x 0 +' 0 ])< for 0 jk 0 ad + j ([x 0 &' 0, x 0 +' 0 ])=0 for k 0 < jk (if k 0 <k). The, there is a positive costat c 1 with c 1 f (k 0 &1) (x 0 ) & f & W k, p (2, +), for every f # C c ([x 0 &' 0, x 0 +' 0 ]) if ad oly if x 0 (k 0 &1)-regular. is right or left 4. PROOF OF THE RESULTS FOR M First of all, some remarks about the defiitio of the multiplicatio operator. Defiitio 12. We say that the multiplicatio operator is well defied i P k, p (2, +) if give ay sequece [s ] of polyomials covergig to 0 i W k, p (2, +), the [xs ] also coverges to 0 i W k, p (2, +). I this case, if

27 MULTIPLICATION OPERATOR IN SOBOLEV SPACES 183 [q ] # P k, p (2, +), we defie M([q ]):=[xq ]. If we choose aother Cauchy sequece [r ] represetig the same elemet i P k, p (2, +) (i.e. [q &r ] coverges to 0 i W k, p (2, +)), the [xq ] ad [xr ] represet the same elemet i P k, p (2, +) (sice [x(q &r )] coverges to 0 i W k, p (2, +)). This defiitio is as atural as the followig. Defiitio 13. If + is a p-admissible vectorial measure (ad hece W k, p (2, +) is a space of classes of fuctios), we say that the multiplicatio operator is well defied i W k, p (2, +) if give ay fuctio h # V k, p (2, +) with &h& W k, p (2, +)=0, we have &xh& W k, p (2, +)=0. I this case, if [ f ] is a equivalece class i W k, p (2, +), we defie M([ f ]) :=[xf ]. If we choose aother represetative g of [ f ] (i.e., & f& g& W k, p (2, +)=0) we have [xf ]=[xg], sice &x( f &g)& W k, p (2, +)=0. The followig result characterizes the spaces W k, p (2, +) with M well defied i the sese of Defiitio 13 [RARP2, Theorem 5.2]. Theorem C. Let us cosider 1p< ad a p-admissible vectorial measure +. Assume that xf # V k, p (2, +) for every f # V k, p (2, +). The M is well defied i W k, p (2, +) if ad oly if K(2, +)=[0]. Although both defiitios are atural, it is possible for a p-admissible measure + with W k, p (2, +)=P (the closure of P is cosidered with the orm i W k, p (2, +)) that M is well defied i W k, p (2, +) ad ot well defied i P k, p (2, +) (see Corollary 4.4). The followig lemma characterizes the spaces P k, p (2, +) with M well defied. Remark. From ow o we use Defiitio 12 istead of Defiitio 13. Lemma 4.1. Let us cosider 1p< ad +=(+ 0,..., + k ) a vectorial measure i M. The followig facts are equivalet: (1) The multiplicatio operator is well defied i P k, p (2, +). (2) The multiplicatio operator is bouded i P k, p (2, +). (3) There exists a positive costat c such that &xq& W k, p (2, +)c &q& W k, p (2, +), for every q # P. Remark. Whe we say that the multiplicatio operator is bouded i P k, p (2, +), we are assumig implicitly that it is well defied i P k, p (2, +), sice otherwise the boudedess has o sese. Proof. It is clear that coditio (3) implies (1). If we assume (1), we have that the multiplicatio operator M is cotiuous i 0 # (P, &}& W k, p (2, +)).

28 184 JOSE M. RODRI GUEZ Sice M is a liear operator i the ormed space (P, &}& W k, p (2, +)), we kow that M is bouded i (P, &}& W k, p (2, +)), which gives (3). We ow show the equivalece betwee (2) ad (3). Let us cosider a elemet # # P k, p (2, +). This elemet # is a equivalece class of Cauchy sequeces of polyomials uder the orm i W k, p (2, +). Assume that a Cauchy sequece of polyomials [q ] represets #. The orm of # is defied as &#& P k, p (2, +)=lim &q & W k, p (2, +), which obviously does ot deped o the represetative chose. Hece, coditio (2) is equivalet to lim &xq & W k, p (2, +)c lim &q & W k, p (2, +), for every Cauchy sequece of polyomials [q ]. Now the equivalece betwee (2) ad (3) is clear. K We ow deduce the followig particular case. Corollary 4.1. Let us cosider 1p< ad +=(+ 0,..., + k ) a p-admissible vectorial measure i M with W k, p (2, +)=P. If the multiplicatio operator is well defied i P k, p (2, +), the it is well defied ad bouded i W k, p (2, +). Lemma 4.2. Let us cosider 1p< ad +=(+ 0,..., + k ) a fiite vectorial measure with 2 a compact set. The, the multiplicatio operator is bouded i P k, p (2, +) if ad oly if there exists a positive costat c such that &q ( j&1) & L p (2, + j )c &q& W k, p (2, +), for every 1 jk ad q # P. Proof. If M is bouded i P k, p (2, +), we have that &(xq) ( j) & L p (2, + j )&M& &q& W k, p (2, +), for every 1 jk ad q # P. Sice &(xq) ( j) & L p (2, + j )=&xq ( j) + jq ( j&1) & L p (2, + j ) &q ( j&1) & L p (2, + j )&K &q ( j) & L p (2, + j ),