DENSITY OF THE SET OF ALL INFINITELY DIFFERENTIABLE FUNCTIONS WITH COMPACT SUPPORT IN WEIGHTED SOBOLEV SPACES

Transcription

1 Scietiae Mathematicae Japoicae Olie, Vol. 10, (2004), DENSITY OF THE SET OF ALL INFINITELY DIFFERENTIABLE FUNCTIONS WITH COMPACT SUPPORT IN WEIGHTED SOBOLEV SPACES EIICHI NAKAI, NAOHITO TOMITA AND KÔZÔ YABUTA Received October 15, 2003 Abstract. Miller (1982) stated that, without a proof, if 1 < p < ad ω is i Muckehoupt s A p-class, the C comp (R ) is dese i weighted Sobolev spaces L p k (R,ω(x)dx). I this paper, we show this property i the case p = 1 which is the critical case. We also give a proof i the case 1 <p< for coveiece. Geeralizig the case 1 <p<, we ca prove the desity for Orlicz-Sobolev, Loretz-Sobolev ad Herz-Sobolev spaces with A p-weights. 1. Itroductio Let 1 p<. For ay o-egative iteger k, the Sobolev space L p k (R ) is defied as the space of fuctios f, with f L p (R ) ad all α f x α exist ad α f x α Lp (R ) i the weak sese, wheever α k. The Sobolev space L p k (R ) is complete with the orm f L p = α ( f 0 ) f k x α, p x 0 = f, α k where p is the usual orm of L p (R ). It is well kow that C comp (R ), the set of all ifiitely differetiable fuctios with compact support, is dese i L p k (R ). For a weight fuctio ω, the weighted Sobolev space L p k (ω) =Lp k (R,ω(x)dx) is defied by usig L p (ω) =L p (R,ω(x)dx) istead of L p (R ). Miller [8] stated that, without a proof, if 1 <p< ad ω is i Muckehoupt s A p -class, the C comp (R ) is also dese i weighted Sobolev spaces L p k (ω). I this paper, we show this property i the case p = 1 which is the critical case (see [9, 6.6 i p.160]). We also give a proof i the case 1 <p< for coveiece. Geeralizig the case 1 <p<, we ca prove the desity for Orlicz-Sobolev, Loretz-Sobolev ad Herz-Sobolev spaces with A p -weights. We give the defiitio of A p i the ext sectio. Our mai results are the followig: Theorem 1.1. Let 1 p<, ω A p ad k is a o-egative iteger. The C comp (R ) is dese i L p k (ω). I geeral, eve if f L 1 (ω), f( y) is ot ecessarily i L 1 (ω). We show i the ext sectio if f L 1 (ω) with ω A 1 the f( y) isil 1 (ω) a.e. y (see Remark 2.1) Mathematics Subject Classificatio. Primary 46E35, Secodary 46E30. Key words ad phrases. desity, Muckehoupt s A p-weight, weighted Sobolev space, weighted Orlicz- Sobolev space, weighted Loretz-Sobolev space, weighted Herz-Sobolev space.

2 40 EIICHI NAKAI, NAOHITO TOMITA AND KÔZÔ YABUTA Let L 1 loc (R ) be the set of all locally itegrable fuctios o R. The Hardy-Littlewood maximal fuctio of f L 1 loc (R ) is defied by 1 Mf(x) = sup f(y) dy, Q x Q Q where the supremum is take over all cubes Q cotaiig x. The case 1 <p< i Theorem 1.1 ca be geeralized as follows. Let E be a subspace of L 1 loc (R ) equipped with a orm or quasi-orm E. Let E k be the space of all fuctios f E such that α f x α exist i the weak sese ad α f x α space E k is a orm or quasi-orm space with f Ek = α f x α, E α k E wheever α k. The the ( 0 ) f x 0 = f. Theorem 1.2. Let k be a o-egative iteger ad E have the followig properties: 1. The characteristic fuctios of all balls i R are i E. 2. If g E ad f(x) g(x) a.e., the f E. 3. If g E, f j (x) g(x) a.e. (j =1, 2, ) ad f j (x) 0(j + ) a.e., the f j 0(j + ) i E. If the operator M is bouded o E, the Ccomp (R ) is dese i E k. I the ext sectio we prove the theorems. I the third sectio we state applicatios of Theorem 1.2 to Orlicz-Sobolev, Loretz-Sobolev ad Herz-Sobolev spaces with A p -weights. 2. Proof First, we give the defiitio of Muckehoupt s A p -class, 1 p<. A o-egative locally itegrable fuctio ω is said to belog to A p, deoted ω A p,if ( )( p sup ω(x)dx ω(x) dx) 1/(p 1) < + (1 <p< ), Q Q Q Q Q ( 1 (ess sup ω(x)dx) supx Q Q Q ω(x) 1) < + (p =1), Q where the supremum is take over all cubes Q i R. I the case 1 <p<, the operator M is bouded o L p (ω) if ad oly if ω A p. I the case p =1,Mω(x) Cω(x) a.e. x R if ad oly if ω A 1. See [3] for example. For a fuctio ψ o R ad t>0, we defie ψ t (x) = 1 t ψ ( x t To prove the theorems, we state two lemmas. For the proof of the first lemma, see [2, Propositio 2.7] or [9, p.63]. Lemma 2.1. Let ψ be a fuctio o R which is o-egative, radial, decreasig (as a fuctio o (0, )) ad itegrable. The sup (ψ t f)(x) ψ 1 Mf(x), x R. t>0 The followig is a key lemma to prove Theorem 1.1 i the case p =1. ).

3 DENSITY IN WEIGHTED SOBOLEV SPACES 41 Lemma 2.2. Let 1 p< ad ψ be as i Lemma 2.1. If ω A p, the there exists a costat C>0 such that ψ t f L p (ω) C f L p (ω) for f L p (ω), where C is idepedet of t>0. Proof. The case p = 1: Usig ψ(y) =ψ( y), Lemma 2.1, Mω(x) Cω(x) a.e. x R ad Fubii s theorem, we have ψ t f(x) ω(x) dx ψ t (x y) f(y) ω(x) dydx R R R ( ) = f(y) ψ t (y x)ω(x) dx dy R R (2.1) f(y) ψ 1 Mω(y) dy R C ψ 1 R f(y) ω(y) dy. The case 1 <p< : Usig Lemma 2.1 ad the boudedess of M o L p (ω), we have the coclusio. Remark 2.1. From (2.1) it follows that ( ) ( ) ψ t (y) f(x y) ω(x) dx dy = ψ t (y) f(x y) dy ω(x) dx R R R R = ψ t (x y) f(y) ω(x) dydx R R C ψ 1 f L 1 (ω). Let ψ be the Poisso kerel. The ψ(y) > 0 for all y R. Fubii s theorem shows that if f L 1 (ω) with ω A 1 the f( y) L 1 (ω) a.e. y. Proof of Theorem 1.1. Let L p k,comp (ω) be the set of all f Lp k (ω) with compact support. The L p k,comp (ω) is dese i Lp k (ω). Let ψ C comp (R ) be a o-egative, radial ad decreasig fuctio with R ψ(x) dx = 1. We show that, for all f L p k,comp (ω), ψ t f f L p k (ω) 0 as t 0. Sice α (ψ t f) x α = ψ t α f x α, it is eough to show that, for f L p comp (ω), (2.2) ψ t f f L p (ω) 0 as t 0. The case p = 1: Sice L comp (R ) is dese i L 1 (ω), for ɛ>0 we ca take a fuctio g L comp(r ) such that f g L 1 (ω) <ɛ. Usig Lemma 2.2, we have that ψ t f f L 1 (ω) ψ t f ψ t g L 1 (ω) + ψ t g g L 1 (ω) + g f L 1 (ω) C f g L 1 (ω) + ψ t g g L 1 (ω) + g f L 1 (ω) (C +1)ɛ + ψ t g g L 1 (ω). We ote that ψ t g(x) g(x) a.e. x as t 0. From g L comp (R ) it follows that ψ t g g ad that supp ψ t g is icluded i a certai ball for 0 <t<1. Therefore,

4 42 EIICHI NAKAI, NAOHITO TOMITA AND KÔZÔ YABUTA by Lebesgue s covergece theorem, we have ψ t g g L1 (ω) 0ast 0. This shows (2.2). The case 1 <p< : We ote that ψ t f(x) f(x) a.e. x as t 0. From Lemma 2.1 ad the boudedess of M o L p (ω) it follows that ψ t f(x) ψ 1 Mf(x) a.e. x ad Mf L p (ω). Therefore, by Lebesgue s covergece theorem, we have (2.2). Proof of Theorem 1.2. From (1) ad (2) it follows that Ccomp(R ) E k. Let E k,comp be the set of all f E k with compact support. The, usig smooth cut-off fuctios ad the property (3), we have that E k,comp is dese i E k. Sice the operator M is bouded, we ca use the same method as Theorem 1.1 i the case 1 <p<, ad we have the coclusio. 3. Applicatios of Theorem 1.2 Weighted Orlicz, Loretz, Herz spaces, ad their geeralizatios have the properties (1) ad (2). Therefore, i the case that the property (3) holds ad the operator M is bouded, we ca apply Theorem 1.2. For a weight fuctio ω ad a measurable set Ω R, let ω(ω) = Ω ω(x) dx ad let χ Ω be the characteristic fuctio of Ω Weighted Orlicz-Sobolev spaces. A fuctio Φ : [0, + ] [0, + ] is called a Youg fuctio if Φ is covex, lim r +0 Φ(r) = Φ(0) = 0 ad lim r + Φ(r) = Φ(+ ) = +. Ay Youg fuctio is icreasig. A Youg fuctio Φ is said to satisfy 2 coditio, deoted Φ 2, if Φ(2r) CΦ(r) for some costat C>0. If Φ 2, the Φ satisfies 0 < Φ(r) < + for 0 <r<+. I this case Φ is cotiuous ad bijective from [0, + ) to itself. For a Youg fuctio Φ, the complemetary fuctio Φ is defied by Φ(r) = sup{rs Φ(s) :s 0}, r 0. For a Youg fuctio Φ ad a weight fuctio ω, let { } L Φ (ω) =L Φ (R,ω(x)dx) = f : Φ(ɛ f(x) ) ω(x) dx < + for some ɛ>0, R { ( ) } f(x) f L Φ (ω) = if λ>0: Φ ω(x) dx 1. R λ For a Youg fuctio Φ 2, let log h Φ (λ) log h Φ (λ) i(φ) = lim = sup λ 0+ log λ 0<λ<1 log λ, h Φ(λt) Φ(λ) = sup t>0 Φ(t). If Φ(r) =r p,1 p<, the L Φ (ω) =L p (ω), f LΦ (ω) = f Lp (ω) ad i(φ) = p. If f j 0iL Φ (ω) asj +, the Φ( f j (x) ) ω(x) dx 0 as j +. R If ad oly if Φ 2, the coverse is true. I this case L Φ (ω) has the property (3). Let Φ ad Φ satisfy 2 coditio. The the operator M is bouded o L Φ (ω) if ad oly if ω A i(φ) (see [4, Theorem 2.1.1]). The weighted Orlicz-Sobolev space L Φ k (ω) =LΦ k (R,ω(x)dx) is defied by usig L Φ (ω) = L Φ (R,ω(x)dx) istead of L p (R ). Corollary 3.1. Let Φ ad Φ satisfy 2 coditio, ω A i(φ), ad k be a o-egative iteger. The C comp (R ) is dese i L Φ k (ω).

5 DENSITY IN WEIGHTED SOBOLEV SPACES Weighted Loretz-Sobolev spaces. Let ω be a weight fuctio. For a measurable fuctio f, the distributio fuctio ω(f,s) ad the rearragemet f (t) with respect to the measure ω(x)dx are defied by ω(f,s) =ω({x R : f(x) >s}) = ω(x) dx, for s>0, {x R : f(x) >s} f (t) = if{s >0:ω(f,s) t}, for t>0. The weighted Loretz space L (p,q) (ω) =L (p,q) (R,ω(x)dx) is defied to be the set of all f such that f L (p,q) (ω) <, where ( q 1/q t (q/p) 1 (f (t)) dt) q, 0 <p<, 0 <q<, f L (p,q) (ω) = p 0 t 1/p f (t), 0 <p,q=. sup t>0 If 0 <p= q, the L (p,p) (ω) =L p (ω) ad f L (p,p) (ω) = f Lp (ω). If there exists a fuctio g with lim t + g (t) = 0 such that lim f j(x) = 0 ad f j (x) g(x) a.e. x R, j + the lim f j (t) = 0 ad f j (t) g (t), t > 0. j + Actually, for all s>0, we have χ {y R : f j(y) >s}(x) χ {y R : g(y) >s}(x) a.e. x R, ad ω(g, s) < +, i.e. χ {y R : g(y) >s} L 1 (ω). By Lebesgue s covergece theorem, we have ω(f j,s) 0(j + ) for all s>0. If g L (p,q) (ω) with 0 <p<, the lim t + g (t) = 0. Therefore, if 0 <p< ad 0 <q<, the L (p,q) (ω) has the property (3). Let 1 <p<, 1 q<, ω A p, The the operator M is bouded o L (p,q) (ω) (see [1, Theorems 3 ad 4]). The weighted Loretz-Sobolev space L (p,q) k (ω) =L (p,q) k (R,ω(x)dx) is defied by usig L (p,q) (ω) =L (p,q) (R,ω(x)dx) istead of L p (R ). Corollary 3.2. Let 1 <p<, 1 q<, ω A p, ad k is a o-egative iteger. The Ccomp(R ) is dese i L (p,q) k (ω) Weighted Herz-Sobolev spaces. Let B k = B(0, 2 k ) for k Z ad R k = B k \ B k 1. Let α R, 0<p,q, ω 1 ad ω 2 be weight fuctios. The homogeeous weighted Herz space K q (ω 1,ω 2 ) is defied to be the set of all f such that f K q (ω 1,ω 2) <, where ( 1/p f K q (ω = 1,ω 2) ω 1 (B k ) αp/ fχ Rk p L q (ω 2)), k= with the usual modificatios whe p = ad/or q =. The o-homogeeous weighted Herz space Kq (ω 1,ω 2 ) is defied to be the set of all f such that f K q (ω 1,ω 2) <, where ( 1/p f K q (ω 1,ω 2) = ω 1 (B 0 ) αp/ fχ B0 p L q (ω + 2) ω 1 (B k ) αp/ fχ Rk p L q (ω 2)), k=1

6 44 EIICHI NAKAI, NAOHITO TOMITA AND KÔZÔ YABUTA with the usual modificatios whe p = ad/or q =. If α = 0 ad 0 < p = q, the K p (ω 1,ω 2 ) = Kp (ω 1,ω 2 ) = L p (ω 2 ) ad f K p (ω = f 1,ω 2) Kp (ω 1,ω 2) = f L p (ω 2). α,q If 0 <p,q<, the K p (ω 1,ω 2 ) ad Kp α,q (ω 1,ω 2 ) have the property (3). If α, p, q, ω 1 ad ω 2 satisfy the assumptio i the ext corollary, the the operator M α,q is bouded o K p (ω 1,ω 2 ) ad o Kp α,q (ω 1,ω 2 ) (see [5, Theorem 1]). Actually, Professor Lu poited out 1 Mf(x) C sup r>0 r f(x) dy y x <r f(x) C sup r>0 y x <r y x dy f(x) C R y x dy. The weighted Herz-Sobolev space K q,k (ω 1,ω 2 ) ad K q,k (ω 1,ω 2 ) is defied by usig K q (ω 1,ω 2 ) ad Kq (ω 1,ω 2 ), respectively, istead of L p (R ). Corollary 3.3. Let ω 1 A qω1, ω 2 A qω2, 0 <p<, 1 <q< ad k is a o-egative iteger, where ω 1 ad ω 2 satisfy either of the followig, (i) ω 1 = ω 2, 1 q ω1 q ad q ω1 /q < αq ω1 <(1 q ω1 /q), (ii) 1 q ω1 <, 1 q ω2 q ad 0 <αq ω1 <(1 q ω2 /q). The Ccomp (R ) is dese i K q,k (ω 1,ω 2 ) ad i K q,k (ω 1,ω 2 ). I the case ω 1 (x) =ω 2 (x) 1, we deote K q,k (ω 1,ω 2 ) ad K q,k (ω 1,ω 2 )by K q,k (R ) ad K q,k (R ), respectively. It is kow that Ccomp(R ) is dese i K q,k (R ) ad i K q,k (R )if0<p<, 1<q<, 0<α<(1 1/q) ad k is a o-egative iteger ([6, Propositio 2.1]). Refereces [1] H-M. Chug, R. A. Hut ad D. S. Kurtz, The Hardy-Littlewood maximal fuctio o L(p, q) spaces with weights, Idiaa Uiv. Math. J., 31 (1982), [2] J. Duoadikoetxea, Fourier Aalysis, Amer. Math. Soc., Providece, RI, [3] J. García-Cuerva ad J. L. Rubio de Fracia, Weighted orm iequalities ad related topics, North- Hollad Publishig Co., Amsterdam, [4] V. Kokilashvili ad M. Krbec, Weighted iequalities i Loretz ad Orlicz spaces, World Scietific Publishig Co., Ic., River Edge, NJ, [5] S. Lu, K. Yabuta ad D. Yag, Boudedess of some subliear operators i weighted Herz-type spaces, Kodai Math. J., 23 (2000), [6] S. Lu ad D. Yag, Herz-type Sobolev ad Bessel potetial spaces ad their applicatios, Sci. Chia Ser. A, 40 (1997), [7] B. Muckehoupt, Weighted orm iequalities for the Hardy maximal fuctio, Tra. Amer. Math. Soc., 165 (1972), [8] N. Miller, Weighted Sobolev spaces ad pseudodifferetial operators with smooth symbols, Tra. Amer. Math. Soc., 269 (1982), [9] E. M. Stei, Sigular itegrals ad differetiability properties of fuctios, Priceto Uiversity Press, Priceto, NJ, Eiichi Nakai, Departmet of Mathematics, Osaka Kyoiku Uiversity, Kashiwara, Osaka , Japa address: eakai@cc.osaka-kyoiku.ac.jp

7 DENSITY IN WEIGHTED SOBOLEV SPACES 45 Naohito Tomita, Departmet of Mathematics, Osaka Uiversity, Machikaeyama 1-1, Toyoaka, Osaka , Japa address: Kôzô Yabuta, School of Sciece ad Techology, Kwasei Gakui Uiversity, Gakue 2-1, Sada , Japa address: