It Joural of Math Aalyi, Vol 6, 202, o 3, 50-5 Fractioal Itegral Oerator ad Ole Ieuality i the No-Homogeeou Claic Morrey Sace Mohammad Imam Utoyo Deartmet of Mathematic Airlagga Uiverity, Camu C, Mulyorejo Surabaya, 605, Idoeia m oetojo@yahoocom Toto Nuatara Deartmet of Mathematic UM Malag, 6545, Idoeia totouatara@yahoocom Bauki Widodo Deartmet of Mathematic ITS Surabaya, 60, Idoeia b widodo@matematikaitacid Suhariigih Deartmet of Phyic Airlagga Uiverity, Camu C, Mulyorejo Surabaya, 605, Idoeia uhariigih@uairacid Abtract We etablih the eceary ad ufficiet coditio for the boudede of the fractioal itegral oerator I α i the o-homogeeou claic Morrey ace I additio, we alo derive Ole tye ieualitie ivolvig I α I thi aer, we ue the meaure of order which more geeral tha the reviou tudie Coeuetly, the reult of reviou tudie i a articular form of the reult of thi tudy Mathematic Subject Claificatio: 47B38, 42B35, 26A33, 26D0
502 Mohammad Imam Utoyo et al Keyword: Fractioal itegral oerator, o-homogeeou claic Morrey ace, Ole ieuality Itroductio Let R d be euied with a metric ad a Borel meaure µ We ay that a meaure µ atifie the doublig coditio (µ DC ) if there exit a cotat C > 0 uch that for all ball B(a, r), µ(b(a, 2r)) µ(b(a, r)) If µ DC, the R d i called homogeeou ace If µ doe t atify the doublig coditio, the R d i called a o-homogeeou ace The reult of reearch o the boudede of fractioal itegral oerator i homoge ace ca be foud i [, 2, 3, 5, 9, 0] Reearcher have foud that ome reult are till valid eve if µ doe t atify the aumtio of doublig coditio Thee reult ca be ee i [4, 6, 7, 8, ] Our mai object of tudy i thi aer i the fractioal itegral oerator o o-homogeeou ace R d, I α, defied for by the formula f(y) I α f(x) := dy, 0 < α < d R d x y α I thi aer, we aume that µ atifie the growth coditio of order > 0 We ay that a meaure µ atifie the growth coditio (µ GC() ), if there exit a cotat C > 0 uch that for all ball B(a, r), µ(b(a, r)) r I [4, 6, 7, 8, ], reearcher aume that µ GC() with a i the defiitio of I α I thi aer, we will give the eceary ad ufficiet coditio for the boudede of I α i o-homogeeou Lebegue ace L (µ) ad i ohomogeeou claic Morrey ace L,λ (µ) I, additio, we hall alo derive Ole tye ieualitie ivolvig I α We defie a claic Morrey ace a the et of all f L loc (µ) uch that ( f : L,λ (µ) := u B:=B(a,r) f(y) r λ If λ = 0, the L,λ (µ) = L (µ) 2 The boudede of I α i o-homogeeou Lebegue ace Let u begi with ome aumtio ad relevat fact that follow A cutomary the letter C deote cotat, which are ot ecearily the ame from lie B
Fractioal itegral oerator ad Ole ieuality 503 to lie To rove the boudede of I α, we eed the boudede of maximal oerator M i L (µ) The maximum oerator M, defied for > 0, by formula M f(x) := u r>0 f(y) r B(x,r) where f L loc (µ) Theorem 2 The maximal oerator M i boudede i L (µ) Proof Proof thi theorem i imilar to rovig of the boudede of M i L (µ) (ee i [8]) I the ext theorem, we will give the eceary ad ufficiet coditio for the boudede I α from L (µ) to L (µ) Theorem 22 Let < < < The Oerator I α i bouded from L (µ) to L (µ) if ad oly if = ( α) + Proof Neceity Aume that I α i bouded from L (µ) to L (µ) ad B := B(a, r) i be a arbitrary ball i R d Sice χ B : L (µ) = µ(b/, we get χ B L (µ) Therefore I α χ B : L (µ) χ B : L (µ) = µ(b If x, y B, the r α I α χ B (y) Therefore ( r α µ(b+/ = C o B / ( / (r α µ(b)) dµ(x) I α χ B (y) dµ(x) B I α χ B : L (µ) χ B : L (µ) = Cµ(B/, Sice µ GC(), we get = ( α) + µ(b) r ( α) + Sufficiecy Let B := B(x, r) i ball i R d ad f L (µ) Suoe that I α f(x) = I α f (x) + I α f 2 (x) where f (x) = fχ B ad f 2 (x) = fχ B c For f, we have the followig etimate: I α f (x) k= k= 2 k r x y <2 k+ r (2 k r) α+ (2 k+ r) f(y) x y α B(a,2 k+ r) f(y)
504 Mohammad Imam Utoyo et al r α+ M f(x) k= (2 k ) α+ = Cr α+ M f(x), α + > 0 By Hölder ieuality ad the fact that µ GC() where = ( α) +, we have the followig etimate: I α f 2 (x) 2 k r x y <2 k+ r ( (2 k r) α f(y) B(a,2 k+ r) r ( ) α + k= f : L (µ) ( ) α + = Cr f : L (µ), α + f(y) x y α / ( (2 k ) B(a,2 k+ r) ( ) α + ( ) = < 0 ) Combiig the two etimate, we get I α f(x) r α+ (M f(x) + r f : L (µ)) Aumig that f 0 ae, we chooe r = ( ) M f(x) The we have f:l (µ) I α f(x) M f(x) (α +) M f(x) f : L (µ) f : L (µ) (α +) By uig the boudede of M o L (µ), Theorem 22 i comletely roved If we chooe =, the we will get the followig reult which ca be viewed a Hardy-Littlewood-Sobolev tye for o-homogeeou ace Corollary 23 Let < < α ad = α The oerator I α i bouded from L (µ) to L (µ) if ad oly if µ GC() 3 The boudede of I α i o-homogeeou claic Morrey ace Before we reet the boudede of I α i the claic Morrey ace, we have the followig lemma which how articularly that the ace L,λ (µ) i ot emty The lemma will alo be ueful later whe we rove the eceary coditio for the boudede of I α i the claic Morrey ace
Fractioal itegral oerator ad Ole ieuality 505 Lemma 3 If B o := B(a o, r o ), the χ Bo L, λ (µ) where χ Bo i the characteritic fuctio of the ball B o Moreover, there exit a cotat C > 0 uch that χ Bo : L, λ ( λ) (µ) r o Proof Let B o := B(a o, r o ) be a arbitrary ball i R d It i eay to ee that ( χ Bo : L, λ µ(b Bo ) (µ) = ub:=b(a,r) r λ We may uoe that B B o If r r o,we ue fact that µ GC(), r ( λ) o, λ > 0 O the other had, if r > r o, the the µ(b Bo) r λ µ(b B o) r λ µ(bo) r λ o µ(b) r λ r ( λ) o Thi comlete the roof of the lemma I the ext theorem, we will give the eceary ad ufficiet coditio for the boudede I α from L, λ (µ) to L,λ 2 (µ) Theorem 32 Let µ GC() with = ( α), < < < ad + 0 < λ < The oerator I α i bouded from L, λ (µ) to L,λ 2 (µ) if ad oly if λ = λ 2 Proof Neceity Aume that I α i bouded from L, λ (µ) to L,λ 2 (µ) ad B o := B(a o, r o ) i ball i R d If x, y B o, the r α µ(b o ) I α χ Bo (y) Bae o Lemma 3, we have r λ 2 +α o µ(b+/ = C ( r λ 2 o (ro α B o µ(b o )) dµ(x) / o ( r λ 2 o B o I α χ Bo (y) dµ(x) / I α χ Bo : L,λ 2 (µ) χ Bo : L, λ ( λ ) (µ) r o, µ(b o + (r o ) λ + λ 2 + α Sice µ GC() where = ( α), we have + = λ + λ 2 + α Thu, + λ = λ 2 Sufficiecy For a R d ad r > 0, let B := B(a, r), B := B(a, 2r), ad f L, λ (µ) Suoe that I α f(x) = I α f (x) + I α f 2 (x) where f (x) = fχ B
506 Mohammad Imam Utoyo et al ad f 2 (x) = fχ Bc Sice f : L (µ) = (2r) λ (2r) λ B f(y) <, we get f L (µ) Bae o Theorem 22 ad the fact that λ = λ 2 we have ( r λ 2 B(a,r) I α f (y) r λ 2 I α f (y) (R d r λ 2 + λ (2r) λ B f(y) f : L, λ (µ) Now we oberve that if x B(a, r) ad y B c the x y > r Hece Hölder ieuality ad the fact that µ GC() with = ( α) ad 0 < + λ < yield I α f 2 (x) B c f(y) x y f(y) x y >r α x y α (2 k r) α 2 k r x y <2 k+ r f(y) ( (2 k r) α f(y) B(x,2 k+ r) ( B(x,2 k+ r) ) (2 k r) α + λ + ( ) (2 k+ r) λ B(x,2 k+ r) f(y) r α + λ + ( ) f : L, λ (µ) 2 k(α + λ + ( ) ) r α + λ + ( ) f : L, λ λ ( ) (µ), α + + < 0
Fractioal itegral oerator ad Ole ieuality 507 Bae o the fact that = ( α) + ad λ = λ 2, we have Therefore ( λ 2 B(a,r) λ 2 + α + λ ( ) + + = 0 I α f 2 (x) r λ 2 +α + λ + ( ) + f : L, λ (µ) = C f : L, λ (µ) By Mikowki ieuality, Theorem 32 i comletely roved If we chooe =, the we will get the followig reult which ca be viewed a Sae tye for o-homogeeou ace Corollary 33 Let < <, = α, 0 < λ α <, ad µ GC() The oerator I α i bouded from L,λ (µ) to L,λ 2 (µ) if ad oly if λ = λ 2 We ue the followig lemma whe we rove the eceary coditio for the boudede I α from L,λ (µ) to L,λ (µ) The roof of thi lemma i imilar with Lemma 3 Lemma 34 Let 0 < λ < If B o := B(a o, r o ), the χ Bo L,λ (µ) Moreover, there exit a cotat C > 0 uch that χ Bo : L,λ (µ) r λ o I the ext theorem, we will give the eceary ad ufficiet coditio for the boudede I α from L,λ (µ) to L,λ (µ) Theorem 35 Let < < < ad 0 < λ < The Oerator I α i bouded from L,λ (µ) to L,λ (µ) if ad oly if = α + λ Proof Neceity Aume that I α i bouded from L,λ (µ) to L,λ (µ) ad B o := B(a o, r o ) i a arbitrary ball i R d By uig the ame roce a i theorem 32 we get µ(b o + (r o ) λ + λ + α Sice µ GC(), we have + = λ + λ + α Thu, = α + λ
508 Mohammad Imam Utoyo et al Sufficiecy Let B := B(x, r) i ball i R d ad f L,λ (µ) Suoe that I α f(x) = I α f (x) + I α f 2 (x) where f (x) = fχ B ad f 2 (x) = fχ B c For f, we have the followig etimate: I α f (x) k= r α+ M f(x) k= 2 k r x y <2 k+ r (2 k r) α+ (2 k+ r) k= f(y) x y α B(a,2 k+ r) f(y) (2 k ) α+ = Cr α+ M f(x), α + > 0 By Hölder ieuality ad the fact that µ GC() ad we have the followig etimate: = α + λ, I α f 2 (x) 2 k r x y <2 k+ r ( (2 k r) α f(y) B(a,2 k+ r) (2 k r) ( ) α + + λ ( (2 k+ r) λ f(y) x y α / ( B(a,2 k+ r) B(a,2 k+ r) f(y) ) / r ( ) α + k= + λ f : L,λ (µ) (2 k ) ( ) α + + λ ( ) α + + = Cr λ f : L,λ ( ) (µ), α + + λ = λ < 0 Combiig the two etimate, we get I α f(x) r α (r M f(x) + r ( )+λ f : L,λ (µ)) Aumig that f 0 ae, we chooe r = ( I α f(x) M f(x) (α +) λ M f(x) f:l,λ (µ) f : L (µ) (α +) lambda ) λ The, we have
Fractioal itegral oerator ad Ole ieuality 509 M f(x) f : L (µ) By uig the boudede of M o L (µ), Theorem 35 i comletely roved If we chooe =, the we will get the followig reult which ca be viewed a Adam tye for o-homogeeou ace Corollary 36 Let < <, 0 < λ < α, ad µ GC() The α oerator I α i bouded from L,λ (µ) to L,λ (µ) if ad oly if = 4 Ole Tye Ieualitie α λ I tudyig a Schrödiger euatio with erturbed otetial W o R articularly, for = 3, Ole roved the followig theorem [5] Theorem 4 (Ole) Let < < ad 0 < λ < α If W α L λ α,λ (R ), the the oerator W i bouded o L,λ (R ) Moreover, there exit a cotat C > 0 uch that WI α f : L,λ (R ) W : L λ α,λ (R ) f : L,λ (R ) The reult of tudie about the boudede of WI α o R ca be ee i [7, 0] I thi aer, we will reet here boudede of WI α i ohomogeeou Lebegue ace ad o-homogeeou claic Morrey ace µ GC(), > 0 Theorem 42 Let > 0 ad < < If W L α + (µ), the the oerator W i bouded i L (µ) Moreover, there exit a cotat C > 0 uch that WI α f : L (µ) W : L α + (µ) f : L (µ) Proof Let atify > ad = ( α) By Hölder ieuality, we + have ( R d WI α f(x) dµ(x) ( ) W (x) dµ(x) R d ( f(x) dµ(x) R d ( ) α + ( W (x) α + dµ(x) f(x) dµ(x) R d R d By the boudede of I α from L (µ) to L (µ) (Theorem 22), we get WI α f : L (µ) W : L α + (µ) f : L (µ) Thi comlete the roof of the theorem
50 Mohammad Imam Utoyo et al Theorem 43 Let < < <, = ( α) ad 0 < λ + < If W L α + (µ), the the oerator W i bouded i L, λ (µ) Moreover, there exit a cotat C > 0 uch that WI α f : L, λ (µ) W : L α + (µ) f : L, λ (µ) Proof Let λ 2 atify λ = λ 2 By Hölder ieuality, we have ( r λ R d WI α f(x) dµ(x) ( ) r λ + λ ( 2 W (x) dµ(x) R d ( W (x) R d ) α + ( α + dµ(x) r λ 2 r λ 2 R d f(x) dµ(x) R d f(x) dµ(x) By the boudede of I α from L, λ (µ) to L,λ 2 (µ) (Theorem 32), we get WI α f : L, λ (µ) W : L α + (µ) f : L, λ Thi comlete the roof of the theorem (µ) Theorem 44 Let > 0, 0 < λ < ad < < If W L λ α + (µ), the the oerator W i bouded i L,λ (µ) Moreover, there exit a cotat C > 0 uch that WI α f : L,λ (µ) W : L λ α + (µ) f : L,λ (µ) Proof Let atify > ad = α + By Hölder ieuality, we λ have (fracr λ WI α f(x) dµ(x) R d ) lec (fracr ( λ W (x) dµ(x) f(x) dµ(x) R d r λ R d ( ) α + ( W (x) r λ α + dµ(x) f(x) dµ(x) R d r λ R d By the boudede of I α from L,λ (µ) to L,λ (µ) (Theorem 35), we get WI α f : L,λ (µ) W : L λ α + (µ) f : L,λ (µ) Thi comlete the roof of the theorem
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