The Australian Journal of Mathematical Analysis and Applications

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1 The Australia Joural of Mathematical Aalysis ad Applicatios Volume 13, Issue 1, Article 9, pp 1-10, 2016 THE BOUNDEDNESS OF BESSEL-RIESZ OPERATORS ON GENERALIZED MORREY SPACES MOCHAMMAD IDRIS, HENDRA GUNAWAN AND ERIDANI Received 23 May, 2016; accepted 22 Jue, 2016; published 4 July 2016, 2016 DEPARTMENT OF MATHEMATICS, BANDUNG INSTITUTE OF TECHNOLOGY, BANDUNG 40132, INDONESIA mochidris@studetsitbacid DEPARTMENT OF MATHEMATICS, BANDUNG INSTITUTE OF TECHNOLOGY, BANDUNG 40132, INDONESIA hguawa@mathitbacid URL: DEPARTMENT OF MATHEMATICS, AIRLANGGA UNIVERSITY, SURABAYA 60115, INDONESIA eridaidiadewi@gmailcom ABSTRACT I this paper, we prove the boudedess of Bessel-Riesz operators o geeralized Morrey spaces The proof uses the usual dyadic decompositio, a Hedberg-type iequality for the operators, ad the boudedess of Hardy-Littlewood maximal operator Our results reveal that the orm of the operators is domiated by the orm of the kerels Key words ad phrases: Bessel-Riesz operators, Hardy-Littlewood maximal operator, geeralized Morrey spaces, Boudedess, Kerels 2000 Mathematics Subject Classificatio Primary 42B20; Secodary 26A33, 42B25, 26D10, 47G10 ISSN electroic): c 2016 Austral Iteret Publishig All rights reserved The first ad secod authors are supported by ITB Research & Iovatio Program 2015

2 2 MOCHAMMAD IDRIS, HENDRA GUNAWAN AND ERIDANI 1 INTRODUCTION We begi with the defiitio of Bessel-Riesz operators For γ 0 ad 0 < α <, we defie I α,γ f x) := K α,γ x y) f y) dy, R for every f L p loc R ), p 1, where K α,,γ x) := x α 1+ x ) γ Here, I α,γ is called Bessel-Riesz operator ad K α,γ is called Bessel-Riesz kerel The ame of the kerel resembles the product of Bessel kerel ad Riesz kerel [13] While the Riesz kerel captures the local behaviour, the Bessel kerel take cares the global behaviour of the fuctio The Bessel-Riesz kerel is used i studyig the behaviour of the solutio of a Schrödiger type equatio [8] For γ = 0, we have I α,0 = I α, kow as fractioal itegral operators or Riesz potetials[12] Aroud 1930, Hardy ad Littlewood [5, 6] ad Sobolev [11] have proved the boudedess of I α o Lebesgue spaces via the iequality I α f L q C f L p, for every f L p R ), where 1 < p <, ad 1 = 1 α Here C deotes a costat which α q p may deped o α, p, q, ad, but ot o f For 1 p q, the classical) Morrey space L p,q R ) is defied by L p,q R ) := {f L p loc R ) : f L p,q < }, ) 1/p where f L p,q := sup r 1/q 1/p) r>0,a R fx) p dx For these spaces, we have a iclusio property which is preseted by the followig theorem Theorem 11 For 1 p q, we have L q R ) = L q,q R ) L p,q R ) L 1,q R ) O Morrey spaces, Spae [10] has show that I α is bouded form L p 1,q 1 R ) to L p 2,q 2 R ) for 1 < p 1 < q 1 <, 1 α p 2 = 1 p 1 α, ad 1 q 2 = 1 q 1 α Furthermore, Adams [1] ad Chiareza ad Frasca [2] reproved it ad obtaied a stroger result which is preseted below Theorem 12 [Adams, Chiareza-Frasca] If 0 < α <, the we have I α f L p 2,q 2 C f L p 1,q 1, for every f L p 1,q 1 R ) where 1 < p 1 < q 1 <, 1 α p 2 = 1 p 1 1 αq 1 ), ad 1 q 2 = 1 q 1 α For φ : R + R + ad 1 p <, we defie the geeralized Morrey space where f L p,φ := sup r>0,a R 1 φr) L p,φ R ) := {f L p loc R ) : f L p,φ < }, 1 1/p dx) r fx) p Here we assume that φ is almost decreasig ad t /p φt) is almost decreasig, so that φ satisfies the doublig coditio, that is, there exists a costat C such that 1 φr) C wheever 1 r 2 C φv) 2 v I 1994, Nakai [9] obtaied the boudedess of I α from L p1,φ R ) to L p2,ψ R ) where 1 < p 1 <, 1 α p 2 = 1 p 1 α ad v α 1 φv)dv Cr α φr) Cψr) for every r > 0 Nakai s r result may be viewed as a extesio of Spae s Later o, i 2009, Guawa ad Eridai [3] exteded Adams-Chiareza-Frasca s result Theorem 13 [Guawa-Eridai] If φv) dv Cφ r), ad φr) r v Crβ for every r > 0, β < α, 1 < p 1 <, 0 < α <, the we have α p 1 I α f L p 2,ψ C f L p 1,φ for every f L p 1,φ R ) where p 2 = βp 1 α+β ad ψr) = φr)p 1/p 2, r > 0, Vol 13, No 1, Art 9, pp 1-10, 2016

3 THE BOUNDEDNESS OF BESSEL-RIESZ OPERATORS ON MORREY SPACES 3 The proof of the boudedess of I α o Lebesgue spaces, Morrey spaces, or geeralized Morrey spaces, usually ivolves Hardy-Littlewood maximal operator, which is defied by 1 Mf x) := sup f y) dy, B x B for every f L p loc R ) where B deotes the Lebesgue measure of the ball B = Ba, r) cetered at a R with radius r > 0) It is well kow that the operator M is bouded o L p R ) for 1 < p [12, 13] ad also o Morrey spaces L p,q for 1 < p q [2] Next, we kow that I α,γ is guarateed to be bouded o geeralized Morrey spaces because K α,γ x) K α x) for every x R The boudedess of I α,γ o Lebesgue spaces ca also be proved by usig Youg s iequality, as show i [7] Theorem 14 [7] For γ > 0 ad 0 < α <, we have K α,γ L t R ) wheever Accordigly, we have α I α,γ f L q K α,γ L t f L p B < t < +γ α for every f L p R ) where 1 p < t, < t < +γ α, = α q p t Usig the boudedess of Hardy-Littlewood maximal operator, we also kow that I α,γ is bouded o Morrey spaces Theorem 15 [7] For γ > 0 ad 0 < α <, we have I α,γ f L p 2,q 2 C K α,γ L t f L p 1,q 1 for every f L p 1,q 1 R ) where 1 < p 1 < q 1 < t, 1 q = 1 q t < t < +γ α, 1 α p = 1 p 1 + 1, ad t I 1999, Kurata et al [8] proved the boudedess of W I α,γ o geeralized Morrey spaces where W is a multiplicatio operator A similar result to Kurata s ca be foud i [3] I the ext sectio, we shall reprove the boudedess of I α,γ o geeralized Morrey spaces usig a Hedberg-type iequality ad the boudedess of Hardy-Littlewood maximal operator o these spaces Theorem 16 Nakai) For 1 < p, we have for every f L p,φ R ) Mf L p,φ C f L p,φ, Our results show that the orm of Bessel-Riesz operators is domiated by the orm of their kerels o geeralized) Morrey spaces 2 MAIN RESULTS For γ > 0 ad 0 < α <, oe may observe that the kerel K α,γ belogs to Lebesgue spaces L t R ) wheever < t <, where +γ α α k Z 2 k R) α )t k R) γt K α,γ t L t see [7]) With this i mid, we obtai the boudedess of I α,γ o geeralized Morrey spaces as i the followig theorem, Vol 13, No 1, Art 9, pp 1-10, 2016

4 4 MOCHAMMAD IDRIS, HENDRA GUNAWAN AND ERIDANI Theorem 21 Let γ > 0 ad 0 < α < If φr) Cr β for every r > 0, αt p 1 1 < p 1 < t, ad < t <, the we have +γ α α I α,γ f L p 2,ψ C K α,γ L t f L p 1,φ, for every f L p 1,φ R ) where p 2 = βp 1 α+β, ad ψr) = φr)p 1/p 2 Proof Let γ > 0 ad 0 < α < Suppose that φr) Cr β for every r > 0, αt p 1 1 < p 1 < t, < t < Take f +γ α α Lp 1,φ R ) ad write I α,γ f x) := I 1 x) + I 2 x), x y α fy) 1+ x y ) γ dy ad I 2 x) := x y R for every x R where I 1 x) := x y <R R > 0 Usig dyadic decompositio, we have the followig estimate for I 1 : x y α f y) I 1 x) 1 + x y ) γ dy C 1 C 2 Mf x) 2 k R ) α k R) γ 2 k R ) α +/t 2 k R ) /t k R) γ We the use Hölder s iequality to get 1 2 k R ) ) α )t+ 1/t I 1 x) C 2 Mf x) k R) γt Because we have 1 f y) dy 2 k R ) ) 1/t 2 k R ) ) α )t+ 1/t 2 k R ) ) α )t+ 1/t k R) γt k R) γt K α,γ Lt, k Z we obtai I 1 x) C 3 K α,γ L t Mf x) R /t To estimate I 2, we use Hölder s iequality agai: 2 k R ) α I 2 x) C k R) γ It follows that C 4 2 k R ) α k R) γ 2 k R ) /p 1 I 2 x) C 5 f L p 1,φ C 6 f L p 1,φ f y) dy 2 k R ) α +/t k R) γ φ 2 k R ) 2 k R ) /t 2 k R ) α +/t k R) γ k R ) β+/t f y) p 1 dy) 1/p1 β < α, β < α, x y α fy) 1+ x y ) γ dy,, Vol 13, No 1, Art 9, pp 1-10, 2016

5 THE BOUNDEDNESS OF BESSEL-RIESZ OPERATORS ON MORREY SPACES 5 Aother use of Hölder s iequality gives 2 k R ) ) α )t+ 1/t I 2 x) C 6 f L p 1,φ 2 k k R) γt R ) βt + Because βt + < 0 ad 2 k R) α )t+ K 1+2 k R) γt α,γ t L, we obtai t Summig the two estimates, we obtai I 2 x) C 7 K α,γ L t f L p 1,φ R β R /t I α,γ fx) C 8 K α,γ L t ) Mfx)R /t + f L p 1,φR /t +β, ) 1/t for every x R Now, for each x R, choose R > 0 such that R β = Mfx) Hece we get f L p 1,φ a Hedberg-type iequality for I α,γ f, amely I α,γ fx) C 9 K α,γ L t f α/β L p 1,φ Mfx) 1+α/β Now put p 2 := βp 1 For arbitrary a α+β R ad r > 0, we have ) 1/p2 I α,γ fx) p 2 dx C 9 K α,γ Lt f 1 p 1/p 2 L p 1,φ Mfx) p 1 dx) 1/p2 We divide both sides by φr) p 1/p 2 r /p 2 to get I α,γfx) p 2 dx) 1/p2 Mfx) p 1 dx) 1/p2 ) ψr)r /p 2 C 9 K α,γ L t f 1 p 1/p 2 L p 1,φ φr) p 1/p 2r /p 2, where ψr) := φr) p 1/p 2 Takig the supremum over a R ad r > 0, we obtai I α,γ f L p 2,ψ C 10 K α,γ L t f 1 p 1/p 2 L p 1,φ Mf p 1/p 2 L p 1,φ By the boudedess of the maximal operator o geeralized Morrey spaces Nakai s Theorem), the desired result follows: I α,γ f L p 2,ψ C K α,γ L t f L p 1,φ We ote that from the iclusio property of Morrey spaces, we have K α,γ L s,t K α,γ L t,t = K α,γ L t wheever 1 s t, < t < Now we wish to obtai a more geeral result for the +γ α α boudedess of I α,γ by usig the fact that the kerel K α,γ belogs to Morrey spaces Theorem 22 Let γ > 0 ad 0 < α < If φr) Cr β for every r > 0, αt p 1 1 < p 1 < t, < t <, the we have +γ α α β < α, I α,γ f L p 2,ψ C K α,γ L s,t f L p 1,φ, for every f L p 1,φ R ) where 1 s t, p 2 = βp 1 α+β, ad ψ v) = φ v)p 1/p 2, Vol 13, No 1, Art 9, pp 1-10, 2016

6 6 MOCHAMMAD IDRIS, HENDRA GUNAWAN AND ERIDANI Proof Let γ > 0 ad 0 < α < Suppose that φr) Cr β for every r > 0, αt p 1 β < α, 1 < p 1 < t, < t < As i the proof of Theorem 21, we have I +γ α α α,γfx) := I 1 x) + I 2 x) for every x R Now, we estimate I 1 usig dyadic decompositio as follow: I 1 x) C 1 = C 2 Mf x) 2 k R ) α k R) γ x y α f y) 1 + x y ) γ dy 2 k R ) α +/s 2 k R ) /s k R) γ, where 1 s t By Hölder s iequality, 1 2 k R ) ) α )s+ 1/s I 1 x) C 2 Mf x) k R) γs f y) dy 2 k R ) ) 1/s We also have 1 2 k R) α )s+ 1+2 k R) γs 0< x <R Ks α,γ x) dx, so that ) 1 I 1 x) C 3 Mf x) Kα,γ s s x) dx R /s C 3 K α,γ L s,t Mf x) R /t 0< x <R Next, we estimate I 2 by usig Hölder s iequality As i the proof of Theorem 21, we obtai 2 k R ) α I 2 x) C k R) γ k R ) 1/p1 /p 1 f y) p 1 dy) It thus follows that I 2 x) C 5 f L p 1,φ C 6 f L p 1,φ 2 k R ) ) 1/s α φ2 k R) 2 kr x y <2 k+1r dy k R) γ 2 k R) /s φ2 k R) 2 k R ) /t 2kR x y <2 k+1r Ks α,γx y)dy 2 k R) /s /t Because φr) Cr β ad R Ks α,γ x y)dy)1/s we get From the two estimates, we obtai ) 1/s 2 k R) /s /t K α,γ L s,t for every k = 0, 1, 2,, I 2 x) C 7 K α,γ L s,t f L p 1,φ I α,γ fx) C 9 K α,γ L s,t 2 k R ) β+/t C 8 K α,γ L s,t f L p 1,φ R β R /t ) Mfx)R /t + f L p 1,φR /t +β, for every x R Now, for each x R, choose R > 0 such that R β = Mfx) Hece we get f L p 1,φ I α,γ f x) C 9 K α,γ L s,t f α/β L p 1,φ Mf x) 1+α/β, Vol 13, No 1, Art 9, pp 1-10, 2016

7 THE BOUNDEDNESS OF BESSEL-RIESZ OPERATORS ON MORREY SPACES 7 Put p 2 := βp 1 For arbitrary a α+β R ad r > 0, we have ) 1/p2 I α,γ f x) p 2 dx C 9 K α,γ f 1 p 1/p 2 Ls,t L p 1,φ Mf x) p 1 dx) 1/p2 Divide both sides by φ r) p 1/p 2 r /p 2 ad take the supremum over a R ad r > 0 to get I α,γ f L p 2,ψ C 10 K α,γ L s,t f 1 p 1/p 2 L p 1,φ Mf p 1/p 2 L p 1,φ, where ψr) := φr) p 1/p 2 With the boudedess of the maximal operator o geeralized Morrey spaces Nakai s Theorem), we obtai as desired I α,γ f L p 2,ψ C p1,φ K α,γ L s,t f L p 1,φ, Note that by Theorem 22 ad the iclusio of Morrey spaces, we recover Theorem 21: I α,γ f L p 2,ψ C K α,γ L s,t f L p 1,φ C K α,γ L t f L p 1,φ We still wish to obtai a better estimate The followig lemma presets that the Bessel-Riesz kerels belog to geeralized Morrey space L s,σ R ) for some s 1 ad a suitable fuctio σ Lemma 23 Suppose that γ > 0 ad 0 < α < If σ : R + R + satisfies r α )s+ 1 dr Cσ s R)R 0<r R for every R > 0, the K α,γ L s,σ R ) Proof Suppose that the hypothesis holds It is sufficiet to evaluate the itegral aroud 0 We observe that Kα,γx)dx s x α )s = dx C r α )s+ 1 dr Cσ s R)R x R x R 1 + x ) γs 0<r R We divide both sides of the iequality by σ s R)R ad take s th -root to obtai ) 1/s x R Ks α,γx)dx C 1/s σr)r /s Now, takig the supremum over R > 0, we have Hece K α,γ L s,σ R ) sup R>0 x R Ks α,γx)dx ) 1/s σr)r /s < By the hypothesis of Lemma 23 we also obtai R 2 k R< x 2 k+1 R Ks ρ,γ x)dx)1/s σ2 k R)2 k R) /s K α,γ L s,σ for every iteger k ad R > 0 Moreover, P k= 1 Ks α,γ2 k R)2 k R) ) 1/s σr)r) /s K α,γ L s,σ holds for every R > 0 Oe may observe that 1 s l R 1 l σr 1 ) for every R 1 > 1 For σr) = R /t, this iequality reduces to 1 s t We shall ow use the lemma to prove the followig theorem, Vol 13, No 1, Art 9, pp 1-10, 2016

8 8 MOCHAMMAD IDRIS, HENDRA GUNAWAN AND ERIDANI Theorem 24 Suppose that σ : R + R + satisfies the doublig coditio ad σr) Cr α for every r > 0, so that K α,γ L s,σ R ) for 1 s <, where 0 < α < ad γ > 0 If α φr) Cr β for every r > 0, where p 1 < β < α, the we have I α,γ f L p 2,ψ C p1,φ K α,γ L s,σ f L p 1,φ for every f L p 1,φ R ), where 1 < p 1 < α, p 2 = βp 1 β+ α ad ψr) = φr)p 1/p 2 Proof Let γ > 0 ad 0 < α < Suppose that σ : R + R + satisfies the doublig coditio ad σr) Cr α for every r > 0, such that K α,γ L s,σ R ) for 1 s < Suppose also α that φr) Cr β for every r > 0, where p 1 < β < α, 1 < p 1 < As i the proof of α Theorem 21, we write I α,γ f x) := I 1 x) + I 2 x) for every x R As usual, we estimate I 1 by usig dyadic decompositio: x y α f y) I 1 x) 1 + x y ) γ dy C 1 = C 2 Mf x) 2 k R ) α k R) γ 2 k R ) α +/s 2 k R ) /s k R) γ By usig Hölder iequality, we obtai 1 2 k R ) ) α )s+ 1/s I 1 x) C 2 Mf x) k R) γs But 1 2 k R) α )s+ 1+2 k R) γs 0< x <R Ks α,γ x) dx, ad so we get I 1 x) C 2 Mf x) Next, we estimate I 2 as follows: 2 k R ) α I 2 x) C k R) γ C 4 C 5 f L p 1,φ C 6 f L p 1,φ 0< x <R C 2 K α,γ L s,σ Mf x) σ R) R C 3 K α,γ L s,σ Mf x) R α 2 k R ) α k R) γ 2 k R ) /p 1 f y) dy 2 k R ) ) 1 Kα,γ s s x) dx R /s f y) dy ) 1/s ) 1/p1 f y) p 1 dy 2 k R ) α φ 2 k R ) 2 k R ) ) 1/s 2 kr x y <2 k+1r dy k R) γ 2 k R ) α+β 2 k R) /s 2kR x y <2 k+1r Ks α,γ x y) dy σ 2 k R) 2 k R) /s ) 1/s, Vol 13, No 1, Art 9, pp 1-10, 2016

9 THE BOUNDEDNESS OF BESSEL-RIESZ OPERATORS ON MORREY SPACES 9 Because R Ks α,γx y)dy σ2 k R)2 k R) K α,γ s Ls,σ for every k = 0, 1, 2,, we obtai I 2 x) C 6 K α,γ L s,σ f L p 1,φ 2 k R ) α+β C 7 K α,γ L s,σ f L p 1,φ R α+β It follows from the above estimates for I 1 ad I 2 that I α,γ f x) C 8 K α,γ L s,σ Mf x) R α + f L p1,φ R α+β) for every x R Now, for each x R, choose R > 0 such that R β = Mfx) f L p 1,φ, whece I α,γ f x) C 9 K α,γ L s,σ f α )/β L p 1,φ Mf x) 1+ α)/β Put p 2 := βp 1 For arbitrary a β+ α R ad r > 0, we have ) 1/p2 I α,γ f x) p 2 dx C 9 K α,γ f 1 p 1/p 2 Ls,σ L p 1,φ Divide the both sides by φ r) p 1/p 2 r /p 2 to get ) 1/p2 ) Mf x) p 1 dx I α,γfx) p 2 dx) 1/p2 I α,γfx) p 2 dx) 1/p2 ψ r) r /p 2 = φr) p 1/p 2r /p 2 Mfx) p 1 dx) 1/p2 ) C 9 K α,γ L s,σ f 1 p 1/p 2 L p 1,φ φr) p 1/p 2r /p 2, where ψr) := φr) p 1/p 2 Fially, take the supremum over a R ad r > 0 to obtai I α,γ f L p 2,ψ C 10 K α,γ L s,σ f 1 p 1/p 2 Mf p 1/p 2 L p 1,φ L p 1,φ Because the maximal operator is bouded o geeralized Morrey spaces Nakai s Theorem), we coclude that I α,γ f L p 2,ψ C p1,φ K α,γ L s,σ f L p 1,φ 3 CONCLUDING REMARKS The results preseted i this paper, amely Theorems 21, 22, ad 24, exted the results o the boudedess of Bessel-Riesz operators o Morrey spaces [7] Similar to Guawa-Eridai s result for I α, Theorems 21, 22, ad 24 esures that I α,γ : L p1,φ R ) L p 2,φ p 1 /p 2 R ) Notice that if we have σ : R + R + such that for t every R > 0, the Theorem 24 gives a better estimate tha Theorem 22 Now, if we defie σr) := 1 + R 1) /t R /t for some t 1 > t, the K α,γ L s,σ < K α,γ L s,t By Theorem 22 ad the iclusio property of Morrey spaces, we obtai, +γ α I α,γ f L p 2,ψ C K α,γ L s,σ f L p 1,φ < C K α,γ L s,t f L p 1,φ C K α,γ L t f L p 1,φ ), the α R /t < σr) holds for We ca therefore say that Theorem 24 gives the best estimate amog the three Furthermore, we have also show that, i each theorem, the orm of Bessel-Riesz operators o geeralized Morrey spaces is domiated by that of Bessel-Riesz kerels, Vol 13, No 1, Art 9, pp 1-10, 2016

10 10 MOCHAMMAD IDRIS, HENDRA GUNAWAN AND ERIDANI REFERENCES [1] DR ADAMS, A ote o Riesz potetials, Duke Math J, 42, 1975), pp [2] F CHIARENZA ad M FRASCA, Morrey spaces ad Hardy-Littlewood maximal fuctio, Red Mat, 7, 1987), pp [3] H GUNAWAN ad ERIDANI, Fractioal itegrals ad geeralized Olse iequalities, Kyugpook Math J, 1, 2009) 49 [4] L GRAFAKOS, Classical Fourier Aalysis, Graduate Texts i Matemathics, Vol 249, Spriger, New York, 2008 [5] GH HARDY ad JE LITTLEWOOD, Some properties of fractioal itegrals I, Math Zeit, 27, 1927), pp [6] GH HARDY ad JE LITTLEWOOD, Some properties of fractioal itegrals II, Math Zeit, 34, 1932), pp [7] M IDRIS, H GUNAWAN, J LINDIARNI, ad ERIDANI, The boudedess of Bessel-Riesz operators o Morrey spaces, submitted to the Proceedigs of ISCPMS 2015 [8] K KURATA, S NISHIGAKI ad S SUGANO, Boudedess of itegral operator o geeralized Morrey spaces ad its aplicatio to Schrödiger operator, Proc Amer Math Soc, 128, 1999), pp [9] E NAKAI, Hardy-Littlewood maximal operator, sigular itegral operator ad Riesz potetials o geeralized Morrey spaces, Math Nachr, 166, 1994), pp [10] J PEETRE, O the theory of L p,λ spaces, J Fuct Aal, 4, 1969), pp [11] SL SOBOLEV, O a theorem i fuctioal aalysis Russia), Math Sob, 46, 1938), pp [Eglish traslatio i Amer Math Soc Trasl ser 2, 34, 1963), pp 39-68] [12] EM STEIN, Sigular Itegral ad Differetiability Properties of Fuctios, Priceto Uiversity Press, Priceto, New Jersey, 1970 [13] EM STEIN, Harmoic Aalysis: Real Variable Methods, Orthogoality, ad Oscillatory Itegrals, Priceto Uiversity Press, Priceto, New Jersey, 1993, Vol 13, No 1, Art 9, pp 1-10, 2016