ON SINGULAR INTEGRAL OPERATORS

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1 ON SINGULAR INTEGRAL OPERATORS DEJENIE ALEMAYEHU LAKEW Abstract. I this paper we study sigular itegral operators which are hyper or weak over Lipscitz/Hölder spaces ad over weighted Sobolev spaces de ed o ubouded smooth domais i the stadard -D Euclidea space R, where 1. The operator i this case is oe of the hypersigular itegral operators which has bee studied extesibly tha other hyper sigular itegral operators. It will be show the cotrol of sigularity of hyper sigular itegral operators that are de ed iterms of Cauchy geeratig kerels by workig o weighted fuctio spaces such as W p;k ; k x k + dx for some > 0 ad, some positive iteger. The latter spaces usually are termed as weighted Sobolev spaces. 1. Sigular Itegral Operators I this short ote we discuss few poits about super sigular itegral operators, weak(or sub) sigular ad just sigular itegral operators by showig few examples ad preset some results. We therefore itroduce geeral sigular itegral operators i terms of itegrals with Cauchy geeratig kerels ad some other geeral sigular itegral operators with out kerels. The calculus versios of sigular itegral operators are improper itegrals, itegrals with ubouded itegrads or itegrals with ubouded itervals of itegratios. To start our work, let be some bouded domai i the Euclidea space R ad be some itegrable fuctio over ad x 0 2 it, Date: August 19, Mathematics Subject Classi catio. Primary 30G35,35A22. Key words ad phrases. Hyper sigular operaors, Cauchy kerels, weighted Sobolev spaces. This paper is i al form ad o versio of it will be submitted for publicatio elsewhere. 1

2 2 DEJENIE ALEMAYEHU LAKEW iterior of the domai with the property that j (x) j= 1 x!x 0 which i this case x 0 is a sigular poit of the fuctio. The itegral give by (x) dx is called a sigular itegral of the fuctio over the domai with a sigularity poit x 0. We evaluate such sigular itegrals by evaluatig the Cauchy pricipal value of the sigular itegral which is give as follows. Let > 0 ad cosider the ball B (x 0 ; ) ad de e := B (x 0 ; ). The we cosider the itegral over the deleted sub domai by (x) dx which avoids the sigularity x 0 : If the it : (x) dx called the Cauchy pricipal value(c.p.v.) exits, the we de e the value of the sigular itegral as: (x) dx := (x) dx Examples of elemetary sigular itegral operators are give below: I the uidimesioal Euclidea space R 1 : let = ( 1; 1) ad de e the fuctio by (x) =j x j ; for 0 < < 1 The the fuctio has a sigularity at 0, sice j (x) j= 1 x Therefore, the itegral give by (x) dx is a weakly sigular itegral

3 Let > 0 ad cosider ON SINGULAR INTEGRAL OPERATORS 3 itegral at x 0 = B (0; ) = ( 1; 1) ( ; ) : The the itegral (x) dx is o more a sigular ad therefore has a ite itegral as log as the fuctio is itegrable o the domai. Therefore, (x) dx = j x j dx is a fuctio of ad ad if we deote this fuctio by I (; ), the we have I (; ) = which is a ite value i terms of ad. The takig the c.p.v. of the above itegral : (x) dx = = I (; ) 2 = 1 j x j dx as 1 > 0. Whe = 1; the fuctio is 1 (x) =j x j 1 ad this fuctio geerates a itegral 1 (x) dx called a sigular itegral. For = 1 +, > 0, the itegral sigular itegral. Besides (x) dx = = I (; ) j x j dx (x) dx is called a hyper = = 1

4 4 DEJENIE ALEMAYEHU LAKEW Therefore the improper itegral is diverget. We therefore costruct the classical sigular itegral operators which are obtaied from geeratig Kerels. Let us begi with oe of the most commo geeratig kerels give by the fuctio: x K (x) =! j x j which is called the Cauchy kerel whose sigularity is at zero. This kerel gives sigular itegral operator o the space of fuctios such that the covolutio is ite over the domai, which is give by ( ) (x) = K (x y) (y) d y From the classi catio of sigular itegrals, we will see that is ideed a weak sigular itegral: let 2 R >0, K (x) = x! j x j = ( 1) K(x) which gives that K is a homogeeous fuctio of expoet 1 which is less tha : The sigular itegral operator give above i literature is called the Teodorescu trasform. It is a importat trasform i Sobolev spaces with a regularity augmetatio property by oe : : W p;k ()! W p;k+1 () : We ca further study the fuctio spaces where the weak sigular itegral works. I the sequel, we use the followig set up: For > 0; cosider B (x; ) ; the ball cetered at x ad radius ad cosider the puctured domai = B (x; ) : Propositio 1. If is ubouded ad smooth domai i R, the K (x) is p itegrable over for 1 < p < 1. Proof. x kk (x) k = k! j x j k = r1!

5 ON SINGULAR INTEGRAL OPERATORS 5 for kxk = r ad usig polar coordiates, we have the followig orm estimates: 1 kk (x) k p dx c () r p(1 )+ 1 dr r p(1 )+ = c ()!1 p(1 ) + j p(1 )+ p(1 )+ = c()!1 p(1 ) + p(1 ) + p(1 )+ ad this is ite ad equals c() ; if p( 1) p(1 ) + < 0 That is 1 < p < 1 which proves the propositio. If the domai is a bouded smooth oe, the we cosider a sigularity at a ite poit ad the expoet of itegrability will be di eret. Now, as we see that K is i the Sobolev space W p;k ( ) for p >, 1 we ca determie the fuctio space where we ca work with this fuctio as a geeratig kerel for sigular itegral operators. Propositio 2. The covolutio K j f is well de ed ad ite over W q;k ( ) for 1 < q <. Proof. From Hölder s iequality, the product Kf 2 W 1;k ( ) whe K 2 W p;k ( ) ad f 2 W q;k ( ) such that p 1 + q 1 = 1. Therefore as p 2 ( ; 1), we have q 2 (1; ) which is the required 1 result. Corollary 1. Whe is a 2 D domai, the Sobolev idex p should strictly be greater tha 2. Therefore the geeratig kerel does ot work over W 2;k ().

6 6 DEJENIE ALEMAYEHU LAKEW Propositio 3. Let be a smooth, ubouded domai i R p 2 ( ; 1) ad q be the cojugate idex of p. The we have : 1 ad k Kf k W 1;k ( )k K k W p;k ( )k f k W q;k ( )!k K k W p;k ( )k f k W ;k ( ) as q %. Proof. From Hölder s iequality, we have k Kf k W 1;k ( )= j Kf jk K k W p;k ( ) : k f k W q;k ( ) The takig the itig orm o the idices p ad q with p 1 +q 1 = 1 we have : k K k W q% p;k ( ) : k f k W q;k ( ) =k K kw p;k ( ) : k f k W ;k ( ) sice p & 1 ) q % ad that ishes the argumet. The ext sigular itegral we cosider is the oe geerated from the fudametal solutio of the Laplacia operator!! 2 = = e j e 2 j=1 j j=1 1 which is give by 2: (x) =! kxk +2 ad the correspodig sigular itegral associated is give by 2; () = 2: (x y) (y) d y We ivestigate i which geeralized Lebesgue space is 2: over ubouded domai R : j=1 Propositio 4. Let be a smooth ad ubouded domai i R for 1. The 2: 2 W p;k ( ; Cl ) for p 2 ( ; 1). +2 Proof. Cosider the itegral j 2: j p dx, usig polar coordiates, the itegral becomes : c (;! ) 1 r (+2)p+ 1 dr ad it will be ite

7 ON SINGULAR INTEGRAL OPERATORS 7 towards the boudary of the domai whe p >, where c (;! +2 ) is a costat that depeds o ad the surface area! of the uit sphere S 1 : 2. Weighted Sobolev Spaces If we try to d Sobolev spaces i which the kerel 2: works, we might ed up i workig with a dual spaces whose cojugate idices are egative. For istace i the itig cases : q! as p &, which shows 2 +2 that q has a egative itig idex which is goig to be a cojugate idex of a itig idex of p i some sese. To remedy this, we itroduce a weight o the Lebesgue volume measure dx so that we avoid dual spaces with egative idices. The weight fuctio that we choose stretches the Lebesgue volume measure so that the sigularity from the kerel is better maaged ad made more cotrolled. We choose a radial weight fuctio give by w(x) =k x k 2+, where is some positive costat ad we ivestigate the itegral : 2:(x)d(x) where d (x) = w(x)dx. Propositio 5. Over ubouded domai R, 2: 2 W p;k ( ) for < p < 1. Proof. We see from the propositio that the iterval for the idex p is much improved ad the cojugate space will be a dual space with positive idex.

8 8 DEJENIE ALEMAYEHU LAKEW Therefore, k 2: k p d (x) = c (;! ) = c (;! )!1 1 r (+2)p+2++ r (+2)p++2+ ( + 2) p j 1 dr < 1 whe ( + 2) p < 0 which implies that p > 2++ which 2+ is the required result. Next, we determie the Sobolev space W q;k () i which the product 2: is itegrable or the covolutio 2: j is ite. Propositio 6. Over ubouded domai R, ad for 1 + < +2 p < 1, with respect to the weighted measure d (x) =k x k 2+ dx we have 2: 2 W 1;k (; k x k 2+ dx) whe 2 W q;k (; k x k 2+ dx) for 1 < q < Proof. From the previous propositio, for p > 2++ ; we proved that 2+ 2: 2 W p;k (; k x k 2+ dx). Therefore, if is a fuctio i W q;k (; k x k 2+ dx) such that p 1 + q 1 = 1, we have the itegral estimate : j 2: jk 2: k W p;k (;kxk 2+ dx) : k k W q;k (;kxk 2+ dx) where 1 < q < Propositio 7. Let R be a ubouded smooth domai, ad 1 + < p < 1, the we have the followig orm estimates: +2 ad k 2: k W p;k (;kxk 2+ dx) : k kw ( );k (;kxk 2+ dx) k 2: : k k kw (1+ +2) ;k W q;k (;kxk 2+ dx) (;kxk 2+ dx)

9 ON SINGULAR INTEGRAL OPERATORS 9 q%(1+ +2 ) k 2: k W p;k (;kxk 2+ dx) : k k W q;k (;kxk 2+ dx) = k 2: k W p;k (;kxk 2+ dx) : k +2 (1+ kw ) ;k (;kxk 2+ dx) Proof. Note that the Sobolev orm used here is with respect to th weighted measure d (x) =k x k 2+ dx. The rst part of the propositio follows from the decreasig mootoic ature of Lebesgue orm with +2 (1+ ) respect to the icrease i the idex sice q % 1 ad the secod follows from the geeral theory of cotiuity of Lebesgue orm. Corollary 2. Whe = 2, we have : k 2: k W ;pk (;kxk 2+ dx) : k k W ( 1+ 4 );k (;kxk 2+ dx) k 2: k W (1+ 4 );k (;kxk 2+ dx) : k k W q;k (;kxk 2+ dx) ad q%(1+ 4 ) k 2: k W p;k (;kxk 2+ dx) : k k W q;k (;kxk 2+ dx) = k 2: k W p;k (;kxk 2+ dx) : k k W (1+ 4 ) ;k (;kxk 2+ dx) 3. Geeratig kerels: l; (x) I this sectio, we extrapolate the idea of costructig sigular itegral operators as covolutios with fudametal solutios of the Dirac operator to the oce geerated by fudametal solutios of higher iterates of the Dirac operator. Some kerels geerate hyper sigular itegral operators ad others form weaker sigular itegral operators. It is therefore iterestig to look at di ereces of these formatios from the very costructios of the operators. These fuctios are costructed by recursive ( or iterative ) way from the fudametal solutios of the Dirac operator ad its higher iterates ad are give below:

10 10 DEJENIE ALEMAYEHU LAKEW where l <. 8 < l; (x) = : (; l) x! kxk ; if l is odd l+1 (;l)! kxk l+1 ; if l is eve For a detail study of the costructios of these fuctios ad their applicatio for costructig complete family of fuctios ad miimal family of fuctios, oe ca see [5],[6] Propositio 8. For 8 l < ad ubdd; smooth R, the fuctio l; 2 < < p < 1; whe l is odd W p;k l ( ; Cl ) for : : < p < 1; whe l is eve. +1 l Proof. For ubouded ad smooth with = B (x; ) for > 0, usig polar coordiates, the itegral k l; (x) k p dx is domiated by the itegral C (; ;! ) 1 r p( l)+ 1 dr for l odd with ite itegral whe the idex p satis es the iequality l < p < 1 ad whe l is eve, it is domiated by the itegral: 1 C (; ;! ) r p(+1 l)+ 1 dr which agai is coverget for the idices which satisfy the iequality: + 1 l < p < 1 where C (; ;! ) is some costat that depeds o ; ad!. Thus for l : odd, whe we work with this geeratig kerels, we have the idices p that depeds o l ad ad the cojugate idex q has the followig itig values: as p! ; we have : q!. l l

11 ON SINGULAR INTEGRAL OPERATORS 11 Thus, as l; 2 W p;k (; Cl ) for < p < 1; the workig Sobolev l spaces for these kerels are W q;k (; Cl ) for 1 < q < such that l p 1 + q 1 = 1. Therefore for 2 W q;k (; Cl ), we have the covergece of the sub-sigular or i the literature termiology weak sigular itegral operators : l; (x) (x) dx with the usual itegral iequality: 0 k l; (x) (x) kdxa k l; ( x) k p dx k ( x) k q dx For l eve, we have the cojugate idex q! as p # ad l 1 +1 l sice l < ; we have that > 1 ad therefore, the above iequality l 1 holds agai. The as covolutio, we have : Propositio 9. For 1 < q <,or 1 < q <, the itegral operator l l 1 y) (y) dy is a weak-sigular itegral operator from l; (x W q;k (; Cl )! W q;k+1 (; Cl ) : Proof. First, as l; 2 W p;k (; Cl ) for 1 < p < 1, we have that for 2 W q;k (; Cl ), for 1 < q < ( for l odd) or for 1 < q < l (for l eve) with p 1 + q 1 = 1 l 1 such that the itegral l; (x y) (y) dy is coverget but sigular with out the pucture. The covolutio is the usual Teodorescu trasform which has the mappig property : l; : W q;k (; Cl )! W q;k+1 (; Cl ) :

12 12 DEJENIE ALEMAYEHU LAKEW Propositio 10. I the usual 3 D Euclidea space, if l = 3, the we ca ot work o the usual geeralized Hilbert space W 2;k (; Cl ). Proof. For such a settig, we have that 3 < p < 1 ad therefore the workig fuctio spaces will have cojugate Sobolev idices with rage 1 < q < 3, i which the idex 2 is ot icluded. 2 Therefore the Sobolev space of idex 2 which is the geeralized Hilbert space W 2;k (; Cl ) is o more a viable space. Refereces [1] K. G :: urlebeck, U. K :: ahler, J. Rya ad W. Spr :: oessig, Cli ord Aalysis Over Ubouded Domais, Adv. Appl. Math. 19(1997), [2] K. G :: urlebeck ad W. Spr :: ossig, Quaterioic Aalysis ad Elliptic Boudary Value Problems, Birkhauser, Basel [3], Quaterioic ad Cli ord Aalysis for Physicists ad Egieers, Joh Wiley Sos, Cichester, [4] Dejeie A. Lakew, W 2;k Best Approximatio of a Regular Fuctio, J. Appl. Aal., Vol. 13, No. 2 (2007) pp [5] Dejeie A. Lakew ad Joh Rya, Cli ord Aalytic Complete Fuctio Systems for Ubouded Domais, Math. Meth. Appl. Sci. 2002;25; (with Joh Rya). [6], Complete Fuctio Systems ad Decompositio Results Arisig i Cli ord Aalysis, Comp. Meth. Fuc. Theory, No. 1(2002) (with Joh Rya). [7] S.G. Mikhli, S. Prossdorf, Sigular Itegral Operators, Academic Verlag, Berli (1980). [8] Joh Rya, Itrisic Dirac Operators i C, Advaces i Mathematics 118, (1996). [9], Applicatios of Complex Cli ord Aalysis to the Study of Solutios to Geeralized Dirac ad Klei-Gordo Equatios with Holomorphic Potetials, J. Di. Eq., 67, (1987). [10] K.T. Smith, Primier of Moder Aalysis, Udergraduate Texts i Mathematics, Spriger Verlag, New York (1983). [11] H. Triebel, Iterpolatio Theory, Fuctio Spaces, Di eretial Operators, North-Hollad Mathematical Library, 1978.About this Shell Coloial Heights, Virgiia, address: dalemayehu@hotmail.com URL: