On Weighted Estimates of High-Order Riesz Bessel Transformations Generated by the Generalized Shift Operator

Transcription

1 Acta Mathematica Sinica, Engish Series Feb., 25, Vo.21, No.1, Pubished onine: June 21, 24 DOI: 1.17/s Htt:// On Weighted Estimates of High-Order Riesz esse Transformations Generated by the Generaized Shift Oerator Ismai EKINCIOGLU Deartment of Mathematics, Dumuınar University, Kütahya, Turkey E-mai: Ayhan SERETCI Deartment of Mathematics, Ankara University, 61 Tandogan-Ankara, Turkey E-mai: Abstract In this aer, we estabish sufficient conditions on weights which ensure that high-order Riesz esse transformations generated by the generaized shift oerator act boundedy from one weighted L -sace into another. Keywords Generaized shift oerator, Riesz esse transformations MR2 Subect Cassification 47G1; 45E1, Introduction Mutidimensiona singuar integras generated by the generaized shift oerator in were investigated by Kyuchantsev [1] and by Kiriyanov and Kyuchantsev [2], where weighted L - estimates were obtained for them. Aiev and Gadzhiev [3] studied the boundedness of singuar integras generated by a generaized shift oerator in L -saces with radia weights. Here we get certain sufficient conditions for the boundedness of high-order Riesz esse transformations generated by a generaized shift oerator in weighted L -saces with radia weights under certain restrictions on the growth of the weight functions. Furthermore, we suose that the weights do not vanish anywhere excet erhas at the origin and at infinity. The methods of roof used here are coser to that in the aer of Aiev and Gadzhiev [3]. Stein [4] started the theory of weighted L -estimates for the cassica Caderon Zygmund singuar integra oerators with the radia ower weight x α, x R n. For further detais see [5]. 2 Notations and ackground Let = {x : x =x 1,x 2,...,x n,x n k+1,...,x n, 1 k n}, and define { k L,ν = f : f L,ν fx x 2ν 1 } <, Received November 27, 22, Revised Ari 24, 23, Acceted June 13, 23

2 54 Ekinciogu I. and Serbetci A. where ν >, =1, 2,...,k are fixed arameters, 1 <, anddx = dx 1 dx n. Denote by L,ν v theweightedsace L,ν v = { f : f L,ν v R k n fxv x k x 2ν 1 } <, where the radia weight v x is generated by a non-negative amost everywhere ositive function vt, t<. The generaized shift oerator is defined by π π [ T y fx =c ν... f x y, x 2 n k+1 + y2 n k+1 2x n k+1y n k+1 cos α 1, where..., ] k x 2 n + yn 2 2x n y n cos α k sin 2ν 1 α dα, c ν = π k 2 Γν + 1 2, Γν x =x,x n k+1,...,x n, y =y,y n k+1,...,y n, and x,y R n k. The shift T y generates the corresonding convoution f 1 f 2 y = f 1 x [ T x f 2 y ] k. Note that this convoution satisfies the roerty f 1 f 2 =f 2 f 1. We need the foowing artition of unity : Let ψt C R 1 be such that ψt > for 1 2 <t<2, and ψt =for the remaining t R 1.Let ϕx =ψx / ψ2 x, x R n. = Since suψ2 t=[2 1, 2 +1 ], the sum in the denominator contains at most two non-zero terms for each x. Wesetϕ x =ϕ2 x beow. Obviousy: eow we aso use the function a ϕ x C R n ; b suϕ x ={x : 2 1 x 2 +1 }; c ϕ x = 1 for any x. = ϕ x = +2 m= 2 ϕ m x. It is cear that su ϕ x ={x :2 3 x 2 +3 }. The foowing generaized theorem of Hardy see [5], [6] ays an imortant roe in roving our main resut: Theorem 2.1 such that Let 1 q. There exists a constant C indeendent of the function g t q 1 µt q 1 gτdτ dt c vtgt dt, 2.1

3 On Weighed Estimates of Singuar Integras 55 if and ony if 1 H 1 =su µτ q t 1 q dτ vτ dτ <, 2.2 t> t where + =. Moreover, if C is the best constant in 2.1, thenh 1 C H 1 q 1 q q 1, and H 1 = C if =1or q =. Theorem 2.2 such that if and ony if Let 1 q. There exists a constant C indeendent of the function g µt gτdτ q dt t 1 q 1 C vtgt dt, 2.3 t 1 H 2 =su µt q 1 q dτ vτ dτ <. 2.4 t> t Moreover, the best constant C in 2.3 satisfies the estimates H 2 C H 2 q 1 q q 1. The main goa of this aer is to estabish weighted L -estimates for the norms of the high-order Riesz esse transformations generated by a generaized shift oerator: R k fx.v. c P k θ [ k T y fx ] k y n+k+2 ν n k+ dy c k im ε {y : y >ɛ} P k θ [ T y fx ] k y n+k+2 ν n k+ dy imr k ε ɛ fx, 2.5 where θ = y/ y, the characteristic P k θ beongs to some function sace on the hemishere S k = {x : x =1} and satisfies the canceation condition P k θ θ 2ν n k+dθ =, S k ν =ν 1,...,ν k is a muti-index consisting of fixed ositive numbers, ν = ν ν k,and 1 c k =2 n+2 ν n + k +2 ν k 2 Γ Γ, k =1, 2, Here P k x = P k x 1,...,x n k,x 2 n k+1,...,x2 n is a homogeneous oynomia with order k which satisfies k P k =. We reca that the Laace esse equation k is defined by n 2 k 2ν k = +, 1 k n. x n k+ x 2 x n k+ The existence of the imit 2.5 for a x and for Schwartz test functions fxcanberoved in a standard way if we take into account the we-known estimate T y fx fx cx y. The theorem beow is known about the behavior of the singuar integra oerator R k in L, ν : Theorem 2.3 Suose that the characteristic P k θ of the singuar integra 2.5 satisfies the

4 56 Ekinciogu I. and Serbetci A. conditions P k θ S k for some q>1. Then see [7] and [8]. θ 2ν n k+ dθ =, S k Pk θ q R k f L,ν CK q f L,ν, 1 <<. θ 2ν 1 n k+ dθ q = K q <, 3 Weighted Estimates for the Oerator R k The aim of this aer is to make the foowing assertion about the behavior of the singuar integra oerator 2.5 in weighted saces with radia weights. To rove the reated theorem we first give the foowing two emmas: Lemma 3.1 {u Rkn: u >ɛ} P k u u where c ν = π k 2 k u n+k+2 ν [ T u fx ] k u 2ν n k+ du = c ν f y, y 2n k+1 + y2n k+2,..., yn+k y2 n+k Γν k Γν x ỹ >ɛ k P k θ x ỹ n+k+2 ν 1 n k+2 dỹ, and x =x,x n k+1,...,x n,,...,,x }{{} =x 1,...,x n k, ỹ = k-terms y,y n k+1,...,y n,y n+1,...,y n+k and x θ y x n k+1 y n k y 2 = x ỹ, n k+2 x n y n+k yn+k 2,..., x ỹ x ỹ Proof We denote the first art by I I = π k Γν P k u u 2 k Γν f f u >ɛ u n+k+2 ν π π... x u, x 2n k+1 + u2n k+1 2x n k+1u n k+1 cos α 1,..., x 2n + u 2n 2x n u n cos α k u 2ν n k+ sin2ν 1 α dα du = π k 2 Γν k Γν P k u u u >ɛ u n+k+2 ν π. π... x u, x 2 n k+1 2x n k+1u n k+1 cos α 1 +u n k+1 cos α 1 2 +u n k+1 sin α 1 2,..., x 2 n 2x n u n cos α k +u n cos α k 2 +u n sin α k 2 k = c ν P k u u s >ɛ u n+k+2 ν π... π u 2ν n k+ sin2ν 1 α dα du f x u, x n k+1 u n k+1 cos α 1 2 +u n k+1 sin α 1 2,

5 On Weighed Estimates of Singuar Integras 57..., x n u n cos α k 2 +u n sin α k 2 k u 2ν n k+ sin2ν 1 α dα du. Now, we ass to the new variabes x =x,x n k+1,..., x n,,...,, ỹ =y,y n k+1,..., y n,y n+1,...,y n+k :x u = y,y n k+2 1 =x n k+ u n k+ cos α,y n k+2 =u n k+ sin α, α <πand u n k+ >, =1, 2,...,k. Since the Jacobian of the transformation is equa to u n k+1.u n k+2...u n 1 we have P k θ I = x ỹ >ɛ x ỹ f y, y 2n k+1 + n+k+2 ν y2n k+2,..., yn+k y2 n+k 1 n k+2 dỹ. This roves the emma. Lemma 3.2 fu where c ν = π k 2 u 2ν n k+ du = c ν k Γν k Γν. +k f y, y 2n k+1 + y2n k+2,..., yn+k y2 n+k 1 n k+2 dydy n+1 dy n+k, The roof is straightforward by substituting y = u,y n k+2 1 = u n k+ cos α,y n k+2 = u n k+ sin α, α <πand u n k+ >,, 2,...,k. Theorem 3.3 Suose that the characteristic P k θ of the singuar integra oerator 2.5 satisfies the conditions P k θ θ 2ν n k+dθ =, S k su Pk θ <. θ S k 3.1 Assume that the non-negative functions wt and vt,t, are reated by wt K 1 vt, and that, in addition, the foowing conditions hod : su wt K 2 inf wt, =, 1,...; t t 2 +1 r [ su wtt n+k+2 ν / ] dt 1 [ vtt n+k+2 ν / ] dt 1 = K 3 < ; 3.3 r> t r t 1 r su r> r [ wtt n+k+2 ν / ] dt t [ vtt n+k+2 ν / ] dt t 1 = K 4 <, 3.4 where 1 <<, = + and K 1, K 2, K 3, K 4 are ositive constants. Then : a There exists a constant K 5, indeendent of f and ɛ such that R k ɛf K 5 f L,ν v. L,ν w and b The imit im ɛ R k ɛf, which wi be denoted by R k f, exists in the sense of L,νw, R k f L,ν w K 5 f L,ν v.

6 58 Ekinciogu I. and Serbetci A. Proof We note that the coefficients c k in the estimates beow deend in genera on the arameters n, and ν, but not the function f nor the arameter ɛ>. We first rove art a of the theorem. Part b foows from art a. Without oss of generaity we assume that fx is an infinitey differentiabe function. We define the characteristic function χ ɛ x, for x,by { 1, x >ɛ, χ ɛ x =, x <ɛ. Then we have R k ɛf = w P k y y x χ ɛ y [ T y fx ] k L,ν w y n+k+2 ν n k+ dy = w x T y [χ ɛ x P k x x P k x x x n+k+2 ν ] fy. For simicity we et K ɛ x =χ ɛ x. Using the functions ϕ x n+k+2 ν and ϕ introduced in 2 in connection with a artition of unity, we get R k ɛf w x ϕ x T y K ɛ xfy ϕ s y L,ν w = { c w x ϕ x w x ϕ x w x ϕ x s= T y K ɛ x ϕ yfy T y K ɛ x s 3 T y K ɛ x s 3 ϕ s yfy ϕ s yfy } = c 1 {I 1 + I 2 + I 3 }. We first consider I 1. y the condition 3.2 with resect to w x, we have I 1 su w x ϕ x T y K ɛ x ϕ yfy n k+ dy 2 1 x 2 +1

7 On Weighed Estimates of Singuar Integras 59 c 2 inf w x 2 1 x 2 +1 Using Theorem 2.3, we get I 1 c 3 inf w x 2 1 x 2 +1 T y K ɛ x ϕ yfy ϕ xfx From the condition 3.2, there exists a constant c 4 such that Therefore k. inf 2 1 t 2 +1 wt c 4 inf 2 3 t 2 +3 wt =, 1, 2,... I 1 c 5 ϕ xfxw x If we use here the obvious inequaity for >1, ϕ ϕ =5 and the condition wt K 1 vt, we have I 1 c 6 fxv x k k. = c 6 fx L,ν v. Secondy we consider the integra I 2 : I 2 = w x ϕ x T y K ɛ x ϕ s yfy n k+ dy s 3 x ỹ >ɛ = w P k y y x ϕ x χ ɛ y R y T y k ϕ k n+k+2 ν s xfx n k+ dy n s 3 If we use Lemma 3.1 here, we get I 2 c 7 w P k θ x ϕ x x ỹ n+k+2 ν f s 3 ϕ s y, y 2n k+1 + y2n k+2,..., yn+k y2 n+k y, y 2n k+1 + y2n k+2,..., yn+k y2 n+k We consider the integra I 2 = 1 n k+2 dỹ w x ϕ x s 3 ϕ s y,. x ỹ >ɛ P k θ x ỹ n+k+2 ν y 2n k+1 + y2n k+2,..., yn+k y2 n+k..

8 6 Ekinciogu I. and Serbetci A. f y, y 2n k+1 + y2n k+2,..., yn+k y2 n+k 1 n k+2 dỹ. Since 2 1 x 2 +1, ỹ 2 2 and x = x, it foows that x ỹ x ỹ =2 2. Using this and the condition su θ S k P k θ <, weget I 2 c 8 w x ϕ x2 2 n+k+2 ν {Rkn+kỹ f = c 9 y, { s 3 ϕ s y, y 2n k+1 + y2n k+2,..., yn+k y2 n+k y 2n k+1 + y2n k+2,..., y 2 n+k 1 + y2 n+k 1 n k+2 dỹ } w x ϕ x2 2 n+k+2 ν s 3 ϕ s u fu k k u 2ν n k+ du }, in view of Lemma 3.2. Passing to oar coordinates, we have I 2 c n+k+2 ν su w t ϕ x 2 1 t r n+k+2 ν 1 dr frθ θ 2ν n k+ dθ. S k Since su wt K 2 inf wt K 2 w2 fort [2 1, 2 +1 ], we get from Höder s inequaity that { I 2 c 11 2 n+k+2 ν 1 w 2 2 frθ θ 2ν 1 n k+ dθ. r dr} n+k+2 ν 1 S k Now et Ωr = frθ ρ θ 2ν 1 ρ n k+ dθ, Wρ = Ωrr n+k+2 ν 1 dr. S k Then I 2 = c 12 I 2 c 13 w 2 2 n+k+2 ν 1 1 W = If the ast factor 2 is written as the difference , and the series is written as the sum of two terms = and =1, then it is easy to see, by using the monotonicity of the non-negative function W ρ and the condition 3.2 on the function wt, that 3.5 can be

9 On Weighed Estimates of Singuar Integras 61 estimated by the sum of the two integras 1...and... Thus 1 1 I 2 c 14 + w tw tt n+k+2 ν 1 1 dt We want to rove that 1 = c 14 w tw tt n+k+2 ν 1 1 dt = c 14 I 2. I 2 c 15 v tω tt n+k+2 ν 1 dt = c 15 f L,ν v. In other words, we shoud rove that w tt n+k+2 ν 1 1 t dt Ωττ n+k+2 ν 1 dτ c16 v tt n+k+2 ν 1 Ω tdt. ut the atter is a secia case of the inequaity 2.1 of Hardy if there we substitute = q, gt =t n+k+2 ν 1 Ωt, µ t =w t n+k+2 ν 1 1 and ν = v tt 1 n+k+2 ν 1. It is easy to see that in this case the condition 2.2 becomes the condition 3.4 of our theorem. Thus, we have roved that I 2 c 17 f L,ν v. Finay, we estimate I 3.LetuswriteI 3 = c 18 = I 3, where I 3 = w x ϕ x = w x ϕ x Using Lemma 3.1, we have I 3 = f T y K ɛ x s 3 ϕ s yfy P y k y χ ɛ y y T y k ϕ n+k+2 ν s xfx w x ϕ x s 3 ϕ s y, x ỹ >ɛ s 3 P k θ x ỹ n+k+2 ν y 2n k+1 + y2n k+2,..., yn+k y2 n+k y, y 2n k+1 + y2n k+2,..., yn+k y2 n+k 1 n k+2 dỹ. Since 2 1 x 2 +1, ỹ 2 +2,and x = x, it foows that ỹ 2 x from this x 1 2 ỹ, and hence x ỹ x ỹ = ỹ x 1 2 ỹ. Thus, { I 3 c 19 w 1 x ϕ x +k ỹ ỹ n+k+2 ν f s 3 y, ϕ s y, y 2n k+1 + y2n k+2,..., yn+k y2 n+k y 2n k+1 + y2n k+2,..., y 2 n+k 1 + y2 n+k.

10 62 Ekinciogu I. and Serbetci A. = c 2 1 n k+2 dỹ } s 3 { w 1 x ϕ x fu u u n+k+2 ν ϕ s u u 2ν n k+ du }, in view of Lemma 3.2. Passing to oar coordinates, and then using Höder s inequaity and the condition 3.2 on wt, we get I 3 c 21 su w x ϕ x 2 1 x 2 +1 { r 1 dr frθ 2 S k θ 2ν 1 } n k+ dθ Letting c 22 inf w x 2 n+k+2 ν 2 1 x 2 +1 { frθ θ 2ν 1 } n k+ dθ. S k 2 Ωt = frθ S k θ 2ν 1 n k+ dθ, W 1 t = t Ωτ dτ τ, we have the estimate I 3 c 22 w 2 2 n+k+2 ν W 1 2. Thus I 3 c 23 = w 2 2 n+k+2 ν W As in the estimate of I 2 it can be seen that I 3 c 24 W 1 tw tt n+k+2 ν 1 dt. We rove that the right-hand side can be estimated from above by the quantity c 25 v tt n+k+2 ν 1 Ω tdt c 26 f L,ν v. In other words, we must show that Ωτ dτ w tt n+k+2 ν 1 dt c 27 Ω tv tt n+k+2 ν 1 dt. t τ ut this is a secia case of the dua Hardy inequaity 2.3, where one shoud substitute q =, gτ =Ωτ/τ, µ t =w tt n+k+2 ν 1 and ν t =v tt n+k+2 ν + 1. Then the condition 2.4 becomes the condition 3.3 of our theorem.

11 On Weighed Estimates of Singuar Integras 63 Summing the estimates obtained for I 1, I 2 and I 3,wegettheinequaity R k f ɛ L,ν c 28 f L,ν v, 3.6 w which cometes the roof of the first art of the theorem. Now et us roceed to the second k art. We rove that the imit im ɛ R ɛ f = Rk f exists in the sense of L,νw and hence the estimate 3.6 hods for R k f. It suffices to rove that the imit exists for functions that have comact suort, are smooth, and are even with resect to the variabes x n k+1,...,x n. Indeed, reresenting any function f in L,ν v in the form of a sum f = f 1 + f 2,wheref 1 is a function that has comact suort, is smooth, and is even with resect to x n k+1,...,x n, and f 2 is such that f 2 L,ν v is sufficienty sma, we have from the equaity R k f = ɛ k R ɛ f 1 + R k ɛ f 2 and 3.6 that R k ɛ f R k c 29 f 2 L,ν v δ, L,ν w ɛ f 1 where δ is a sufficienty sma number. k Therefore, it suffices to rove the existence of the imit im ɛ R ɛ f = Rk f in the sense of L,ν w for smooth comacty suorted functions that are even with resect to the variabes x n k+1,...,x n.takingfx as such a function and using the canceation condition 3.1, we have k R ɛ 2 f R k P k θ ɛ 1 f = y T y fx n+k+2 ν n k+ dy = ɛ 1 < y <ɛ 2 ɛ 1 < y <ɛ 2 P k θ [ T y fx fx ] k y n+k+2 ν n k+ dy, where x, y and θ = y y. y using the Tayor Desarte formua [9] for T y fx itisnothardtoshowthat Therefore, R k f ɛ R k 2 ɛ 1 f T y fx fx L,ν v c 3 y. L,ν w ɛ 1 < y <ɛ 2 c 31 y y n+k+2 ν n k+ dy c 32ɛ 2 ɛ 1. k Since the sace L,ν w is comete, this imies that the imit im ɛ R f exists and ɛ beongs to L,ν w. Thus the roof is comete. Note that Theorem 3.3 is vaid for ower weights. It is not hard to show that the singuar integra oerator R k acts boundedy in L,νw fortheowerweight,wt =t α, n + k + 2 ν / < α, β < n + k +2 ν /. Acknowedgments We fee great easure to have the oortunity to areciate Professor A. D. Gadzhiev s excetiona advice and comments on this aer. References [1] Kyuchantsev, M. I., On singuar integras generated by the generaized shift oerator, I.. Sibirsk. Math. Zh., 11, Engish transation: in Siberian Math. J., 11, 197

12 64 Ekinciogu I. and Serbetci A. [2] Kiriyanov, I., A., Kyuchantsev, M. I.: On singuar integras generated by the generaized shift oerator, II, Sibirsk. Mat. Zh., 11, , Engish transation: in Siberian Math. J., 11, 197 [3] Aiev, I. A., Gadzhiev, A. D.: Weighted Estimates for Mutidimensiona Singuar Integras Generated by The Generaized Shift Oerator. Matematika Sbornik Rossiyskaya Akad. Nauk., 1839, , Engish transation: Russian Acad. Sci. Sb. Math., 771, [4] Stein, E. M.: Note on Singuar Integras. Proc. Amer. Math. Soc., 8, [5] Dyk in, E. M., Osienker,. P.: Weighted Estimates of Singuar Integras And Aications. J. Soviet Math., 33, 1985 [6] Maz ya, V., G.: Soboev Saces, Sringer-Verag, erin, 1985 [7] Stein, E. M.: Singuar Integras And Differentiabiity Proerties of Functions, Princeton Univ. Press, Princeton, N. J., 197 [8] Ekinciogu, İ., Serbetci, A.: On the singuar integra oerators generated by the generaized shift oerator. Int. J. A. Math., 11, [9] Levitan,. M., esse function exansions in series and fourier integras. Usekhi Mat. Nauk., 6, no.242, [1] Aiev, I. A.: On Riesz Transformations Generated by A Generaized Shift Oerators. Izvestiya Acad. of Sciences of Azerbaydian, 1, [11] Dzhaneidze, O. P.: On boundedness of a mutidimentiona singuar oerator acting in a sace with a weight. Soobshch. Akad. Nauk Gruzin. USSR., 87, [12] Guseinov, E. G.: Singuar integras in saces of functions that are integrabe with a monotone weight. Math. Sb., 174, ; Engish transation: in Math. USSR Sb., 6, 1988 [13] Yung-Ming, C.: Theorems of Asymtotic Aroximations. Math. Ann., 14,