Sobolev-Il in Inequality for a Class of Generalized Shift Subadditive Operators

Transcription

1 Nonlinear Analyi and Differential Equation, Vol. 5, 217, no. 2, HIKAI Ltd, Sobolev-Il in Inequality for a Cla of Generalized Shift Subadditive Operator S.K. Abdullayev Baku State Univerity Intitute of Mathematic and Mechanic of ANAS, Azerbaijan E.A. Mammadov Baku State Univerity, Azerbaijan Copyright c 216 S.K. Abdullayev and E.A. Mammadov. Thi article i ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. Abtract We tudy a problem of etablihment of Sobolev-Il in inequalitie type trong and weak inequalitie for ubadditive operator with majorizing operator from certain cla of iez potential type integral convolution with almot monotone kernel, generated by both ordinary and generalized hift operator, aociated with Laplace-Beel differential operator. Keyword: Sobolev-Il in inequalitie, ubadditive operator, majorizing operator, iez potential, monotone kernel, generalized hift operator, Laplace-Beel differential operator. 1 Introduction Partial equation containing the Laplace-Beel differential operator Bmk,k, uing the Fourier-Beel many dimenional tranform wa firt tudied in the paper of I.A. Kipriyanov ee [1]. For further invetigation they introduced the weight pace L p,ν. Contruction of fundamental olution of B- elliptic equation wa given in the paper of I.A. Kipriyanov and L.A. Ivanov [2], where it i proved that the

2 76 S.K. Abdullayev and E.A. Mammadov olution of the equation Bmk,k u x f x i a cylindrical potential type integral operator u x IB α f x y α n γk.mk T y f x y γ k,mk dy, mk,k < α < m k γ k,mk when α 2 called the iez potential, that contain the tranformation T y, in one-dimenional cae introduced by B.M. Levitan [3] and called the generalized or Beel hift operator. In [4], I.A. Kipriyanov in fact reduced the problem of obtaining a priori etimate to etimate of iez generalized potential and their appropriate derivative. Etimate of Hardy-Littlewood-Sobolev and Sobolev-Il in type inequalitie generalizing the one-dimenional Hardy-Littlewood inequalitie for potential type integral i one of the main element of integral repreentation method developed firt by S.L. Sobolev ee [5]. Hardy-Littlewood-Sobolev- type inequalitie for the iez B-potential IB α in the cale of L p,ν pace were obtained in the paper of A.D. Gadjiev and I.A. Aliyev [6]. A pecial place in etablihing Hardy-Littlewood-Sobolev and Sobolev-Il in type etimate for integral operator of B-harmonic analyi in different metric i occupied by the work of V.S. Guliyev and hi follower ee [7], [8]. For the firt time, in the paper of S.K. Abdullayev and Z.A. Damirova [9], S.K. Abdullayev and B.K. Agarzayev [1], thee etimated were extended for the cae of the iez potential, with nonpower kernel of the form IB ω f x T y f x ω y y mk v dµ y mk,k in the cae of ordinary and generalized hift T y, repectively. In the paper, for ubadditive operator majorized with operator of certain cla of integral convolution of iez potential I α B, with almot monotone kernel, we prove the validity of Sobolev-Il in type etimate. Note that for thi cla of ubadditive operator, the Hardy-Hittlewood-Sobolev type etimate were etablihed in [11]. A i known, the iez-beel generalized potential even ordinary iez potential, ee Nakai [2] with nonpower kernel don t act, generally peaking in the cale of L p,ν pace. 2 Some deignation and preliminary information Let l be Euclidean pace of dimenion lm, k be integer, n m k 1, p 1,

3 Sobolev-Il in inequality for a cla of generalized mk,k { x 1,..., x mk mk : x mi >, i 1,..., k }, c ν π m, m T y γ u x π... u x y, x m1, y m1 α1,..., x mk, y mk αk in γ m1 1 α 1... in γ mk 1 α k dα 1...dα k be a generalized hift operator generated by the Laplace-Beel operator where Bmk,k x m 2 j1 x 2 j mk jm1 2 x 2 j γ j x j x mk,k, γ m1 >,..., γ mk >, x j x, y m, x x, x m1,..., x mk, y y, y m1,..., y mk, x mi, y mi αi i 1,..., k, C ν i a normalizing factor. Further we aume y γ x 2 mi 2x mi y mi co α i y 2 mi γ,...,, γ m1,..., γ mk mk,k, k γ γ mi, a n γ, i1 mk i1 y γ i i y γ m1 m1...y γ mk mk, if y mk,k. In deignation γ the index n indicate the dimenion of thi vector, the index k the amount of it poitive coordinate; γn, k,..., m if k. L Φ γ i Orlicz pace [12] determined by the N-function Φ: L Φ γ {f izm. : f L Φ γ {λ inf > : Φ ε f x dµ γ y <, ε > } f x Φ dµ γ y 1, λ dµ y y γ dy y γ m1 m1...y γ mk mk dy 1...dy mk., }

4 78 S.K. Abdullayev and E.A. Mammadov The function Φ : [,, ] i aid to be N-function if there exit a non-decreaing left-continuou function q : [,, ] uch that q, q t t and Φ r r q t dt. In the cae Φ t t p, t > and 1 p < we denote the pace L Φ γ by L p,γ -pace of function integrable in p-th degree with the weight y ν m1 m1...y ν mk mk : L p,γ f izm. : f Lp,γ 1/p f y p dµ γ y <. Definition 1. The poitive function g t almot decreae almot increae on the et X, if there exit a contant uch that for any c d > c l >, and alo the relation f g x X mean that there exit a contant C > uch that C 1 f x g x Cf x, x X. Let Ω p,α Ωp,α p 1, α > be a union of the function ω :,, uch that ω t increae almot increae, t { α/p}ε ω t decreae almot decreae for mall ε > and the integral t 1 ω t dt converge. Obviouly, Ω p,α Ω 1,α and Ω p,α Ω 1,α and alo if ω Ω p,α then ω 2t Cω t, where C i independent of t,. Definition 2. Let p 1, α >. It i aid that the ubadditive operator A belong to the cla Kγ p, Ωp,α if 1. Af x exit for almot all x, when L p,γ n.k and 2. there exit ω Ω p,α and C > that for almot all x, Af x C nk T y γ f x ω y y α dµ γ y. Denote α n γ. In the cae when ω Ω p,α, it directly follow from the definition ee [12] that the generalized iez potential IB ω f x T y γ f x ω y y α dµ γ y,

5 Sobolev-Il in inequality for a cla of generalized the Beel potential JBf ω x mk,k T y γ f x G ω γ y dµ γ y, ω δ 1/2 G ω γ x c ω δ γ δ n γ /2 e 4π x 2 π dδ δ δ, and the generalized B-fractional-maximum function ω B, r 1/α Mγ ω γ f x up Tγ y r> B, r f x dµ γ y, γ B,r B, r γ dµ γ y, B,r belong to the cla K γ p, Ωp,α. Note that when ω t t, < < α, IB ω i the iez potential of order, JB ω i the Beel potential of order, while Mγ ω f x i the B fractional maximum function Mγf x. 3 Hardy-Littlewood-Sobolev- inequalitie The following theorem, where Hardy-Littlewood-Sobolev etimate are etablihed for the operator from the cla K γ p, Ωp,α, wa proved in the paper [11]. Theorem 1. Let 1 p < and A K γ p, Ωp,α,k. Then there exit the N -function Φ uch that Φ 1 r a r r a/p ω t t 1 dt r >, where Φ 1 i the invere of Φ, ω i a function from definition 2 correponding to the operator A, and a if p > 1, then C >, f L p,γ Af L Φ ν C f L p,γ b if p 1, then C >, f L 1,γ, β > {x: Afx >2β} 1 dµ x Φ c β f L 1,γ emark 1. Thi theorem i exact for the generalized iez potential I ω B when ω Ω p,α ee [12]. We alo note that thi theforem cover the cae of ordinary hift in all variable, if we put k that i neceary in etablihing the Sobolev-Il in inequalitie in general cae. 1.

6 8 S.K. Abdullayev and E.A. Mammadov 4 Sobolev-Il in-type theorem Let n m k 2 and {1,..., m k 1}. Then we partition the pace mk,k of the point x x 1,..., x mk into the direct um of the pace,k of the point x x n1,..., x n with coordinate x n1,..., x n where k rang {n 1,..., n } {m 1,..., m k} and 1 n 1 <... < n m k and the pace n,k k of the point x uch that x x, x for denotation ee [5]. Note that at certain choice of the vector γ,...,, γ m1,..., γ mk even at one and the ame value of the parameter, m, k, the expanion x x, x i determined nonuniquely. Thi circumtance doe not influence of final reult, but the in pecific cae the coordinate x n1,..., x n are fixed. Aume m rang {n 1,..., n } {1,..., m}. Then m, k are integer uch that m m, k k and m k. If m > k >, we aume then obviouly, and {n 1,..., n } {1,..., m} {j 1,..., j m }, j 1 <... < j m, {n 1,..., n } {m 1,..., m k} {m i 1,..., m i k }, And alo, if k >, then i 1 <... < i k, y y j1,..., y jm, y mi1,..., y mik d y dy j1...dy jm dy mi1...dy mik. γ,k,...,, γ i1,..., γ ik,k, γ,k γ i1... γ ik, y γ,k y γ i 1 mi 1...y γ i k mi k, dµ,k y y γ,k d y. In thee denotation we aume alo m m m, k k k,,k m k,k, n,k k m k,k n,k. Further, y, y n,k, y γ n,k and dµ n,k y are determined from the equalitie y y, y γ γ,k, γ n,k, y γ y γ m1 m1...y γ mk mk y γ,k y γ n,k

7 Sobolev-Il in inequality for a cla of generalized and dµ y dµ,k y dµ n,k y, repectively. And alo if m, then {j 1,..., j m }, k and y y mi1,..., y mik, dµ,k y y γ i 1 mi 1...y γ i k mi k dy mi1...dy mik y γ,k d y. Note that at certain choice of the parameter γ,...,, γ m1,..., γ mk even at one and the ame value of the parameter, m, k the expanion x x, x i determined non-uniquely. Thi circumtance doen t influence on final reult, but in pecific cae, the coordinate x n1,..., x n are fixed. Introduce the denotation ee the denotation α mk,k a,k γ,k, a n,k n γ n,k, a,k a n,k α mk,k, ω p,a,k t ω t t a,k/p, ω p,an,k t ω t t a n,k/p, and note ome propertie of the operator T y that we will repeatedly ue in the equel: T 1 T y 1 1; T 2 T y Cf CT y y, C ; T 3 if f g, then T y f T y g ; T 4 T f p T f p ; T 5 mk,k T y f x p dµ y 1/p f p,ν ; T 6 T y,y γ f x, x Tγ y,k T y γ n,k f x, x T y γ n,k T y γ,k f x, x. 5 Main reult. Theorem 4 main. Let 1 p <, k, m, n m k 2, A K γ p, Ωp,α and ω an be appropriate function. If {1,..., m k 1}, m m, k k k m and ω p,an,k Ω p,a,k, then there exit the N-function Φ Φ p, uch that r Φ 1 p, r a,k r a,k /p ω p,an,k t t 1 dt r > and a if p > 1, then there exit C > uch that for any function

8 82 S.K. Abdullayev and E.A. Mammadov f L p,γ mk,k and Af, x L Φp, γ,k,k C f L p,γ mk,k, b there exit C > uch that for any function f L 1,γ for any β > and x n,k, {x: Af, x >2β} 1 dµ x Φ c 1, β f L 1,γ Let ω :,,. Let introduce the ubadditive operator I ω γ : f I ω γ f, where Iγ ω f x T y γ ω x x α f y dµγ y. The following lemma i a tarting point for proving the analoge of Sobolev econd theorem on potential. Lemma C. Let k, S {1,..., m k 1}, k k, m m, k m, 1 p < and ω Ω p,n γ. Then there c 1 > exit uch that for any function f L loc 1,γ and any x x, x inequality i valid I ω γ f x C 1 Iω p,an,k γ,k f p, x, 1. the following where f p, t f t, L p,γn,k, t,k n,k,k T y ω p,a n,k x x y,k I ω p,an,k γl,k f p, x Proof Let B t ω t t α. Denote F x, y n,k f y, Lp,γn,k dµ k, y. n,k T y γ B x f y, y dµ n,k y. 1

9 Sobolev-Il in inequality for a cla of generalized Then taking into account the equality dµ nk y dµ,k y dµ n,k y, by uing the Foubini theorem, we have Thu,,k Iγ ω f x n,k Iγ ω f x T y γ B x f y dµ γ y Tγ y B x f y, y dµ n,k y dµ,k y.,k T y B x f y dµ y F x, y dµ,k y. 2 Let p 1. Taking into account the propertie T 1 T 6 of the generalized hift operator T y and almot decreae of B t ω t t n γ, we have Whence, allowing for we get C T y γ B x T y,y γ B x, x Tγ y T γ y B x, x n,k n,k T y γ n,k T y γ,k CB x CT y γ,k B x. B t ω t t α mk,k ω t t α n,k t a,k ωp,an,k t t a,k.,k,k Iγ ω f x T,k γ,k B x T y γ B x f y y γ dy n,k f y, y dµ n,k y dµ,k y T γ,k B x f y, L1γn,k dµ,k y I ωp,α,k γl f 1, x. By thi, in the cae p 1 we proved lemma C. Let p > 1. Uing 1, having applied the Holder inequality, we get F x, y Tγ y B x Lp,γ n,k n,k f y, Lp,γ,k,k. 3

10 84 S.K. Abdullayev and E.A. Mammadov Etimate from above T y γ B x Lp,γ n,k. n,k Allowing for the propertie T 1 T 6 of the generalized hift operator T y, we get Tγ y B x T y,y γ B x, x Tγ y,k Tγ y B x, x n,k where ȳ Then π π... Tγ y B ỹ, x k in gγ ij 1 α il dα il, n,k l1 x j1 y j1,..., x jm y jm, x mi1, y mi1 αi1,..., x m ik, y mik x mi, mi αi x 2 mi 2x mi mi co α i 2 mi, ỹ, x ỹ 2 x 2 1/2, m ỹ x ji y ji 2 k x 2 mi1 2x mil y mil co α il ymi 2 l n,k i1 π π... T y l1 T y γ B x Lp,γn,k γ B ỹ, x k in γ ij 1 α il dα il n,k Having applied the Minkovky inequality, from the lat one we get l1 Tγ y B x π π C... J L p,γ n,k n,k k α ik 1/2 p dµ n,k in γ il 1 α il dα il, l1., y 1/p. J p Tγ y B ỹ, x n,k 1/p y γ n,k d y. 4 n,k Taking into account the property T 5, etimate J J p Tγ y B ỹ, x n,k y γ n,k d y 1/p n,k

11 Sobolev-Il in inequality for a cla of generalized B ỹ, y p y γ n,k d y 1/p n,k n,k ω ỹ, y ỹ, y p n γ ε 1 ỹ, y ny p ε Hence, allowing for the condition ω Ω p,n γ k,n and p y γ n,k d y 1 p. ỹ, y ỹ, y ỹ 2 y 2 1/2 ỹ, we get D n,k J C ỹ ω ỹ n y k,n ε p D 1 ỹ, y nγk,n p ε p y γ p n,k d y. 5 Etimate D D n,k C D 1 [ ỹ y 2] 1 [ γ εp ] 2 y γ n,k n,k 1 y n γ y γ n,k d y ỹ [n γ εp ] [n γ εp ] ỹ p CD 1 y p, n,k d y 1 p 1 y 1 n γ p y γ n,k d y. 6 ỹ Etimate D 1 from upper. Making change of variable y ỹ z y z, we have y γ k,n d y γ ỹ n,k ỹ n z γ n,k d z. 1 p Whence D 1 ỹ n γ n,k p D 11,

12 86 S.K. Abdullayev and E.A. Mammadov D 11 1 z n γ z γ n,k d z. 7 1 p n,k Paing to pherical coordinate z tθ, we get D 11 d and thi integral converge, a the function 1 t n γ t γn,k [n 1]dt 1 p γn,k [n 1] 1 t n γ t i of order γn,k [n 1] > a t and n γ γn,k [n 1] 1 2 a t. Thu, taking equentially into account 7, 6 and 5, we get n γ n,k D 1 C ỹ p, [n γ εp n ] γ n,k [n γ D CD 1 ỹ p εp ] C ỹ p ỹ p whence J C C ỹ [ γ εp ] [ γ,k ] ỹ p ỹ p ε, ω ỹ ỹ n γ k,n ε p ω ỹ D C ỹ ω ỹ n γ k,n ε p ω ỹ C n γ k,n γ,k ỹ p p h γ n,k p In the lat paage we take into account 1 ỹ γ,k C n γ γ,k p p n γn,k γn,k γ p [ γ,k ] ỹ p ε ω p,an,k ỹ. ỹ γ,k γ,k p

13 Sobolev-Il in inequality for a cla of generalized n γn,k γ,k γ,k p p p n γn,k γ,k. p Subtituting the obtained repreentation in 4, we have Tγ y B x π π C... J L p,γ n,k n,k C π π ω ỹ... ỹ β k i1 in γ mi 1 α i dα i CT y x k in γ il 1 α il dα il, l1 ω x. x γ,k We take into account the lat etimation in 3 and get: F x, y CTγ y ω x,k f y, x γ,k Lp,γk,k. Taking thi into account in 2, we eaily get: C,k T y γ,k Thu, the inequality I ω γ f x ω x f y, x γ,k Lp,γ,k dµ,k y,k CI ω p,a n,k γ,k f p, x. I ω γ f x C 1 I ω p,a n,k γ,k f p, x and Lemma C are proved in the cae if p > 1, a well. Theorem 2 directly follow from Lemma C. By applying. Theorem 1 and that from the relation ω p,α Ω p,α it follow ω Ω p,αmk,k. Note that if ω t α, then all the reult of the paper [8] Theorem 2 and Theorem 3 belonging to relating to Sobolev-Il in etimation for the iez potential IB ω f x y α n γ k,mk T y γk,mk f x y y k,mk dy, mk,k < α < m k γ k,mk, where the cae < m < m, < k < k wa not conidered at all, follow from Theorem 2.

14 88 S.K. Abdullayev and E.A. Mammadov eference [1] I.A. Kipriyanov, Singular Elliptic Boundary Value Problem, M. Nauka, Fizmatlith, [2] I.A. Kipriyanov, L.A. Ivanov, Obtaining fundamental olution for homogeneou equation with ingularitie in everal variable, Proc. of S.L. Sobolev Workhop, Novoibirk, , [3] B.M. Levitan, Expanion in erie by Beel function, and Fourier integral, Upekhi Mat. Nauk, , no. 2, [4] I.A. Kipriyanov, M.I. Klyuchanchev, Etimation of urface potential generated by generalized hift operator, Dokl. AN SSS, , no. 5, [5] S.L. Sobolev, On one functional analyi theorem, Math. Sb., , no. 3, [6] A.D. Gadjiev, I.A. Aliyev, On clae of potential type operator generated by generalized hift, In eport of eminar of I.N. Vekua intitute of Aplied mathematic, Tbilii, 5 199, no. 2, [7] V.S. Guliyev, Sobolev theorem for the iez-potential, Doklady AN, , no. 4, [8] V.S. Guliyev, N.N. Garakhanova, The Sobolev-Il in theorem for the B-iez potential, Siberian Matem. Journal, 5 29, no. 1, [9] S.K. Abdullayev, B.K. Agarzayev, On one property of iez generalized potenial, Tranaction iue Mat. and Mech. Serie of Phyical-Technical and Matematical Science, XXIV 4 Baku-25, ELM. [1] S.K. Abdullayev, B.K. Agarzayev, Sobolev-Il in theorem for iez potential with generalized hift and almot monotone kernel, Ucheniye Zapiki Orlovky Go. Univ., , no. 3, [11] E. Nakai, H. Sumitomo, On generalized iez potential and pace of ome mooth function, Sci. Math. Jpn., 54 21, no. 3, [12] M.A. Kanoelkiy, Ya.B. utitkiy, Convex Function and Orilicz Space, Fizmatgiz, Mocow, eceived: December 23, 216; Publihed: January 25, 217