Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 36503, 8 pages doi:10.1155/2007/36503 Research Article On the (p,q)-Boundedness of Nonisotropic Spherical Riesz Potentials Mehmet Zeki Sarikaya and H ¨ useyin Yildirim Received 20 November 2006; Accepted 1 March 2007 Recommended by Shusen Ding We introduced the concept of nonisotropic spherical Riesz potential operators generated by the λ-distance of variable order on λ-sphere and its (p,q)-boundedness were investi- gated. Copyright © 2007 M. Z. Sarikaya and H. Yildirim. This is an open access article distrib- uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let R n =  x =  x 1 ,x 2 , ,x n  : x i ∈ R,1≤ i ≤ n  . (1.1) In R n spaces, L p and L ∞ are defined as follows: L p = L p  Ω n,λ  =  f (x): f  p =   Ω n,λ   f (x)   p dx  1/p < ∞  ,1≤ p<∞ L ∞ = L ∞  Ω n,λ  =  f (x): f  ∞ = esssup x∈Ω n,λ   f (x)   < ∞  , (1.2) where Ω n,λ is the n-dimensional unite λ-sphere of R n which is dependent on the λ- distance. The λ-distance between points x = (x 1 , ,x n )andy = (y 1 , , y n )isdefined by the following formula given in [1–10]: |x − y| λ :=    x 1 − y 1   1/λ 1 +   x 2 − y 2   1/λ 2 + ···+   x n − y n   1/λ n  |λ|/n , (1.3) 2 Journal of Inequalities and Applications where x, y ∈ Ω n,λ , λ = (λ 1 ,λ 2 , ,λ n ), λ k > 0, k = 1,2, ,n, |λ|=λ 1 + λ 2 + ···+ λ n .Note that this distance has the following properties of homogeneity for any positive t,    t λ 1 x 1   1/λ 1 + ···+   t λ n x n   1/λ n  |λ|/n = t |λ|/n |x| λ . (1.4) This equality give us that nonisotropic λ-distance is the order of a homogeneous function |λ|/n. So the nonisotropic λ-distance has the following properties: (1) |x| λ = 0 ⇔ x = θ, (2) |t λ x| λ =|t| |λ|/n |x| λ , (3) |x + y| λ ≤ 2 (1+1/λ min )|λ|/n (|x| λ + |y| λ ). Here we consider λ-spherical coordinates by the following formulas: x 1 =  ρcosθ 1  2λ 1 , ,x n =  ρsinθ 1 sinθ 2 ···sinθ n−1  2λ n . (1.5) We obtained that |x| λ = ρ 2|λ|/n . It can be seen that the Jacobian J λ (ρ,ϕ) of this t ransforma- tion is J λ (ρ,θ) = ρ 2|λ|−1 W λ (θ), where W λ (θ) is the bounded function, which only depends on angles θ 1 ,θ 2 , ,θ n−1 . It is clear that if λ 1 = λ 2 =··· = λ n = 1/2, then the λ-distance is the Euclidean distance. We define an gle cos |x − y| λ = x · y, (1.6) where x and y are vectors on the n-dimensional unite λ-sphere. For f ∈ L(Ω n,λ ), 0 <α(x) <n, we will consider the following nonisotropic spherical Riesz potential operator generated by the λ-distance of variable order: I α(x) λ f (x) =  Ω n,λ |x − y| α(x)−n λ f (y)dy, x ∈ Ω n,λ . (1.7) The aim of this paper to show that the well-known properties of classical Riesz p oten- tials may be formulated for our generalization (1.7). We will study the (p,q)-boundedness of operators (1.7). Note that our results are the generalization of corresponding results for classical Riesz potentials, given in [11]. The important properties of the nonisotropic Riesz potentials and theirs generalizations were studied by many authors. We refer to pa- pers [1–9, 12]. The nonisotropic spherical Riesz potential generated by λ-distance is the classical Riesz potential for λ i = 1/2, i = 1,2, , n and α(x) = α.Hereparticularimpor- tance of the nonisotropic kernel is that it does not have the classical t riangle inequality. It is well known that the classical Riesz potentials I α ϕ = ϕ ∗|x| α−n are bounded oper- ators from L p (R n )toL q (R n )for1/q = 1/p− α/n,0<α<n,1≤ p<q<∞ [10]. Lemma 1.1. Let J λ (x) =  Ω 1 n,λ f (x)K(x, y)dy, x ∈ Ω 2 n,λ , k 1 = sup y∈Ω 1 n,λ   Ω 2 n,λ |K(x, y)| q dx  1/r < ∞, k 2 = sup x∈Ω 2 n,λ   Ω 1 n,λ |K(x, y)| q dy  1/q−1/r < ∞ (1.8) M. Z. Sarikaya and H. Yildirim 3 and the following condit i ons are carried out: 1 ≤ p ≤ r ≤∞, 1 − 1/p +1/r = 1/q, f ∈ L p (Ω 1 n,λ ). Then   J λ   L p (Ω 1 n,λ ) ≤   f   L r (Ω 2 n,λ ) k 1 k 2 . (1.9) Proof. Let λ, μ, ν be positive numbers such that 1/λ+1/μ+1/ν = 1. We write J λ (x) =  Ω 1 n,λ f p(1/p−1/μ) (x) f p/μ (x) K q(1/q−1/ν) (x, y)K q/ν (x, y)dy. (1.10) By H ¨ older’s inequality with exponents λ, μ,andν,weobtain J λ (x) ≤   Ω 1 n,λ f (y) pλ(1/p−1/μ) K(x, y) λq(1/q−1/ν) dy  1/λ   Ω 1 n,λ f (y) p dy  1/μ   Ω 1 n,λ K(x, y) q dy  1/ν . (1.11) Sincewewanttohave f p and K q in the integrand above, we note that we can choose λ, μ, ν in such a way 1 λ =  1 p − 1 μ  , 1 λ =  1 q − 1 ν  , 1 λ = 1 r . (1.12) With these choices of λ, μ and ν, we can rewrite expression last inequality, J λ (x) ≤f  p/μ L p (Ω 1 n,λ )   Ω 1 n,λ K(x, y) q dy  1/ν   Ω 1 n,λ f (y) p K(x, y) q dy  1/λ . (1.13) Takin g rth powers and integrating in x,  Ω 2 n,λ   J λ (x)   r dx ≤f  rp/μ L p (Ω 1 n,λ )  Ω 2 n,λ   Ω 1 n,λ K(x, y) q dy  r/ν   Ω 1 n,λ f (y) p K(x, y) q dy  r/λ dx ≤f  rp/μ L p (Ω 1 n,λ ) sup x∈Ω 2 n,λ   Ω 1 n,λ K(x, y) q dy  r/ν  Ω 2 n,λ sup y∈Ω 1 n,λ K(x, y) q   Ω 1 n,λ f (y) p dy  dx ≤f  rp/μ L p (Ω 1 n,λ )  f  p L p (Ω 1 n,λ ) sup y∈Ω 1 n,λ   Ω 2 n,λ K(x, y) q dx  sup x∈Ω 2 n,λ   Ω 1 n,λ K(x, y) q dy  r/ν . (1.14) Hence   J λ (x)   r L r (Ω 2 n,λ ) ≤f  p(r/μ+1) L p (Ω 1 n,λ ) sup y∈Ω 1 n,λ   Ω 2 n,λ K(x, y) q dx  sup x∈Ω 2 n,λ   Ω 1 n,λ K(x, y) q dy  r/ν . (1.15) 4 Journal of Inequalities and Applications Takin g rth roots, we have the following inequality:   J λ (x)   L r (Ω 2 n,λ ) ≤f  p(1/μ+1/r) L p (Ω 1 n,λ ) sup y∈Ω 1 n,λ   Ω 2 n,λ K(x, y) q dx  1/r sup x∈Ω 2 n,λ   Ω 1 n,λ K(x, y) q dy  1/ν ≤f  L p (Ω 1 n,λ ) sup y∈Ω 1 n,λ   Ω 2 n,λ K(x, y) q dx  1/r sup x∈Ω 2 n,λ   Ω 1 n,λ K(x, y) q dy  1/q−1/r ≤f  L p (Ω 1 n,λ ) k 1 k 2 . (1.16)  Theorem 1.2 (Riesz-Therin interpolation theorem, [13]). Suppose T is simultaneously of weak types (p 0 ,q 0 ) and (p 1 ,q 1 ), 1 ≤ p i ,q i ≤∞.If0 <t<1 and 1/p t = (1 − t)/p 0 + t/p 1 , 1/q t = (1 − t)/q 0 + t/q 1 , then T is of type (p t ,q t ),and T (p t ,q t ) ≤T 1−t (p 0 ,q 0 ) T t (p 1 ,q 1 ) . (1.17) The following theorem gives the condition of absolute convergence of the potential I α(x) λ f . Theorem 1.3. Let 0 <m ≤ α(x) <n, f ∈ L 1 (Ω n,λ ). Then the integral (1.7)isabsolutely convergent for almost every x. Proof. Let L y,θ ={x ∈ Ω n,λ : y · x = cosθ}, |L y,θ |=|Ω n−1,λ |sin 2|λ|−1 θ.Hencewehave  Ω n,λ   I α(x) λ f (x)   dx ≤  Ω n,λ   f (y)   |x − y| n−α(x) λ dydx =  Ω n,λ   f (y)    Ω n,λ 1 |x − y| n−α(x) λ dxdy =  Ω n,λ   f (y)     π 0   L y,θ 1 θ (2|λ|/n)(n−α(x)) dL y,θ (x)  dθ  dy =  Ω n,λ   f (y)     1 0 +  π 1   L y,θ 1 θ (2|λ|/n)(n−α(x)) dL y,θ (x)  dθ  dy ≤  Ω n,λ   f (y)     1 0   L y,θ 1 θ (2|λ|/n)(n−m) dL y,θ (x)  dθ +  π 1   L y,θ dL y,θ (x)  dθ  dy ≤  Ω n,λ   f (y)     1 0   Ω n−1,λ   sin 2|λ|−1 θ θ (2|λ|/n)(n−m) dθ +  π 1   Ω n−1,λ   sin 2|λ|−1 θdθ  dy ≤   Ω n−1,λ    Ω n,λ   f (y)     1 0 1 θ 1−(2|λ|/n)m dθ+  π 1 dθ  dy ≤ M f  1 < ∞. (1.18) The proof is completed.  M. Z. Sarikaya and H. Yildirim 5 Theorem 1.4. Let 0 <m ≤ α(x) <n, 1 ≤ p<∞. Then I α(x) λ f is of type (p, p), that is,   I α(x) λ f   p ≤ M f  p , (1.19) where the constant M is depe ndent on λ, m,andn. Proof. Let S θ f (x) = 1   L x,θ    L x,θ f (y)dL x,θ (y). (1.20) Thus we have   S θ f   p ≤ M f  p . (1.21) By the Minkowsky inequality for integrals, we have the following inequality:   Ω n,λ   I α(x) λ f (x)   p dx  1/p =   Ω n,λ      Ω n,λ   f (y)   |x − y| n−α(x) λ dy     p dx  1/p ≤   Ω n,λ      π 0   Ω n−1,λ   sin 2λ|−1 θ θ (2|λ/n)(n−α(x)) 1   L x,θ    L x,θ f (y)dL x,θ dθ     p dx  1/p ≤ M   Ω n,λ      π 0 1 θ 1−(2|λ|/n)α(x)   S θ ( f )   dθ     p dx  1/p ≤ M   1 0 +  π 1    Ω n,λ 1 θ (1−(2|λ|/n)α(x))p   S θ ( f )   p dx  1/p dθ ≤ M  1 0 1 θ 1−(2|λ|/n)α(x)   Ω n,λ   S θ ( f )   p dx  1/p dθ + M  π 1   Ω n,λ   S θ ( f )   p dx  1/p dθ ≤ M  1 0 1 θ 1−(2|λ|/n)α(x)   S θ ( f )   p dθ + M  π 1   S θ ( f )   p dθ ≤ M f  p   1 0 dθ θ 1−(2|λ|/n)α(x) +  π 1 dθ  ≤ M f  p . (1.22) The proof is completed.  The following theorem is an expanded form of Theorem 1.4. Theorem 1.5. Let 0 <m ≤ α(x) <n, 1 <p≤ r, n/p − n/r < m. Then I α(x) λ f is of type (p,r), that is,   I α(x) λ f   r ≤ M f  p , (1.23) where the constant M is depe ndent on λ, m,andn. 6 Journal of Inequalities and Applications Proof. Let q = pr/(pr + p − r), 1/q +1/q  = 1. We show that I α(x) λ f is of type (1,q)and (q  ,∞). By the Minkowsky inequality for integrals, we have the following inequality:   I α(x) λ f   q ≤   Ω n,λ      Ω n,λ   f (y)   |x − y| n−α(x) λ dy     q dx  1/q ≤  Ω n,λ   Ω n,λ   f (y)   q |x − y| (n−α(x))q λ dx  1/q dy =  Ω n,λ   f (y)     Ω n,λ 1 |x − y| (n−α(x))q λ dx  1/q dy ≤  Ω n,λ   f (y)     π 0   L y,θ 1 θ (2|λ|/n)(n−α(x))q dL y,θ (x)  dθ  dy ≤  Ω n,λ   f (y)     1 0   L y,θ 1 θ (2|λ|/n)(n−m)q dL y,θ (x)  dθ +  π 1   Ω n−1,λ   sin 2|λ|−1 θdθ  dy ≤  Ω n,λ   f (y)     1 0   Ω n−1,λ   sin 2|λ|−1 θ θ (2|λ|/n)(n−m)q dθ +M  dy ≤  Ω n,λ   f (y)      Ω n−1,λ    1 0 1 θ (2|λ|/n)(n−m)q−2|λ|+1 dθ +M  dy ≤ M f  1 . (1.24) Thus the last integral is convergence where pr pr + p − r < n n − m for n p − n r <m, q< n n − m =⇒ 2|λ| n (n − m)q +1− 2|λ| < 1. (1.25) This shows that I α(x) λ f is of ty pe (1,q). On the other hand, from H ¨ older’s i nequality, we have   I α(x) λ f   <  Ω n,λ   f (y)   |x − y| n−α(x) λ dy ≤   Ω n,λ   f (y)   q  dy  1/q    Ω n,λ 1 |x − y| (n−α(x))q λ dy  1/q ≤ M f  q  . (1.26) Therefore we have   I α(x) λ f   ∞ ≤ M f  q  . (1.27) This shows that I α(x) λ f is of type (q  ,∞). Let t = q(1−1/p), then from Theorem 1.2, I α(x) λ f is of type (p,r)where1/p=(1−t)/1+ t/q  ,1/r = (1 − t)/q  .Theproofiscompleted.  M. Z. Sarikaya and H. Yildirim 7 Theorem 1.6. Let 0 <m ≤ α(x) <n, 1 <p<r, n/p − n/r = m. Then I α(x) λ f is of type (p,r). Proof. Firstly, for a constant m we will consider the α(x) = m. Thus, by using Lemma 1.1 for K(x, y) =|x − y| m−n λ , we obtain the following inequality:   I m λ f   r <M f  p . (1.28) This shows that I m λ is of (p,r)type. Let Ω n,λ,x =  y ∈ Ω n,λ : |x − y| λ ≥ 1  , Ω n,λ,x = Ω n,λ \Ω n,λ,x . (1.29) Then   I α(x) λ f   ≤  Ω n,λ   f (y)   |x − y| n−α(x) λ dy ≤  Ω n,λ,x   f (y)   dy+  Ω n,λ,x   f (y)   |x − y| n−m λ dy ≤  Ω n,λ,x   f (y)   dy+ I m λ f (x) ≤ M   f  p + I m λ f (x). (1.30) Therefore we have   I α(x) λ f   r ≤   M   f  p + I m λ f (x)   r ≤ M   f  p +   I m λ f (x)   r ≤ M   f  p + M f  p = C f  p . (1.31) Thus I α(x) λ f is of type (p, r). The proof is completed.  References [1] O. V. Besov, V. P. Il’in, and P. I. Lizorkin, “The L p -estimates of a certain class of non-isotropically singular integrals,” Doklady Akademii Nauk SSSR, vol. 169, pp. 1250–1253, 1966. [2] ˙ I. C¸ ınar, “The Hardy-Littlewood-Sobolev inequality for non-isotropic Riesz potentials,” Turkish Journal of Mathematics, vol. 21, no. 2, pp. 153–157, 1997. [3] ˙ I. C¸ ınar and H. Duru, “The Hardy-Littlewood-Sobolev inequality for (β,γ)-distance Riesz po- tentials,” Applied Mathematics and Computation, vol. 153, no. 3, pp. 757–762, 2004. [4] A. 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Mehmet Zeki Sarikaya: Department of Mathematics, Faculty of Science and Arts, Kocatepe University, 03200 Afyon, Turkey Email address: sarikaya@aku.edu.tr H ¨ useyin Yildirim: Department of Mathematics, Faculty of Science and Arts, Kocatepe University, 03200 Afyon, Turkey Email address: hyildir@aku.edu.tr . Corporation Journal of Inequalities and Applications Volume 2007, Article ID 36503, 8 pages doi:10.1155/2007/36503 Research Article On the (p ,q)-Boundedness of Nonisotropic Spherical Riesz Potentials Mehmet. results are the generalization of corresponding results for classical Riesz potentials, given in [11]. The important properties of the nonisotropic Riesz potentials and theirs generalizations were. by Shusen Ding We introduced the concept of nonisotropic spherical Riesz potential operators generated by the λ-distance of variable order on λ-sphere and its (p ,q)-boundedness were investi- gated. Copyright