On the -Boundedness of Nonisotropic Spherical Riesz Potentials

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 Hüseyin 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 investigated. Copyright © 2007 M. Z. Sarikaya and H. Yildirim. This is an open access article distributed 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     Rn = x = x1 ,x2 ,... ,xn : xi ∈ R, 1 ≤ i ≤ n . (1.1) In Rn spaces, L p and L∞ are defined as follows:    L p = L p Ωn,λ = f (x) :  f  p =    Ωn,λ 1/ p p f (x) dx <∞ , 1≤ p<∞ (1.2)  L∞ = L∞ Ωn,λ = f (x) :  f ∞ = esssup f (x) < ∞ , x∈Ωn,λ where Ωn,λ is the n-dimensional unite λ-sphere of Rn which is dependent on the λdistance. The λ-distance between points x = (x1 ,...,xn ) and y = (y1 ,..., yn ) is defined by the following formula given in [1–10]: |x − y |λ := x1 − y 1 1/λ1 + x2 − y 2 1/λ2 + · · · + xn − y n 1/λn |λ|/n , (1.3) 2 Journal of Inequalities and Applications where x, y ∈ Ωn,λ , λ = (λ1 ,λ2 ,...,λn ), λk > 0, k = 1,2,...,n, |λ| = λ1 + λ2 + · · · + λn . Note that this distance has the following properties of homogeneity for any positive t, t λ1 x 1 1/λ1 + · · · + t λn x n 1/λn |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/λmin )|λ|/n (|x|λ + | y |λ ). Here we consider λ-spherical coordinates by the following formulas:  x1 = ρ cosθ1 2λ1  ,...,xn = ρ sinθ1 sinθ2 · · · sinθn−1 2λn . (1.5) We obtained that |x|λ = ρ2|λ|/n . It can be seen that the Jacobian Jλ (ρ,ϕ) of this transformation is Jλ (ρ,θ) = ρ2|λ|−1 Wλ (θ), where Wλ (θ) is the bounded function, which only depends on angles θ1 ,θ2 ,...,θn−1 . It is clear that if λ1 = λ2 = · · · = λn = 1/2, then the λ-distance is the Euclidean distance. We define angle cos |x − y |λ = x · y, (1.6) where x and y are vectors on the n-dimensional unite λ-sphere. For f ∈ L(Ωn,λ ), 0 < α(x) < n, we will consider the following nonisotropic spherical Riesz potential operator generated by the λ-distance of variable order: Iλα(x) f (x) =  α(x)−n Ωn,λ |x − y |λ x ∈ Ωn,λ . f (y)d y, (1.7) The aim of this paper to show that the well-known properties of classical Riesz potentials 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 papers [1–9, 12]. The nonisotropic spherical Riesz potential generated by λ-distance is the classical Riesz potential for λi = 1/2, i = 1,2,...,n and α(x) = α. Here particular importance of the nonisotropic kernel is that it does not have the classical triangle inequality. It is well known that the classical Riesz potentials Iα ϕ = ϕ ∗ |x|α−n are bounded operators from L p (Rn ) to Lq (Rn ) for 1/q = 1/ p − α/n, 0 < α < n, 1 ≤ p < q < ∞ [10].  Lemma 1.1. Let Jλ (x) = Ω1n,λ f (x)K(x, y)d y, x ∈ Ω2n,λ ,  k1 = sup y ∈Ω1n,λ Ω2n,λ  1/r |K(x, y)|q dx < ∞, k2 = sup x∈Ω2n,λ Ω1n,λ |K(x, y)|q d y 1/q−1/r <∞ (1.8) M. Z. Sarikaya and H. Yildirim 3 and the following conditions are carried out: 1 ≤ p ≤ r ≤ ∞, 1 − 1/ p + 1/r = 1/q, f ∈ L p (Ω1n,λ ). Then   Jλ  L p (Ω1n,λ ) ≤   f Lr (Ω2n,λ ) k1 k2 . (1.9) Proof. Let λ, μ, ν be positive numbers such that 1/λ + 1/μ + 1/ν = 1. We write  Jλ (x) = f p(1/ p−1/μ) (x) f p/μ (x)K q(1/q−1/ν) (x, y)K q/ν (x, y)d y. Ω1n,λ (1.10) By Hölder’s inequality with exponents λ, μ, and ν, we obtain  Jλ(x) ≤ f (y) pλ(1/ p−1/μ) K(x, y)λq(1/q−1/ν) d y 1 1/λ 1/μ  f (y) p d y 1 Ωn,λ 1/ν K(x, y)q d y 1 Ωn,λ Ωn,λ . (1.11) Since we want to have f p and K q in the integrand above, we note that we can choose λ, μ, ν in such a way   1 1 1 = − , λ q ν 1 1 1 = − , λ p μ 1 1 = . λ r (1.12) With these choices of λ, μ and ν, we can rewrite expression last inequality,  p/μ 1/ν   K(x, y)q d y 1 Jλ (x) ≤  f L p (Ω1 ) Ωn,λ n,λ 1/λ Ω1n,λ f (y) p K(x, y)q d y . (1.13) Taking rth powers and integrating in x,  Ω2n,λ r Jλ (x) dx r p/μ ≤  f L p (Ω1 ) n,λ   r/ν   q K(x, y) d y 1 Ω2n,λ Ωn,λ  r p/μ ≤  f L p (Ω1 ) sup n,λ r p/μ x∈Ω2n,λ K(x, y)q d y 1 Ωn,λ r/ν  Ω2n,λ y ∈Ω1  p n,λ y ∈Ω1n,λ r/λ dx  sup K(x, y)q n,λ ≤  f L p (Ω1 )  f L p (Ω1 ) sup n,λ Ω1n,λ f (y) p K(x, y)q d y Ω1n,λ f (y) p d y dx  K(x, y)q dx sup 2 Ωn,λ x∈Ω2n,λ K(x, y)q d y 1 r/ν . Ωn,λ (1.14) Hence   Jλ (x)r p(r/μ+1) Lr (Ω2n,λ ) ≤  f L p (Ω1n,λ ) sup y ∈Ω1n,λ   K(x, y)q dx sup 2 Ωn,λ x∈Ω2n,λ K(x, y)q d y 1 Ωn,λ r/ν . (1.15) 4 Journal of Inequalities and Applications Taking rth roots, we have the following inequality:   Jλ (x) Lr (Ω2n,λ ) p(1/μ+1/r) ≤  f L p (Ω1 ) n,λ  K(x, y)q dx 2 sup Ωn,λ y ∈Ω1n,λ  Ω2n,λ y ∈Ω1n,λ  K(x, y) dx 1/ν K(x, y)q d y 1 sup Ωn,λ x∈Ω2n,λ  1/r q ≤  f L p (Ω1n,λ ) sup 1/r 1/q−1/r q sup Ω1n,λ x∈Ω2n,λ (1.16) K(x, y) d y ≤  f L p (Ω1n,λ ) k1 k2 .  Theorem 1.2 (Riesz-Therin interpolation theorem, [13]). Suppose T is simultaneously of weak types (p0 , q0 ) and (p1 , q1 ), 1 ≤ pi , qi ≤ ∞. If 0 < t < 1 and 1/ pt = (1 − t)/ p0 + t/ p1 , 1/qt = (1 − t)/q0 + t/q1 , then T is of type (pt , qt ), and T (pt ,qt ) ≤ T 1(p−0t,q0 ) T t(p1 ,q1 ) . (1.17) The following theorem gives the condition of absolute convergence of the potential Iλα(x) f . Theorem 1.3. Let 0 < m ≤ α(x) < n, f ∈ L1 (Ωn,λ ). Then the integral (1.7) is absolutely convergent for almost every x. Proof. Let L y,θ = {x ∈ Ωn,λ : y · x = cosθ }, |L y,θ | = |Ωn−1,λ | sin2|λ|−1 θ. Hence we have  Ωn,λ Iλα(x) f (x) dx  ≤  =  =  =  ≤ f (y) n−α(x) d y dx Ωn,λ |x − y | λ  Ωn,λ Ωn,λ Ωn,λ Ωn,λ f (y) L y,θ θ 0 dL y,θ (x) dθ d y (2|λ|/n)(n−α(x))   1  π   f (y) + 0 0 f (y) ≤ Ωn−1,λ  Ωn,λ 0  1 L y,θ θ 1  1  f (y)  1 Ωn,λ  1 f (y)  ≤ 1 n−α(x) dx d y Ωn,λ |x − y | λ  π  (2|λ|/n)(n−α(x)) dL y,θ (x) dθ d y  π  1 L y,θ θ dL y,θ (x) dθ + (2|λ|/n)(n−m) π 1 L y,θ  dL y,θ (x) dθ d y  Ωn−1,λ sin2|λ|−1 θ dθ + Ωn−1,λ sin2|λ|−1 θdθ d y θ (2|λ|/n)(n−m) 1 f (y)  1 0 θ 1 dθ+ 1−(2|λ|/n)m π 1  dθ d y ≤ M  f 1 < ∞. (1.18) The proo (...truncated)