The Restriction and the Continuity Properties of Potentials Depending on l-distance

Turk J Math 30 (2006) , 263 – 275. c TÜBİTAK The Restriction and the Continuity Properties of Potentials Depending on λ-distance M. Zeki Sarıkaya, Hüseyin Yıldırım Abstract In this study we establish theorems on the restriction and continuity of the generalized Riesz potentials with the non-isotropic kernels depending on λ-distance. Key Words: Riesz Potential, Non-Isotropic distance. 1. Introduction It is well known that the classical Riesz Potentials Iα ϕ = ϕ ∗ |x|α−n are bounded operators from Lp (Rn ) to Lq (Rn ) for 1q = 1p − αn , 0 < α < n, 1 ≤ p < q < ∞ [1]. For these potentials, Y. Mizuta showed continuity and restriction properties [2],[3]. In this article we define the non-isotropic generalized Riesz potential generated by λ-distance and study the restriction and continuity properties of these potentials. The generalized Riesz potential generated by λ-distance is the classical Riesz potential for λi = 12 , i = 1, 2, ..., n. Here particular importance of the non-isotropic kernel is that it doesn’t have the classical triangle inequality. 2. Preliminaries The λ-distance between points x = (x1 , ..., xn) and y = (y1 , ..., yn) is defined by the following formula given in [4]: 2000 Mathematics Subject Classification: 31B10, 44A15 and 47B37. 263 SARIKAYA, YILDIRIM 1 1 1 |λ| |x − y|λ := (|x1 − y1 | λ1 + |x2 − y2 | λ2 + ... + |xn − yn | λn ) n , where 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:  1 1 tλ1 x1 λ1 + ... + tλn xn λn  |λ| n |λ| = t n |x|λ , t > 0. This equality gives us that the non-isotropic λ-distance has order of homogeneous function |λ| n . So the non-isotropic λ-distance has the following properties: 1 |x|λ = 0 ⇔ x = θ, θ = (0, 0, ..., 0); 2. tλ x λ = |t| n |x|λ ; |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 cos ϕ1 )2λ1 , ..., xn = (ρ sin ϕ1 sinϕ2 ... sin ϕn−1 )2λn . 2|λ| We obtain that |x|λ = ρ n . It can be seen that the Jacobian Jλ (ρ, ϕ) of this transformation is Jλ (ρ, ϕ) = ρ2|λ|−1 Ωλ (ϕ), where Ωλ (ϕ) is the bounded function, which only depends on angles ϕ1 , ϕ2 , ..., ϕn−1. It is clear that, if λi = 12 , i = 1, ..., n, the λ-distance is Euclidean distance. Now for 0 < α < n, we shall consider the generalized Riesz potential with the nonisotropic kernel depending on λ-distance Z α−n (2.1) Iα,λ f(x) = |x − y|λ f(y)dy, Rn where x ∈ Rn . Equality (2.1) is a well-known classical Riesz potential for λi = 12 , i = 1, ..., n. For a positive r and any x ∈ Rn , we denote the open λ-ball Bλ (x, r) with radius r and a center x as Bλ (x, r) = {y : |y − x|λ < r }. 264 SARIKAYA, YILDIRIM In this article we need the following Theorems given in [3]. Theorem 2.1 (Young’s inequality): Let 1 ≤ p, q ≤ ∞ and 1r = 1p + 1q − 1 ≥ 0. If f ∈ Lp (Rn ) and g ∈ Lq (Rn ), then kf ∗ gkr ≤ kfkp kgkq . Theorem 2.2 (Hardy’s inequalities): If f is a nonnegative measurable function on R+ and r > 0, then  ∞ x R R 0 p f(y)dy x −r−1  p1 dx ≤ 0 p r ∞ R p −r−1 p r−1 [yf(y)] y  p1 dy 0 and  ∞ ∞ R R 0 p f(y)dy  1p x r−1 dx ≤ x p r ∞ R [yf(y)] y  1p dy . 0 There are various ways of proving restriction and continuity of classical Riesz potentials [3]. In this paper we study the restriction and continuity properties of generalized Riesz potentials with the non-isotropic kernel depending on λ-distance for functions in Lp . 3. Restriction properties Our main aim is to give a proof of restriction of Iα,λ . |λ| Theorem 3.1 Let 0 < nλ1 (α − 1) < 1p . Then    Z Z Rn−1 |x0 −y 0 |λ <1  1p |Iα,λ f(0, x0 ) − Iα,λ f(0, y0 )| |λ| p n−2−( nλ (α−1)+1)p 1 |x0 − y0 |λ  dx0 dy0  ≤ M kfkp where x ∈ Rn and x = (x1 , ..., xn) = (x1 , x0), x0 = (x2 , ..., xn). In order to prove the Theorem 3.1, we need the following Lemmas. 265 SARIKAYA, YILDIRIM Lemma 3.1 Let 0 < α < n. Then there is the following inequality. α−n |x − y| λ α−n − |y − z|λ α−n−1 ≤ M r |x − y|λ where y ∈ Rn − Bλ (x, 2r) and M is a constant independent of x and y. Proof. Let r = |x − z|λ , |x − y|λ = a, |y − z|λ = b and a 6= 0, b 6= 0. Thus 0 < a − r < b < a + r. Now we consider f(t) = t1β , where t ∈ [a, b] (or t ∈ [b, a]) , n − α = β > 0. Then function f(t) is continuous and continuously differentiable in [a, b] (or [b, a]) . Therefore, there is the following equality from Lagrange Theorem, that 0 |f(b) − f(a)| = f (ξ) |b − a| ξ ∈ [a, b] [or ξ ∈ [b, a]] . Here, |b − a| < r we have the inequality 1 1 1 1 − β = −β β+1 |b − a| ≤ β β+1 r. bβ a ξ ξ If a < ξ < b, then we have 1 1 1 α−n−1 − β ≤ β β+1 r ≤ M r |x − y|λ . bβ a a If b < ξ < a, ξ ∈ (a − r, a), ξ = a − θr, 0 < θ < 1, then we have 1 1 1 α−n−1 − β =β r ≤ M r |x − y|λ . bβ a (a − θr)β+1 2 The proof is completed. |λ| α < 1, then Lemma 3.2 If nλ 1 Z |(z1 , z 0 )|λ α−n |λ| dz 0 ≤ M |z1 | nλ1 α−1 . The proof of this Lemma can be easily seen with change of variable λ2 λ λn λ z2 = t2 z1 1 , ..., zn = tn z1 1 and using λ-spherical coordinates in the integral. 266 SARIKAYA, YILDIRIM Lemma 3.3 If Z |λ| (α − 1) < 1, then nλ1 |(z1 , x0 + h0 )|λ α−n − |(z1 , x0 )|λ α−n |λ| dx0 ≤ M |h0 |λ |z1 | nλ1 (α−1)−1 . (3.2) |(z1 , x0 )|λ dx0 . {x0 : |x0 |λ >2|z1 |} Proof. From Lemma 3.1 we have the inequality Z |(z1 , x0 + h0 )|λ α−n −(z| 1 , x0 )|λ α−n Z dx0 ≤ |h0 |λ {x0 : |x0 |λ >2|z1 |} α−n−1 {x0 : |x0 |λ >2|z1 |} 2 Thus from Lemma 3.2 we obtain (3.2). Proof of Theorem 3.1 We will adapt to our paper the proof given by Mizuta [3] for the classical Riesz potential. Note that Iα,λ f(0, x0 ) = Z Z |(−z1 , x0 − z 0 )|λ α−n f(z1 , z 0 )dz1 dz 0 R1 Rn−1 and |Iα,λ f(0, x0 + h0 ) − Iα,λ f(0, x0 )| ! R R 0 0 0 α−n 0 0 α−n 0 0 ≤ |(z1 , x + h − z )|λ − |(−z1 , x − z )|λ |f(z1 , z )| dz dz1 . R1 Rn−1 Hence by Young’s inequality we have the inequality 0 kIα,λ f(0, . + h ) − Iα,λ f(0, .)kp ≤ R R R1 Rn−1 × R Rn−1 ! 0 0 α−n α−n |(−z1 , x + h )|λ − |(−z1 , x0 )|λ dx0 ! p1 0 p |f(z1 , z )| dz 0 dz1 . 267 SARIKAYA, YILDIRIM In case α < 1, and in view of Lemma 2.2 and Lemma 2.3, we have kIα,λ f(0, . + h0 ) − Iα,λ f(0, .)kp R |λ| |h0 |λ |z1 | nλ1 ≤ M ≤ M |h0 |λ R1 +M R |λ| |z1 |<|h0 |λ R (α−1)−1 |z1 | nλ1 kf(z1 , z 0 )kp dz1 (α−1)−1 kf(z1 , z 0 )kp dz1 |λ| |z1 | nλ1 (α−1) kf(z1 , z 0 )kp dz1 |z1 |≥|h0 |λ M [I1 (h0 ) + I2 (h0 )]. = Passing to the λ-spherical coordinates, we obtain R [I1 (h0 )]P |λ| (n−2+( (α−1)+1)p) nλ1 Rn−1 |h0 | λ dh0 R = |λ| (2−n−( nλ (α−1)+1)p) Rn−1 |h0 |λ " × M |h0 |λ 1 #p R |λ| nλ1 (α−1)−1 |z1 | |z1 |<|h0 |λ R∞ 2|λ0 | (2−n− |λ| (α−1)p)−1 n−1 nλ kf(z1 , z 0 )kp dz1 1 = M r 0  2|λ| p r Rn |λ| (α−1)−1 ×  |z1 | nλ1 kf(z1 , z 0 )k dz1  dr. p 0 2|λ0 | Here for u = r n−1 , we have Z∞ u =M |λ| 2− nλ (α−1)p 1 0  u p Z |λ|  |z1 | nλ1 (α−1)−1 kf(z1 , z 0 )k dz1  du. p 0 By Hardy (...truncated)