Selçuk J. Appl. Math. Selçuk Journal of
Vol. 5. No. 1. pp. 49-57, 2004 Applied Mathematics
Local Estimation of Riesz-type Potential Operator on Weighted Spaces of Summation Functions
Hüseyin Y¬ld¬r¬m and Umut Mutlu Özkan
Department of Mathematics, Faculty of Science and Arts, Kocatepe University, Ahmet N. Sezer Campus, Afyon-Turkey;
e-mail: hyildir@aku.edu.tr; umut_ozkan@aku.edu.tr
Received: January 11, 2004
Summary. In this work, after de…ning and obtaining the fractional integrals generated by the Riesz-type potentials, their local estimations are stated. Key words:Fractional Integral, Riesz-type Potential, Maximal Function, BMO Spaces, Estimations.
1991 Mathematics Subject Classi…cation: Primary 42B20, 42B25, 46E30
1. Introduction
The classical Riesz Potentials for local estimations are studied by N.K. Kara-petyants and A.I. Ginsburg [1]. Moreover, it is well known that the fractional integral I+' = t+ ' provides the following conditions [2].
a) From Lp(0; 1) to Lq(0; 1), 1 q q where q = p(1 p) 1 for p < 1.
b) From Lp(0; 1) to BM O(0; 1), for p = 1.
c) From Lp(0; 1) to H
1
p(0; 1) for p > 1.
On the solution of integral equations of the …rst type I+' = f it will be clear that, it is necessary to have more de…nite information on the image of operator I+. For this reason, information on this local characteristic is introduced [3]:
(1) (f; I) = 0 @ 1 Z jf(x)jr dx 1 A 1 r ; I (0; 1); 1 r 1
and its asymptotic character is considered as jIj ! 0. These estimations give necessary conditions for the representation of the function f by fractional in-tegral. The local characteristics (1) are average of the absolute value of the function f of r th order. At the same time, these characteristics are related to the maximal function of Hardy-Littlewood. These local characteristics on the other hand for fractional integrals and Riesz potentials were studied in [1-3].
In this paper, we will introduce and study similar characteristics on the weighted space Lp(0; 1) with weight (x) = (1 x) , > 0 which is
generaliza-tion of the results given in [4].
We will show the operators of Riemann-Liouville fractional integrals by I+, which is (I+')(x) = x Z 0 '(t)(x t) 1dt; 0 < x < 1:
Furthermore, let 0 < < 1 , 1 p 1 , p0 = p(p 1) 1and p = p
p p, for p < 1. Then (2) (r;0)(f; ") = 0 @1 " " Z 0 jf(x)jrdx 1 A 1 r and (3) (r;a)(f; ") = 0 @1 2" a+" Z a " jf(x)jrdx 1 A 1 r :
Moreover, in the last case we assume that a " 2 (0; 1). For example a > 2" Note that by the aid of this characteristics, it is possible to give interesting characteristics of the space L1(0; 1).
Lemma: Let f 2 Lloc1 (0; 1). If one of the following expresssion is bounded then the others are automatically bounded and there are equations among them that is norm: kfk1= ess sup x2(0;1) jf(x)j , kfk11= sup I Z I jf(x)j dx and
kfkq1= 0 @sup I Z I jf(x)j q dx 1 A 1 q ; 1 q < 1. Theorem 1. Let f = I+', 0 < < 1, ' 2 Lp(0; 1), 1 p 1, (x) = (1 x) ; > 0, p0< 1. Then, (4) (r;0)( f ; ") = Cp" 1 p 0 @ " Z 0 j 'jpdx 1 A 1 p where, if 1 < p < 1, 1 r p ; if 1 < p 1, 1 r 1; if p = 1, 1 r < p ; if p = 1, 1 r < 1. According to these conditions,
(5) (r;a)( f ; ") = CpWp; (") k 'kp where, W ;p( ) = 8 > > > < > > > : " 1p f or 1 < p < 1; 1 r p " 1 f or p = 1 1 r < p (1 + jln "j)pp1 f or p = 1 1 r < 1 1 f or 1 < p 1 1 r 1
Proof. We will prove the theorem by several steps for > 0. Case 1: a = 0: In this case, we have
(r;0)( f; ") = 0 @1 " " Z 0 j( f)(x)jrdx 1 A 1 r = 0 @1 " " Z 0 (1 x) x Z 0 '(y)(x y) 1dy r dx 1 A 1 r 0 @1 " " Z Zx 1 x 1 y 1 (y)(x y) 1dy r dx 1 A 1 r
+ 0 @1 " Z 0 x Z 0 (y)(x y) 1dy r dx 1 A 1 r (6) = (r;0) I+ 1 x 1 y 1 ; " + (r;0)(I+ ; ") where, (y) = (1 y) '(y) 2 Lp(0; 1):If 1 p
1
, 1 r < p and by changing the variables x" for x and y" for y in the above stated expression,
A( ; ") = (r;0) I+ 1 x
1 y 1 ; " : Then, we can also obtain
Ar( ; ") =
1
Z
0
jI(x; ")jrdx:
Dividing the set of integration in the interior integral I(x; ") to interval (xak+1; xak), where, ak = 2
k
, k = 0; 1; 2; :::, we represent I(x; ") in the form
I(x; ") = " x Z xa1 ("y) 1 "x 1 "y 1 (x y) 1dy +" 1 X k=1 xak Z xak+1 ("y) 1 "x 1 "y 1 (x y) 1dy = I1+ I2:
Note that in the integral I1,
1 "x
1 "y 1 1: Therefore we have jI1j "
x
Z
xa1
j ("y)j (x y) 1dy
where, we are applying Hölder’s inequality for the factors j ("y)jpr (x y)s 1r
and j ("y)j1 pr (x y)s p01 with degree p
1= r, p2= r p r p, p3= p0. Moreover, 2s = 1 r 1 p .
jI1j C:" ( 1 p 1 r)k k1 p r p 0 B @ x Z xa1 j ("y)jp(x y)rs 1dy 1 C A 1 r : Then, we obtain A1( ; ") 0 @ 1 Z 0 jI1( ; ")jrdx 1 A 1 r C" (1p 1r)k k1 p r p 0 @ x Z xa1 j ("y)jpdy 1 Z 0 (x y)rs 1dx 1 A 1 r C" (p1 1r)k k p:
Now, let us consider the estimation for I2. By p0 < 1, we have
jI2j C" x 1 1 X k=1 (2k+1 1) xak Z xak+1 j ("y)j dy C" x 1k kp 1 X k=1 (2k+1 1) 0 B @ xak Z xak+1 dy 1 C A 1 p0 C" 1px 1 pk k p 1 X k=1 (2k+1 1)2 kp0 C" p1x 1 pk k p:
Therefore, we write the following inequality A2( ; ") C"
1
pk k
p for r < p :
In the (r;0)(f; ") (p;0)(f; "); r p inequality, it is seen that 1 r < p < 1. In the case 1 r 1 and 1 < p 1 we have
(7) I+ 1 x 1 y 1 ; " Cx 1 p 0 @ x Z 0 j (y)jpdy 1 A 1 p :
Our obtained estimations are similar to estimations which are obtained for
(r;0)( f; ") . Finally, for r = p and 1 < p <
1
we can show that I+; is a bounded operator from Lp(0; 1) to Lp(0; 1) by applying Hardy-Littlewood
theorem [2]. Since 1 p = 1 r , we have 1 p = 1 p p = 1 p :
Case 2: a > 0: Firstly consider the cases 1 r 1 , 1 < p 1 . Using estimation (7), we …nd (by using p > 1 )
(r;a)( f; ") C k kp 0 @1 2" a+" Z a " x( 1p)rdx 1 A 1 r and a > 2"; x 2 (0; 1) we have, (r;a)( f; ") C k kp:
Thus, we have he following inequality
(r;a) h I+[(1 x 1 y) 1] ; " i 1 2" a+"R a " x R a " [(1 x 1 y) 1] (y)(x y) 1dy r dx !1 r + 0 @1 2" a+" Z a " a " Z 0 1 x 1 y 1 (y)(x y) 1dy r dx 1 A 1 r (8) = I1+ I2:
If we change the variables = x a + "; = y a + " in the integral I1, then
we obtain I1= 0 B @2"1 2" Z 0 Z 0 1 ( + a ") 1 ( + a ") 1 ( ) 1 ( + a ") d r d 1 C A 1 r (r;a) I+ 1 (x + a ") 1 (y + a ") v 1 ( + a " ; ) ; 2" C" 1pk k p
In the case of a = 0, we have showen that above estimations are similar. For estimation I2, we use the following inequality
I2 0 B @2"1 a+" Z a " a " 2 Z 0 1 x 1 y v 1 (y) (x y) 1dy r dx 1 C A 1 r + 0 B @2"1 a+" Z a " a " Z a " 2 1 x 1 y v 1 (y) (x y) 1dy r dx 1 C A 1 r = I20+ I200: for I20, we have y < 1 x 2 and therefore 1 x y 2 1 x: Then, I20 C k kp 1 2" (1 a + ") (v+ 1)r+1 (1 a ")(v+ 1)r+1 1 r (1 a + ") v+p01 C k kp(1 a + ") v+ 1 (1 a + ") v+p01 C k kp" 1 p where, we used " < a 2, 1 < p < 1 and 1 r 1.
For estimation I200 we use
1 x
1 y C for 2" < a. Then, we have
I20 C 0 B @2"1 a+" Z a " 0 B @ a " Z a " 2 (x y) 1j (y)jdy 1 C A r dx 1 C A 1 r C" 1pk k p:
Finally, consider the case p = 1. The estimations for I1 in (8) can be
realized in the similar way as mentioned above. For estimation I2, consider I20
and I200. Moreover, it is easy to see that I0
2 C k kp. Then, we can …nd, I20 C 0 B @2"1 a+" Z a "Z (x y) p01j (y)jdy r dx 1 C A 1 r
C k kp 0 B B @ 1 2" a+" Z a " dx 0 B @ a " Z a " 2 1 x ydy 1 C A r p01 C C A 1 r k kp 0 @1 2" a+" Z a " ln x a " 2 ln (x (a ")) r p0 dx 1 A 1 r C k kp ln a " 2" 1 p0 where, ln x a " 2 ln (x (a ")) ln a " 2" and 2" < a . Then, we have I2 Ck kp(1 + jln "j) 1 p0 :
This completes the proof for the case v > 0. Case 3: v < 0 The case is easy, since
(r;0)( f; ") 0 @1 2" " Z 0 0 @ x Z 0 j (y)j (x y)1 dy 1 A r dx 1 A 1=r (r;0)(j j; ") : By analogy (r;a)( f; ") (r;a)(j j; ")
and it is possible to use “without weight” estimations.
Case 4: As …nal remarks, …rst of all, note that the constant cris independent
of a in the Theorem 1 and for p = 1, it the estimation cr= O r
1
p0 as r ! 1: holds
Moreover, the estimation of Theorem 1 is the best possible in the scale of loga-ritmic degree.
Theorem 2. Under conditions of Theorem 1 for 1 p < 1 it’s true that
(r;0)( f ; ") = o "
1
p ; " ! 1:
By analogy in the case 1 p < 1, we have
We take the opportunity to express deep thanks to Prof. Dr. N. K. Karapetyans for their support’s in the paper.
References
1. Karapetyans, N. K., Ginsburg, A. I. (1995): Local estimates for Riesz potentials on the ball, T. J. of Mat., 19, 19-29.
2. Samko, S. G., Kilbas A. A., Marichev, O. A. (1993): Fractional integrals and derivatives, Amsterdam.
3. Karapetyans, N. K., Rubin, V. S. (1986): The local properties of fractional integrals and the spaces BMO on a segmet real axis, preprint, Rostov, Gos. Univ., Rostov-on-Don, 1985 Manuscript No: 869-B, deposited at VINITI, (Russian).
4. Karapetyans, N. K., Ginsburg, A. I. (1996): Local estimates for fractional integrals in the weighted space of summable functions, Rostov-on-Don, DGTU, 73-79.
