Selçuk J. Appl. Math. Selçuk Journal of Vol. 7. No. 1. pp. 17-23, 2005 Applied Mathematics
On The Non-Isotropic Fractional Integrals Generated By The−Distance M. Zeki Sarıkaya And Hüseyin Yıldırım
Department of Mathematics, Faculty of Science and Arts, Kocatepe University,Afyon, Turkey
e-mail: sarikaya@ aku.edu.tr, hyildir@ aku.edu.tr
Received: August 1, 2005
Summary. The main purpose of this paper is to investigate the behaviour of non-isotropic fractional integrals generated by the −distance associated to a measure on a quasimetric space satisfying just a mild growth condition, namely that the measure of each −cube is controlled by a fixed power of its radius. Keywords. Fractional integral, Quasimetric, −Distance.
2000 Mathematics Subject Classification:31B10, 44A15 and 47B37. 1. Introduction
The success of the spaces of homogeneous type as the natural setting for a large portion of Harmonic Analysis, mainly the Calderon-Zygmund theory, led to the firm belief by almost all specialists in the field, that we had achieved the right level of generality to do analysis. A measure of this success is the fact that the setting chosen by E.M. Stein[1] in his book to develop the basic theory is, essentially, that of a space of homogeneous type.
In this article we have defined the non-isotropic fractional integrals associated to the measure generated by the −distance and we see that, for 1 , non-isotropic fractional integral maps () continuously into (), where 1
= 1 −
, with a substitute weak-type result for = 1 This generalizes the classical Hardy-Littlewood-Sobolev theorem. We also observe that condition (1.1) is actually necessary for this Hardy-Littlewood-Sobolev theorem to holds. These results, under similar conditions for classical fractional on the metric space, were also studied by J.G-Cuerva and A.E. Gatto[3].
(X ) will be a quasimetric measure space that is, is a −distance on X and is a Borel measure on X We define the open −cube ( ) with a center and radius as
( ) = { ∈ X : ( ) 2||} ∈ X 0 We have
(1) (( )) ≤ 2||
where || = 1+ + and is a constant independent of and We define a quasimetric (a non-isotropic quasi-distance) in X by
( ) := (|1− 1| 1 1 + |2− 2| 1 2 + + |− | 1 )|| where = (1 2 ) 0 = 1 2 , || = 1+ 2+ + . This quasimetric is named non-isotropic quasi-distance [2]. Note that this distance has the following properties of homogeneity for any positive
( ) = ³¯ ¯1 1− 11 ¯ ¯11 + +¯¯ − ¯ ¯1 ´ || = ||( ) 0
This equality give us that non-izotropic −distance is the order of a homoge-neous function || . So a quasimetric in X has the following properties:
1 ( ) = 0 ⇔ ( ) = 2 ( ) = || || ( ) 3 ( ) ≤ (( ) + ( )) where ∈ X and = 2 1+ 1 min ||
Here we consider −spherical coordinates by the following formulas :
1= ( cos 1)21 2= ( sin 1cos 2)22 = ( sin 1sin2 sin −1)2 We obtained that ||=
2||
It can be seen that the Jacobian ( ) of this
transformation is ( ) = 2||−1Ω() where Ω() is the bounded function, which only depend on angles 1 2 −1 It is clear that if 1 = 2 = = = 12 then quasimetric(the non-isotropic -distance) is the Euclidean distance.
Let 0 . The non-isotropic fractional integral associated to the measure will be defined, for appropriate functions on X as
(2) () = Z X () ( )2||(1− ) ()
Lemma 1. For every 0 (3) Z () () ( )2||(1− ) ≤ 2||
Proof. If ≤ inequality (3) follows immediately from (1). If , we obtain R () () ()2||(1− ) = ∞ P =0 R 2−−1≤()2− () ()2||(1− ) ≤ P∞ =0 1 (2−−1)2||(1−)(( 2−)) ≤ P∞ =0 (2−)2|| (2−−1)2||(1−) = P∞ =0 2−2|| 2|| = 2||
Lemma 2. For every 0 (4) Z X\() () ( )2||(1+ ) ≤ −2|| Proof. We have R X\() () ()2||(1+ ) = ∞ P =0 R 2≤()2+1 () ()2||(1+ ) ≤ P∞ =0 1 (2)2||(1+)(( 2 +1)) ≤ P∞ =0 2−2|| − 2|| = −2||
According to Lemma 1, for fixed the function 7→ 1
()2||(1−
) is locally
integrable with respect to Therefore (2) makes perfectly good sense when is bounded and has bounded support. However, in many cases we will need to define the non-isotropic fractional integral for larger classes of functions. In such case, we will explain in detail how to carry out the corresponding extensions.
Theorem 1. For 1 ≤ and 1 = 1 − , we have (5) ({ ∈ X : | ()| }) ≤ Ã kk() !
Proof. We will adapt the proof given by Stein for R[1] to our paper. We can take ≥ 0 () = R () () ()2||(1− )() + R X\() () ()2||(1− )() = + By Hölder’s inequality we have
|| ≤ Ã R X\() |()|() !1 Ã R X\() 1 ()2||(1− )0() !1 0 ≤ kk() Ã R X\() 1 ()2||(1− )0() !1 0 2 || (1 − )0 = 2 || (1 + ), where = (0− 1) − 0, so that 0 = (1 − 1
0) − 0 Therefore from Lemma 2 we have
|| ≤ kk() ³ −2|| ´1 0 = kk()−2||( 1 − )
which hold even for = 1 We can assume that kk()= 1. Also, for 0
we choose so that −2||(1− )= 2 Then { ∈ X : | ()| } ⊂ n ∈ X : || 2o∪n ∈ X : || 2o By the relation between and , the second of these sets is empty. From Hölder’s inequality and Lemma 1 we have
|| ≤ Ã R () |()| ()2||(1− )() !1 Ã R () () ()2||(1− ) !1 0 ≤ −2||(1− ) Ã R () |()| ()2||(1− )() !1
By applying Tchebichev’s inequality we have ({ ∈ X : | ()| }) ≤ ¡© ∈ X : || 2 ª¢ ≤ −2||0 −R X R () |()| ()2||(1− )()() = −2||0 −R X R () () ()2||(1− )|()| () ≤ −2||0 −2|| = 2||− since = −2||(−1 )
Corollary For 1 ≤ and 1= 1−, we have (6) k k()≤ kk()
This can also be proved by a direct application of Marcinkiewicz’s interpo-lation theorem.
The next result reveals that condition (1) is the minimal requirement one must have in order for (6) or (5) to be valid. Note that we consider only measure without atoms, so that the non-isotropic fractional integral given by (2) is well defined.
Theorem 2. For a measure finite over −cube and not having any atoms, condition (1) is necessary for the Hardly-Littlewood-Sobolev theorem to hold.
Proof. Suppose that (6) holds. Let be a −cube of radius If () = 0 then (1) is trivially true. Let () 6= 0 Then for each ∈ we have
() = R () () ()2||(1− ) ≥ () (2)2||(1−) From (6) we have () 1+ 1 (2)2||(1−) ≤ Ã R () ¯ ¯ ¯ ¯ () !1 ≤ °° ° ° () = () 1 which is equivalent to (7) ()1+ 1 −1 ≤ (2||)1− Since 1 +1 − 1 = 1 −
inequality (7) is condition (1). If we assume (5), then similar argument works.
Now we will associate to our fixed −dimensional measure, a family of non-isotropic fractional kernels and the coressponding non-non-isotropic fractional inte-gral operators depending on -distace.
Definition 1. Let 0 and 0 ≤ 1 A function : XX → C
is said to be a non-isotropic fractional kernel of order and regularity if it satisfies the following two conditions;
(8) |( )| ≤
( )2||(1−
or (9) |( ) − (0 )| ≤ (0 ) 2|| ( ) 2|| (−+)
for ( ) ≥ 2(0 ) The corresponding operator which will be called ” non-isotropic fractional operator generated by −distance”, will be given by
(10) ( )() =
Z
X
( ) ()()
By (8), is well defined for ∈ () 1 ≤ and (6) or (5) are also valid for it, as for . Next we see that non-isotropic fractional integral generated by −distance is an example of non-isotropic fractional integral operator with a kernel having regularity 1
Lemma 3. Let ∈ X be such that 2( ) ≤ ( ) Then, ¯ ¯ ¯ ¯( )1 − − 1 ( )− ¯ ¯ ¯ ¯ ≤ ( )( )−+1 where is a constant independent of and
Proof.Let = ( ) ( ) = and ( ) = Thus 0 − + Now we consider () =1 where ∈ [ ] [or ∈ [ ]] − = 0
Then function () has continuous and continuous derivatives in [ ] [or [ ]] Therefore, from Lagrange Theorem
|() − ()| =¯¯¯0()¯¯¯ | − | ∈ [ ] [or ∈ [ ]] From | − | we have the following inequality
¯ ¯ ¯ ¯ 1 − 1 ¯ ¯ ¯ ¯ = ¯ ¯ ¯ ¯− 1 +1 ¯ ¯ ¯ ¯ | − | ≤ ¯ ¯ ¯ ¯ 1 +1 ¯ ¯ ¯ ¯ If then we have the following inequality
¯ ¯ ¯ ¯ 1 − 1 ¯ ¯ ¯ ¯ ≤ 1 +1 ≤ ( ) ( )−+1
If ∈ ( − ), = − 0 1 then we have the following inequality ¯ ¯ ¯ ¯ 1 − 1 ¯ ¯ ¯ ¯ = 1 ( − )+1 ≤ ( ) ( )−+1 The proof is completed.
Definition 2. Let be a non-isotropic fractional kernel of order and regularity ∈ () and − We define (11) ˜( )() = Z X {( ) − (0 )} ()()
where 0 is some fixed point of X.
We observe that the integral in (11) converges both localy and at ∞ as a consequence of (8), (9) and Hölder’s inequality of course the function just defined depends on the election of 0 But the difference between any two functions obtained in (11) for different elections of 0 is just a constant.
4. References
1. E. M. Stein (1993): Harmonic Analysis. Real-Variable methods, orthogonality, and oscillatory integrals. Princeton, N.J.
2. E. B.Fabes, N.M Riviere (1967): Symbolic calculus of kernels with mixed homo-genety, Proceding of Symposion in Pure Math. P:107-127.
3. J. Garcia-Cuerva , A. E. Gatto (2002): Boundedness properties of fractional integral operators assaciated to non-doubling measures. Prepirint.
