C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 67, N umb er 2, Pages 188–194 (2018) D O I: 10.1501/C om mua1_ 0000000873 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
POTENTIAL OPERATORS ON CARLESON CURVES IN MORREY SPACES
AHMET EROGLU AND IRADA B. DADASHOVA
Abstract. In this paper we study the potential operator I in the Morrey space Lp; and the spaces BM O de…ned on Carleson curves . We prove that for 0 < < 1, I is bounded from the Morrey space Lp; ( )to Lq; ( )on simple Carleson curves if (and only if in the in…nite simple Carleson curve ) 1=p 1=q = =(1 ), 1 < p < (1 )= , and from the spaces L1; ( )to W Lq; ( )if (and only if in the in…nite case) 1 1q =1 :
1. Introduction
Morrey spaces were introduced by C. B. Morrey [11] in 1938 in connection with certain problems in elliptic partial di¤erential equations and calculus of varia-tions. Later, Morrey spaces found important applications to Navier-Stokes and Schrödinger equations, elliptic problems with discontinuous coe¢ cients, and poten-tial theory.
The main purpose of this paper is to establish the boundedness of potential operator I in Morrey spaces Lp; de…ned on Carleson curves . We prove
Sobolev-Morrey inequalities for the operator I . In particular, we get the analog of the theorem by D.R. Adams [1] regarding the inequality for the Riesz potentials in Morrey spaces de…ned on Carleson curves. We emphasize that in the in…nite case of the derived conditions are necessary and su¢ cient for appropriate inequalities. Note that the results we obtain here the potential operators are valid not only on Carleson curves, but also in a more general context of metric spaces or homogeneous type spaces at least under the condition (B(x; r)) rd (see [4, 5, 8, 12]).
The paper is organized as follows. In Section 2, we present some de…nitions and auxiliary results. In Section 3, we establish the main result of the paper: We prove that for 0 < < 1, I is bounded from the Morrey space Lp; ( ) to
Received by the editors: June 14, 2017, Accepted: July 27, 2017. 2010 Mathematics Subject Classi…cation. 42B20, 42B25, 42B35.
Key words and phrases. Carleson curve, Morrey space, potential operator, Sobolev-Morrey inequalty.
c 2 0 1 8 A n ka ra U n ive rsity. C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rs ity o f A n ka ra -S e rie s A 1 M a t h e m a tic s a n d S t a tis tic s . C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra -S é rie s A 1 M a t h e m a tic s a n d S t a tis t ic s .
Lq; ( ) on simple Carleson curves if (and only if in the in…nite simple Carleson
curves) 1=p 1=q = =(1 ), 1 < p < (1 )= , and from the spaces L1; ( ) to
W Lq; ( ) if (and only if in the in…nite case) 1 1q = 1 :
2. Preliminaries
Let = ft 2 C : t = t(s); 0 s l 1g be a recti…able Jordan curve in the complex plane C with arc-length measure (t) = s; here l = = lengths of : We denote
(t; r) = \ B(t; r); t 2 ; r > 0; where B(t; r) = fz 2 C : jz tj < rg.
A recti…able Jordan curve is called a Carleson curve if the condition (t; r) c0r
holds for all t 2 and r > 0, where the constant c0> 0 does not depend on t and
r. Let Lp( ), 1 p < 1 be the space of measurable functions on with …nite
norm kfkLp( )= Z jf(t)jpd (t) 1=p :
Let 1 p < 1,0 1. We denote by Lp; ( ) the Morrey space as the set of
locally integrable functions f on with the …nite norm kfkLp; ( )= sup
t2 ; r>0
r pkfk
Lp( (t;r)):
Note that Lp;0( ) = Lp( ); and if < 0 or > 1, then Lp; ( ) = ; where is
the set of all functions equivalent to 0 on .
We denote by W Lp; ( ) the weak Morrey space as the set of locally integrable
functions f with …nite norm kfkW Lp; ( )= sup >0 sup r>0; t2 r Z f 2 (t;r): jf ( )j> g d ( ) !1=p : Let f 2 Lloc1 ( ). The maximal operator M and the potential operator I on
are de…ned by Mf(t) = sup t>0j (t; r)j 1 Z (t;r)jf( )jd ( ); and I f(t) = Z f ( )d ( ) jt j1 ; 0 < < 1; respectively.
Maximal operators and potential operators in various spaces de…ned on Carleson curves has been widely studied by many authors (see, for example [2, 3, 6, 7, 8, 9, 10, 12]).
N. Samko [12] studied the boundedness of the maximal operator M de…ned on quasimetric measure spaces, in particular on Carleson curves in Morrey spaces Lp; ( ):
Theorem A. Let be a Carleson curve, 1 < p < 1, 0 < < 1 and 0 < 1. Then M is bounded from Lp; ( ) to Lp; ( ).
V. Kokilashvili and A. Meskhi [9] studied the boundedness of the potential op-erator de…ned on quasimetric measure spaces, in particular on Carleson curves in Morrey spaces and proved the following:
Theorem B.Let be a Carleson curve, 1 < p < q < 1, 0 < < 1, 0 < 1<pq,
1
p = q2 and 1 p
1
q = . Then the operator I is bounded from the spaces Lp; 1( )
to Lq; 2( ).
3. Sobolev-Morrey inequality for potential operator on Carleson curves
In this section we prove Sobolev-Morrey inequalities for the potential operators in Morrey space de…ned on Carleson curves.
Theorem 1. Let be a simple Carleson curve, 0 < < 1, 0 < 1 and 1 p < 1 .
1) If 1 < p < 1 , then the condition 1 p
1
q = 1 is su¢ cient and in the
in…nite case also necessary for the boundedness of I from Lp; ( ) to Lq; ( ).
2) If p = 1, then the condition 1 1q = 1 is su¢ cient and in the in…nite case also necessary for the boundedness of I from L1; ( ) to W Lq; ( ).
Proof. 1) Su¢ ciency. Let be a simple Carleson curve, 0 < < 1, 0 < 1 , f 2 Lp; ( ) and 1 < p < 1 . Then I f(t) = Z (t;r) + Z n (t;r) ! f ( )jt j 1d ( ) A(t; r) + C(t; r): (1) For A(t; r) we have
jA(t; r)j Z (t;r)jf( )jjt j 1d ( ) 1 X j=1 2 jr 1 Z (t;2 j+1r)n (t;2 jr)jf( )jd ( ) 1 X j=1 2 jr 1 (t; 2 j+1r) Mf(t) 2c0r Mf(t) 1 X j=1 2 j :
Hence
jA(t; r)j C1r Mf(t) with C1=
2c0
2 1: (2)
For C(t; r) by the Hölder’s inequality we have jC(t; r)j Z n (t;r)jt j jf( )j pd ( ) !1=p Z n (t;r)jt j (p+ 1)p0d ( ) !1=p0 = J1 J2:
Let < < 1 p. For J1 we get
J1= 1 X j=0 Z (t;2j+1r)n (t;2jr)jf( )j p jt j d ( ) 1=p 2pr p kfk Lp; ( ) 1 X j=0 2( )j 1=p= C2r p kfkLp; ( ); (3) where C2= 2 2 1 1=p . For J2 we obtain J2= 0 @X1 j=1 Z (t;2j+1r)n (t;2jr)jt j (p+ 1)p 0 d ( ) 1 A 1=p0 0 @X1 j=1 2jr (p+ 1)p 0 (t; 2j+1r) 1 A 1=p0 0 @c0 1 X j=1 2jr (p+ 1)p0+1 1 A 1=p0 C3rp+ 1 p; (4) where C3= c 1 p0 0 1 2 1 p .
Then from (3) and (4) we have jC(t; r)j C4r
Q
p + kfk
Lp; ( ); (5)
where C4= C2 C3.
Thus, from (2) and (5) we have
jI f(t)j C1r Mf(t) + C4r
1
q kfk
Minimizing with respect to r; at t = h (Mf(t)) 1kfkLp; ip=(1 ) we arrive at jI f(t)j C5(Mf(t))p=qkfk 1 p=q Lp; ( ); where C5= C1+ C4.
Hence, by Theorem B, we have Z (t;r)jI f(t)j q d ( ) C5kfkqLp;p( ) Z (t;r)(Mf(t)) p d ( ) C5Cp; r kfkqLp;p( )kfkpLp; ( )= C6r kfkqLp; ( ); where C6= C5 Cp; . Therefore I f 2 Lq; ( ) and kI fkLq; ( ) C6kfkLp; ( ):
Necessity. Let be an in…nite simple Carleson curve, 1 < p < 1 and I bounded from Lp; ( ) to Lq; ( ). De…ne fr( ) =: f (r ). Then kfrkLp; ( )= r 1 p sup r1>0; 2 r1 Z (t;rr1) jf( )jpd ( ) !1=p = r 1p kfk Lp; ( ) and I fr(t) = r I f(rt); kI frkLq; ( )= r sup r1>0; t2 r1 Z (t;r1) jI f(rt)jqd (t) !1=q = r 1q sup r1>0; t2 r1 Z (t;rr1) jI f(t)jqd (t) !1=q = r 1q kI fk Lq; ( ):
By the boundedness I from Lp; ( ) to Lq; ( )
kI fkLq; ( ) Cp;q; r +1 q 1 p kfk Lp; ( );
where Cp;q; depends only on p, q and .
If 1p <1q +1 ; then for all f 2 Lp; ( ), we have kI fkLq; = 0 as r ! 0.
Similarly, if 1p > 1q +1 ; then for all f 2 Lp; ( ), we obtain kI fkLq; ( )= 0
as r ! 1
2) Su¢ ciency. Let f 2 L1; ( ). We have
f 2 (t; r) : jI f( )j > 2 g f 2 (t; r) : jA( ; r)j > g + f 2 (t; r) : jC( ; r)j > g :
Taking into account inequality (2) and Theorem A we have f 2 (t; r) : jA( ; r)j > g 2 (t; r) : Mf( ) > C
1r
C7r
r kfkL1; ( );
where C7 = C1 C1; and thus if C4r
1 q kfk L1; ( ) = , then jC( ; r)j and consequently, j f 2 (t; r) : jC( ; r)j > g j = 0: Finally f 2 (t; r) : jI f( )j > 2 g C7r r kfkL1; ( )= C8r kfkL1; ( ) !q ; where C8= C7 C4q 1.
Necessity. Let I bounded from L1; ( ) to W Lq; ( ). We have
kI frkW Lq; = sup >0 sup r1>0; 2 r1 Z f 2 (t;r1) :jI fr( )j> g d ( ) !1=q = r sup >0 r sup r1>0; 2 Z f 2 (t;r1) :jI f (r )j> r g d ( ) !1=q = r 1q sup >0 r sup r1>0; 2 r (r1r) Z f 2 (t;rr1) :jI f ( )j> r g d ( ) !1=q = r 1q kI fk W Lq; :
By the boundedness I from L1; ( ) to W Lq; ( )
kI fkW Lq; C1;q; r
+1q (1 )
kfkL1; ( );
where C1;q; depends only on q and .
If 1 < 1q +1 ; then for all f 2 L1; ( ), we have kI fkW Lq; = 0 as r ! 0.
Similarly, if 1 > 1
q +1 ; then for all f 2 L1; ( ), we obtain kI fkW Lq; = 0
as r ! 1. Therefore 1 = 1q +1 .
References
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Current address : Ahmet Eroglu (Corresponding author): Omer Halisdemir University, De-partment of Mathematics, Nigde, Turkey
E-mail address : [email protected]
ORCID Address: http://orcid.org/0000-0002-3674-6272
Current address : Irada B. Dadashova: Baku State University, Applied Mathematics and Cybernetics, Baku, Azerbaijan
E-mail address : [email protected]
