The Boundedness of Fractional Integral Operators in Local and Global Mixed Morrey-type Spaces
aa r X i v : . [ m a t h . F A ] F e b The Boundedness of Fractional Integral Operators inLocal and Global Mixed Morrey-type Spaces ∗ Houkun Zhang, Jiang Zhou † College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046People’s Republic of China
Abstract:
In this paper, we introduced the local and global mixed Morrey-typespaces, and some properties of these spaces are also studied. After that, the necessaryconditions of the boundedness of fractional integral operators I α are studied respectivelyin mixed-norm Lebesgue spaces and the local mixed Morrey-type spaces. Last but notleast, the boundedness of I α is obtained by Hardy operators’ boundedness in weightedLebesgue spaces. As corollaries, we acquire the boundedness of I α in mixed Morreyspaces(Corollary 5.6) and mixed Lebesgue spaces(Corollary 5.7) respectively. Particu-larly, Corollary 5.7 obtains a sufficient and necessary condition. Keywords:
Local and global mixed Morrey-type spaces; Fractional integral operators;Hardy operators If E is a nonempty measurable subset on R n and f is a measurable function on E ,then we put k f k L p ( E ) := k f χ E k L p ( R n ) = (cid:18)Z R n | f ( y ) | p χ E dy (cid:19) p . For x ∈ R n and r >
0, let Q ( x, r ) denote the open cube centered at x of length 2 r and Q ( x, r ) ∁ denote the set R n \ Q ( x, r ) . Let ~p = ( p , p , · · · , p n ) , ~q = ( q , q , · · · , q n ) , ~s = ( s , s , · · · , s n ) · · · are n-tuples and0 < p i , q i , s i < ∞ , i = 1 , , · · · , n . We define that ~p = ( p , p , · · · , p n ) and ~p < ~q means p i < q i holds for each i . In particular, ~p < r means p i < r hold for each i . ∗ The research was supported by the National Natural Science Fundation of China(12061069). † Corresponding author E-mail address: [email protected]. f ∈ L loc ( R n ). The fractional integral operators I α were defined by I α f ( x ) = Z R n f ( y ) | x − y | n − α dy, < α < n. These operators I α play an essential role in real and harmonic analysis [1, 2].In partial differential equations, Morrey spaces M p,λ are widely used to investigate thelocal behavior of solutions to elliptic and parabolic differential equations. In 1938, Morreyintroduced these spaces [3] and they were defined as following: For 0 ≤ λ ≤ n, ≤ p ≤ ∞ ,we say that f ∈ M p,λ if f ∈ L locp ( R n ) and k f k M p,λ = k f k M p,λ ( R n ) = sup x ∈ R n , r> r − λp k f k L p ( Q ( x,r )) < ∞ . It is obvious that if λ = 0 then M p, = L p ( R n ); if λ = n , then M p,n = L ∞ ( R n ); if λ < λ > n , then M p,λ = Θ, where Θ is the set of all functions almost everywhere equivalentto 0 on R n .For nearly two decades, due to the more precise structure of mixed-norm functionspaces than the corresponding classical function spaces, the mixed-norm function spacesare widely used in the partial differential equations [20–23].In 1961, the mixed Lebesgue spaces L ~p ( R n ) were studied by Benedek and Panzone[15]. These spaces were natural generalizations of the classical Lebesgue spaces L p . Thedefinition was stated as following: Let f is a measurable function on R n and 0 < ~p ≤ ∞ ;we say that f belongs to the mixed Lebesgue spaces L ~p ( R n ), if the number obtainedafter taking successively the p -norm for x , the p -norm for x , · · · , the p n -norm for x n , and in that order, is finite; the number obtained will be denoted by k f k L ~p ( R n ) or k · · · k k f k L p ( R ) k L p ( R ) · · · k L pn ( R ) .When ~p < ∞ , we can write k f k L ~p ( R n ) as following, k f k L ~p ( R n ) = Z R · · · (cid:18)Z R | f ( x ) | p dx (cid:19) p p · · · dx n ! pn < ∞ . If p = p = · · · = p n = p , then L ~p ( R n ) are reduced to classical Lebesgue spaces L p and k f k L ~p ( R n ) = (cid:18)Z R n | f ( x ) | p dx (cid:19) p . Particularly, if p = p = · · · = p n = ∞ , then the result is similar: L ~p = L ∞ .After that, a host of function spaces with mixed norm were introduced, such as mixed-norm Lorentz spaces [24], mixed-norm Lorentz-Marcinkiewicz spaces [27], mixed-norm Or-licz spaces [25], anisotropic mixed-norm Hardy spaces [26], mixed-norm Triebel-Lizorkinspaces [28] and weak mixed-norm Lebesgue spaces [29]. More information can be foundin [33]. 2oreover, combining mixed Lebesgue spaces and Morrey spaces, Nogayama, in 2019,introduced mixed Morrey spaces [4, 5] and these spaces were defined as following: let ~q = ( q , q , · · · , q n ) ∈ (0 , ∞ ] n and p ∈ (0 , ∞ ] satisfy n X j =1 q j ≥ np ;the mixed Morrey spaces M p~q ( R n ) were defined to be the set of all measurable functions f such that their quasi-norms k f k M p~q ( R n ) := sup (cid:26) | Q | p − n ( P nj =1 1 qj ) k f χ Q k L ~q ( R n ) : Q is a cube in R n (cid:27) are finite. It is obvious that if q = q = · · · = q n = q , then M p~q = M pq = M q,λ and if p = n P nj =1 1 q j , then M p~q = L ~q . In particular, before these definitions of mixed Morrey spaces, another definitions weregiven by [30] in 2017. Their norms were defined by taking successively the M p q -norm for x , the M p q -norm for x . It is easy to prove that k f k M p~q ( R ) ≤ (cid:13)(cid:13)(cid:13) k f k M p q ( R ) (cid:13)(cid:13)(cid:13) M p q ( R ) , if p = P i =1 1 p i .In 1981, Adams introduced a variant of Morrey-typle spaces M pθ,λ [19]. In 2004, Bu-renkov and Guliyev developed these spaces and defined local Morrey-type spaces LM pθ,ω and global Morrey-type spaces GM pθ,ω with their some properties were studied [6]. Theyare defined as follows: let 0 < p, θ ≤ ∞ and let ω be a non-negative measurable functionon (0 , ∞ ). We say f belong to the local Morrey-type spaces LM pθ,ω , if the quasi-norms k f k LM pθ,ω = k f k LM pθ,ω ( R n ) = k ω ( r ) k f k L p ( Q (0 ,r )) k L θ (0 , ∞ ) . are finite and f belong to the global Morrey-type spaces GM pθ,ω , if the quasi-norms k f k GM pθ,ω = sup x ∈ R n k f ( x + · ) k LM pθ,ω . are finite.Note that GM pθ,r − λ/p = M pθ,λ , ≤ λ ≤ n and GM p ∞ ,r − λ/p = M p,λ , ≤ λ ≤ n .This paper will give the definitions of local and global mixed Morrey-type spaces andtheir some properties.In 1955, Plessis studied the boundedness of fractional integral operators [7] in classi-cal Lebesgue spaces. Adams in 1975 investigated boundedness of the fractional integraloperators in Morrey spaces [8]. If ω , which is a positive measurable function defined on30 , ∞ ), replace the power function r − λ/p in the definitions of M p,λ , then these become theMorrey-type spaces M p,ω . Mizuhara [9], Nakai [10], and Guliyev [11] generalized Adams’result respectively. The boundedness of fractional integral operators in local Morrey-typespaces was also studied in [12–14].In [15], Benedek and Panzone proved the boundedness of fractional integral oper-ators in mixed lebesgue spaces and it was stated as follows: Let X = E n , and ~α =( α , α , · · · , α n ) an n-tuple of real numbers, 0 < α i <
1. If ~p and ~q are such that ~p − ~q = ~α, < ~p < ~α , then k I β f k L ~q ≤ C k f k L ~p holds for every f ∈ L ~p , where β = P ni =1 α i and C = C ( ~α, ~p ).In 1974, Adams and David studied the boundedness of the fractional integral operators I α in the mixed lebesgue spaces L ~p ( R ) [31]. Their results are stated as the following: Let ~p = ( p , p ) , ~q = ( q , q ) , ≤ p ≤ q ≤ ∞ , < p < q < ∞ , and α = P i =1 1 p i − P i =1 1 q i ,we have k I α f k L ~q ≤ C k f k L ~p ;let ~p = ( p , p ) , ~q = ( q , q ) , < p < q < ∞ , < p = q < ∞ , and α = p − q , then k I α f k L ~q ≤ C k f k L ~p . It is unlucky that we fail to prove that the second result is also right when n > p , q are replaced by p ′ = ( p l +1 , · · · , p n ) , q ′ = ( q l +1 , · · · , q n )( n − l ≥
2) respectively because ofthe application of Fubini’s theorem.In [4], Toru Nogayama proved the boundedness of fractional integral operators inmixed Morrey spaces and it was stated as following: Let 0 < α < n, < ~q, ~s < ∞ and0 < p, r < ∞ ; assume that np ≤ P nj =1 1 q j and nr ≤ P nj =1 1 s j ; if1 r = 1 p − αn , ~qp = ~sr ;then, for f ∈ M p~q ( R n ), k I α f k M r~s ( R n ) ≤ C k f k M p~q ( R n ) . It is obvious that this result can not be reduced to the result of [4], even if p = n P nj =1 1 q j , r = n P nj =1 1 s j .Let ~p = ( p , p , · · · , p n ) , ~p = ( p , p , · · · , p n ), and 0 < p i , p i < ∞ , i − , , · · · , n . Moreover, we define that f ∈ L loc~p means that f χ E ∈ L ~p , if E is compactset. If E is a nonempty measurable subset on R n and f is a measurable function on E ,then we put k f k L ~p ( E ) := k f χ E k L ~p ( R n ) . By A . B ( B & A ), we denote that A ≤ CB where C > A ∼ B means that A . B and A & B . We define that t + = t if t > t + = 0 if t <
0. 4he paper is organized as follows. We start with defining the local and global mixedMorrey-type spaces, and some properties of these spaces are also studied in Section 2. InSection 3, We will give the necessary conditions of the boundedness of fractional integraloperators I α in mixed-norm Lebesgue spaces and the local mixed Morrey-type spaces. Inorder to acquire relationship between fractional integral operators I α and Hardy operators H , L ~p -estimates of I α over the cube Q ( x, r ) is investigated in the Section 4. In the lastsection, according to results of Section 4, the inequality k I α f k LM ~p θ ,ω . k Hg ~p k Lθ ,ν (0 ,, ∞ ) is proved. As corollaries, we acquire the boundedness of fractional integral operators inglobal mixed Morrey-type spaces, the boundedness of fractional maximal operators in localand global mixed Morrey-type spaces, and the other two new results of the boundednessof fractional integral operators in mixed Lebesgue spaces and mixed Morrey spaces. In this section, we will give the definitions of local and global mixed Morrey-typespaces.From [15], we know H¨older’s inequality and Minkowski’s inequality in mixed Lebesguespaces. In [4], Fatou’s property for L ~p was also given. Lemma 2.1 (H¨older’s inequality for L ~p ). Let 1 ≤ ~p ≤ ∞ and ~p + ~p ′ = 1. Then forany f ∈ L ~p and g ∈ L ~p ′ , Z R n f ( x ) g ( x ) dx ≤ k f k L ~p k g k L ~p ′ holds. Remark 2.2.
In particular, if 0 < ~r, ~p, ~q < ∞ , ~p + ~q = ~r , f ∈ L ~p and g ∈ L ~q , then k f g k L ~r ≤ k f k L ~p k g k L ~q also holds.They are easily proved by successive applications of classical H¨older’s inequality. Lemma 2.3 (Minkowski’s inequality for L ~p ). Let 1 ≤ ~p ≤ ∞ . If f, g ∈ L ~p , then k f + g k L ~p ≤ k f k L ~p + k g k L ~p . It is easy to prove it by successive applications of classical Minkowski’s inequality.
Lemma 2.4 (Fatou’s property for L ~p ). Let 0 < ~p ≤ ∞ . Let { f i } ∞ i =1 be a sequence ofnon-negative measurable functions on R n . Then (cid:13)(cid:13)(cid:13) lim inf i →∞ f i (cid:13)(cid:13)(cid:13) ≤ lim inf i →∞ k f i k . Remark 2.5.
By classical Fatou’s property, it also can be proved thatlim sup i →∞ k f i k ≤ (cid:13)(cid:13)(cid:13)(cid:13) lim sup i →∞ f i (cid:13)(cid:13)(cid:13)(cid:13) .
5e state the Lebesgue differential theorem in the setting of mixed-norm Lebesguespaces as the following lemma.
Lemma 2.6.
Let f ∈ L loc~p and 0 < ~p < ∞ , thenlim r → k χ Q ( x,r ) k − L ~p ( R n ) k f k L ~p ( Q ( x,r )) = | f ( x ) | a.e. x ∈ R n . Proof:
Let ~p = ( p , p , · · · , p n ). Without less of generality, we assume n = 2.It is easy to know that k χ Q ( x,r ) k L ~p = | r | P i =1 1 pi , where x = ( x , x ) . Therefore, by the classical Lebesgue differential theorem, we get thatfor any ǫ >
0, there exist δ >
0, such that r < δ and k χ Q ( x,r ) k − L ~p ( R n ) k f k L ~p ( Q ( x,r )) = | r | − P i =1 1 pi (cid:18) Z R (cid:16) Z R | f ( y , y ) | p χ Q ( x,r ) ( y , y ) dy (cid:17) p p dy (cid:19) p = (cid:18) r Z I (cid:16) r Z I | f ( y , y ) | p dy (cid:17) p p dy (cid:19) p ≤ (cid:16) ( | f ( x , x ) | p + ǫ ) p p + ǫ (cid:17) p a.e. x ∈ R n . | f ( x , x ) | + ǫ p + ǫ p where I = ( x − r, x + r ) , I = ( x − r, x + r ). By the definition of limit,lim r → k χ Q ( x,r ) k − L ~p ( R n ) k f k L ~p ( Q ( x,r )) = | f ( x ) | a.e. x ∈ R n . The proof is complete. (cid:4)
Based on the definition of local and global Morrey-type spaces and mixed Morreyspaces, the definitions of local and global mixed Morrey-type spaces are introduced.
Definition 2.7.
Let 0 < ~p, θ ≤ ∞ and let ω be a non-negative measurable function on(0 , ∞ ). We denote the local mixed Morrey-type spaces and the global mixed Morrey-typespaces by LM ~pθ,ω , GM ~pθ,ω respectively. For any functions f ∈ L loc~p ( R n ), we say f ∈ LM ~pθ,ω when the quasi-norms k f k LM ~pθ,ω = k f k LM ~pθ,ω ( R n ) = k ω ( r ) k f k L ~p ( Q (0 ,r )) k L θ (0 , ∞ ) ≤ ∞ ;we say f ∈ GM ~pθ,ω when the quasi-norms k f k GM ~pθ,ω = sup x ∈ R n k f ( x + · ) k LM ~pθ,ω ≤ ∞ . In particular, we say f ∈ LM [ x ] ~pθ,ω , if k f ( x + · ) k LM ~pθ,ω = k ω ( r ) k f k L ~p ( Q ( x,r )) k L θ (0 , ∞ ) < ∞ . emark 2.8. Note that if ω ( r ) = r nq − P nj =1 1 pj , θ = ∞ , then GM ~pθ,ω = M q~p .Next, some properties of local and global mixed Morrey-type spaces are given. Theorem 2.9.
Let 0 < ~p, θ ≤ ∞ and let ω be a non-negative measurable function on(0 , ∞ ).(1) If for all t > k ω ( r ) k L θ ( t, ∞ ) = ∞ , (2 . LM ~pθ,ω = GM ~pθ,ω = Θ . (2) If for all t > k ω ( r ) r P nj =1 1 pi k L θ (0 ,t ) = ∞ , (2 . f (0) = 0 for all f ∈ LM ~pθ,ω continuous at 0 and GM ~pθ,ω = Θ for all 0 < ~p < ∞ . Proof: (1) Suppose that (2.1) holds true for all t ∈ (0 , ∞ ) and f is not the elementzero. Then there is t such that A = k f k L ~p ( Q (0 ,t )) >
0. Hence, k f k GM ~pθ,ω ≥ k f k LM ~pθ,ω ≥ k ω ( r ) k f k L ~p ( Q (0 ,r )) k L θ ( t , ∞ ) ≥ A k ω ( r ) k L θ ( t , ∞ ) = ∞ . Therefore, k f k GM ~pθ,ω = k f k LM ~pθ,ω = ∞ . (2) Suppose that (2.2) holds true for all t ∈ (0 , ∞ ) and f ∈ LM ~pθ,ω . By Lemma 2.6 thereexists lim r → r − P nj =1 1 pi k f k L ~p ( Q (0 ,r )) = B = C | f (0) | a.e. x ∈ R n . where C = 2 P nj =1 1 pi . If B >
0, then there exists t > r − P nj =1 1 pi k f k L ~p ( Q (0 ,r )) ≥ B a.e. x ∈ R n holds for all 0 < r ≤ t . Consequently, k f k LM ~pθ,ω ≥ k ω ( r ) k f k L ~p ( Q (0 ,r )) k L θ (0 ,t ) ≥ B k ω ( r ) r P nj =1 1 pi k L θ (0 ,t ) = ∞ a.e. x ∈ R n . Here, k f k LM ~pθ,ω = ∞ , so f / ∈ LM ~pθ,ω and we have arrived at a contradiction.Next let 0 < ~p < ∞ , f ∈ GM ~pθ,ω . By lemma2.1, we know thatlim r → r − P nj =1 1 pi k f k L ~p ( Q ( x,r )) = C | f ( x ) | a.e. x ∈ R n , where C = 2 P nj =1 1 pi . By the above argument, f ( x ) = 0 a.e. x ∈ R n . (cid:4) Due to the above argument, the sets Ω θ and Ω ~p,θ are defined as following. Definition 2.10.
Let 0 < ~p, θ ≤ ∞ . We denote by Ω θ the set of all functions ω whichare non-negative, measurable on (0 , ∞ ), not equivalent to 0, and such that for some t > k ω k L θ ( t, ∞ ) < ∞ . ~p,θ the set of all functions ω which are non-negative, measurableon (0 , ∞ ), not equivalent to 0, and such that for some t , t > k ω k L θ ( t , ∞ ) < ∞ , k ω ( r ) r P nj =1 1 pi k L θ (0 ,t ) = ∞ . The following Theorem tells us that LM ~pθ,ω and GM ~pθ,ω are complete, when 1 ≤ ~p, θ < ∞ . Theorem 2.11. (1) Let 1 ≤ ~p, θ < ∞ , and ω ∈ Ω θ . Suppose f n ∈ LM ~pθ,ω , for all n ∈ N + and ∞ X n =1 k f n k LM ~pθ,ω < ∞ . Then P ∞ n =1 f n exist. If f = P ∞ n =1 f n , then f ∈ LM ~pθ,ω and k f k LM ~pθ,ω ≤ ∞ X n =1 k f n k LM ~pθ,ω . Hence, LM ~pθ,ω is compte.(2) Let 1 ≤ ~p, θ < ∞ , and ω ∈ Ω ~pθ . Suppose f n ∈ GM ~pθ,ω , for all n ∈ N + and ∞ X n =1 k f n k GM ~pθ,ω < ∞ . Then P ∞ n =1 f n exist. If f = P ∞ n =1 f n , then f ∈ GM ~pθ,ω and k f k GM ~pθ,ω ≤ ∞ X n =1 k f n k GM ~pθ,ω . Hence, GM ~pθ,ω is compte. Proof: (1) It is easy to see that for any
R > k ω k L θ ( R, ∞ ) k f k L ~p ( Q (0 ,R )) ≤ k f k LM ~pθ,ω . Thus ∞ X n =1 k f n k L ~p ( Q (0 ,R )) ≤ C ∞ X n =1 k f n k LM ~pθ,ω . Since L ~p ( Q (0 , r )) is complete [15], then P ∞ n =1 f n converges a.e. to some f ∈ L loc~p ( R n ) ∞ X n =1 f n = f and k f k L ~p ( Q (0 ,R )) ≤ ∞ X n =1 k f n k L ~p ( Q (0 ,R )) . k f k LM ~pθ,ω = k ω ( r ) k f k L ~p ( Q (0 ,r )) k L θ (0 , ∞ ) ≤ k ω ( r ) ∞ X n =1 k f n k L ~p ( Q (0 ,r )) k L θ (0 , ∞ ) ≤ ∞ X n =1 k ω ( r ) k f n k L ~p ( Q (0 ,r )) k L θ (0 , ∞ ) = ∞ X n =1 k f n k LM ~pθ,ω Now, the complete of LM ~pθ,ω will be proved. let { f n } is Cauchy sequence in LM ~pθ,ω .Without less of generality, assume { f n } satisfies that ∞ X n =1 k f n − f n − k LM ~pθ,ω < ∞ , where f = 0.Thanks to the above argument, there exist f = P ∞ n =1 ( f n − f n − ) = lim n →∞ f n and f ∈ LM ~pθ,ω . It is easy via Fatou’s property to obtain thatlim sup n →∞ k f − f n k LM ~pθ,ω ≤ k lim sup n →∞ | f − f n |k LM ~pθ,ω = 0Hence, the complete of LM ~pθ,ω is proved.(2) We can prove (2) through the same method as (1), so we omit the proof. (cid:4) Definition 2.12. [34] The intersection of a family function space { X α } α ∈ A is a Banachspace X such that(a) X ֒ → X α , α ∈ A ;(b) if for a certain Banach Space Y we have Y ֒ → X α , α ∈ A then Y ֒ → X .It is easy to prove the following prosperity. Prosperity 2.13. GM ~pθ,ω is the interaction of { LM [ x ] ~pθ,ω } x ∈ R n . I α In [15,31], the boundedness of fractional integral operators is proved in mixed Lebesguespaces. Next, a sufficient and necessary condition of fractional integral operators’ bound-9dness will be given in mixed norm Lebesgue spaces.
Lemma 3.1.
Let 0 < α < n, < ~p < ~q < ∞ . Then k I α f k L ~q . k f k L ~p (3 . α = n X j =1 p j − n X j =1 q j . Proof:
Without less of generality, let n = 2. Duce to | y − z | = | ( y , y ) − ( z , z ) | ≥| y i − z i | , i = 1 ,
2, it is easy to acquire that | y − z | α − ≤ Y i =1 | y i − z i | α i − , where α i = p i − q i , α i ∈ (0 , I α inclassical Lebesgue spaces, It is easy to acquire that k I α f k L ~q ( R = (cid:18) Z R (cid:18) Z R (cid:12)(cid:12)(cid:12)(cid:12) Z R Z R f ( z , z ) | x − z | − α dz dz (cid:12)(cid:12)(cid:12)(cid:12) q dy (cid:19) q q dy (cid:19) q = (cid:18) Z R (cid:18) Z R | x − z | − α (cid:18) Z R (cid:12)(cid:12)(cid:12)(cid:12) Z R f ( z , z ) | x − z | − α dz (cid:12)(cid:12)(cid:12)(cid:12) q dy (cid:19) q dz (cid:19) q dy (cid:19) q = (cid:18) Z R (cid:18) Z R k f ( · , z ) k L p ( R ) | x − z | − α dz (cid:19) q dy (cid:19) q = k f k L ~p ( R ) . Hence, sufficiency is proved.Let δ t f ( x ) = f ( tx ). Then δ t − I α δ t = t − α I α , k δ t f k L ~p ( R ) = t − P i =1 1 pi k f k L ~p ( R ) . k δ t − I α f k L ~q ( R ) = t P i =1 1 pi k I α f k L ~q ( R ) . Assume (3.1) is satisfied. Then k I α f k L ~q ( R ) = t α k δ t − I α δ t f k L ~q ( R ) = t α + P i =1 1 qi k I α δ t f k L ~q ( R ) ≤ Ct α + P i =1 1 qi k δ t f k L ~p ( R ) = Ct α + P i =1 1 qi − P i =1 1 pi k f k L ~p ( R ) . Therefor, α = P i =1 1 p i − P i =1 1 q i is obtained. (cid:4) Remark 3.2.
By translation invariance of I α , we can prove that ~p ≤ ~q is a necessarycondition for the boundedness of fractional integral operators in mixed Lebesgue spaces.10et τ ih f ( x ) = f ( x , · · · , x i − , x i + h, x i +1 , · · · , x n )( i = 1 , , · · · , n ) and k I α f k L ~q ≤ C k f k L ~p . By translation invariant of I α , k I α f + τ ih I α f k L ~q = k I α ( f + τ ih f ) k L ~q ≤ C k f + τ ih f k L ~p . Duce to Lemma 2.3 of [32], let h → ∞ , then k I α f k L ~q ≤ pi − qi C k f k L ~p . Therefore, ~p ≤ ~q .The following theorem states the necessity of I α in local mixed Morrey-type spaces. Theorem 3.3. (1) Let 1 ≤ ~p ≤ ∞ , < ~p ≤ ∞ , < α < n, < θ , θ < ∞ , ω ∈ Ω θ , ω ∈ Ω θ . Moreover, let ω ∈ L θ (0 , ∞ ). Then α ≥ (cid:18) n X i =1 p i − n X i =1 q i (cid:19) + is necessary for the boundedness of I α from LM ~p θ ,ω to LM ~p θ ,ω .(2) Let 1 ≤ ~p ≤ ∞ , < ~p ≤ ∞ , < α < n, < θ , θ < ∞ , ω ∈ Ω ~p θ , ω ∈ Ω ~p θ .Moreover, let ω ∈ L θ (0 , ∞ ). Then α ≥ (cid:18) n X i =1 p i − n X i =1 q i (cid:19) + is necessary for the boundedness of I α from GM ~p θ ,ω to GM ~p θ ,ω . Proof:
Suppose that k I α f k LM ~p θ ,ω ≤ k f k LM ~p θ ,ω ∀ f ∈ LM ~p θ ,ω . Let f ∈ L ~p ⊆ LM ~p θ ,ω , and f is not almost everywhere 0. Then for t > k δ t f k L ~p i ( Q (0 ,r )) = t − P ni =1 1 p i k f k L ~p ( Q (0 ,tr )) ,δ − t I α δ t = t − α I α , k δ − t I α f k L ~p ( Q (0 ,r )) = t P ni =1 1 p i k I α f k L ~p ( Q (0 ,r/t )) . So, it is easy for us to acquire that k I α f k LM ~p θ ,ω = t α k δ − t I α ( δ t f ) k LM ~p θ ,ω = t α + P ni =1 1 p i k ω ( r ) k I α ( δ t f ) k L ~p ( Q (0 ,r/t )) k L θ (0 , ∞ ) ≤ t α + P ni =1 1 p i k ω ( r ) k I α ( δ t f ) k L ~p ( Q (0 ,r )) k L θ (0 , ∞ ) ≤ Ct α + P ni =1 1 p i k ω ( r ) k δ t f k L ~p ( Q (0 ,r )) k L θ (0 , ∞ ) ≤ Ct α + P ni =1 1 p i − P ni =1 1 p i k ω ( r ) k f k L ~p ( Q (0 ,tr )) k L θ (0 , ∞ ) ≤ Ct α + P ni =1 1 p i − P ni =1 1 p i k ω ( r ) k L θ (0 , ∞ ) k f k L ~p ( R n ) . P ni =1 1 p i ≤ P ni =1 1 p i , then 0 < α < n is necessary because of definition of I α . If P ni =1 1 p i > P ni =1 1 p i , then P ni =1 1 p i − P ni =1 1 p i < α < n is necessary because of the abovediscussion.(2) We can prove (2) through the same method as (1), so we omit the proof. (cid:4) L ~p -Estimates of I α Over the Cube Q ( x, r ) We consider the following “partia” fractional integral operators I α,r f ( x ) = I α ( f χ Q ( x,r ) )( x ) = Z Q ( x,r ) | f ( y ) || x − y | n − α dy. ¯ I α,r f ( x ) = I α ( f χ Q ( x,r ) ∁ )( x ) = Z Q ( x,r ) ∁ | f ( y ) || x − y | n − α dy. Theorem 4.1.
Let 0 < ~p ≤ ∞ , < α < n and f ∈ L loc ( R n ). Then for any Q ( x, r ) ⊆ R n k I α ( | f | ) k L ~p ( Q ( x,r )) ∼ k I α ( | f | χ Q ( x, r ) ) k L ~p ( Q ( x,r )) + r P ni =1 1 pi ¯ I α, r f ( x ) . Proof:
Clearly k I α ( | f | ) k L ~p ( Q ( x,r )) ≤ k I α ( | f | χ Q ( x, r ) ) k L ~p ( Q ( x,r )) + k I α ( | f | χ Q ( x, r ) ∁ ) k L ~p ( Q ( x,r )) . If y ∈ Q ( x, r ) , z ∈ Q ( x, r ) ∁ , then | y − z | ∼ | x − z | . therefor k I α ( | f | χ Q ( x, r ) ∁ ) k L ~p ( Q ( x,r )) = k Z Q ( x, r ) ∁ | f ( z ) || y − z | n − α dz k L ~p ( Q ( x,r )) ∼ k Z Q ( x, r ) ∁ | f ( z ) || x − z | n − α dz k L ~p ( Q ( x,r )) = Z Q ( x, r ) ∁ | f ( z ) || x − z | n − α dz k χ Q ( x,r ) k L ~p ( R n ) ∼ r P ni =1 1 pi ¯ I α, r f ( x ) . For the left-hand side inequalities, in the one hand, k I α ( | f | χ Q ( x, r ) ) k L ~p ( Q ( x,r )) ≤ k I α ( | f | ) k L ~p ( Q ( x,r )) ;in the other hand, if y ∈ Q ( x, r ) and z ∈ Q ( x, r ) ∁ , then k I α ( | f | ) k L ~p ( Q ( x,r )) ≥ k I α ( | f | χ Q ( x, r ) ∁ ) k L ~p ( Q ( x,r )) ∼ r P ni =1 1 pi ¯ I α, r f ( x ) . (cid:4) Theorem 4.2.
Let1 < ~p ≤ ~p < ∞ , ~p = ~p n X i =1 p i − n X i =1 p i ≤ α < n, (4 . < ~p < ~p ≤ ∞ , ~p > , < α < n, (4 . < ~p < ∞ , < ~p < ∞ , n X i =1 (cid:18) p i − p i (cid:19) + < α < n, (4 . k I α ( | f | χ Q ( x, r ) ) k L ~p ( Q ( x,r )) . r α − (cid:0) P ni =1 1 p i − P ni =1 1 p i (cid:1) k f k L ~p ( Q ( x, r )) . Proof: (1) When 1 < ~p < ~p < ∞ and n X i =1 p i − n X i =1 p i ≤ α < n, it is easy to calculate that I α ( | f | χ Q ( x, r ) )( y ) = Z Q ( x, r ) | f ( z ) || y − z | n − α dz . r α − (cid:0) P ni =1 1 p i − P ni =1 1 p i (cid:1) Z Q ( x, r ) | f ( z ) || y − z | n − β dz = r α − (cid:0) P ni =1 1 p i − P ni =1 1 p i (cid:1) I β ( f χ Q ( x, r ) )( y ) , where β = P ni =1 1 p i − P ni =1 1 p i and y ∈ Q ( x, r ). Duce to Lemma 3.1, It is obtained that k I α ( | f | χ Q ( x, r ) ) k L ~p ( Q ( x,r )) . r α − (cid:0) P ni =1 1 p i − P ni =1 1 p i (cid:1) k I β ( f χ Q ( x, r ) ) k L ~p ( Q ( x,r )) . r α − (cid:0) P ni =1 1 p i − P ni =1 1 p i (cid:1) k f k L ~p ( Q ( x, r )) (2) If 1 < ~p ≤ ~p < ∞ , we take ~s = ( s , s , · · · , s n ) such that 1 < ~s < ~p ≤ ~p < ∞ and n X i =1 s i − n X i =1 p i ≤ α < n. Thanks to (1) and H¨older’s inequality, it is easy to acquire that k I α ( | f | χ Q ( x, r ) ) k L ~p ( Q ( x,r )) . r α − (cid:0) P ni =1 1 si − P ni =1 1 p i (cid:1) k f k L ~s ( Q ( x, r )) . r α − (cid:0) P ni =1 1 p i − P ni =1 1 p i (cid:1) k f k L ~p ( Q ( x, r )) . < ~p ≤ ~p < ∞ , ~p >
1, then0 < α < n.
By H¨older’s inequality, k I α ( | f | χ Q ( x, r ) ) k L ~p ( Q ( x,r )) . r P ni =1 1 p i − P ni =1 1 P i k I α ( | f | χ Q ( x, r ) ) k L ~p ( Q ( x,r )) . Duce to Minkowski’s inequality, k I α ( | f | χ Q ( x, r ) ) k L ~p ( Q ( x,r )) . (cid:13)(cid:13)(cid:13)(cid:13)Z Q ( x, r ) | f ( z ) || · − z | n − α dz (cid:13)(cid:13)(cid:13)(cid:13) L ~p ( Q ( x,r )) ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z B (0 , √ nr ) | ( f χ Q ( x, r ) )( · − z ) || z | n − α dz (cid:13)(cid:13)(cid:13)(cid:13) L ~p ( Q ( x,r )) ≤ Z B (0 , √ nr ) | z | n − α dz k ( f χ Q ( x, r ) ) k L ~p ( R n ) . r α k f k L ~p ( Q ( x, r )) . Hence, k I α ( | f | χ Q ( x, r ) ) k L ~p ( Q ( x,r )) . r α − ( P ni =1 1 p i − P ni =1 1 p i ) k f k L ~p ( Q ( x, r )) . (4) For other cases that there are i = j such that p i < p i , p j > p j hold, the proof is given as following.To simplify the process of the proof, let ~p = (¯ p , p ′ ) , ¯ p = ( p , , p , , · · · , p ,l ) and p ′ = ( p ,l +1 , p ,l +2 , · · · , p ,n ). Similarly, let ~p = (¯ p , p ′ ) , x = (¯ x, x ′ ) , and z = (¯ z, z ′ ).Moreover, ¯ p ≤ ¯ p , p ′ > p ′ . k I α ( | f | χ Q ( x, r ) ) k L ~p ( Q ( x,r )) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z Q ( x ′ , r ) Z Q (¯ x, r ) | ( f χ Q ( x, r ) )(¯ z, z ′ ) || · − (¯ z, z ′ ) | n − α d ¯ z dz ′ (cid:13)(cid:13)(cid:13)(cid:13) L ¯ p ( Q (¯ x,r )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ′ ( Q ( x ′ ,r )) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z Q ( x ′ , r ) | · − z ′ | n − l − α × (cid:13)(cid:13)(cid:13)(cid:13)Z Q (¯ x, r ) | ( f χ Q ( x, r ) )(¯ z, z ′ ) || · − ¯ z | l − α d ¯ z (cid:13)(cid:13)(cid:13)(cid:13) L ¯ p ( Q (¯ x,r )) dz ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ′ ( Q ( x ′ ,r )) . r α − ( P li =1 1 p i − P li =1 1 p i ) × (cid:13)(cid:13)(cid:13)(cid:13)Z Q ( x ′ , r ) k ( f χ Q ( x ′ , r ) )( · , z ′ ) k L ¯ p ( Q (¯ x, r )) | · − z ′ | n − l − α dz ′ (cid:13)(cid:13)(cid:13)(cid:13) L p ′ ( Q ( x ′ ,r )) . r α − ( P ni =1 1 p i − P ni =1 1 p i ) kk f k L ¯ p ( Q (¯ x, r )) k L p ′ ( Q ( x ′ , r )) . r α − ( P ni =1 1 p i − P ni =1 1 p i ) k f k L ~p ( Q ( x, r )) , where α = α + α and they satisfy that l X i =1 p i − l X i =1 p i < α < l, < α < n − l. The proof is complete. (cid:4)
Before starting the next theorem, the following lemma [16] is given.
Lemma 4.3.
Let f is be a non-negative measurable function, then for any r > Z | x | >r f ( x ) | x | β dx = β Z ∞ r Z r ≤| x |≤ t f ( x ) dx dtt β +1 . Theorem 4.4.
Let (4.1) or (4.2) or (4.3) is satisfied. Then k I α ( | f | ) k L ~p ( Q ( x,r )) . r P ni =1 1 p i Z ∞ r k f k L ~p ( Q ( x,t )) dtt σ +1 , where σ = P ni =1 1 p i − α . Proof:
Note that if P ni =1 1 p i ≤ α and f is not almost everywhere equivalent to 0 on R n , then σ + 1 ≤ Z ∞ r k f k L ~p ( Q ( x,t )) dtt σ +1 ≥ k f k L ~p ( Q ( x,r )) Z ∞ r dtt σ +1 = ∞ . α < P ni =1 1 p i . By Theorem 4.1 and Theorem 4.2 k I α ( | f | ) k L ~p ( Q ( x,r )) ∼ k I α ( | f | χ Q ( x, r ) ) k L ~p ( Q ( x,r )) + r P ni =1 1 p i ¯ I α, r f ( x ) . r α − (cid:0) P ni =1 1 p i − P ni =1 1 p i (cid:1) k f k L ~p ( Q ( x, r )) + r P ni =1 1 p i ¯ I α, r f ( x )= (I) + (II) . In the one hand, (I) = r α − (cid:0) P ni =1 1 p i − P ni =1 1 p i (cid:1) k f k L ~p ( Q ( x, r )) ∼ r P ni =1 1 p i Z ∞ r dtt σ +1 k f k L ~p ( Q ( x, r )) ≤ r P ni =1 1 p i Z ∞ r k f k L ~p ( Q ( x,t )) dtt σ +1 . In the other hand, thanks to Lemma 4.3 and H¨older’s inequality(II) = r P ni =1 1 p i ¯ I α, r f ( x )= r P ni =1 1 p i Z Q ( x, r ) ∁ | f ( z ) || x − z | n − α dz = ( n − α ) r P ni =1 1 p i Z ∞ r Z r ≤| x − y |≤ t | f ( y ) | dy dtt n − α +1 . r P ni =1 1 p i Z ∞ r k f k L ( Q ( x,t )) dtt n − α +1 . r P ni =1 1 p i Z ∞ r k f k L ~p ( Q ( x,t )) dtt σ +1 . The proof is complete. (cid:4) I α In this section, the boundedness of the fractional Integration is obtained by Hardyoperators’ boundedness in weighted Lebesgue spaces. According to [17, 18], the bounded-ness of Hardy operators was obtained. Hence the sufficient condition of the boundednessof I α and its corollaries are given.Let H be the Hardy operators,( Hg )( t ) := Z t g ( r ) dr, < t < ∞ , g ( t ) is non-negative measurable function on (0 , ∞ ). Theorem 5.1.
Let (4.1) or (4.2) or (4.3) is satisfied. Moreover, let 0 < θ ≤ ∞ and ω ∈ Ω θ . Then k I α f k LM ~p θ ,ω . k Hg ~p k Lθ ,ν (0 ,, ∞ ) , where g ~p = k f k L ~p ( Q ( x,t − σ )) , σ = n X i =1 p i − α.ν ( r ) = ω ( r − σ ) r − σ P ni =1 1 p i − θ σ − θ . Proof:
Duce to Theorem 4.4, k I α f k LM ~p θ ,ω = k ω ( r ) k I α f k L ~p ( Q ( x,r )) k L θ (0 , ∞ ) . k ω ( r ) r P ni =1 1 p i Z ∞ r k f k L ~p ( Q ( x,t )) dtt σ +1 k L θ (0 , ∞ ) ∼ k ω ( r ) r P ni =1 1 p i Z r − σ k f k L ~p ( Q ( x,t − σ )) dt k L θ (0 , ∞ ) ∼ k ω ( r − σ ) r − σ P ni =1 1 p i − θ σ − θ Hg ~p ( r ) k L θ (0 , ∞ ) = k Hg ~p k Lθ ,ν (0 ,, ∞ ) . The proof is complete. (cid:4)
Theorem 5.2.
Let (4.1) or (4.2) or (4.3) is satisfied. Moreover, let 0 < θ , θ ≤ ∞ , ω ∈ Ω θ and ω ∈ Ω θ . Then if k Hg ~p k Lθ ,ν (0 ,, ∞ ) . k g ~p k Lθ ,ν (0 ,, ∞ ) , where υ is same as theorem 5.1 and ν ( r ) = ω ( r − σ ) r − θ σ − θ ,I α is bounded from LM ~p θ ,ω to LM ~p θ ,ω . Proof:
Assume that the operator H is boundedness from L θ ,υ (0 , ∞ ) to L θ ,υ (0 , ∞ ).By Theorem 5.1, k I α f k LM ~p θ ,ω . k Hg ~p k Lθ ,ν (0 , ∞ ) . k g ~p k L θ ,ν (0 ,, ∞ ) . Note that k g ~p k Lθ ,ν (0 ,, ∞ ) = k ν ( r ) k f k L ~p ( Q ( x,r − σ ) k L θ (0 , ∞ ) = k ω ( r − σ ) r − θ σ − θ k f k L ~p ( Q ( x,r − σ ) k L θ (0 , ∞ ) ∼ k ω ( r ) k f k L ~p ( Q ( x,r )) k L θ (0 , ∞ ) = k f k LM ~p θ ,ω . (cid:4) According to Theorem 5.2 and [17, 18], we get the following Theorem.
Theorem 5.3.
Let (4.1) or (4.2) or (4.3) is satisfied. Moreover, let 0 < θ , θ ≤ ∞ , ω ∈ Ω θ and ω ∈ Ω θ .Then I α is bounded from LM ~p θ ,ω to LM ~p θ ,ω , if(1) If 1 < θ ≤ θ < ∞ , then A := sup t> (cid:18) Z ∞ t ω θ ( r ) r θ ( α − ( P ni =1 1 p i − P ni =1 1 p i )) dr (cid:19) θ (cid:18) Z ∞ t ω θ ( r ) dr (cid:19) − θ < ∞ , and A := sup t> (cid:18) Z t ω θ ( r ) r θ P ni =1 1 p i d r (cid:19) θ (cid:18) Z ∞ t ω θ ( r ) r θ ′ ( α − P ni =1 1 p i ) (cid:0) R ∞ r ω θ ( ρ ) dρ (cid:1) − θ ′ dr (cid:19) θ ′ < ∞ . (2) If 1 < θ ≤ , < θ ≤ θ < ∞ , then A < ∞ and A ; = sup t> t α − P ni =1 1 p i (cid:18) Z t ω θ ( r ) r θ P ni =1 1 p i d r (cid:19) θ (cid:18) Z ∞ t ω θ ( r ) dr (cid:19) − θ < ∞ . (3) If 1 < θ < ∞ , < θ < θ < ∞ , θ = 1, then A := Z ∞ (cid:18) R ∞ t ω θ ( r ) r θ ( α − ( P ni =1 1 p i − P ni =1 1 p i )) dr R ∞ r ω θ ( r ) dr (cid:19) θ θ − θ × ω θ ( t ) t θ ( α − ( P ni =1 1 p i − P ni =1 1 p i )) ! θ − θ θ θ < ∞ and A := Z ∞ "(cid:0) Z t ω θ ( r ) r θ P ni =1 1 p i d r (cid:19) θ (cid:18) Z ∞ t ω θ ( r ) r θ ′ ( α − P ni =1 1 p i ) (cid:0) R ∞ r ω θ ( ρ ) dρ (cid:1) − θ ′ dr (cid:19) θ − θ θ θ θ − θ × ω θ ( t ) t θ ′ ( α − P ni =1 1 p i ) (cid:0) R ∞ t ω θ ( ρ ) dρ (cid:1) − θ ′ dt ! θ − θ θ θ < ∞ . (4) If 1 = θ < θ < ∞ , then A < ∞ and A := Z ∞ (cid:18) R ∞ t ω ( r ) r α − ( P ni =1 1 p i − P ni =1 1 p i ) dr + t α − P ni =1 1 p i R t ω θ ( r ) r θ P ni =1 1 p i dr R ∞ t ω θ ( r ) dr (cid:19) θ ′ − × t α − P ni =1 1 p i Z t ω θ ( r ) r θ P ni =1 1 p i dr dtt ! θ ′ < ∞ . < θ < θ ≤
1, then A < ∞ and A := Z ∞ sup t ≤ s< ∞ s α − P ni =1 1 p i θ θ θ − θ (cid:18) R ∞ s ω θ ( ρ ) dρ (cid:19) θ θ − θ (cid:18) Z t ω θ ( r ) r θ P ni =1 1 p i dr (cid:19) θ θ − θ × ω θ ( t ) t θ P ni =1 1 p i dt ! θ − θ θ θ < ∞ . (6) If 0 < θ ≤ , θ = ∞ , then A := ess sup
If the conditions of Theorem 5.3 are satisfied. Moreover, ω ∈ Ω ~p θ and ω ∈ Ω ~p θ . Then I α are bounded from GM ~p θ ,ω to GM ~p θ ,ω .It is well-known that M α f < I α ( | f | ), then the following corollary is obtained. Corollary 5.5.
If the conditions of Theorem 5.3 are satisfied, then M α are boundedfrom LM ~p θ ,ω to LM ~p θ ,ω . Moreover, if ω ∈ Ω ~p θ and ω ∈ Ω ~p θ . Then M α are alsobounded from GM ~p θ ,ω to GM ~p θ ,ω .If ω ( r ) = r nq − P ni =1 1 p i , ω ( r ) = r nq − P nj =1 1 p i , θ = θ = ∞ and , then GM ~p θ ,ω = M q ~p GM ~p θ ,ω = M q ~p . Moreover, if q = q − αn , then A < ∞ is satisfied. Corollary 5.6.
Let nq ≤ P ni =1 1 p i , nq ≤ P ni =1 1 p i and (4.1) or (4.2) or (4.3) is satisfied.Moreover, if 1 q = 1 q − αn , then for f ∈ M q ~p ( R n ), k I α f k M q ~p ( R n ) ≤ C k f k M q ~p ( R n ) . Corollary 5.7.
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