The Boundedness of the Hardy-Littlewood Maximal Operator and Multilinear Maximal Operator in Weighted Morrey Type Spaces

The aim of this paper is to prove the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces and multilinear maximal operator on multiple weighted Morrey spaces. In particular, the result includes the Komori-Shirai theorem and the Iida-Sato-Sawano-Tanaka theorem for the Hardy-Littlewood maximal operator and multilinear maximal function.

Hence it is natural to consider multiple weighted Morrey spaces.
Definition 5 (multiple weighted Morrey spaces).Let 0 <  1 , . . .,   < ∞ and 1/ 0 ≤ 1/ 1 + ⋅ ⋅ ⋅ + 1/  .Let ⃗  = ( 1 , . . .,   ) be a multiple weight.Let  be a weight.The multiple weighted Morrey space is defined by the quantity for vector valued function ⃗  = ( In this framework, we investigate the well-known results.The following theorem which is elementary result was discovered by Muckenhoupt [1]. Theorem A. Let  be a weight and 1 <  < ∞:           () ≤            () (9) if and only if Komori and Shirai [2] introduced the weighted Morrey spaces and proved the following theorem.The following theorem gives us the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces.
On the other hand, in [3], the following theorem was proved.The following result gives also the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces.
Theorem C. Let  be a weight and Moreover we can extend Theorem C to the multilinear version (see [3,4]).
Then one has where ⃗ Because the ways of the evaluation by the weighted Morrey norm are different, Theorems B and C are independent.Therefore, it is natural to consider unifying Theorems B and C. The question is not settled yet.In this paper, we unify Theorems B-D.

Main Results
Theorem 6.Let 1 <  ≤  0 < ∞ and let  1 and  2 be weights.Additionally assume that  1 satisfies the doubling condition: there exists  > 0 such that Assume that the following condition holds: Then, one has  1 ()  = , then Theorem 8 corresponds with Theorem B. In fact, condition (19) corresponds with  ∈   (R  ).We state the detail: Theorem 10.Let  1 ,  2 , and V be weights.Additionally assume that  1 satisfies the doubling condition.Let 1 <  ≤  0 < ∞,  > 1, and Then one has . (25) We can extend Theorem 10 as follows.
Theorem 11.Let  1 ,  2 , and V be weights.Additionally assume that  1 satisfies the doubling condition.Let 1 <  ≤  0 ≤  0 < ∞,  > 1, and Then one has By considering the multilinear version, we obtain the following theorems.
Lemma 14.Let 0 <  < ∞.Let V be a weight.For every cube  0 ⊂ R  , one has where Moreover we can extend Lemma 14 to the multilinear maximal function.
Lemma 15.Let 0 <  < ∞.Let V be a weight.For every cube  0 ⊂ R  , one has where Remark 16.In [3], there is not the restriction  ⊂  0 in the maximal function M 0  ( ⃗ , V).However, reexamining the argument of the proof in [3], we can remove the restriction.Hence the norm inequalities in Lemmas 14 and 15 are sharper than the norm inequalities in [3].