Boundedness of composition operators on Morrey spaces and weak Morrey spaces
Naoya Hatano, Masahiro Ikeda, Isao Ishikawa, Yoshihiro Sawano
aa r X i v : . [ m a t h . F A ] A ug BOUNDEDNESS OF COMPOSITION OPERATORS ON MORREYSPACES AND WEAK MORREY SPACES
NAOYA HATANO, MASAHIRO IKEDA, ISAO ISHIKAWA, AND YOSHIHIRO SAWANO
Abstract.
In this study, we investigate the boundedness of composition op-erators acting on Morrey spaces and weak Morrey spaces. The primary aim ofthis study is to investigate a necessary and sufficient condition on the bound-edness of the composition operator induced by a diffeomorphism on Morreyspaces. In particular, detailed information is derived from the boundedness,i.e., the bi-Lipschitz continuity of the mapping that induces the compositionoperator follows from the continuity of the composition mapping. The idea ofthe proof is to determine the Morrey norm of the characteristic functions, andemploy a specific function composed of a characteristic function. As the spe-cific function belongs to Morrey spaces but not to Lebesgue spaces, the resultreveals a new phenomenon not observed in Lebesgue spaces. Subsequently, weprove the boundedness of the composition operator induced by a mapping thatsatisfies a suitable volume estimate on general weak-type spaces generated bynormed spaces. As a corollary, a necessary and sufficient condition for theboundedness of the composition operator on weak Morrey spaces is provided. Introduction
In this study, we investigate the boundedness of composition operators on Mor-rey spaces and weak Morrey spaces. The composition operator C ϕ induced by amapping ϕ is a linear operator defined by C ϕ f ≡ f ◦ ϕ , where f ◦ ϕ representsthe function composition. The composition operator is also called the Koopmanoperator in the fields of dynamical systems, physics, and engineering [12]. Recently,it has attracted attention in various scientific fields [11, 10].Let ( X, µ ) be a σ -finite measure space, and L ( X, µ ) be the set of all µ -measurablefunctions on X . We provide a precise definition of the composition operators in-duced by a measurable map ϕ : X → X . Definition 1.1 (Composition operator) . Let ϕ : X → X be a measurable map,and assume that ϕ is nonsingular, namely, µ ( ϕ − ( E )) = 0 for each µ -measurablenull set E . Let V and W be function spaces contained in L ( X, µ ). The compositionoperator C ϕ is the linear operator from W to V such that its domain is D ( C ϕ ) ≡{ h ∈ W : h ◦ ϕ ∈ V } , and C ϕ f ≡ f ◦ ϕ for f ∈ D ( C ϕ ).Subsequently, we employ the result obtained by Singh [16] for the boundedness ofthe composition operator on the Lebesgue space L p ( X, µ ). Henceforth, we denoteby L ( X, µ ) the space of all µ -measurable functions.Singh [16] provided the following necessary and sufficient condition for the map ϕ to generate a bounded mapping acting on Lebesgue spaces: Mathematics Subject Classification.
Primary 42B35; Secondary 47B33.
Key words and phrases.
Composition operators, Boundedness, Morrey spaces, weak Morreyspaces.
Theorem 1.2 ([16]) . Let < p < ∞ . Then, the composition operator C ϕ inducedby ϕ : X → X is bounded on the Lebesgue space L p ( X, µ ) if and only if there existsa constant K = K ( ϕ ) such that for all µ -measurable sets E in R n , µ ( ϕ − ( E )) ≤ Kµ ( E ) . In this case, the operator norm is given by (1.1) k C ϕ k L p → L p = sup <µ ( E ) < ∞ (cid:18) µ ( ϕ − ( E )) µ ( E ) (cid:19) p . The boundedness of the composition operator on L ∞ ( X, µ ) easily follows fromthe definition. Theorem 1.2 was extended to several important function spaces,such as Lorentz spaces [1, 5], Orlicz spaces [2, 6, 13], mixed Lebesgue spaces [7],Musielak-Orlicz spaces [14] and reproducing kernel Hilbert spaces [9]. However,there are no previous results on the boundedness of composition operators actingon Morrey spaces and weak Morrey spaces.The first aim of this strudy is to investigate a necessary and sufficient conditionon the boundedness of the composition operator C ϕ on Morrey spaces. Subse-quently, we discuss the boundedness of the operator on weak Morrey spaces.Hereafter, we consider the Euclidean space R n ; µ is the Lebesgue measure dx.The set of all measurable functions is denoted by L ( R n ). We denote by | E | thevolume of a measurable set E ⊂ R n . Let χ A : R n → R ≥ be an indicator functionfor a subset A ⊂ R n , which is defined as χ A ( x ) = 1 if x ∈ A and χ A ( x ) = 0,otherwise.Now, we recall the definition of Morrey spaces on R n . Definition 1.3 (Morrey space) . Let 0 < q ≤ p < ∞ . The Morrey space M pq ( R n )is a quasi-Banach space defined by M pq ( R n ) ≡ { f ∈ L ( R n ) : k f k M pq < ∞} , endowed with the quasi-norm k f k M pq ≡ sup Q ∈Q | Q | p − q k f χ Q k L q , where Q denotes the family of all cubes parallel to the coordinate axis in R n .From the H¨older inequality, we observe that the Lebesgue space L p ( R n ) is em-bedded into the Morrey space M pq ( R n ), where 0 < q ≤ p < ∞ . Remark . Let 0 < q ≤ p < ∞ . Then, we have L p ( R n ) = M pp ( R n ) ⊂ M pq ( R n ) . Moreover, L p ( R n ) is not dense in M pq ( R n ) [15].We now state the main results of the present paper. The following theoremprovides a sufficient condition on the boundedness of the composition operator C ϕ on the Morrey space M pq ( R n ). Theorem 1.5.
Let < q ≤ p < ∞ . Then, the composition operator C ϕ inducedby ϕ : R n → R n is bounded on the Morrey space M pq ( R n ) , if ϕ is a Lipschitz mapthat satisfies the volume estimate (1.2) | ϕ − ( E ) | ≤ K | E | , OUNDEDNESS OF COMPOSITION OPERATORS ON MORREY SPACES 3 for all measurable sets E in R n and some constant k independent of E . In partic-ular, we obtain (1.3) k C ϕ k M pq →M pq ≤ (max(1 , √ nL )) − np + nq sup E :0 < | E | < ∞ (cid:18) | ϕ − ( E ) || E | (cid:19) q , where L > is a Lipschitz constant of ϕ , and E in the supremum moves over allthe Lebesgue measurable sets E satisfying < | E | < ∞ . Conversely, as stated in the following theorem, if ϕ : R n → R n is a diffeomor-phism, then the M pq ( R n )-boundedness of the composition operators C ϕ and C ϕ − indicates that ϕ is bi-Lipschitz. Note that any bi-Lipschitz mapping satisfies theassumption of Theorem 1.5. Theorem 1.6.
Let n ∈ N , and ϕ : R n → R n be a diffeomorphism in the sensethat ϕ and its inverse ϕ − are differentiable. Suppose < q < p < ∞ , or q = p and n = 1 . If the composition operators C ϕ and C ϕ − induced by maps ϕ and ϕ − ,respectively, are bounded on M pq ( R n ) , then ϕ is bi-Lipschitz.Remark . In the case of p = q and n = 1, Theorem 1.6 reduced to Theorem1.2 according to [16]. Unless n = 1, condition q < p is essential in the followingsense. If n ≥ q = p , then the same conclusion as in Theorem 1.6 fails. Wepresent a counterexample in Example 3.3 in Section 4. Noting that the Morreyspace M pp ( R n ) coincides with the Lebesgue space L p ( R n ) (see Remark 1.9), weobserve a new phenomenon from Theorem 1.6.Subsequently, we investigate the characterization of the boundedness of the com-position operators acting on weak Morrey spaces, which are defined as follows: Definition 1.8 (Weak Morrey space) . Let 0 < q ≤ p < ∞ . The weak Morreyspace W M pq ( R n ) is a quasi-Banach space defined byW M pq ( R n ) ≡ { f ∈ L ( R n ) : k f k W M pq < ∞} endowed with the quasi-norm k f k W M pq ≡ sup λ> λ k χ { x ∈ R n : | f ( x ) | >λ } k M pq . The weak Morrey space W M pq ( R n ) has the following basic properties: Remark . Let 0 < q < p < ∞ . Then, we have M pq ( R n ) ⊂ W M pq ( R n ) , W M pp ( R n ) = W L p ( R n ) , where W L p ( R n ) is the weak Lebesgue space (see [8, Chapter 1] for more).The following theorem provides a necessary and sufficient condition on theboundedness of the composition operator on weak Morrey spaces. Theorem 1.10.
Let < q ≤ p < ∞ , and let ϕ : R n → R n be a measurablefunction. Then, ϕ generates the composition operator C ϕ which is bounded on theweak Morrey space W M pq ( R n ) if and only if there exists a constant K such that forall measurable sets E in R n , the estimate k χ ϕ − ( E ) k M pq ≤ K k χ E k M pq . NAOYA HATANO, MASAHIRO IKEDA, ISAO ISHIKAWA, AND YOSHIHIRO SAWANO holds. In particular, we obtain k C ϕ k W M pq → W M pq = sup E k χ ϕ − ( E ) k M pq k χ E k M pq , where the supremum is taken over all the measurable sets E in R n with < k χ E k M pq < ∞ .Remark . (1) Theorem 1.10 indicates that the composition operator C ϕ isbounded on weak Morrey spaces, once it is bounded on Morrey spaces (seeSection 4 for more).(2) The conclusion of cases q = p in this theorem was provided in [3].(3) Theorem 1.10 is a special case of Theorem 1.13 below.In fact, we will establish the boundedness of the composition operator in a moregeneral framework. Definition 1.12.
Let ( B ( R n ) , k · k B ) be a linear subspace of L ( R n ) such that k| f |k B = k f k B for all f ∈ B ( R n ). The weak-type space (W B ( R n ) , k · k W B ) of B isdefined by W B ( R n ) ≡ { f ∈ L ( R n ) : k f k W B < ∞} , endowed with the quasi-norm k f k W B ≡ sup λ> λ k χ { x ∈ R n : | f ( x ) | >λ } k B . Now, we can rewrite Theorem 1.10 as follows:
Theorem 1.13.
Let ( B ( R n ) , k · k B ) be a normed space. Then, the compositioninduced by ϕ is bounded on the weak-type space (W B ( R n ) , k · k W B ) if and only ifthere exists a constant K such that for all measurable sets E in R n , the estimate (1.4) k χ ϕ − ( E ) k B ≤ K k χ E k B . holds. In particular, we obtain (1.5) k C ϕ k W B → W B = sup E k χ ϕ − ( E ) k B k χ E k B , where the supremum is taken over all the measurable sets E in R n with < k χ E k B < ∞ . The remainder of this paper is organized as follows: In Section 2, we proveTheorems 1.5 and 1.6. In Section 3, we present some examples and counterexamplesof the mapping that induces the composition operator to be bounded on Morreyspaces. In Section 4, we prove Theorem 1.13.2.
Proof of Theorems 1.5 and 1.6
In this section, we prove Theorems 1.5 and 1.6. The proof of Theorem 1.5 isprovided in Subsection 2.1. However, Theorem 1.6 is more difficult to prove. InSubsection 2.2, we reduce matters to the linear setting. We divide its proof intotwo steps: we consider case p ≤ nq in Subsection 2.3 and case nq ≤ p in Subsection2.4. OUNDEDNESS OF COMPOSITION OPERATORS ON MORREY SPACES 5
Proof of Theorem 1.5.
Proof of Theorem 1.5.
A cube, Q ∈ Q , is fixed. We note that, according to theLipschitz continuity of ϕ , the estimatesdiam( ϕ ( Q )) := sup x, ˜ x ∈ Q | ϕ ( x ) − ϕ (˜ x ) | ≤ L sup x, ˜ x ∈ Q | x − ˜ x | = √ nLℓ ( Q ) , hold; thus, there exist cubes Q , Q ∈ Q such that Q ⊃ Q, Q ⊃ ϕ ( Q ) , | Q | = | Q | = (max(1 , √ nL )) n | Q | . As ϕ satisfies condition (1.2), we can apply the L q ( R n )-boundedness of the compo-sition operators (Theorem 1.2) to obtain | Q | p − q (cid:18) ˆ Q | f ( ϕ ( x )) | q d x (cid:19) q ≤ | Q | p − q (cid:18) ˆ R n | f ( ϕ ( x )) | q χ ϕ ( Q ) ( ϕ ( x )) d x (cid:19) q ≤ | Q | p − q · k C ϕ k L q → L q (cid:18) ˆ R n | f ( x ) | q χ ϕ ( Q ) ( x ) d x (cid:19) q ≤ ((max(1 , √ nL )) − n | Q | ) p − q · k C ϕ k L q → L q (cid:18) ˆ R n | f ( x ) | q χ Q ( x ) d x (cid:19) q ≤ (max(1 , √ nL )) − np + nq k C ϕ k L q → L q k f k M pq , which indicates that the composition operator C ϕ is bounded on M pq ( R n ). More-over, by applying the equation (1.1), we obtain (1.3), which completes the proof ofthe theorem. (cid:3) Reduction of the diffeomorphism to the linear setting.
In the following,for a differentiable vector-valued function ϕ = ( ϕ , . . . , ϕ n ) T on R n , we denote by Dϕ the Jacobi matrix of ϕ , that is, Dϕ ≡ (cid:18) ∂ϕ i ∂x j (cid:19) ≤ i,j ≤ n ≡ ( ϕ i,j ) ≤ i,j ≤ n . In this subsection, by applying the following lemma (Lemma 2.1), we reduce thediffeomorphism ϕ : R n → R n in Theorem 1.6 to the linear mapping Dϕ : R n → M n ( R ). By the estimate of the singular values of the Jacobi matrix Dϕ , we willshow that ϕ is bi-Lipschitz (see Proposition 2.6 below). Lemma 2.1.
Let < q ≤ p < ∞ . Suppose that a diffeomorphism ϕ : R n → R n induces a bounded composition operator C ϕ from M pq ( R n ) to itself. Then, thereexists a positive constant k > such that k C Dϕ ( x ) f k M pq = k f ( Dϕ ( x ) · ) k M pq ≤ K k f k M pq for all x ∈ R n and f ∈ M pq ( R n ) . In particular, the operator norm of k C Dϕ ( x ) k isbounded above by a constant independent of x .Proof of Lemma 2.1. Set K ≡ k C ϕ k M pq →M pq < ∞ . First, we prove the assertionfor f ∈ C ∞ c ( R n ), where C ∞ c ( R n ) is the set of all smooth functions with compact NAOYA HATANO, MASAHIRO IKEDA, ISAO ISHIKAWA, AND YOSHIHIRO SAWANO support. Let t >
0. We calculate (cid:13)(cid:13)(cid:13)(cid:13) f (cid:18) ϕ ( x + t · ) − ϕ ( x ) t (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) M pq = t − np (cid:13)(cid:13)(cid:13)(cid:13) f (cid:18) ϕ ( · ) − ϕ ( x ) t (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) M pq ≤ Kt − np (cid:13)(cid:13)(cid:13)(cid:13) f (cid:18) · − ϕ ( x ) t (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) M pq = K k f k M pq . By letting t →
0, we obtain the desired result for f ∈ C ∞ c ( R n ).Let f ∈ L ∞ c ( R n ), where L ∞ c ( R n ) is the set of all L ∞ ( R n )-functions with compactsupport. Then, for any p ∈ (0 , ∞ ), we can choose a sequence { f j } ∞ j =1 of compactlysupported smooth functions such that f j converges to f in L p ( R n ) as j → ∞ . Bypassing to a subsequence, we may assume that f j converges to f , almost everywherein R n as j → ∞ . Thus, by the Fatou lemma, the inequality k f ( Dϕ ( x ) · ) k M pq ≤ lim inf j →∞ k f j ( Dϕ ( x ) · ) k M pq holds. As we have proved the assertion for f j , we have k f j ( Dϕ ( x ) · ) k M pq ≤ K k f j k M pq . As L p ( R n ) is embedded into M pq ( R n ) (see Remark 1.9), f j converges to f in M pq ( R n ) as j → ∞ . Consequently,lim inf j →∞ k f j k M pq = k f k M pq . By combining these observations, the following estimate holds: k f ( Dϕ ( x ) · ) k M pq ≤ K k f k M pq . Finally, let f ∈ M pq ( R n ). For k ∈ N , we set f k ≡ f χ [ − k,k ] n χ [0 ,k ] ( | f | ) ∈ L ∞ c ( R n ).Then, we have k f k ( Dϕ ( x ) · ) k M pq ≤ K k f k k M pq ≤ K k f k M pq according to the previous paragraph. By using the Fatou lemma again, we obtain k f ( Dϕ ( x ) · ) k M pq ≤ K k f k M pq , as required. (cid:3) Let ϕ : R n → R n be a diffeomorphism and Dϕ : R n → M n ( R ) be the Jacobimatrix of ϕ . For x ∈ R n , the Jacobi matrix Dϕ ( x ) can be decomposed by theSingular value decomposition (see Lemma 2.2 below) as(2.1) Dϕ ( x ) = U AV, where A = A ( x ) = diag( α ( x ) , . . . , α n ( x )) is a diagonal matrix with havingpositive components satisfying α ( x ) ≤ · · · ≤ α n ( x ), and U = U ( x ) and V = V ( x ) are orthogonal matrices. Lemma 2.2 (Singular value decomposition) . Let A be an n × n real regular ma-trix, and α , . . . , α n > be the singular values of A . Then, there exist orthogonalmatrices U and V such that U AV = diag( α , . . . , α n ) . OUNDEDNESS OF COMPOSITION OPERATORS ON MORREY SPACES 7
Now, by the definition of the Morrey norm k · k M pq , and a simple computation,we have the equivalence n np − nq k C A ( x ) k M pq →M pq ≤ k C Dϕ ( x ) k M pq →M pq ≤ n − np + nq k C A ( x ) k M pq →M pq . Here, the operator norms k · k M pq →M pq of the composition operators induced by theorthogonal matrices are bounded above by a constant independent of the selectionof the rotation matrices. Therefore, we have the following lemma: Lemma 2.3.
Let < q ≤ p < ∞ . Suppose that a diffeomorphism ϕ : R n → R n ,induces a bounded composition operator C ϕ on M pq ( R n ) . Let α ( x ) , . . . , α n ( x ) bethe singular values of Dϕ ( x ) , and let us denote A ( x ) := diag( α ( x ) , . . . , α n ( x )) .Then, the operator norm of C A ( x ) is bounded above by a constant independent of x . Hereafter, we use shorthand A ( x ) . B ( x ) to denote estimate A ( x ) ≤ CB ( x )with some constant C > x . Notation A ( x ) ∼ B ( x ) represents A ( x ) . B (x) and B ( x ) . A ( x ). Proposition 2.4.
Let < q ≤ p < ∞ , x ∈ R n , and ϕ : R n → R n be dif-feomorphism. If the composition operators C ϕ and C ϕ − induced by ϕ and ϕ − ,respectively are bounded on the Morrey space M pq ( R n ) , then we have α ( x ) × · · · × α n ( x ) ∼ , where α ( x ) , . . . , α n ( x ) are the singular values of Dϕ ( x ) . This proposition can be proved by combining Lemma 2.1 and Lemma 2.5 below.
Lemma 2.5.
Let < q ≤ p < ∞ and { a , . . . , a n } ⊂ R > be a positive sequenceand set D ≡ diag( a , . . . , a n ) . Then, the following estimate holds: a × · · · × a n ≥ k C D k − p M pq →M pq . Proof.
We introduce matrix W ∈ M n ( R ) corresponding to the transform( x , x , . . . , x n ) ( x , x , . . . , x n , x ) . Then, by a simple computation, for any k ∈ { , · · · , n } , we observe that identities W − k DW k = diag( a n − k +1 , a n − k +2 , . . . , a n , a , a , . . . , a n − k ) and k C W − k DW k k M pq →M pq = k C D k M pq →M pq hold. Noting that the identity n Y k =1 W − k DW k = a a . . . a n E holds, where E ∈ M n ( R ) denotes the identity matrix, the equality (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C n Q k =1 W − k DW k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M pq →M pq = ( a a × . . . × a n ) − np holds. By combining this and identity C n Q k =1 W − k DW k = n Q k =1 C W − k DW k , the conclu-sion of this lemma is proved. (cid:3) To obtain the bi-Lipschitz continuity of ϕ , we use the following proposition,which is obtained using the mean value theorem. NAOYA HATANO, MASAHIRO IKEDA, ISAO ISHIKAWA, AND YOSHIHIRO SAWANO
Proposition 2.6.
Let ϕ : R n → R n be a diffeomorphism. Let α ( x ) be a minimalsingular value. If there exists a positive constant C > independent of x such thatfor all x ∈ R n , (2.2) α ( x ) ≥ C, then the inverse function ϕ − of ϕ is Lipschitz.Proof of Proposition 2.6. x, ˜ x ∈ R n are fixed. As mapping ϕ − is differentiable onthe line segment between x and ˜ x , by the mean value theorem, we can considerpoint x on the line segment between x and ˜ x and obtain | ϕ − ( x ) − ϕ − (˜ x ) | = k Dϕ − ( x ) k F | x − ˜ x | , where the quantity k A k F is a Frobenius norm defined by p tr( A T A ) for matrix A .Now, using the decomposition (2.1), we can calculate k Dϕ − ( x ) k F = n X j =1 (cid:12)(cid:12)(cid:12)(cid:12) α j ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ √ nC . Consequently, we obtain the Lipschitz continuity of ϕ − . (cid:3) According to this proposition, to obtain Theorem 1.6, it suffices to show thatthere exists a positive constant
C > x ∈ R n , the estimate(2.2) holds. We divide the proof of (2.2) into the two cases p ≤ nq and nq ≤ p .2.3. Proof of (2.2) in the case p ≤ nq . To obtain the estimate (2.2), we estimatethe operator norm of the diagonal matrices A ( x ) in the decomposition (2.1) asfollows using Lemma 2.7 and Proposition 2.8 below. Lemma 2.7.
Let n ≥ m ≥ , < q ≤ nm q ≤ p ≤ nm − q < ∞ and ≤ a ≤ · · · ≤ a n − . Then, we have k χ [0 , × [0 ,a ] ×···× [0 ,a n − ] k M pq = a q · · · a q m − a np − mq m − . Proof.
As we have to consider only cubes of form [0 , R ] n for R > k · k M pq , we have identity k χ [0 , × [0 ,a ] ×···× [0 ,a n − ] k M pq = sup R> R np − nq { min(1 , R ) min( a , R ) · · · min( a n − , R ) } q By considering the case of R = 1 , a , . . . , a n − , we can determine the supremum onthe right-hand side of the above identity as follows:(2.3) k χ [0 , × [0 ,a ] ×···× [0 ,a n − ] k M pq = max (cid:18) , a np − q , a q a np − q , . . . , a q · · · a q n − a np − n − q n − (cid:19) . Here, according to assumption p ≤ nm − q , we observe that a q · · · a q m − a np − mq m − ≥ a q · · · a q m − a np − m − q m − ≥ · · · ≥ a np − q ≥ . OUNDEDNESS OF COMPOSITION OPERATORS ON MORREY SPACES 9
According to the assumption nm q ≤ p , we calculate a q · · · a q m − a np − mq m − ≥ a q · · · a q m a np − m +1 q m ≥ · · · ≥ a q · · · a q n − a np − nq n − , to conclude that quantity a q · · · a q m − a np − mq m − is the largest when taking the maxi-mum in the equation (2.3). Hence, this is the desired result. (cid:3) Proposition 2.8.
Let n ≥ m ≥ , < q ≤ nm q ≤ p ≤ nm − q < ∞ and ≤ a ≤ · · · ≤ a n − . Then, we have k C diag(1 ,a ,...,a n − ) k M pq →M pq ≥ a − q · · · a − q m − a − np + mq m − . Proof.
Them, we use k χ [0 , × [0 ,R ] ×···× [0 ,R n − ] k M pq = R q · · · R q m − R np − mq m − , and k χ [0 , × [0 ,a − R ] ×···× [0 ,a − n − R n − ] k M pq = ( a − R ) q · · · ( a − m − R m − ) q ( a − m − R m − ) np − mq for 1 ≤ R ≤ · · · ≤ R n − with 1 ≤ a − R ≤ · · · ≤ a − n − R n − . (cid:3) Now, we prove the estimate (2.2). It suffices to consider case nm q < p ≤ nm − q ,for each m = 2 , . . . , n . According to Lemma 2.3 and Proposition 2.8, we calculate1 & k C A ( x ) k M pq →M pq (2.4) ≥ α ( x ) − np Y i ∈ I (cid:18) α i ( x ) α ( x ) (cid:19) − q · (cid:18) α j ( x ) α ( x ) (cid:19) − np + mq = α ( x ) Y i ∈ I α i ( x ) ! − q α j ( x ) − np + mq , where I is a subset of { , . . . , n } such that ♯I = m − j ∈ I . Combining theestmate (2.4) and Proposition 2.4, we have1 & Y I ⊂{ ,...,n } ♯I = m − Y j ∈ I α ( x ) Y i ∈ I α i ( x ) ! − q α j ( x ) − np + mq = α ( x ) − m − q ( n − m − )( α ( x ) · · · α n ( x ))( − np + q )( n − m − ) ∼ α ( x ) − m − q ( n − m − ) + ( np − q )( n − m − ) = α ( x )( np − nq )( n − m − )and then α ( x ) & . Proof of (2.2) in the case nq ≤ p . Let nq ≤ p . Using Lemmas 2.3 and 2.10,we obtain the estimate (2.2). To prove Lemma 2.10, we will use Lemma 2.9 below. Lemma 2.9.
Let < q < nq ≤ p < ∞ . Then, χ [0 , × R n − ∈ M pq ( R n ) . Proof.
We calculate k χ [0 , × R n − k M pq = sup R> R np − nq min(1 , R ) q R n − q = sup R> R np − q min(1 , R ) q = 1 < ∞ . (cid:3) Lemma 2.10.
Let < a ≤ · · · ≤ a n . We assume that D = diag( a , a , . . . , a n ) induces a bounded composition operator on the Morrey space M pq ( R n ) with theoperator norm M . Moreover, we assume that < q < nq ≤ p < ∞ . Then, a ≥ M − pn .Proof. Note that χ [0 , × R n − ◦ ( a E ) = χ [0 ,a − ] × R n − = χ [0 , × R n − ◦ D . Usingscaling, we calculate a − np k χ [0 , × R n − k M pq = k χ [0 , × R n − ◦ ( a n E ) k M pq = k χ [0 , × R n − ◦ D k M pq ≤ M k χ [0 , × R n − k M pq . Thus, according to Lemma 2.9, this is the desired result. (cid:3) Examples
In this section, we present some examples and counterexamples. In Example3.1, the mapping inducing the composition operator satisfies the assumption inTheorem 1.5. In Example 3.2, the mapping inducing the composition operatordoes not satisfy the assumption in Theorem 1.5; however, the composition operatoris bounded on the Morrey spaces. Example 3.3 presents a counterexample of cases n ≥ q = p in Theorem 1.6. Example 3.1.
The affine map ϕ , written as ϕ ( x ) = Ax + b for some A ∈ GL( n ; R )and b ∈ R n induces the composition operator C ϕ bounded on the Morrey space M pq ( R n ) whenever 0 < q ≤ p < ∞ . This follows from the fact that mapping ϕ satisfies the assumption of Theorem 1.5. Example 3.2.
Let n = 1 and 1 < p < ∞ . Then, the composition operator inducedby ϕ : R → R , ϕ ( x ) ≡ ( e x − , if x ≥ ,x, if x < , is bounded on M p ( R ) and ϕ satisfies the volume estimate (1.2); however, ϕ is notLipschitz.Here, we prove that the composition mapping C ϕ induced by ϕ : R → R isbounded on M p ( R ). It suffices to show that, for all a ≥ b > b − a ) p − ˆ ba | C ϕ f ( x ) | d x . k f k M p Now, we check inequality (3.1). If 0 < b − a ≤
1, through the change of variablesas y = e x − e b − e a ∼ e a ( b − a ), we obtain( b − a ) p − ˆ ba | C ϕ f ( x ) | d x = ( b − a ) p − ˆ e b − e a − | f ( y ) | d yy + 1 ≤ { e a ( b − a ) } p − · e − ap ˆ e b − e a − | f ( y ) | d y . k f k M p . OUNDEDNESS OF COMPOSITION OPERATORS ON MORREY SPACES 11
Furthermore, when b − a >
1, or equivalently, ( b − a ) − <
1, we calculate( b − a ) p − ˆ ba | C ϕ f ( x ) | d x ≤ ∞ X j =0 ˆ j +1 − j − | f ( y ) | d yy + 1 ≤ ∞ X j =0 − jp · (2 j ) p − ˆ j +1 − j − | f ( y ) | d y . k f k M p as desired. Example 3.3.
Let ϕ ( x , x ) ≡ (cid:18) x + x , x x + 1 (cid:19) be a diffeomorphism on R . Let us consider the boundedness of C ϕ on M pq ( R ).In the case of p = q , C ϕ is bounded, Dϕ ( x , x ) has determinant 1. In contrast, inthe case of p > q , C ϕ is not bounded; in fact, the first component is not Lipschitz.4. Boundedness of composition operators on weak type spaces
To prove Theorem 1.13, we use the following identity.
Remark . Through a simple calculation, we have k χ E k W B = k χ E k B for allmeasurable sets E in R n . Proof.
First, we assume that the composition operator C ϕ is bounded on W B ( R n ),that is, there exists a constant K such that the estimate k C ϕ f k W B ≤ K k f k W B holds for any f ∈ W B ( R n ). Then, upon choosing f = χ E , the estimates k χ ϕ − ( E ) k W B = k C ϕ χ E k W B ≤ K k χ E k W B hold. By using Remark 4.1, we conclude that k χ ϕ − ( E ) k B ≤ K k χ E k B . Second, we assume the condition (1.4). Considering E = { x ∈ R n : | f ( x ) | > λ } , wehave k C ϕ f k W B = sup λ> λ k χ ϕ − ( { x ∈ X : | f ( x ) | >λ } ) k B ≤ K sup λ> λ k χ { x ∈ X : | f ( x ) | >λ } k B = k f k W B . Finally, the equation(4.1) k C ϕ k W B → W B ≤ sup E k χ ϕ − ( E ) k B k χ E k B is trivial. According to the definition of the operator norm k · k W B → W B ,(4.2) k C ϕ k W B → W B ≥ sup E k χ ϕ − ( E ) k B k χ E k B Combining these estimates (4.1) and (4.2), we obtain equation (1.5). (cid:3)
The weak type spaces generated by the Banach lattice are essential.
Definition 4.2.
We say that a Banach space ( B ( R n ) , k · k B ) contained in L ( R n )is a Banach lattice if the inequality k f k B ≤ k g k B holds for all f, g ∈ B ( R n ) thatsatisfies | f | ≤ | g | , a.e.. Remark . If B ( R n ) is a Banach lattice (see Definition 4.2), then B ( R n ) is em-bedded in W B ( R n ).Now, as the special case of the Morrey space B ( R n ) = M pq ( R n ), in Theorem1.13, we have Theorem 1.10.In Theorem 1.10, through real interpolation, it is known thatW M pq ( R n ) = [ M prqr ( R n ) , L ∞ ( R n )] − r, ∞ (see [4]). Here, as the L ∞ ( R n )-boundedness of the composition operators is trivialand the M pq ( R n )-boundedness and M prqr ( R n )-boundedness of composition opera-tors, for r >
0, are equivalent owing to the fact that | C ϕ f | r = C ϕ [ | f | r ] for map-ping ϕ , then we obtain that the boundedness “ C ϕ : M pq ( R n ) → M pq ( R n ) implies C ϕ : W M pq ( R n ) → W M pq ( R n )”. Acknowledgement . This work was supported by a JST CREST Grant (Num-ber JPMJCR1913, Japan). This work was also supported by the RIKEN JuniorResearch Associate Program. The second author is supported by a Grant-in-Aidfor Young Scientists Research (No.19K14581), Japan Society for the Promotion ofScience. The fourth author is supported by a Grant-in-Aid for Scientific Research(C) (19K03546), Japan Society for the Promotion of Science.
References [1] S. C. Arora, G. Datt, S. Verma, Composition operators on Lorentz spaces, Bull. Austral.Math. Soc. 76 (2007), no. 2, 205–214.[2] Y. Cui, H. Hudzik, R. Kumar and L. Maligranda, Composition operators in Orlicz spaces,(English summary) J. Aust. Math. Soc. 76 (2004), no. 2, 189–206.[3] R. E. Castillo, F. A. Vallejo Narvaez and J. C. Ramos Fernndez, Multiplication and compo-sition operators on weak L p spaces, Bull. Malays. Math. Sci. Soc. 38 (2015), no. 3, 927–973.[4] M. Cwikel and A. Gulisashvili, Interpolation on families of characteristic functions, StudiaMath. 138 (2000), no. 3, 209–224.[5] N. A. Evseev, A bounded composition operator on Lorentz spaces, (Russian) Mat. Zametki102 (2017), no. 6, 836843; translation in Math. Notes 102 (2017), no. 5–6, 763–769.[6] N. Evseev and A. Menovschikov, Bounded operators on mixed norm Lebesgue spaces, Com-plex Anal. Oper. Theory 13 (2019), no. 5, 2239–2258.[7] N. A. Evseev and A. V. Menovshchikov, The composition operator on mixed-norm Lebesguespaces, (Russian) Mat. Zametki 105 (2019), no. 6, 816–823; translation in Math. Notes 105(2019), no. 5–6, 812–817[8] L. Grafakos, Classical Fourier Analysis, Texts in Mathmatics, Springer, New York, Thirdedition (2014).[9] M. Ikeda, I. Ishikawa and Y. Sawano, Composition operators on reproducing kernel Hilbertspaces with analytic positive definite functions, arXiv:1911.11992.[10] I. Ishikawa, K. Fujii, M. Ikeda, Y. Hashimoto and Y. Kawahara Metric on Nonlinear Dy-namical Systems with Perron–Frobenius Operators, in Advances in Neural Information Pro-cessing Systems 31 (2018), 911919.[11] Y. Kawahara, Dynamic Mode Decomposition with Reproducing Kernels for Koopman Spec-tral Analysis, Neural Information Processing Systems, Advances in Neural Information Pro-cessing Systems 29 (NIPS 2016), 9 pages.[12] B.O. Koopman. Hamiltonian systems and transformation in Hilbert space. Proceedings ofthe National Academy of Sciences, 17(5) (1931), 315318.[13] R. Kumar, Composition operators on Orlicz spaces, Integral Equations Operator Theory 29(1997), no. 1, 17–22.[14] K. Raj and S. K. Sharma, Composition operators on Musielak-Orlicz spaces of Bochnertype, Math. Bohem. 137 (2012), no. 4, 449–457.[15] Y. Sawano, A non-dense subspace in M pq with 1 < q < p < ∞ . Trans. A. Razmadze Math.Inst. 171 (2017), no. 3, 379–380. OUNDEDNESS OF COMPOSITION OPERATORS ON MORREY SPACES 13 [16] R. K. Singh, Composition operators induced by rational functions, Proc. Amer. Math. Soc.59 (1976), no. 2, 329–333.(Naoya Hatano)
Department of Mathematics, Faculty of Science and Technology,Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan/Center for Ad-vanced Intelligence Project, RIKEN, Japan, and Department of Mathematics, ChuoUniversity, 1-13-27, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan, (Masahiro Ikeda)
Center for Advanced Intelligence Project, RIKEN, Japan/Departmentof Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi,Kohoku-ku, Yokohama 223-8522, Japan (Isao Ishikawa)
Center for Advanced Intelligence Project, RIKEN, Japan, and De-partment of Engineering for Production and Environment, Graduate School of Scienceand Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, Ehime 790-8577, Japan, (Yoshihiro Sawano)(Yoshihiro Sawano)