Boundedness of composition operators on Morrey spaces and weak Morrey spaces

In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces. The primary aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator induced by a diffeomorphism on Morrey spaces. In particular, detailed information is derived from the boundedness, i.e., the bi-Lipschitz continuity of the mapping that induces the composition operator follows from the continuity of the composition mapping. The idea of the proof is to determine the Morrey norm of the characteristic functions, and employ a specific function composed of a characteristic function. As the specific function belongs to Morrey spaces but not to Lebesgue spaces, the result reveals a new phenomenon not observed in Lebesgue spaces. Subsequently, we prove the boundedness of the composition operator induced by a mapping that satisfies a suitable volume estimate on general weak-type spaces generated by normed spaces. As a corollary, a necessary and sufficient condition for the boundedness of the composition operator on weak Morrey spaces is provided.


Introduction
In this study, we investigate the boundedness of composition operators on Morrey spaces and weak Morrey spaces. The composition operator C ϕ induced by a mapping ϕ is a linear operator defined by C ϕ f ≡ f • ϕ, where f • ϕ represents the function composition. The composition operator is also called the Koopman operator 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 0 (X, µ) be the set of all µ-measurable functions on X. We provide a precise definition of the composition operators induced 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 −1 (E)) = 0 for each µ-measurable null set E. Let V and W be function spaces contained in L 0 (X, µ). The composition operator 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 of the composition operator on the Lebesgue space L p (X, µ). Henceforth, we denote by L 0 (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: Theorem 1.2 ( [16]). Let 0 < p < ∞. Then, the composition operator C ϕ induced by ϕ : X → X is bounded on the Lebesgue space L p (X, µ) if and only if there exists a constant K = K(ϕ) such that for all µ-measurable sets E in R n , µ(ϕ −1 (E)) ≤ Kµ(E).
In this case, the operator norm is given by The boundedness of the composition operator on L ∞ (X, µ) easily follows from the 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 acting on Morrey spaces and weak Morrey spaces.
The first aim of this strudy is to investigate a necessary and sufficient condition on the boundedness of the composition operator C ϕ on Morrey spaces. Subsequently, 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 0 (R n ). We denote by |E| the volume of a measurable set E ⊂ R n . Let χ A : R n → R ≥0 be an indicator function for a subset A ⊂ R n , which is defined as Now, we recall the definition of Morrey spaces on R n .
where Q denotes the family of all cubes parallel to the coordinate axis in R n .
From the Hölder inequality, we observe that the Lebesgue space [15]. We now state the main results of the present paper. The following theorem provides a sufficient condition on the boundedness of the composition operator C ϕ on the Morrey space M p q (R n ). Theorem 1.5. Let 0 < q ≤ p < ∞. Then, the composition operator C ϕ induced by ϕ : R n → R n is bounded on the Morrey space M p q (R n ), if ϕ is a Lipschitz map that satisfies the volume estimate for all measurable sets E in R n and some constant k independent of E. In particular, we obtain where L > 0 is a Lipschitz constant of ϕ, and E in the supremum moves over all the Lebesgue measurable sets E satisfying 0 < |E| < ∞.
Conversely, as stated in the following theorem, if ϕ : R n → R n is a diffeomorphism, then the M p q (R n )-boundedness of the composition operators C ϕ and C ϕ −1 indicates that ϕ is bi-Lipschitz. Note that any bi-Lipschitz mapping satisfies the assumption of Theorem 1.5. Theorem 1.6. Let n ∈ N, and ϕ : R n → R n be a diffeomorphism in the sense that ϕ and its inverse ϕ −1 are differentiable. Suppose 0 < q < p < ∞, or q = p and n = 1. If the composition operators C ϕ and C ϕ −1 induced by maps ϕ and ϕ −1 , respectively, are bounded on M p q (R n ), then ϕ is bi-Lipschitz. Remark 1.7. In the case of p = q and n = 1, Theorem 1.6 reduced to Theorem 1.2 according to [16]. Unless n = 1, condition q < p is essential in the following sense. If n ≥ 2 and q = p, then the same conclusion as in Theorem 1.6 fails. We present a counterexample in Example 3.3 in Section 4. Noting that the Morrey space M p p (R n ) coincides with the Lebesgue space L p (R n ) (see Remark 1.9), we observe a new phenomenon from Theorem 1.6.
Subsequently, we investigate the characterization of the boundedness of the composition operators acting on weak Morrey spaces, which are defined as follows: The weak Morrey space WM p q (R n ) has the following basic properties: Remark 1.9. Let 0 < q < p < ∞. Then, we have [8,Chapter 1] for more).
The following theorem provides a necessary and sufficient condition on the boundedness of the composition operator on weak Morrey spaces. Theorem 1.10. Let 0 < q ≤ p < ∞, and let ϕ : R n → R n be a measurable function. Then, ϕ generates the composition operator C ϕ which is bounded on the weak Morrey space WM p q (R n ) if and only if there exists a constant K such that for all measurable sets E in R n , the estimate holds. In particular, we obtain where the supremum is taken over all the measurable sets E in R n with 0 < χ E M p q < ∞. Remark 1.11.
(1) Theorem 1.10 indicates that the composition operator C ϕ is bounded on weak Morrey spaces, once it is bounded on Morrey spaces (see Section 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 more general framework.
Now, we can rewrite Theorem 1.10 as follows: be a normed space. Then, the composition induced by ϕ is bounded on the weak-type space (WB(R n ), · WB ) if and only if there exists a constant K such that for all measurable sets E in R n , the estimate holds. In particular, we obtain where the supremum is taken over all the measurable sets E in R n with 0 < χ E B < ∞.
The remainder of this paper is organized as follows: In Section 2, we prove Theorems 1.5 and 1.6. In Section 3, we present some examples and counterexamples of the mapping that induces the composition operator to be bounded on Morrey spaces. In Section 4, we prove Theorem 1.13.

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 is provided in Subsection 2.1. However, Theorem 1.6 is more difficult to prove. In Subsection 2.2, we reduce matters to the linear setting. We divide its proof into two steps: we consider case p ≤ nq in Subsection 2.3 and case nq ≤ p in Subsection 2.4.

Proof of Theorem 1.5.
Proof of Theorem 1.5. A cube, Q ∈ Q, is fixed. We note that, according to the Lipschitz continuity of ϕ, the estimates diam(ϕ(Q)) := sup hold; thus, there exist cubes Q 1 , Q 2 ∈ Q such that As ϕ satisfies condition (1.2), we can apply the L q (R n )-boundedness of the composition operators (Theorem 1.2) to obtain Moreover, by applying the equation (1.1), we obtain (1.3), which completes the proof of the theorem.

2.2.
Reduction of the diffeomorphism to the linear setting. In the following, for a differentiable vector-valued function ϕ = (ϕ 1 , . . . , ϕ n ) T on R n , we denote by Dϕ the Jacobi matrix of ϕ, that is, In this subsection, by applying the following lemma (Lemma 2.1), we reduce the diffeomorphism 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 will show that ϕ is bi-Lipschitz (see Proposition 2.6 below).
Suppose that a diffeomorphism ϕ : R n → R n induces a bounded composition operator C ϕ from M p q (R n ) to itself. Then, there exists a positive constant k > 0 such that for all x 0 ∈ R n and f ∈ M p q (R n ). In particular, the operator norm of C Dϕ(x0) is bounded above by a constant independent of x 0 .
Proof of Lemma 2.1. Set K ≡ C ϕ M p q →M p q < ∞. First, we prove the assertion for f ∈ C ∞ c (R n ), where C ∞ c (R n ) is the set of all smooth functions with compact support. Let t > 0. We calculate is the set of all L ∞ (R n )-functions with compact support. Then, for any p ∈ (0, ∞), we can choose a sequence {f j } ∞ j=1 of compactly supported smooth functions such that f j converges to f in L p (R n ) as j → ∞. By passing to a subsequence, we may assume that f j converges to f , almost everywhere in R n as j → ∞. Thus, by the Fatou lemma, the inequality holds. As we have proved the assertion for f j , we have By combining these observations, the following estimate holds: according to the previous paragraph. By using the Fatou lemma again, we obtain Let ϕ : R n → R n be a diffeomorphism and Dϕ : R n → M n (R) be the Jacobi matrix of ϕ. For x 0 ∈ R n , the Jacobi matrix Dϕ(x 0 ) can be decomposed by the Singular value decomposition (see Lemma 2.2 below) as x 0 ), . . . , α n (x 0 )) is a diagonal matrix with having positive components satisfying α 1 (x 0 ) ≤ · · · ≤ α n (x 0 ), and U = U (x 0 ) and V = V (x 0 ) are orthogonal matrices.

Lemma 2.2 (Singular value decomposition). Let
A be an n × n real regular matrix, and α 1 , . . . , α n > 0 be the singular values of A. Then, there exist orthogonal matrices U and V such that U AV = diag(α 1 , . . . , α n ). Now, by the definition of the Morrey norm · M p q , and a simple computation, we have the equivalence Here, the operator norms · M p q →M p q of the composition operators induced by the orthogonal matrices are bounded above by a constant independent of the selection of the rotation matrices. Therefore, we have the following lemma: Suppose that a diffeomorphism ϕ : R n → R n , induces a bounded composition operator C ϕ on M p q (R n ). Let α 1 (x 0 ), . . . , α n (x 0 ) be the singular values of Dϕ(x 0 ), and let us denote A(x 0 ) := diag(α 1 (x 0 ), . . . , α n (x 0 )). Then, the operator norm of C A(x0) is bounded above by a constant independent of x 0 .

Hereafter, we use shorthand
Proposition 2.4. Let 0 < q ≤ p < ∞, x 0 ∈ R n , and ϕ : R n → R n be diffeomorphism. If the composition operators C ϕ and C ϕ −1 induced by ϕ and ϕ −1 , respectively are bounded on the Morrey space M p q (R n ), then we have This proposition can be proved by combining Lemma 2.1 and Lemma 2.5 below.
Then, by a simple computation, for any k ∈ {1, · · · , n}, we observe that identities W −k DW k = diag(a n−k+1 , a n−k+2 , . . . , a n , a 1 , a 2 , . . . , a n−k ) and Noting that the identity n k=1 W −k DW k = a 1 a 2 . . . a n E holds, where E ∈ M n (R) denotes the identity matrix, the equality holds. By combining this and identity C n k=1 W −k DW k = n k=1 C W −k DW k , the conclusion of this lemma is proved.
To obtain the bi-Lipschitz continuity of ϕ, we use the following proposition, which is obtained using the mean value theorem. Proposition 2.6. Let ϕ : R n → R n be a diffeomorphism. Let α 1 (x 0 ) be a minimal singular value. If there exists a positive constant C > 0 independent of x 0 such that for all x 0 ∈ R n , then the inverse function ϕ −1 of ϕ is Lipschitz.
Proof of Proposition 2.6.
x,x ∈ R n are fixed. As mapping ϕ −1 is differentiable on the line segment between x andx, by the mean value theorem, we can consider point x 0 on the line segment between x andx and obtain where the quantity A F is a Frobenius norm defined by tr(A T A) for matrix A. Now, using the decomposition (2.1), we can calculate Consequently, we obtain the Lipschitz continuity of ϕ −1 .
According to this proposition, to obtain Theorem 1.6, it suffices to show that there exists a positive constant C > 0 such that for each x 0 ∈ 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 estimate the operator norm of the diagonal matrices A(x 0 ) in the decomposition (2.1) as follows using Lemma 2.7 and Proposition 2.8 below.
Lemma 2.7. Let n ≥ m ≥ 2, 0 < q ≤ n m q ≤ p ≤ n m − 1 q < ∞ and 1 ≤ a 1 ≤ · · · ≤ a n−1 . Then, we have Proof. As we have to consider only cubes of form [0, R] n for R > 0 as the candidates for supremum in the Morrey norm · M p q , we have identity By considering the case of R = 1, a 1 , . . . , a n−1 , we can determine the supremum on the right-hand side of the above identity as follows: Here, according to assumption p ≤ n m − 1 q, we observe that 1 ≤ a 1 ≤ · · · ≤ a n−1 . Then, we have Proof. Them, we use where I is a subset of {2, . . . , n} such that ♯I = m − 1 and j ∈ I.

2.4.
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.
Proof. We calculate Lemma 2.10. Let 0 < a 1 ≤ · · · ≤ a n . We assume that D = diag(a 1 , a 2 , . . . , a n ) induces a bounded composition operator on the Morrey space M p q (R n ) with the operator norm M . Moreover, we assume that 0 < q < nq ≤ p < ∞. Then, Proof. Note that Using scaling, we calculate Thus, according to Lemma 2.9, this is the desired result.

Examples
In this section, we present some examples and counterexamples. In Example 3.1, the mapping inducing the composition operator satisfies the assumption in Theorem 1.5. In Example 3.2, the mapping inducing the composition operator does not satisfy the assumption in Theorem 1.5; however, the composition operator is bounded on the Morrey spaces. Example 3.3 presents a counterexample of cases n ≥ 2 and 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 p q (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 induced by ϕ : R → R, is bounded on M p 1 (R) and ϕ satisfies the volume estimate (1.2); however, ϕ is not Lipschitz.
Here, we prove that the composition mapping C ϕ induced by ϕ : R → R is bounded on M p 1 (R). It suffices to show that, for all a ≥ 0 and b > 0, Now, we check inequality (3.1). If 0 < b − a ≤ 1, through the change of variables as y = e x − 1 and the fact that e b − e a ∼ e a (b − a), we obtain Furthermore, when b − a > 1, or equivalently, (b − a) −1 < 1, we calculate x 2 3x 1 2 + 1 be a diffeomorphism on R 2 . Let us consider the boundedness of C ϕ on M p q (R 2 ). In the case of p = q, C ϕ is bounded, Dϕ(x 1 , x 2 ) has determinant 1. In contrast, in the case of p > q, C ϕ is not bounded; in fact, the first component is not Lipschitz.

Boundedness of composition operators on weak type spaces
To prove Theorem 1.13, we use the following identity.
Remark 4.1. Through a simple calculation, we have χ E WB = χ E B for all measurable sets E in R n .
Proof. First, we assume that the composition operator C ϕ is bounded on WB(R n ), that is, there exists a constant K such that the estimate C ϕ f WB ≤ K f WB holds for any f ∈ WB(R n ). Then, upon choosing f = χ E , the estimates χ ϕ −1 (E) WB = C ϕ χ E WB ≤ K χ E WB hold. By using Remark 4.1, we conclude that Second, we assume the condition (1.4). Considering E = {x ∈ R n : |f (x)| > λ}, we have Finally, the equation is trivial. According to the definition of the operator norm · WB→WB , Combining these estimates (4.1) and (4.2), we obtain equation (1.5).
The weak type spaces generated by the Banach lattice are essential.
Definition 4.2. We say that a Banach space (B(R n ), · B ) contained in L 0 (R n ) is a Banach lattice if the inequality f B ≤ g B holds for all f, g ∈ B(R n ) that satisfies |f | ≤ |g|, a.e.. Now, as the special case of the Morrey space B(R n ) = M p q (R n ), in Theorem 1.13, we have Theorem 1.10.
In Theorem 1.10, through real interpolation, it is known that WM p q (R n ) = [M pr qr (R n ), L ∞ (R n )] 1−r,∞ (see [4]). Here, as the L ∞ (R n )-boundedness of the composition operators is trivial and the M p q (R n )-boundedness and M pr qr (R n )-boundedness of composition operators, for r > 0, are equivalent owing to the fact that |C ϕ f | r = C ϕ [|f | r ] for mapping ϕ, then we obtain that the boundedness "C ϕ : M p q (R n ) → M p q (R n ) implies C ϕ : WM p q (R n ) → WM p q (R n )".