Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 82, pp. 1–17. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.82 HARDY OPERATORS AND COMMUTATORS ON GENERALIZED CENTRAL FUNCTION SPACES LE TRUNG NGHIA Communicated by Jesus Ildefonso Diaz Abstract. In this article, we study the boundedness of operators of Hardy type on generalized central function spaces, such as the generalized central Hardy space HAp,r φ (Rn), the generalized central Morrey space Ṁp,r φ (Rn), and the generalized central Campanato space ˙CMO p,r φ (Rn), with p ∈ (1,∞), and φ(t) : (0,∞) → (0,∞). We first show that HAp′,r′ φ (Rn) is the predual of ˙CMO p,r φ (Rn). After that, we investigate the boundedness of operators of Hardy type on those spaces. By duality, we obtain the boundedness characterization of function b ∈ ˙CMO p,r φ (Rn) via the Ṁ p,r φ (Rn)-boundedness of commutator [b,H∗]. 1. Introduction and main results Firstly, we introduce a singular solution outside Sobolev spaces and its description via central Morrey and Campanato spaces. Problem setting. We consider the semilinear elliptic equation on the unit ball B := B1(0) ⊂ Rn (with n ≥ 3) −∆u− µ |x|2 u = uq in B, u > 0, u ∈ H1 loc(B \ {0}) (1.1) with • µ ∈ ( 0, ( n−2 2 )2) (subcritical Hardy potential), • q > 1 subcritical, • u = 0 on ∂B (in weak sense). Explicit singular solution. A classical singular approximate solution is given by u(x) = C|x|−γ , where γ = n− 2 2 − √(n− 2 2 )2 − µ. (1.2) This function is • Not in H1(B) because∫ B |∇u|2dx ∼ ∫ 1 0 rn−1−2(γ+1)dr = ∞ • In H1 loc(B \ {0}), with singularity at the origin. 2020 Mathematics Subject Classification. 42B20, 42B35, 42B30, 46A20. Key words and phrases. Hardy operators; commutator; generalized central function space; central atomic space. ©2025. This work is licensed under a CC BY 4.0 license. Submitted July 2, 2025. Published August 8, 2025. 1 2 L. T. NGHIA EJDE-2025/82 Why the solution is independent of q. Let us verify whether u(x) = C|x|−γ solves the full nonlinear equation −∆u− µ |x|2 u = uq. (1.3) We compute ∆(|x|−γ) = γ(γ + 2− n)|x|−γ−2, so that −∆u− µ |x|2 u = C [γ(γ + 2− n)− µ] |x|−γ−2. We compare this with uq = Cq|x|−γq, and for equality we must have −γ − 2 = −γq ⇒ γ(q − 1) = 2 ⇒ γ = 2 q − 1 . Hence, the singular function u(x) = |x|−γ solves the full nonlinear equation only when γ = 2 q−1 . Thus, u(x) = |x|−γ is a solution to the *nonlinear* problem only when 2 q − 1 = n− 2 2 − √(n− 2 2 )2 − µ. But in our example, we fixed: γ = n− 2 2 − √(n− 2 2 )2 − µ, which only matches 2 q−1 for a specific q. Therefore, u is not a solution to the nonlinear equation for general q, but it serves as a model to study the singularity and local behavior, independently of the nonlinearity. It serves as a model to study the singular behavior of more general solutions, particularly near the origin. Membership in central Morrey spaces. We test whether u ∈ Lp,λ(B): ∥u∥Lp,λ := sup r<1 r−λ (∫ B(0,r) |u(x)|pdx )1/p <∞. (1.4) Let u(x) = |x|−γ , then∫ B(0,r) |x|−pγdx ∼ ∫ r 0 ρn−1−pγdρ = rn−pγ ⇒ ∥u∥Lp,λ ∼ r−λ+n p −γ . So u ∈ Lp,λ for λ < n p − γ. (1.5) Oscillation near zero: Campanato spaces. Consider the Campanato seminorm ∥f∥2L2,λ Camp := sup r<1 r−λ ∫ B(0,r) |f(x)− fB(0,r)|2dx. (1.6) For f(x) = |x|−γ , the mean oscillation behaves like r−λ ∫ B(0,r) |f(x)− fB(0,r)|2dx ∼ rn−2γ−λ. (1.7) So f ∈ L2,λ Camp(B) for λ < n− 2γ. (1.8) In particular, since BMO corresponds to λ = n, the function f(x) = |x|−γ is not in BMO, but lies in a Campanato space with smaller λ. In summary. the function u(x) = |x|−γ • is not in the Sobolev space H1(B), • is in the central Morrey space Lp,λ(B) for suitable λ, • is in the Campanato space L2,λ(B) for λ < n− 2γ, • is not in BMO, EJDE-2025/82 HARDY OPERATORS AND COMMUTATORS 3 • serves as a barrier function that captures the singularity of the Hardy potential, indepen- dent of the nonlinearity. This demonstrates how generalized central spaces such as Morrey and Campanato capture the behavior of singular solutions to elliptic PDEs with Hardy-type potentials, even when the nonlin- earity is not presented. Building on this observation, our studies are the following twofold. First, we study some generalized central function spaces, such as Ṁp,r φ (Rn), ˙CMO p,r φ (Rn), and HAp,r φ (Rn), where p ∈ (1,∞). Through the paper, we always assume that φ(t) is non-increasing on (0,∞), and t n p φ(t) is nondecreasing on (0,∞). Then, we demonstrate that HAp′,r′ φ (Rn) is the predual of ˙CMO p,r φ (Rn). Second, we investigate the boundedness of operators of Hardy type on those spaces. By duality, we obtain the boundedness characterization of function b in ˙CMO p,r φ (Rn) by means of the boundedness of commutators [b,H] and [b,H∗] in the above central function spaces. Notation: For q ∈ (1,∞), we denote q′ the conjugate exponent, 1 q + 1 q′ = 1. With |Ω| we denote the Lebesgue measure of a measurable set Ω in Rn, and Bt is the ball centered at 0 ∈ Rn with radius t. As usual, we denote a constant by C, which may depend on p, n and is probably different at different occurrences. Finally, we denote A ≲ B if there exists a constant C > 0 such that A ≤ CB. The Hardy operator is defined by H(f)(x) = 1 νn|x|n ∫ |y|<|x| f(y) dy, x ∈ Rn \ {0} , (1.9) and its dual form is H∗(f)(x) = 1 νn ∫ |y|≥|x| f(y) |y|n dy, x ∈ Rn \ {0} , (1.10) where νn = πn/2 Γ(1+n/2) is the volume of unit ball in Rn. In the pioneering work, when n = 1, Hardy [17] established the integral inequality∫ ∞ 0 ( 1 x ∫ x 0 f(t) dt )p dx ≤ ( p p− 1 )p ∫ ∞ 0 f(x)p dx (1.11) for all non-negative f ∈ Lp(R+), with 1 < p <∞. Note that the constant p p−1 is sharp. By considering two-sided averages of f instead of one-sided, (1.11) can be equivalently formu- lated as ∥H(f)∥Lp(R) ≤ p p− 1 ∥f∥Lp(R) . (1.12) Christ-Grafakos [4] extended (1.12) to n-dimension. Furthermore, a sharp bound of weak type (p, p) of H was obtained by the authors in [12]. Specifically, for any 1 ≤ p ≤ ∞ we have ∥H(f)∥Lp,∞ ≤ ∥f∥Lp,r for all f ∈ Lp(Rn). In addition, ∥H∥Lp→Lp,∞ = 1 . It is known that the inequalities of Hardy type play important roles in many areas of mathematics such as analysis, probability and partial differential equations (see, e.g., [1, 2, 4, 11, 16, 18, 21, 23] and the references therein). For example, a slight modification of (1.11) by setting F (x) =∫ x 0 f(t) dt provides us ∫ ∞ 0 F (x)p xp dx ≤ ( p p− 1 )p ∫ ∞ 0 F ′(x)p dx . The analogue of this inequality in Rn for n > 1 is∫ Rn ∣∣f(x) x ∣∣p ≤ ( p n− p )p ∫ Rn |∇f(x)|p dx (1.13) where ∇f is the gradient of f as usual; this holds for all f ∈ C∞ 0 (Rn \ {0}) if n < p <∞, and for all f ∈ C∞ 0 (Rn) if 1 ≤ p < n. The constant is sharp and equality can only be attained by functions f = 0 a.e. 4 L. T. NGHIA EJDE-2025/82 Since the Hardy operators are centrosymmetric, the function spaces, which are characterized by the boundedness of H and H∗ are central ones. For example, Shi-Lu, [25] established the boundedness of H and H∗ in the central Morrey spaces Ṁ p,λ φ (Rn) (see Definition 1.2). Theorem 1.1 (Shi-Lu [25]). Let 1 < p < ∞ and λ ∈ (0, np ). Then H (resp. H∗) is a bounded operator from Ṁp,λ φ (Rn) → Ṁp,λ φ (Rn). Moreover, the boundedness characterization of operators of Hardy type in the homogeneous Herz spaces has been studied by the authors in [13]. Inspired by the above results, we would like to study the boundedness of operators of Hardy type in generalized central function spaces. Therefore, it is convenient for us to introduce the notions of those spaces. Definition 1.2. A real-valued function f is said to belong to the generalized central Morrey space Ṁp,r φ (Rn) provided the following norm is finite: ∥f∥Ṁp,r φ = sup Bt ∥f∥Lp,r(Bt) |Bt|1/pφ(t) , where the supremum is taken over all the balls Bt in Rn. Remark 1.3. A canonical example is φ(t) = t−λ, λ ∈ (0, np ). In this case, we denote Ṁp,r φ (Rn) by Ṁp,λ(Rn). Next, let us define the φ-central Campanato space ˙CMO p,r φ (Rn). Definition 1.4. A function f ∈ Lp loc(Rn) is said to belong to ˙CMO p,r φ (Rn) if ∥f∥ ˙CMO p,r φ := sup t>0 ∥f − fBt∥Lp,r(Bt) |Bt|1/pφ(t) <∞ , with fB = 1 |B| ∫ B f(y) dy, for set B in Rn. Remark 1.5. When φ(t) ≡ 1, we denote ˙CMO p,r φ (Rn) by ˙CMO p,r (Rn) for short. And, if φ(t) = t−λ, λ ∈ (0, np ], we denote ˙CMO p,r φ (Rn) by ˙CMO p,λ (Rn). Remark 1.6. If there exists a constant D0 ∈ (0, 1) such that φ(2t) ≤ D0φ(t) for all t > 0, then by using the same argument as in [30], we also obtain Ṁp,r φ (Rn) = ˙CMO p,r φ (Rn) . (1.14) In particular, we have Ṁp,λ(Rn) = ˙CMO p,λ (Rn), with λ ∈ (0, np ]. Remark 1.7. Obviously, for 1 ≤ p1 < p2 we have ˙CMO p2,r φ (Rn) ⊂ ˙CMO p1,r φ (Rn) . (1.15) Moreover, it is known that BMO(Rn) ⊊ ˙CMO p2,r (Rn) ⊊ ˙CMO p1,r (Rn) . (1.16) We emphasize that ˙CMO p,r (Rn) depends on p. Therefore, there is no analogy of the famous John-Nirenberg inequality of BMO(Rn) for the space ˙CMO p,r (Rn). Our last interested central function space is the generalized central Hardy space. To define this space, we first point out the definition of a central (1, q, φ)-atom. Definition 1.8. Let 1 < p ≤ ∞, and φ(t) : (0,∞) → (0,∞). A function a(x) is called a central (1, p, r, φ)-atom, if there exists a ball Bt in Rn such that (i) supp(a) ⊂ Bt, (ii) ∫ Bt a(x) dx = 0, (iii) ∥a∥Lp,r ≤ 1 |Bt|1/q′φ(t) . Now, we are ready to define HAp,r φ (Rn) (see definition Hp,φ(Rn) by Zorko [30]). EJDE-2025/82 HARDY OPERATORS AND COMMUTATORS 5 Definition 1.9. Let 1 < p < ∞, and let φ(t) : (0,∞) → (0,∞). We denote, by HAp,r φ (Rn), the family of distributions h that, in the sense of distributions, can be written as h = ∞∑ j=0 λjaj , where aj , j ≥ 0 are central (1, p, r, φ)-atoms, and ∑∞ j=0 |λj | <∞. It is clear that HAp,r φ (Rn) is a vector space. In addition, we denote ∥h∥HAp,r φ = inf { ∞∑ j=0 |λj | } , where the infimum is taken over all possible decompositions of h as above. Then ( HAp,r φ (Rn), ∥ · ∥HAp,r φ ) becomes a normed space. Such a space of this type has been studied by the authors in [3, 15, 14] and in the references cited therein when φ(t) ≡ 1. In fact, Chen–Lau, [3] studied a theory of Hardy spaces HAp,r φ (R) associated with the Beurling algebras Ap, 1 < p < ∞, the space consisting of functions f on Rn for which ∥f∥Ap = ∞∑ k=0 2 kn p′ ∥fχk∥Lp <∞ , where χk is the characteristic function on the set { x ∈ Rn : 2k−1 < |x| ≤ 2k } , k ≥ 1. For convenience, we recall here the definition of HAp,r φ (R) via the Beurling algebras Ap. Definition 1.10. Let f∗ be the vertical maximal function, defined by f∗(x) = sup t>0 ∣∣(f ∗ ψt)(x) ∣∣ , where ψt(x) = t−nψ(x/t), and ψ is an integrable function on Rn such that ∫ Rn ψ(x) dx = 1. Then, we define HAp,r φ (R) by the set of functions f such that ∥f∗∥Ap is finite. Moreover, if we set ∥f∥HAp,r φ = ∥f∗∥Ap , then ∥ · ∥HAp,r φ is a norm. The most interesting aspect of the theory constructed by Chen-Lau is the atomic decomposition of HAp,r(R), for 1 < p ≤ 2. Thanks to this decomposition, they obtained the duality HAp′,r′ φ (Rn)∗ = ˙CMO p,r (Rn) . (1.17) After that, Garćıa-Cuerva [15] extended their results for all p ∈ (1,∞) by using the character- izations via the grand maximal functions. Moreover, the associated spaces HAq,p, 0 < q < 1, 1 < p ≤ ∞ was investigated by the authors in [14]. Remark 1.11. Obviously, for any 1 < p1 < p2 ≤ ∞ we have HAp2,r φ (Rn) ⊂ HAp1,r φ (Rn) . (1.18) It is interesting to emphasize that when φ(t) ≡ 1 the inclusion in (1.18) is strictly according to (1.16) and (1.17). This observation is different from the point of view of the classical Hardy spaces. That is H1,∞(Rn) = H1,q(Rn) (1.19) for 1 < q < ∞, see Theorem A, [6]. By (1.19), one can define H1(Rn) (the real Hardy space) to be any one of the spaces H1,q(Rn) for 1 < q ≤ ∞. Next, we discuss the commutators of Hardy operators. For any operator T , let us define [b, T ](f) := bT (f)− T (bf) . Note that b is called the symbol function of [b, T ]. When T is an operator of Hardy type, the study of [b, T ] has been investigated by many authors in [12, 20, 13, 24, 27, 26, 25, 23, 22], and the references therein. In [22], Long–Wang proved Hardy’s integral inequalities for commutators [b,H] and [b,Hβ ] (the fractional Hardy operator), β ∈ (0, 1), with b belongs to the one-sided dyadic functions ˙CMO p,r (R+). Moreover, Fu et al., [13] obtained some characterizations of ˙CMO p,r (Rn) for 1 < p <∞ via the Lp-boundedness of [b,H] and [b,H∗] in the following theorem. 6 L. T. NGHIA EJDE-2025/82 Theorem 1.12 (Fu et al. [13]). Let b ∈ ˙CMO max{p,p′},r (Rn). Then both [b,H] and [b,H∗] are bounded on Lp. Conversely, (a) if [b,H] is bounded on Lp, then b ∈ ˙CMO p′,r (Rn); (b) if [b,H∗] is bounded on Lp, then b ∈ ˙CMO p,r (Rn). We also mention that Komori, [20] obtained a characterization of function b ∈ ˙CMO p,r (R+) by means of the Lp-boundedness of [b,H] and [b,H∗]. Note that his argument can be adapted for the setting of the Euclidean space Rn instead of R+. Lu-Zhao, [24] extended Theorem 1.12 to the space ˙CMO max{p,q′},λ (Rn) as follows. Theorem 1.13 (Lu-Zhao, [24]). Let 1 < q < p < ∞ be such that 0 < λ = 1 q − 1 p < 1 n . Then b ∈ ˙CMO max{p,q′},λ (Rn) ⇐⇒ [b,H], [b,H∗] : Lq(Rn) → Lp(Rn). 1.1. Main results. As mentioned at the beginning, our first result is the following duality. Theorem 1.14. Let 1 < p <∞, and φ(t) : (0,∞) → (0,∞). Then, we have HAp′,r′ φ (Rn)∗ = ˙CMO p,r φ (Rn) . Remark 1.15. As a consequence of Theorem 1.14, we observe that ˙CMO p,r φ (Rn) is a Banach space. Next, we extend Theorem 1.1 to Ṁ p φ(Rn). Theorem 1.16. Let 1 < p <∞. Assume that there is a constant D0 ∈ (0, 1) such that φ(2t) ≤ D0φ(t), ∀t > 0 . (1.20) Then, H (resp. H∗) is a bounded operator from Ṁp,r φ (Rn) → Ṁp,r φ (Rn). In addition, we have ∥H(f)∥Ṁp,r φ ≤ ( p p− 1 ) ∥f∥Ṁp,r φ (1.21) for f ∈ Ṁp,r φ (Rn); and there is a constant C = C(n, p) > 0 such that ∥H∗(f)∥Ṁp,r φ ≤ C∥f∥Ṁp,r φ (1.22) for f ∈ Ṁp,r φ (Rn). Remark 1.17. We emphasize that condition (1.20) can be relaxed in the Ṁp,r φ -boundedness of H, see the proof of Theorem 1.16. This means that one can take φ(t) ≡ C > 0 in (1.21). As a consequence of Theorem 1.16, Remark 1.6, and Theorem1.14, we have the following corollary. Corollary 1.18. Theorem 1.16. Then, the following statements hold (a) H and H∗ are bounded operators from ˙CMO p,r φ (Rn) → ˙CMO p,r φ (Rn); (b) H and H∗ are bounded operators from HAp′,r′ φ (Rn) → HAp′,r′ φ (Rn). Concerning the boundedness of commutators of Hardy operator, we have the following theorem. Theorem 1.19. Assume hypotheses of Theorem 1.16. If b ∈ ˙CMO max{p,p′},r (Rn), then the fol- lowing statements hold (a) [b,H] (resp. [b,H∗]) is a bounded operator from Ṁp,r φ (Rn) → Ṁp,r φ (Rn); (b) [b,H] (resp. [b,H∗]) is a bounded operator from Ṁp′,r φ (Rn) → Ṁp′,r φ (Rn). Remark 1.20. Similarly as in Remark 1.17, (1.20) can be relaxed for conclusion (a) of Theorem 1.19. By duality, we have the following result. EJDE-2025/82 HARDY OPERATORS AND COMMUTATORS 7 Corollary 1.21. Assume hypotheses in Corollary 1.18. If b ∈ ˙CMO max{p,p′},r (Rn), then [b,H] (resp. [b,H∗]) is a bounded operator on ˙CMO p,r φ (Rn) and HAp′,r′ φ (Rn). Typical examples for the Corollaries 1.18, 1.21 are φ(t) = t−λ, and φ(t) = ( 1 log(1+t) )λ , for λ ∈ (0, n/p]. Our last result is a characterization of function b in ˙CMO p,r (Rn) by means of the boundedness of [b,H∗] in Ṁp,r φ (Rn). Theorem 1.22. Assume hypotheses in Theorem 1.16. If b ∈ Lp loc(Rn), and [b,H∗] is a bounded operator on Ṁp,r φ (Rn), then b ∈ ˙CMO p,r (Rn). Furthermore, there exists a constant C > 0 depend- ing on n, p such that ∥b∥ ˙CMO p,r ≤ C∥[b,H∗]∥Ṁp,r φ →Ṁp,r φ . (1.23) By duality, we have the following corollary. Corollary 1.23. Assume hypotheses in Theorem 1.22. If b ∈ Lp loc(Rn), and [b,H] is a bounded operator on HAp′,r′ φ (Rn), then b ∈ ˙CMO p,r (Rn). In addition, there exists a constant C > 0 depending on n, p such that ∥b∥ ˙CMO p,r ≤ C∥[b,H]∥ HAp′,r′ φ →HAp′,r′ φ . (1.24) As a consequence of Theorem 1.22 and Corollary 1.23, we have the following result. Corollary 1.24. Assume hypotheses in Theorem 1.22. Suppose that t min{n p , n p′ }φ(t) is nondecreas- ing on (0,∞), and b ∈ L max{p,p′} loc (Rn). Then, the following statements hold: (a) If [b,H∗] is a bounded operator on Ṁp,r φ (Rn) and Ṁp′,r φ (Rn), then b ∈ ˙CMO max{p,p′},r (Rn). In addition, there exists a constant C = C(n, p) > 0 such that ∥b∥ ˙CMO max{p,p′},r ≤ C ( ∥[b,H∗]∥Ṁp,r φ →Ṁp,r φ + ∥[b,H∗]∥ Ṁp′,r φ →Ṁp′,r φ ) . (1.25) (b) If [b,H] is a bounded operator on HAp′,r′ φ (Rn) and HAp,r φ (Rn), then b ∈ ˙CMO max{p,p′},r (Rn). In addition, there exists a constant C = C(n, p) > 0 such that ∥b∥ ˙CMO max{p,p′},r ≤ C ( ∥[b,H∥ HAp′,r′ φ →HAp′,r′ φ + ∥[b,H]∥HAp,r φ →HAp,r φ ) . (1.26) Typical examples of functions satisfying Corollary 1.24 are φ(t) = t−λ, and φ(t) = ( 1 log(1+t) )λ , for λ ∈ (0,min{n/p, n/p′}]. Our paper is organized as follows. We study the generalized central Hardy space, and prove Theorem 1.14 in the next section. The last section is devoted to the proof of Theorems 1.16-1.22, and of Corollary 1.18-1.24. 2. Space HAp′,r′ φ (Rn) as the predual of ˙CMO p,r φ (Rn) For any ball B in Rn we denote Lp,r 0 (B) by the subspace of Lp(B) of functions having mean value zero. It is not difficult to verify that Lp,r 0 (B)∗ = Lp′,r′(B)/C(B) , (2.1) where C(B) is the set of the functions, which are constant on B. Then, we have the following embedding result. Proposition 2.1. For any τ > 0, and for f ∈ Lp 0(Bτ ), we have ∥1Bτ f∥HAp,r φ ≤ |Bτ |1/p ′ φ(τ)∥f∥Lp,r(Bτ ) . Proof. Let us set a(x) = 1Bτ f(x) |Bτ |1/p′φ(τ)∥ft∥Lp,r(Bτ ) . Since ∫ Bτ f(x) dx = 0, then it is not difficult to verify that a is a central (1, p, r, φ)-atom. Therefore, the desired result follows from the Definition 1.9. □ 8 L. T. NGHIA EJDE-2025/82 Remark 2.2. As a consequence of Proposition 2.1, if f ∈ HAp,r φ (Rn)∗, then for any τ > 0 we obtain 1Bτ f ∈ Lp,r 0 (Bτ ) ∗ . Proof of Theorem 1.14. Let a be a central (1, p′, r′, φ)-atom with supp(a) ⊂ Bt for some t > 0. Then, for any f ∈ ˙CMO p,r φ (Rn) we have∣∣∣ ∫ Rn f(x)a(x) dx ∣∣∣ = ∣∣∣ ∫ Bt (f(x)− fBt) a(x) dx ∣∣∣ ≤ ∥f − fBt ∥Lp,r(Bt)∥a∥Lp′,r′ (Bt) ≤ ∥f − fBt∥Lp,r(Bt) |Bt|1/pφ(t) ≤ ∥f∥ ˙CMO p,r φ . For every g ∈ HAp′,r′ φ (Rn), one can decompose g = ∑∞ j=0 λjaj , where {aj}j≥0 is a sequence of central (1, p′, r′, φ)-atoms; and ∑∞ j=0 |λj | <∞. Therefore, we deduce from the last inequality that∣∣∣ ∫ Rn f(x)g(x) dx ∣∣∣ = ∣∣∣ ∞∑ j=0 ∫ Rn λjf(x)aj(x) dx ∣∣∣ ≤ ( ∞∑ j=0 |λj | ) ∥f∥ ˙CMO p,r φ ≤ ∥g∥ HAp′,r′ φ ∥f∥ ˙CMO p,r φ . (2.2) This yields ˙CMO p,r φ (Rn) ⊂ HAp′,r′ φ (Rn)∗ . It remains to show that HAp′,r′ φ (Rn)∗ ⊂ ˙CMO p,r φ (Rn) . (2.3) Let F ∈ HAp′,r′ φ (Rn)∗. Thanks to Remark 2.2, we have that 1BτF ∈ Lp′,r′ 0 (Bτ ) ∗ for τ > 0. By (2.1), there exists fτ ∈ Lp,r(Bτ )/C(Bτ ) such that ⟨1Bτ F, g⟩Lp,Lp′ = ∫ Bτ fτ (x)g(x) dx, ∀g ∈ Lp′,r′ 0 (Bτ ) . (2.4) Therefore, for every 0 < τ1 < τ2, we have fτ1(x) = fτ2(x) for a.e. x ∈ Bτ1 , which makes sense by (2.4). Next, let us define f(x) = fτ (x) if x ∈ Bτ . Obviously, we have f ∈ Lp loc(Rn). Now, we demonstrate that f ∈ ˙CMO p,r φ (Rn). Indeed, for any ball Bt in Rn, let us fix τ0 > t. Remind that f(x) = ft(x) ∈ Lp,r(Bt)/C(Bt) for x ∈ Bt. By duality (2.1), we obtain ∥f − fBt ∥Lp,r(Bt) |Bt|1/pφ(t) = 1 |Bt|1/pφ(t) sup ∥h∥ L p′,r′ 0 (Bt) =1 ∣∣∣ ∫ Bt (f(x)− fBt)h(x) dx ∣∣∣ = 1 |Bt|1/pφ(t) sup ∥h∥ L p′,r′ 0 (Bt) =1 ∣∣∣ ∫ Bt f(x) (h(x)− hBt) dx ∣∣∣ = sup ∥h∥ L p′,r′ 0 (Bt) =1 ∣∣∣ ∫ Rn fτ0(x) (h(x)− hBt)1Bt |Bt|1/pφ(t) dx ∣∣∣ . (2.5) Since h ∈ Lp′ 0 (Bt) and ∥h∥Lp′,r′ (Bt) = 1, it follows that (h(x)−hBt)1Bt |Bt|1/pφ(t) is a central (1, p′, r′, φ)-atom (see the proof of Proposition 2.1), and ∥1Bt (h(x)− hBt ) |Bt|1/pφ(t) ∥ HAp′,r′ φ ≤ 1 . EJDE-2025/82 HARDY OPERATORS AND COMMUTATORS 9 With this inequality noted, it follows from (2.5) that ∥f − fBt ∥Lp,r(Bt) |Bt|1/pφ(t) ≤ ∥1Bτ0 F∥ (HAp′,r′ φ )∗ ∥∥1Bt (h(x)− hBt ) |Bt|1/pφ(t) ∥∥ HAp′,r′ φ ≤ ∥F∥ (HAp′,r′ φ )∗ . Since the last inequality holds for every t > 0, we obtain ∥f∥ ˙CMO p,r φ ≤ ∥F∥ (HAp′,r′ φ )∗ , which yields (2.3). Hence, we have completed the proof of Theorem 1.14. □ Next, we use some properties of HAp,r φ (Rn) under certain conditions on φ. Proposition 2.3. Suppose that φ(t) is non-increasing on (0,∞), and there exists τ0 > 0 such that t n p φ(t) is nondecreasing on (τ0,∞). Then HAp′,r′ φ (Rn) is the subspace of L∞ c (Rn)∗. Proof. Let a be a central (1, p′, r′, φ)-atom with supp(a) ⊂ Bt, and let ψ be a test function in L∞ c (Rn) (the space of bounded functions with compact support) with supp(ψ) ⊂ Bt0 . Applying Hölder’s inequality yields∣∣ ∫ Rn a(x)ψ(x) dx ∣∣ ≤ ∥a∥Lp′ (Bt) ∥ψ∥Lp(Bt∩Bt0 ) ≤ ∥ψ∥L∞ |Bt ∩Bt0 |1/p |Bt|1/pφ(t) . If t ≤ max{t0, τ0}, then it follows from the last inequality and the fact φ(t) ≥ min{φ(t0), φ(τ0)} that ∣∣ ∫ Rn a(x)ψ(x) dx ∣∣ ≤ ∥ψ∥L∞ min{φ(t0), φ(τ0)} . Otherwise, we have t n p 0 φ(t0) ≤ t n p φ(t). Therefore,∣∣ ∫ Rn a(x)ψ(x) dx ∣∣ ≤ ∥ψ∥L∞ |B(z0, t0)|1/p |B(z0, t0)|1/pφ(t0) = ∥ψ∥L∞ φ(t0) . By combining the two cases, we obtain∣∣ ∫ Rn a(x)ψ(x) dx ∣∣ ≤ ∥ψ∥L∞ min{φ(t0), φ(τ0)} . (2.6) Now, for each h ∈ HAp′,r′ φ (Rn), we can write h = ∑∞ j=0 λjaj , where aj , j ≥ 0 are (1, p′, r′, φ)- atoms, and ∑∞ j=0 |λj | <∞. Then, it follows from (2.6) that∣∣ ∫ Rn h(x)ψ(x) dx ∣∣ ≤ ∞∑ j=0 |λj | ∣∣ ∫ Rn aj(x)ψ(x) dx ∣∣ ≤ ( ∞∑ j=0 |λj | ) ∥ψ∥L∞ min{φ(t0), φ(τ0)} ≤ ∥h∥ HAp′,r′ φ ∥ψ∥L∞ min{φ(t0), φ(τ0)} . Thus, we obtain the conclusion. □ Remark 2.4. As a consequence of Proposition 2.3, if h ∈ HAp′,r′ φ (Rn), h = ∑∞ j=0 λjaj , then the series converges to h in the norm of L∞ c (Rn)∗. Proposition 2.5. Under the hypotheses in Proposition 2.3, HAp′,r′ φ (Rn) is a Banach space. Proof. Let {fN}N≥1 be a Cauchy sequence in HAp′,r′ φ (Rn). Then, there exists a subsequence {fNk }k≥1 such that ∥fNk − fNk−1 ∥ HAp′,r′ φ (Rn) ≤ 2−k . (2.7) Put f = fN1 + ∑ k≥2 ( fNk − fNk−1 ) . 10 L. T. NGHIA EJDE-2025/82 Note that for each k ≥ 1, we have fNk − fNk−1 = ∑ j≥0 λkj a k j , where {akj }j≥0 is a sequence of central (1, p′, r′, φ)-atoms, and∑ j≥0 |λkj | ≤ ∥∥fNk − fNk−1 ∥∥ HAp′,r′ φ (Rn) + 2−k . With this inequality noted, and by (2.7), we obtain∑ k≥1 ∑ j≥0 |λkj | ≤ ∑ k≥1 21−k <∞ . (2.8) This implies that f can be decomposed into central (1, p′, r′, φ)-atoms. Next, we claim that fNk → f as k → ∞ in the norm of L∞ c (Rn)∗. If this is true, then by (2.8) we can conclude that fN → f in HAp′,r′ φ (Rn) as N → ∞. Since f = fNk0 + ∑ k≥k0+1(fNk −fNk−1 ), it suffices to prove that ∑ k≥k0+1(fNk −fNk−1 ) converges to 0 as k0 → ∞ with respect to the norm of L∞ c (Rn)∗. By (2.6), we obtain∣∣∣ ∫ Rn ∑ k≥k0+1 (fNk − fNk−1 )(x)ψ(x) dx ∣∣∣ ≤ ∑ k≥k0+1 ∑ l≥0 |λkl | ∣∣∣ ∫ Rn akl (x)ψ(x) dx ∣∣∣ ≤ ∑ k≥k0+1 ∑ l≥0 |λkl | ∥ψ∥L∞ min{φ(t0), φ(τ0)} . With this inequality it follows from (2.8) that lim k0→∞ ∥∥ ∑ k≥k0+1 (fNk − fNk−1 ) ∥∥ L∞ c (Rn)∗ = 0 . Therefore, fNk0 → f in L∞ c (Rn)∗ as k0 → ∞. This completes the proof. □ 3. Boundedness of operators of Hardy type in generalized central function spaces 3.1. Hardy operators in generalized central function spaces. Proof of Theorem 1.16. We first prove the Ṁp,r φ -boundedness of H. For each ball Bt in Rn, let us write H(f)(x) = H(f1)(x) +H(f2)(x), ∀x ∈ Rn , with f1 = f1Bt , and f2 = f1Bc t , Bc t = Rn \Bt. For f1, we apply (1.12) to obtain ∥H(f1)∥Lp,r(Bt) |Bt|1/pφ(t) ≤ ( p p− 1 ) ∥f1∥Lp,r |Bt|1/pφ(t) = ( p p− 1 )∥f∥Lp,r(Bt) |Bt|1/pφ(t) ≤ ( p p− 1 ) ∥f∥Ṁp,r φ . (3.1) Next, since f2 = 0 on Bt, for each x ∈ Bt we observe that H(f2)(x) = 1 νn|x|n ∫ |y|<|x| f2(y) dy = 0 . (3.2) A combination of (3.1) and (3.2) yields ∥H(f)∥Lp,r(Bt) |Bt|1/pφ(t) = ∥H(f1)∥Lp,r(Bt) |Bt|1/pφ(t) ≤ ( p p− 1 ) ∥f∥Ṁp,r φ . EJDE-2025/82 HARDY OPERATORS AND COMMUTATORS 11 Since the last inequality holds for any t > 0, we obtain ∥H(f)∥Ṁp,r φ ≤ ( p p− 1 ) ∥f∥Ṁp,r φ . It remains to prove the Ṁp,r φ -boundedness of H∗. We argue similarly as in (3.1) to obtain ∥H∗(f1)∥Lp,r(Bt) |Bt|1/pφ(t) ≲ ∥f∥Ṁp,r φ . (3.3) Next, we observe that |H∗(f2)(x)| = ∣∣∣ 1 νn ∫ |y|≥2t f(y) |y|n dy ∣∣∣ = 1 νn ∣∣∣ ∞∑ k=1 ∫ {2kt≤|y|<2k+1t} f(y) |y|n dy ∣∣∣ ≲ ∞∑ k=1 (2kt)−n ∣∣∣ ∫ {2kt≤|y|<2k+1t} f(y) dy ∣∣∣ ≤ ∞∑ k=1 (2kt)−n ∫ B 2k+1t |f(y)| dy . Thanks to Hölder’s inequality, and (1.20), we obtain |H∗(f2)(x)| ≲ ∞∑ k=1 (2kt)−n∥f∥Lp(B 2k+1t )|B2k+1t|1/p ′ ≲ ∞∑ k=1 ∥f∥Lp(B 2k+1t ) |B2k+1t|1/pφ(2k+1t) φ(2k+1t) ≤ ∞∑ k=1 φ(2k+1t)∥f∥Ṁp,r φ ≤ ∞∑ k=1 Dk+1 0 φ(t)∥f∥Ṁp,r φ ≲ φ(t)∥f∥Ṁp,r φ . Therefore, we deduce that ∥H∗(f2)∥Lp,r(Bt) ≲ |Bt|1/pφ(t)∥f∥Ṁp,r φ . (3.4) Combing (3.3) and (3.4) yields the desired result. The proof is complete. □ Proof of Corollary 1.18. The proof of part (a) follows from Theorem 1.16 and Remark 1.6. It remains to prove (b). Thanks to duality, for every f ∈ HAp′,r′ φ (Rn) we have ∥H(f)∥ HAp′,r′ φ = sup ∥g∥ ˙CMOp,r φ =1 ∣∣∣ ∫ H(f)(x)g(x) dx ∣∣∣ = sup ∥g∥ ˙CMOp,r φ =1 ∣∣∣ ∫ f(x)H∗(g)(x) dx ∣∣∣ ≤ sup ∥g∥ ˙CMOp,r φ =1 ∥f∥ HAp′,r′ φ ∥H∗(g)∥ ˙CMO p,r φ ≲ sup ∥g∥ ˙CMOp,r φ =1 ∥f∥ HAp′,r′ φ ∥g∥ ˙CMO p φ = ∥f∥ HAp′,r′ φ . Hence, we conclude that H maps HAp′,r′ φ (Rn) → HAp′,r′ φ (Rn). Similarly, the conclusion also holds for H∗. Therefore, we complete the proof. □ 12 L. T. NGHIA EJDE-2025/82 3.2. Commutators of Hardy operators in generalized central function spaces. Before proving Theorems 1.19 and 1.22, we recall a fundamental result being useful for our argument later. Lemma 3.1. Let 1 ≤ p <∞, and k ≥ 1. For each ball Bt in Rn, we have∥∥b− bB 2kt ∥∥ Lp,r(Bt) ≤ 2n(k + 1)∥b∥ ˙CMO p,r |Bt|1/p . Proof. For each j ≥ 1, we observe that |bB2j+1t − bB2jt | ≤ 1 |B2jt| ∫ B2jt |b(y)− bB2j+1t | dy ≤ |B2j+1t| |B2jt| 1 |B2j+1t| ∫ B2j+1t ∣∣b(y)− bB2j+1t ∣∣ dy ≤ 2n∥b∥ ˙CMO 1 ≤ 2n∥b∥ ˙CMO p,r . From this inequality, we obtain ∥b− bB 2kt ∥Lp,r(Bt) ≤ ∥b− bBt ∥Lp,r(Bt) + k−1∑ j=0 ∥bB2jt − bB2j+1t ∥Lp,r(Bt) ≤ |Bt|1/p ∥b− bBt ∥Lp,r(Bt) |Bt|1/p + k−1∑ j=0 |bB2jt − bB2j+1t ||Bt|1/p ≤ 2n(k + 1)∥b∥ ˙CMO p,r |Bt|1/p . The proof is complete. □ Next, we estimate ∥1Br ∥Ṁp,r φ for each ball Br in Rn. Lemma 3.2. Suppose that φ(t) is non-increasing, and t n p φ(t) is nondecreasing. Then, for any ball Br in Rn we have ∥1Br∥Ṁp,r φ = 1 φ(r) . Proof. We consider the term I(t) := ∥1Br∥Lp,r(Bt) |Bt|1/pφ(t) , t > 0. If t ≤ r, then since φ(t) is nonincreasing, then we obtain I(t) = |Br ∩Bt|1/p |Bt|1/pφ(t) = |Bt|1/p |Bt|1/pφ(t) ≤ 1 φ(r) . Otherwise, it follows from the monotonicity of |Bt|1/pφ(t) that I(t) ≤ |Br|1/p |Br|1/pφ(r) ≤ 1 φ(r) . Combining the two inequalities yields ∥1Br ∥Ṁp,r φ ≤ 1 φ(r) . (3.5) The reverse of (3.5) is obvious since I(r) = 1 φ(r) . Therefore, the desired result follows. □ Proof of Theorem 1.19. (a) Fix a ball Bt in Rn. We write [b,H](f) = [b,H](f1) + [b,H](f2) , with f1 = f1Bt and f2 = f1Bc t . Since [b,H] maps Lp → Lp, we have ∥[b,H](f1)∥Lp,r(Bt) ≲ ∥b∥ ˙CMO max{p,p′},r∥f1∥Lp = ∥b∥ ˙CMO max{p,p′},r∥f∥Lp,r(B2t) . It follows from the monotonicity of φ that ∥[b,H](f1)∥Lp,r(Bt) |Bt|1/pφ(t) ≲ ∥b∥ ˙CMO max{p,p′},r ∥f∥Lp,r(B2t) |Bt|1/pφ(t) ≤ ∥b∥ ˙CMO max{p,p′},r∥f∥Ṁp,r φ . (3.6) EJDE-2025/82 HARDY OPERATORS AND COMMUTATORS 13 Next, for any x ∈ Bt we observe that [b,H](f2)(x) = 0. A combination of this fact, and (3.6) provides us with ∥[b,H](f)∥Lp,r(Bt) |Bt|1/pφ(t) = ∥[b,H(f1)∥Lp,r(Bt) |Bt|1/pφ(t) ≲ ∥b∥ ˙CMO max{p,p′},r∥f∥Ṁp,r φ . Therefore, we obtain the desired result in part (a). (b) Since [b,H∗] maps Lp → Lp, then we can mimic the proof of (3.6) to obtain ∥[b,H∗](f1)∥Lp,r(Bt) |Bt|1/pφ(t) ≲ ∥b∥ ˙CMO max{p,p′},r∥f∥Ṁp,r φ . (3.7) Concerning f2, we write ∥[b,H∗](f2)∥Lp,r(Bt) = ∥∥ 1 νn ∫ |y|≥2t (b(x)− b(y)) f(y) |y|n dy ∥∥ Lp,r(Bt) ≤ ∥∥ ∞∑ k=0 (2kt)−n ∫ {2kt≤|y|<2k+1t} |b(x)− bB 2k+1t | |f(y)| dy ∥∥ Lp,r(Bt) + ∥∥ ∞∑ k=0 (2kt)−n ∫ {2kt≤|y|<2k+1t} |b(y)− bB 2k+1t | |f(y)| dy ∥∥ Lp,r(Bt) := I1 + I2 . (3.8) We first treat I1. Applying the triangle inequality, Minkowski’s inequality, and the Hölder in- equality yields I1 ≤ ∞∑ k=0 (2kt)−n ∫ {2kt≤|y|<2k+1t} ∥b− bB 2k+1t ∥Lp,r(Bt)|f(y)| dy ≲ ∞∑ k=0 |B2k+1t|−1∥b− bB 2k+1t ∥Lp,r(Bt)∥f∥Lp(B 2k+1t )|B2k+1t| 1 p′ ≤ ∞∑ k=0 ∥b− bB 2k+1t ∥Lp,r(Bt)φ(2 k+1t)∥f∥Ṁp,r φ . Thanks to Lemma 3.1 and (1.20), we obtain from the last inequality that I1 ≲ ∞∑ k=0 2n(k + 2)∥b∥ ˙CMO p,r |Bt|1/pDk+1 0 φ(t)∥f∥Ṁp,r φ ≲ |Bt|1/pφ(t)∥b∥ ˙CMO p,r∥f∥Ṁp,r φ . (3.9) Note that (3.9) was obtained from ∑∞ k=0(k + 2)Dk+1 0 <∞. For I2, we use Hölder’s inequality, and Lemma 3.1 to obtain I2 ≤ ∥∥ ∞∑ k=0 (2kt)−n∥b− bB 2k+1t ∥Lp′,r(B 2k+1t )∥f∥Lp,r(B 2k+1t ) ∥∥ Lp,r(Bt) ≲ ∞∑ k=0 ∥b− bB 2k+1t ∥Lp′,r(B 2k+1t ) |B2k+1t| 1 p′ ∥f∥Lp,r(B 2k+1t ) |B2k+1t|1/pφ(2k+1t) φ(2k+1t)|Bt|1/p ≤ ∞∑ k=0 ∥b∥ ˙CMO p′,r∥f∥Ṁp,r φ Dk+1 0 φ(t)|Bt|1/p ≲ |Bt|1/pφ(t)∥b∥ ˙CMO p′,r∥f∥Ṁp,r φ . (3.10) Combining (3.8), (3.9), and (3.10) yields ∥[b,H∗](f2)∥Lp,r(Bt) |Bt|1/pφ(t) ≲ ∥b∥ ˙CMO max{p,p′},r∥f∥Ṁp,r φ . (3.11) Therefore, the desired result follows from (3.7) and (3.11). The proof is complete. □ 14 L. T. NGHIA EJDE-2025/82 Proof of Corollary 1.21. The proof is similar to the one of Corollary 1.18, then we leave it to the reader. □ Finally we demonstrate Theorem 1.22. The proof follows by way of the following lemma. Lemma 3.3. Let a be a central (1, p′)-atom. Then, there exist two functions f ∈ HAp′,r′ φ (Rn), and g ∈ Ṁp,r φ (Rn) such that a(x) = f(x)H∗(g)(x)− g(x)H(f)(x) , (3.12) ∥f∥ HAp′,r′ φ ∥g∥Ṁp,r φ ≤ 2 n p ln 2 . (3.13) Proof. Suppose that supp(a) ⊂ Bτ for some τ > 0. Let us set f(x) = a(x) φ(τ) ln 2 , and g(x) = φ(τ)1{τ<|x|<2τ}(x) . We first claim that the above construction satisfies (3.12). In fact, if |x| ≥ τ , then it is clear that f(x) = H(f)(x) = 0 since supp(a) ⊂ Bτ , and the cancellation property of a respectively. Therefore, (3.12) is true for all |x| ≥ τ . Otherwise, we have g(x) = 0, and H∗(g)(x) = 1 νn ∫ |y|≥|x| φ(τ)1{τ<|x|<2τ}(y) |y|n dy = φ(τ) νn ∫ 2τ τ νns −nsn−1 ds = φ(τ) ln 2 . This yields the above claim. Now, we demonstrate (3.13). Since a is a central (1, p′)-atom, f is a multiple of central (1, p′, φ)- atom, and ∥f∥ HAp′,r′ φ ≤ 1 ln 2 . (3.14) Moreover, thanks to Lemma 3.2, we obtain ∥g∥Ṁp,r φ = φ(τ)∥1{τ<|x|<2τ}∥Ṁp,r φ ≤ φ(τ) φ(2τ) = 2 n p τ n p φ(τ) (2τ) n p φ(2τ) ≤ 2 n p . (3.15) The last inequality follows from the monotonicity of function t n p φ(t). As a result, (3.13) follows from (3.14) and (3.15). Therefore, we obtain Lemma 3.3. □ Remark 3.4. The above construction demonstrates that g ∈ L∞ c (Rn), and f ∈ Lp′ c (Rn). In addition, the result of Lemma 3.2 can be considered as a HAp′,r′(Rn)∗ factorization. Note that the H1(Rn) factorization by means of the Calderón–Zygmund operators has been studied by the authors in [5, 7, 8, 9, 10, 19, 28, 29] and the references therein. Proof of Theorem 1.22. For thsi purpose, We use a duality argument. Since ˙CMO p,r (Rn) = HAp′,r′(Rn)∗, it follows that for any h ∈ HAp′,r′(Rn), one can decompose h = ∞∑ j=0 λjaj , where {aj}j≥0 is a sequence of central (1, p′)-atoms; and ∑∞ j=0 |λj | <∞. For every j ≥ 0, by applying Lemma 3.3 to aj we have that there exist two functions gj ∈ Ṁp,r φ (Rn), and fj ∈ HAp′,r′ φ (Rn) such that aj(x) = fj(x)H∗(gj)(x)− gj(x)H(fj)(x) , and ∥fj∥HAp′,r′ φ ∥gj∥Ṁp,r φ ≤ 2 n p ln 2 . (3.16) EJDE-2025/82 HARDY OPERATORS AND COMMUTATORS 15 Since b ∈ Lp loc(Rn), and by Remark 3.4, the following integrals are well-defined, and satisfy∣∣∣ ∫ Rn b(x)aj(x) dx ∣∣∣ = ∣∣∣ ∫ Rn b(x)[fj(x)H∗(gj)(x)− gj(x)H(fj)(x)] dx ∣∣∣ = ∣∣∣ ∫ Rn fj(x)[b,H∗](gj)(x) dx ∣∣∣ ≤ ∥fj∥HAp′,r′ φ ∥[b,H∗](gj)∥Ṁp,r φ . (3.17) Note that (3.17) was obtained from Ṁp,r φ (Rn) = ˙CMO p,r φ (Rn) = HAp′,r′ φ (Rn)∗. Since [b,H∗] is a bounded operator on Ṁp,r φ (Rn), it follows from (3.17) and (3.16) that ∣∣ ∫ Rn b(x)aj(x) dx ∣∣ ≤ ∥[b,H∗]∥Ṁp,r φ →Ṁp,r φ ∥gj∥Ṁp,r φ ∥fj∥HAp′,r′ φ ≤ 2 n p ln 2 ∥[b,H∗]∥Ṁp,r φ →Ṁp,r φ . With this inequality, for any h ∈ HAp′,r′ φ (Rn) we obtain ∣∣ ∫ Rn b(x)h(x) dx ∣∣ = ∞∑ j=0 ∣∣λj ∫ Rn b(x)aj(x) dx ∣∣ ≤ ( ∞∑ j=0 |λj | ) 2 n p ln 2 ∥[b,H∗]∥Ṁp,r φ →Ṁp,r φ ≤ 2 n p ln 2 ∥[b,H∗]∥Ṁp,r φ →Ṁp,r φ ∥h∥HAp′,r′ . (3.18) By duality, we obtain ∥b∥ ˙CMO p,r ≤ 2 n p ln 2 ∥[b,H∗]∥Ṁp,r φ →Ṁp,r φ . (3.19) The proof is complete. □ Proof of Corollary 1.23. To obtain the result, we can repeat the proof of Theorem 1.22 with a slight modification in (3.17) as follows∣∣∣ ∫ Rn b(x)aj(x) dx ∣∣∣ = ∣∣∣ ∫ Rn b(x) [fj(x)H∗(gj)(x)− gj(x)H(fj)(x)] dx ∣∣∣ = ∣∣∣ ∫ Rn [b,H](fj)(x)gj(x) dx ∣∣∣ ≤ ∥[b,H](fj)∥HAp′,r′ φ ∥gj∥Ṁp,r φ . (3.20) Since [b,H] maps HAp′,r′ φ → HAp′,r′ φ , we deduce from (3.20) that∣∣ ∫ Rn b(x)aj(x) dx ∣∣ ≤ ∥[b,H]∥ HAp′,r′ φ →HAp′,r′ φ ∥fj∥HAp′,r′ φ ∥gj∥Ṁp,r φ ≤ 2 n p ln 2 ∥[b,H]∥ HAp′,r′ φ →HAp′,r′ φ . By arguing similarly as in (3.18), for any h ∈ HAp′,r′ φ (Rn), we also obtain ∣∣ ∫ Rn b(x)h(x) dx ∣∣ ≤ 2 n p ln 2 ∥[b,H]∥ HAp′,r′ φ →HAp′,r′ φ ∥h∥HAp′,r′ . This yields (1.24). □ Proof of Corollary 1.24. The proof is just a combination of the results in Theorem 1.22 and Corol- lary 1.23. Therefore, we leave it to the reader. □ 16 L. T. NGHIA EJDE-2025/82 4. Applications of Hardy’s inequality We present here several applications of Hardy’s inequality. In [2], Brezis-Vázquez studied the problem −∆u = λf(u) in Ω, u = 0 on ∂Ω, (4.1) where Ω is a bounded domain in Rn, and f is a continuous, positive, increasing and convex function defined for u ≥ 0 with f(0) > 0 and lim s→∞ f(s) s = ∞. The authors established a characterization of the singular H1 extremal solutions and the extremal value λ∗ by a criterion consisting of two conditions: (i) They must be energy solutions, not in L∞. (ii) They must satisfy λ ∫ Ω f ′(u)ϕ2dx ≤ ∫ Ω |∇ϕ|2dx, ∀ϕ ∈ C∞ 0 (Ω). (4.2) Roughly speaking, this formula means that the first eigenvalue of −∆−λf ′(u) is nonnegative, is a version of Hardy’s inequality. To obtain the desired result, they improved a version of the classical Hardy’s inequality. Another application of Hardy’s inequality is to study negative eigenvalues of the self-adjoint operator −∆ − V in L2(Rn), where potential V satisfies V ≥ 0, V ∈ Ln/2(Rn), n ≥ 3. This has important implications in semi-classical spectral analysis, in which the transition between classical and quantum mechanics is studied. Acknowledgements. The author would like to thank Professor Jesus Ildefonso Dı́az for his valuable comments which were very helpful for improving the original manuscript. References [1] Anderson, K.; Muckenhoupt, B.; Weighted weak type Hardy inequalities with application to Hilbert transforms and maximal functions, Studia Math., 72(1) (1982), 9–26. [2] Brezis, H.; Vázquez, J. V.; Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10(2) (1997), 443–469. [3] Chen, Y.; Lau, K.; Some new classes of Hardy spaces, J. Funct. Anal., 84(2) (1989), 255–278. [4] Christ, M.; Grafakos, L.; Best constants for two nonconvolution inequalities, Proc. Amer. Math. Soc., 123(5) (1995), 1687–1693. [5] Coifman, R.; Rochberg, R.; Weiss, G.; Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103(3) (1976), 611–635. [6] Coifman, R.; Weiss, G.; Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83(4) (1977), 569–645. [7] Dao, N. A.; Krantz, S. G.; Lam, N.; Cauchy integral commutators and Hardy factorization on Lorentz spaces, J. Math. Anal. Appl., 498(1) (2021), 124926. [8] Dao, N. A.; Wick, B. D.; Hardy factorization in terms of multilinear Calderón-Zygmund operators using Morrey spaces, submitted. [9] Dao, N. A.; Hardy factorization in terms of fractional commutators in Lorentz spaces, to appear in Front. Math. China, DOI 10.1007/s11464-021-0946-1. [10] Duong, X. T.; Gong, R.; Kuffner, M.-J. S.; Li, J.; Wick, B. D.; Yang, D.; Two weight commutators on spaces of homogeneous type and applications, J. Geom. Anal., 31(1) (2021), 980–1038. [11] Faris, W.; Weak Lebesgue spaces and quantum mechanical binding, Duke Math. J., 43(2) (1976), 365–373. [12] Zhao, F.; Fu, Z.; Lu, S., Endpoint estimates for n-dimensional Hardy operators and their commutators, Sci. China Ser. A, 55(9) (2012), 1977–1990. [13] Fu, Z.; Liu, Z.; Lu, S.; Wang, H.; Characterization for commutators of n-dimensional fractional Hardy opera- tors, Sci. China Ser. A, 50(10) (2007), 1418–1426. [14] Garćıa-Cuerva, J.; Herrero, M.-J. L.; A theory of Hardy spaces associated to the Herz spaces, Proc. London Math. Soc. 69(3) (1994), 605–628. [15] Garćıa-Cuerva, J.; Hardy spaces and Beurling algebras, J. London Math. Soc. 39(3) (1989), 499–513. [16] Ghoussoub, N.; Moradifam, A.; Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann. 349(1) (2011), 1–57. [17] Hardy, G.; Note on a theorem of Hilbert, Math. Z., 6(3) (1920), 314–317. EJDE-2025/82 HARDY OPERATORS AND COMMUTATORS 17 [18] Hardy, G.; Littlewood, J.; Polya, G.; Inequalities, Cambridge University Press, London/New York, 1934. [19] Komori, Y.; Mizuhara, T.; Factorization of functions in H1(Rn) and generalized Morrey spaces, Math. Nachr., 279(6) (2006), 619–624. [20] Komori, Y.; Notes on commutators of Hardy operators, Int. J. Pure Appl. Math., 7(3) (2003), 329–334. [21] Lam, N.; Lu, G.; Zhang, L.; Geometric Hardy’s inequalities with general distance functions, J. Funct. Anal., 279(3) (2020), 108673. [22] Long, S.; Wang, J.; Commutators of Hardy operators, J. Math. Anal. Appl., 274(2) (2002), 626–644. [23] Lu, S.; Function characterizations via commutators of Hardy operator, Front. Math. China, 16(1) (2021), 1–12. [24] Lu, S.; Zhao, F.; A characterization of λ-central BMO space, Front. Math. China, 8(1) (2013), 229–238. [25] Shi, S., Lu, S., Characterization of the central Campanato space via the commutator operator of Hardy type, J. Math. Anal. Appl. 429(1) (2015), 713–732. [26] Shi, S.; Fu, Z.; Lu, S.; On the compactness of commutators of Hardy operators, Pacific J. Math., 307(2) (2020), 239–256. [27] Shi, S.; Lu, S.; Some characterizations of Campanato spaces via commutators on Morrey spaces, Pacific J. Math., 264(1) (2013), 221–234. [28] Tao, J.; Yang, D.; Yang, D.; Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces, Math. Methods Appl. Sci., 42(5) (2019), 1631–1651. [29] Uchiyama, A.; The factorization of Hp on the space of homogeneous type, Pacific J. Math., 92(2) (1981), 453–468. [30] Zorko, C.; The Morrey space, Proc. Amer. Math. Soc., 98(4) (1986), 586–592. Le Trung Nghia Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam. Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam Email address: letrungnghia@tdtu.edu.vn, letrungnghia85@yahoo.com 1. Introduction and main results Problem setting Explicit singular solution Why the solution is independent of q Membership in central Morrey spaces Oscillation near zero: Campanato spaces 1.1. Main results 2. Space HAp',r'(Rn) as the predual of p,r(Rn) 3. Boundedness of operators of Hardy type in generalized central function spaces 3.1. Hardy operators in generalized central function spaces 3.2. Commutators of Hardy operators in generalized central function spaces 4. Applications of Hardy's inequality Acknowledgements References