Abstract
In this paper, we are interested in studying a generalized block space (denoted as \(\textbf{B}^{p,r}_\varphi \)) on a space of homogeneous type. We show that this space is the predual of certain generalized Morrey–Lorentz space. By duality, we obtain the \(\textbf{B}^{p,r}_\varphi \)-bound of operators of Calderón–Zygmund type. In addition, we prove a weak Hardy factorization in terms of commutators of integral operator of Calderón–Zygmund type in block spaces. Thanks to the Hardy factorization result, we obtain a characterization of functions in \(\textrm{BMO}\) via the boundedness of commutators of homogeneous linear Calderón–Zygmund operators in the generalized block space (resp. the generalized Morrey–Lorentz space). Finally, we study a compactness characterization of commutators of Calderón–Zygmund type in generalized Morrey–Lorentz spaces.
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1 Introduction and Main Results
The theory of Calderón–Zygmund operators is a central part of modern harmonic analysis. Many applications to partial differential equations are among the motivations to study Calderón–Zygmund operators on spaces which are beyond the Lebesgue spaces \(L^p\) of the Euclidean spaces. This research direction has been studied extensively and lead to a successful theory of function spaces including Hardy spaces, BMO spaces, Campanato spaces, Morrey–Lorentz spaces on the Euclidean space \({\mathbb {R}}^n\) or more general, on a space of homogeneous type X (see for example [7]) in the last fifty years.
In this paper, we study certain aspects of operators of Calderón–Zygmund type on various function spaces. The aim of this paper is threefold.
(i) Firstly, we study generalized block space \(\textbf{B}^{p,r}_\varphi (X)\) on space of homogeneous type, where we assume that \(p\in (1,\infty )\), \(r\in [1,\infty ]\), and function \(\varphi (t)\) satisfies (1.5) below. Then, we prove that \(\textbf{B}^{p',r'}_\varphi (X)\) is the predual of certain generalized Morrey–Lorentz space \(\textbf{M}^{p,r}_\varphi (X)\).
(ii) Secondly, we investigate the \(\textbf{M}^{p,r}_\varphi \)-bound (resp. \(\textbf{B}^{p',r'}_\varphi \)-bound) of operators of Calderón–Zygmund type, and prove a weak Hardy factorization in terms of commutators of Calderón–Zygmund type in \(\textbf{B}^{p',r'}_\varphi (X)\) and \(\textbf{M}^{p,r}_\varphi (X)\). As a result, we obtain a characterization of functions in \(\textrm{BMO}(X)\) via the \(\textbf{M}^{p,r}_\varphi (X)\) (resp. \(\textbf{B}^{p',r'}_\varphi (X)\)) boundedness of commutators of Calderón–Zygmund type.
(iii) Thirdly, we prove a compactness characterization of commutators of Calderón–Zygmund type in \(\textbf{M}^{p,r}_\varphi (X)\).
Notation: In this paper, we denote by \(\mathcal {C}^\infty _0(X)\) and \(\mathcal {D}^\prime (X)\), the space of infinitely differentiable functions with compact support and the space of distributions respectively. For any \(q\in [1,\infty ]\), we denote \(q'\) the conjugate exponent, \(\frac{1}{q}+ \frac{1}{q'} =1\). Moreover, we denote \(B_t\) by a ball in X with radius \(t>0\).
As usual, we denote a constant by C, which may depend on p, r, n and may change at different lines. We also denote \(A \lesssim B\) if there exists a constant \(C>0\) such that \(A\le C B\). Finally, we denote \(A \approx B\) if \(A \lesssim B\) and \(B \lesssim A\).
Let us recall the definition of a space of homogeneous type, introduced by Coifman and Weiss [7]. Then \((X,d,\mu )\) is a space of homogeneous type if d is a quasi-metric on X and \(\mu \) is a nonzero measure satisfying the doubling condition. A quasi-metric d on a set X is a function \(d: X \times X \rightarrow [0,\infty )\) satisfying
-
(i)
\(d(x,y) = d(y,x)\) for all \(x, y \in X\);
-
(ii)
\(d(x, y) = 0\) if and only if \(x = y\); and
-
(iii)
the quasi-triangle inequality: there is a constant \(A_0\in [1,\infty )\) such that for all \(x, y, z \in X\),
$$\begin{aligned} d(x, y) \le A_0 \left( d(x,z) + d(z, y) \right) \,. \end{aligned}$$(1.1)
We say that a nonzero measure \(\mu \) satisfies the doubling condition if there is a constant \(C_\mu \) only depending on \(\mu \), such that for all \(x \in X\) and \(t > 0\),
where B(x, t) is the quasi-metric ball by \(B(x,t) = \{y \in X: d(x, y) < t\}\) for \(x \in X\) and \(t > 0\). We note that the doubling condition (1.2) implies that there exists a positive constant n (the upper dimension of \(\mu \)) such that for all \(x \in X\), \(\lambda \ge 1\), and \(t > 0\),
for some constant \(A_1>0\).
We emphasize that the ball B(x, t) may not be an open set. However, Macías–Segovia, [30] constructed a quasi-metric \(d'\), which is equivalent to d in the sense that
for some constant \(A_2 > 0\) and for all \(x,y \in X\). In addition, \(d'\) satisfies a regularity estimate of the type
for some constant \(A_3 > 0\), and for some \(\theta \in (0,1]\). Then, the balls associated to \(d'\) are open sets. This fact allows us to work on the quasi-metric d having the same topology with \(d'\).
Throughout this paper, we assume that \(\mu (X) =\infty \) and that \(\mu (\{x_0\}) = 0\) for every \(x_0 \in X\). In addition, we also assume that the function \(\varphi (t):(0,\infty )\rightarrow (0,\infty )\) satisfies the following conditions:
for some constant \(0<D<1\). Note that the last condition implies that \(\varphi (t)\) cannot be a constant function.
Now, we define the generalized Morrey–Lorentz space. A real-valued function f is said to belong to the generalized Morrey–Lorentz space \(\textbf{M}^{p, r}_\varphi (X)\) provided the following norm is finite:
where the supremum is taken over all the balls B(x, t) in X, and \(\Vert f\Vert _{L^{p, r}((B(x,t))}\) denotes the Lorentz norm of f on B(x, t) (see e.g. [11] for more details of Lorentz spaces).
Remark 1.1
When \(r=p\), we denote \(\textbf{M}^{p, r}_\varphi (X)\) as \(\textbf{M}^{p}_\varphi (X)\).
A canonical example is the following case: \(X=\mathbb {R}^n\) equipped with the Lebesgue measure, and \(\varphi (t)=t^{-\alpha }\), \(\alpha \in (0,\frac{n}{p}]\). In this case, \(\textbf{M}^p_\varphi (\mathbb {R}^n)\) is the classical Morrey space.
It is known that Morrey spaces are generalizations of \(L^p\)-spaces, and they play a crucial role in studying the calculus of variations and the theory of elliptic PDE’s (see e.g. [1, 5, 6, 12, 14,15,16, 31,32,33,34,35,36], and the references therein). Later, Campanato [5] extended the classical Morrey spaces by using the modified mean oscillation. As a matter of fact, this family of spaces includes the Morrey spaces, \(\textrm{BMO}\) spaces (the spaces of functions with bounded mean oscillation), and the Lipschitz spaces.
In [42], Zorko studied the generalized Campanato spaces \( \mathcal {L}^p_\varphi (X)\). We say that \(f\in \mathcal {L}^p_\varphi (X)\) if there is a constant \(C_0>0\) such that for every ball B(x, t) in X, we have
where \(\mathcal {P}_k\) is the class of polynomials of degree \(\le k\).
If \(\varphi \) satisfies condition 1.5, then the author showed that the space \( \mathcal {L}^p_\varphi (X)\) is independent of k. Furthermore, \(\mathcal {L}^p_\varphi (\mathbb {R}^n)\) and \(\textbf{M}^p_\varphi (\mathbb {R}^n)\) describe the same space, and \(\textbf{M}^p_\varphi (\mathbb {R}^n)\) is the dual of certain atomic space \(\textbf{H}^p_\varphi (\mathbb {R}^n)\) for \(p\in (1,\infty )\). We note that a celebrated result by Fefferman–Stein [20] is that the dual of \(\textbf{H}^1(\mathbb {R}^n)\) (the classical Hardy space) is \(\textrm{BMO}(\mathbb {R}^n)\). Thus, the Morrey space exhibits some similarity to \(\textrm{BMO}(\mathbb {R}^n)\), concerning duality.
For convenience, we recall here the definition of \(\textrm{BMO}(X)\).
Definition 1.2
A function \(b\in L^1_{\textrm{loc}}(X)\) belongs to \(\textrm{BMO}(X)\) if
where
and the supremum is taken over all balls \(B \subset X\).
Note that Alvarez, [2] defined \(\textbf{H}^{p}_{\varphi }(\mathbb {R}^n)\) in terms of suitable molecules, and used this result to prove the \(\textbf{H}^{p}_{\varphi }\)-bound of linear Calderón–Zygmund operators satisfying the cancellation condition. By duality, he also obtained the boundedness of linear Calderón–Zygmund operators on \(\textbf{M}^p_\varphi (\mathbb {R}^n)\) space. It is known that such a linear operator of Calderón–Zygmund type does not map atoms into atoms, but it maps molecules into molecules. That is a reason why the author introduced the suitable molecules. We emphasize that his proof relies on the cancellation condition, so it cannot be applied to a general linear Calderón–Zygmund operators, such as the Cauchy integrals associated to the Lipschitz curves (see [12, 29, 38]). Then, one of the main purposes in this paper is to extend the boundedness result by Alvarez, [2] to the linear Calderón–Zygmund operators on \(\textbf{M}^{p,r}_\varphi (X)\) (see Theorem 1.11).
For convenience, let us recall the definition of linear Calderón–Zygmund operators.
Definition 1.3
We say that T is a Calderón–Zygmund operator on \((X,d,\mu )\) if T is bounded on \(L^2(X)\) and has the associated kernel K(x, y) such that
for any \(x\notin \textrm{supp}(f)\), and K(x, y) satisfies the following estimates: for all \(x \not = y\),
and for \(d(x,z) \le (2A_0)^{-1} d(x, y)\),
for some \(\eta >0\), where \(V(x,y)=\mu \left( B(x, d(x,y))\right) \).
Note that by the doubling condition we have that \(V(x, y) \approx V(y,x)\).
Definition 1.4
We say that T is homogeneous if for any ball B(y, t) in X, T satisfies
for all \(d(x,y)= M t\), \(M>10 A_0\).
A typical example of such operator is either the Riesz transforms, or the Cauchy integral operators associated with Lipschitz curves.
Next, we define a generalized block space \(\textbf{B}^{p,r}_\varphi (X)\).
Definition 1.5
Let \(p\in (1,\infty )\), \(r\in [1,\infty ]\), and let \(\varphi (t): (0,\infty )\rightarrow (0,\infty )\). A function b(x) is called a \((p,r,\varphi )\)-block, if there exists a ball \(B_t\) in X such that
Next, we define space \(\textbf{B}^{p,r}_ {\varphi }(X)\) via \(( p,r,\varphi )\)-block.
Definition 1.6
Let \(p\in (1,\infty )\), \(r\in [1,\infty ]\), and let \(\varphi (t)\) satisfy (1.5). We denote, by \(\textbf{B}^{p',r'}_{\varphi }(X)\), the family of distributions f that, in the sense of distributions, can be written as
where \(b_k\) is a \((p,r,\varphi )\)-block and \( \{\lambda _k\}_{k\ge 1}\in l^1 \).
It is clear that \(\textbf{B}^{p',r'}_{\varphi }(X)\) is a vector space. In addition, we denote
where infimum is taken over all possible decompositions of f as above.
Then, \(\left( \textbf{B}^{p',r'}_{\varphi }(X), \Vert \cdot \Vert _{\textbf{B}^{p',r'}_{\varphi }} \right) \) becomes a norm space.
For short, we denote \(\textbf{B}^{p,p}_{\varphi }(X)\) by \(\textbf{B}^{p}_{\varphi }(X)\) if \(r=p\).
Remark 1.7
In \(\textbf{B}^{p',r'}_{\varphi }(X)\), the series \(\sum ^\infty _{k=1} \lambda _k b_k\) is convergent in \(L^1\left( B_t\right) \) for any ball \(B_t\) in X, see Lemma 2.1.
This space was introduced by Blasco et al., [4] when \(X=\mathbb {R}^n\), \(r=p\), and \(\varphi (t)=t^{-\alpha }\), \(\alpha \in (0,\frac{n}{p}]\) in order to prove a non-interpolation result. Moreover, the authors also showed that \(\textbf{B}^{p'}_{\varphi }(\mathbb {R}^n)\) is a predual of \(\textbf{M}^{p}_{\varphi }(\mathbb {R}^n)\) for \(p\in (1,\infty )\).
Our first result is the duality between \(\textbf{M}^{p,r}_\varphi (X)\) and \(\textbf{B}^{p',r'}_\varphi (X)\).
Theorem 1.8
Let \(p\in (1,\infty )\), \(r\in [1,\infty ]\), and let \(\varphi (t)\) satisfy (1.5). Then, we have
and
Remark 1.9
As a consequence of Theorem 1.8, we observe that \(\textbf{B}^{p',r'}_\varphi (X)\) is a reflexive Banach space. Moreover, \(\textbf{M}^{p,r}_\varphi (X)\) is a Banach space. This fact will be used to deduce a compactness criterion in \(\textbf{M}^{p,r}_\varphi (X)\) under certain assumptions on X, see Lemma 5.1.
Next, we define a commutator of linear operator of Calderón–Zygmund type.
Definition 1.10
Let T be a linear Calderón–Zygmund operator. Suppose that \(b\in L^1_{\textrm{loc}}(X)\). Then, the commutator [b, T] is defined by
for suitable functions f.
It is known that the theory of commutators has been generalized to other contexts, and it has many important applications to some nonlinear partial differential equations (see e.g. [8, 16, 23] and the references therein). When T is a linear Calderón–Zygmund operator, the \(L^p\) boundedness of [b, T] was first proved by Coifman–Rochberg–Weiss [9]. After that this result has been developed by many authors in [3, 12, 13, 15, 17,18,19, 24, 26, 29, 38, 40] and the references therein. In [40], Uchiyama obtained the compactness of operators of Hankel type. Furthermore, Beatrous–Li [3] proved a boundedness and compactness characterization for [b, T] on \(L^p(X)\), where X is a space of homogeneous type, and some applications to Hankel type operators on Bergman spaces were given by the authors in [10,11,12, 26].
When T is the Cauchy integral operator associated with the Lipschitz curves, the \(L^p\) boundedness and compactness characterization of [b, T] was obtained in [29]. Moreover, Tao et al. [38] extended this result to Morrey spaces (see also [13] for the Lorentz boundedness and compactness characterization of [b, T]). It is obvious that such an operator mentioned above is associated with a kernel K of Calderón–Zygmund type satisfying the homogeneity, i.e:
There exist positive constants \(c_0\) and \(\tilde{C}\) such that for every \(x \in X\) and \(r > 0\), there exists \(y \in B(x,\tilde{C} r) {\setminus } B(x,r)\) satisfying
In this case, Duong et al. [19] established the two weight commutator theorem of Calderón–Zygmund operators in the sense of Coifman–Weiss on spaces of homogeneous type. As applications, they can obtain a two weight commutator theorem for the following Calderón–Zygmund operators: Cauchy integral operator on \(\mathbb {R}\), Cauchy–Szegö projection operator on Heisenberg groups, Szegö projection operators on a family of unbounded weakly pseudoconvex domains, the Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms.
Next, we discuss the Hardy factorization in terms of the commutators. A famous result of Coifman–Rochberg–Weiss [9] is that every \(f\in \textbf{H}^1(\mathbb {R}^n)\) can be written as
with
where \(\mathcal {R}_j\) are the Riesz transform for \(j=1,\dots ,n\).
As a consequence, the authors obtained a characterization of functions b in \(\textrm{BMO}(\mathbb {R}^n)\) through the \(L^2\) boundedness of \([b,\mathcal {R}_j]\), \(j = 1,\dots ,n\). This theory has been studied by many authors in [13, 14, 18, 25, 28, 41], and the references cited therein. For instance, Uchiyama [41] extended the Hardy factorization to \(\textbf{H}^p\) on the space of homogeneous type. In addition, Komori–Mizuhara [25] proved the weak \(\textbf{H}^1\) factorization in terms of the commutators of Calderoón–Zygmund type in generalized Morrey spaces. We do not forget to mention that a weak Hardy factorization for the Bessel operators was obtained by the authors in [18]. Recently, the first author and Wick [14] proved a weak Hardy factorization in terms of multi-linear operator in Morrey spaces.
Inspired by the above results, we would like to generalize the theory of commutators to the generalized Morrey–Lorentz spaces, and the Block spaces. Concerning the \(\textbf{M}^{p,r}_\varphi (X)\) boundedness of commutators in Definition 1.10, we have the following theorem.
Theorem 1.11
Assume the same hypotheses as in Theorem 1.8. If \(b\in \textrm{BMO}(X)\), and T is a linear Calderón–Zygmund operator in Definition 1.3, then [b, T] maps \(\textbf{M}^{p,r}_\varphi (X)\) into itself continuously. Moreover, we have
Remark 1.12
By duality in Theorem 1.8, we have that [b, T] also maps \(\textbf{B}^{p',r'}_\varphi (X) \rightarrow \textbf{B}^{p',r'}_\varphi (X)\) continuously.
Our next result is the weak Hardy factorization in terms of commutators in \(\textbf{B}^{p',r'}_\varphi (X)\) and \(\textbf{M}^{p,r}_\varphi (X)\).
Theorem 1.13
Let \(p\in (1,\infty )\), \(r\in [1,\infty ]\), and let \(\varphi (t)\) satisfy (1.5). Suppose that T is a homogeneous operator of Calderón–Zygmund type. Then, for every function \(f\in \textbf{H}^1(X)\), there exist sequences \(\{ \lambda _{k,j} \} \in l^1\) and functions \(\{g_{k,j}\}, \{ h_{k,j}\} \subset L^\infty _c(X)\) (the space of bounded functions with compact support), such that
in the sense of \(\textbf{H}^1(X)\). In addition, we have that
where the infimum above is taken over all possible representations of f that satisfy (1.13).
We prove this result in Sect. 4.
Remark 1.14
Note that our assumption on the homogeneity of operator T in Theorem 1.13 is weaker than (1.12) used in [19, 25].
As a consequence of Theorem 1.13, we obtain a characterization of functions in \(\textrm{BMO}(X)\) via the \(\textbf{M}^{p,r}_\varphi \) (resp. \(\textbf{B}^{p',r'}_{\varphi }\)) boundedness of commutators of Calderón–Zygmund types.
Corollary 1.15
Let \(p\in (1,\infty )\), \(r\in [1,\infty ]\), and let \(\varphi (t)\) satisfy (1.5). Suppose that T is a linear Calderón–Zygmund operator. If \(b\in \textrm{BMO}(X)\), then the commutator [b, T] maps \(\textbf{M}^{p,r}_\varphi (X)\) into \(\textbf{M}^{p,r}_\varphi (X)\) continuously. Moreover, it holds true that
Conversely, for \(b\in L^1_{\textrm{loc}}(X)\), if T is homogeneous, and [b, T] maps \(\textbf{M}^{p,r}_\varphi (X) \rightarrow \textbf{M}^{p,r}_\varphi (X)\) continuously, then \(b\in \textrm{BMO}(X)\), and
Remark 1.16
By duality, the result of Corollary 1.15 also holds for \(\textbf{B}^{p',r'}_{\varphi }(X)\) in place of \(\textbf{M}^{p,r}_\varphi (X)\).
The last result is a \(\textbf{M}^{p,r}_\varphi \)-compactness characterization in terms of [b, T]. Since our assumptions on the homogeneous space \((X,d,\mu )\) are quite general, then we have to make some additional assumptions to \((X,d,\mu )\). Concerning the compactness, we suppose that the homogeneous space \((X,d,\mu )\) is a vector space, and is a locally compact space such that
and
A typical example of such space is the Euclidean space, equipped with the Lebesgue measure. In addition, we also note that any Ahlfors n-regular metric measure space \((X,d,\mu )\) satisfies (1.15).
Then, we have the following theorem.
Theorem 1.17
Let \(p\in (1,\infty )\), \(r\in [1,\infty ]\), and let \(\varphi (t)\) satisfy (1.5). Assume that \((X,d,\mu )\) is a locally compact space such that (1.14) and (1.15) hold. Then the following statements hold true.
If \(b\in \textrm{CMO}(X)\), and T is a linear Calderón–Zygmund operator, then [b, T] is compact on \(\textbf{M}^{p,r}_\varphi (X)\).
Conversely, for \(b\in L^1_{\textrm{loc}}(X)\), if T is homogeneous, and [b, T] is a compact operator on \(\textbf{M}^{p,r}_\varphi (X)\), then \(b\in \textrm{CMO}(X)\).
To end this section, we list some operators of Calderón–Zygmund type that our results are applicable to: the Cauchy integral operators, the Cauchy-Szegö projection operator on the Heisenberg group \(\mathbb {H}^n\), the Szegö projection operator on a family of unbounded weakly pseudo-convex domains, the Riesz transforms associated with sub-Laplacian on stratified nilpotent Lie groups, the Riesz transform associated with the Bessel operator on \(\mathbb {R}_+\), the Riesz transforms associated with Bessel operators on \(\mathbb {R}^{n+1}_+\). We refer to [19] for the details of these operators.
Finally, we emphasize that our results extend the boundedness and compactness characterization of linear Calderón–Zygmund operators to the generalized Morrey–Lorentz spaces, and Block spaces.
2 Generalied Morrey–Lorentz Space as Dual of Block Space
In this part, we study some properties of the Morrey–Lorentz spaces and the block spaces.
Lemma 2.1
Let \(p\in (1,\infty )\), \(r\in [1,\infty ]\), and let \(\varphi (t)\) satisfy (1.5). Then, we have
Proof of Lemma 2.1
It suffices to show that for any ball \(B(x,t)\subset X\), we have
where constant \(C>0\) is independent of f.
Indeed, let b be a \((p',r',\varphi )\)-block. Suppose that \(\textrm{supp}(b)\subset B(z, \tau )\), for some ball \(B(z, \tau )\) in X. Then, applying Hölder’s inequality in Lorentz spaces yields
If \(\tau \ge t\), then since \(\mu (B(z, \tau ))^\frac{1}{p} \varphi (\tau )\) is nondecreasing, then we deduce from (2.2) that
Otherwise, we have \(\varphi (\tau ) \ge \varphi (t)\). Therefore
As a result, we obtain from (2.2) that
Now, for any \(f\in \textbf{B}^{p',r'}_{\varphi }(X)\), we can write
where \(\{b_k\}_{k\ge 1}\) is a sequence of \((p',r',\varphi )\)-blocks, and \(\sum _{k\ge 1} |\lambda _k|<\infty \).
Thanks to (2.3), we get
This yields (2.1).
Hence, we obtain Lemma 2.1. \(\square \)
Remark 2.2
As a consequence of Lemma 2.1, for every ball \(B(x,t)\subset X\) the series \(\sum ^\infty _{j=1} \lambda _j b_j\) is convergent in \(L^{1}(B(x,t))\) whenever \(\sum ^\infty _{j=1} |\lambda _j|<\infty \), and \(\{b_j\}_{j\ge 1}\) is a sequence of \((p',r',\varphi )\)-blocks.
Proposition 2.3
Let \(p\in (1,\infty )\), \(r\in [1,\infty ]\), and let \(\varphi (t)\) satisfy (1.5). Then, for every ball B(z, t) in X we have
for \(f\in L^{p',r'}_{\textrm{loc}}(X)\).
In addition, we have
and
Proof of Proposition 2.3
The proof of (2.4) is done by letting
Next, we prove (2.5). By (1.6), we can mimic the proof of a) in Lemma 2.1 in order to obtain
On the other hand, it is obvious that
Thus, we obtain the desired result.
Concerning (2.6), it follows from duality and (2.5) that
with \(g_0=c_0 \textbf{1}_{B(z,t)} \varphi (t)\), and \(c_0\) is a normalized constant such that \(\Vert g\Vert _{\textbf{M}^{p,r}_\varphi } = 1\).
With the last inequality noted, and by applying (2.4) with \(f\equiv 1\), we obtain (2.6). \(\square \)
The next result is a dual inequality.
Proposition 2.4
Same hypotheses as in Proposition 2.3. If \(f\in \textbf{M}^{p,r}_\varphi (X)\), and \(g\in \textbf{B}^{p',r'}_{\varphi }(X)\), then
Proof of Proposition 2.4
Since \(g\in \textbf{B}^{p',r'}_{\varphi }(X)\), then we can write
where \(\{\lambda _j\}_{j\ge 1}\in l^1\), and \(\{b_j\}_{j\ge 1}\) are \((p',r',\varphi )\)-blocks.
Assume that \(\textrm{supp}(b_j)\subset B_j\) with its radius \(R_j\), \(j\ge 1\). Then, applying Hölder’s inequality yields
This yields the proof of Proposition 2.4. \(\square \)
Next, we study the Fatou property of block spaces \(\textbf{B}^{p',r'}_{\varphi }(X)\). Such a result was obtained by the authors, [37] for \(\textbf{B}^{p'}_{\varphi }(X)\), \(\varphi (t)=t^{-n+\alpha }\), \(\alpha \in (0,\frac{n}{p})\).
Lemma 2.5
Let \(1< p <\infty \), \(r\in [1,\infty ]\), and let \(\varphi \) satisfy (1.5). Suppose that f and \(f_k\), \(k\ge 1\), are nonnegative, \(\Vert f_k\Vert _{\textbf{B}^{p',r'}_{\varphi }} \le 1\), and \(f_k(x) \uparrow f(x)\) for a.e. \(x\in X\). Then \(f\in \textbf{B}^{p',r'}_{\varphi }(X)\) and \(\Vert f\Vert _{\textbf{B}^{p',r'}_{\varphi }} \le 1\).
Proof
Note that the dyadic cubes were constructed by the authors, [22]. Thus, the proof of Lemma 2.5 follows by using the same argument as in the proof of Theorem 1.2, [37] in that one can replace the \(L^{p}\)-norm by the \(L^{p,r}\)-norm. \(\square \)
Now we have the tools to prove Theorem 1.8.
Proof of Theorem 1.8
We first show that
In fact, thanks to Proposition 2.4, we observe that operator \(\mathcal {T}_f: \textbf{B}^{p',r'}_{\varphi }(X)\rightarrow \mathbb {R}\) defined by
is linear and continuous.
Now, we define \(\mathcal {T}(f)= \mathcal {T}_f\) for \(f\in \textbf{M}^{p,r}_\varphi (X)\). It is obvious that \(\mathcal {T}: \textbf{M}^{p,r}_\varphi (X)\rightarrow \textbf{B}^{p',r'}_{\varphi }(X)^\prime \) is a linear operator. We claim that \(\mathcal {T}\) is injective.
To obtain the result, it suffices to show that if \(\mathcal {T}(f)=0\), then \(f(x)=0\) for a.e \(x\in X\). We argue with a contradiction that there is \(R_0>0\) such that \(f(x)\not =0\) for a.e. \(x\in B_{R_0}=B(0,R_0)\).
Since \(f\in \textbf{M}^{p,r}_\varphi (X)\), then we have \(f\in L^{p,r}(B_{R_0})\). By duality, there exits a function \(\bar{g}\in L^{p',r'}(B_{R_0})\), \(\bar{g}\not =0\) such that
Put
It is clear that g is a \((p',r',\varphi )\)-block. By this fact and (2.8), we obtain
which contradict to \(\mathcal {T}(f)=0\) in \(\textbf{B}^{p',r'}_{\varphi }(X)^\prime \).
Thus, we conclude that linear operator \(\mathcal {T}: \textbf{M}^{p,r}_\varphi (X)\rightarrow \textbf{B}^{p',r'}_{\varphi }(X)^\prime \) is injective. As a result, (2.7) follows.
Therefore, it remains to prove that
Indeed, let \(F \in \left( \textbf{B}^{p',r'}_{\varphi }(X)\right) ^\prime \), and let B be a ball in X. By Proposition 2.3, we have \(F \textbf{1}_{B}\in L^{p',r'}(X)\). Thus, by the duality, there exists \(f_B\in L^{p, r}(X)\) such that
Let \(X = \bigcup _{k\ge 1} B_k\), with \(B_k\Subset B_{k+1}\) for all \(k\ge 1\). Then, we define \(f(x) = f_{B_k}\) if \(x\in B_k\), which makes sense by (2.10). This implies that \(f_{B_k}(x)=f_{B_{k+1}}(x)\) for a.e. \(x\in B_k\).
Thus, it suffices to prove that \(f\in \textbf{M}^{p,r}_{\varphi }(X)\). Indeed, for any ball \(B_t \subset X\), there exists \(k_0\ge 1\) such that \(B_t\subset B_{k_0}\); and by the duality argument we have
Obviously, \(\frac{g\textbf{1}_{B_t} }{\mu \left( B_t \right) ^{ 1/p } \varphi (t)} \) is a \((p', r',\varphi )\)-block. Thus, it follows from the last inequality that
Therefore,
This implies that \(\textbf{M}^{p,r}_\varphi (X)=\textbf{B}^{p',r'}_{\varphi }(X)^\prime \).
Next, we prove (1.11).
It follows from (1.10) that \(\textbf{B}^{p',r'}_{\varphi }(X)\hookrightarrow \textbf{M}^{p,r}_\varphi (X)^\prime = \textbf{B}^{p',r'}_{\varphi }(X)^{\prime \prime }\). Thus, it is enough to show that
To obtain (2.11), we mimic the proof of Theorem 4.1, [37]. Assume that a measurable function f on X satisfies
It is obvious that \(|f(x)| < \infty \) for a.e. \(x \in X\). Assume without loss of generality that \(f \ge 0\) in X, if not we write \(f=f_+-f_{-}\), with \(f_+=\max \{f,0\}\), and \(f_-=\max \{-f,0\}\); and we treat each of them.
For every \(k \ge 1\), let us set \(B_k=B(z_0,k)\) for some \(z_0\in X\), and let \(f_k(x):= \min \{f(x),k\} \textbf{1}_{B_k}(x)\). Note that \(f_k (x) \uparrow f(x)\) for a.e. \(x\in X\). Since \(f_k\in L^\infty _c(X)\) (the space of bounded functions with compact support in X), then it is clear that \(f_k\in \textbf{B}^{p',r'}_{\varphi }(X)\).
Since \(\textbf{M}^{p,r}_\varphi (X)=\textbf{B}^{p',r'}_{\varphi }(X)^{\prime }\), and by (2.12), we apply the Hahn–Banach theorem to obtain
Thanks to Lemma 2.5, we deduce \(\Vert f\Vert _{\textbf{B}^{p',r'}_{\varphi }} \le 1\).
By duality and (2.12), we obtain \(\Vert f\Vert _{(\textbf{M}^{p,r}_\varphi )^\prime } \ge \Vert f\Vert _{\textbf{B}^{p',r'}_{\varphi }} \), which yields (2.11).
Hence, we have completed the proof of Theorem 1.8. \(\square \)
3 Several Lemmas
This part is devoted to the study of the \(\textbf{M}^{p,r}_\varphi (X)\)-bounds, and the \(\textbf{B}^{p,r}_\varphi (X)\)-bounds of the maximal function, the sharp function, and the linear Calderón–Zygmund operators.
Let us first recall the definition of the Hardy–Littlewood maximal function.
For any \(q>0\), we define
for any \(x\in X\), where the supremum is taken over all balls B containing x.
In brief, we denote \(\mathcal {M}=\mathcal {M}_{1}\).
Lemma 3.1
Let \(p\in (1,\infty )\), \(r \in [1,\infty ]\), and let \(\varphi (t)\) satisfy (1.5). Then, for any \(0<q<p\) there is a positive constant \(C=C(q,p,r)\) such that
Proof of Lemma 3.1
Let \(B_t=B(x,t)\) be a ball in X, and we write
So,
We first estimate \(\mathcal {M}_q(f_1)\). Since \(\mathcal {M}_q\) maps \(L^{p,r}(X)\rightarrow L^{p,r}(X)\) (see e.g. [11]), then we get
which yields
Next, since \(f_2=0\) in \(B_t\), then we observe that for any \(z\in B_{t/2}\),
Applying Hölder’s inequality in Lorentz spaces yields
Note that the last inequality follows from the monotonicity of \(\varphi (t)\).
Then, we obtain
This implies that
By combining (3.2) and (3.3), we obtain Lemma 3.1. \(\square \)
As a consequence of Lemma 3.1, and duality, we obtain the \(\textbf{B}^{p',r'}_{\varphi }\)-bound for \(\mathcal {M}\).
Corollary 3.2
Assume hypotheses as in Lemma 3.1. Then, \(\mathcal {M}_q\) maps \(\textbf{B}^{p',r'}_{\varphi }(X)\rightarrow \textbf{B}^{p',r'}_{\varphi }(X)\) continuously.
Proof of Corollary 3.2
For any \(f\in \textbf{B}^{p',r'}_{\varphi }(X)\), we can assume that \(f\ge 0\).
Thanks to Lemma 3.1, applying the Fefferman–Stein inequality (see [21]), and Hölder’s inequality yields
By duality, we obtain from the last inequality that
Hence, we get the conclusion. \(\square \)
Next we prove the \(\textbf{M}^{p,r}_\varphi \)-bound for the sharp maximal function, introduced by Fefferman–Stein, [21]
where the supremum is taken over all balls B containing x.
Lemma 3.3
Let \(p\in (1,\infty )\), \(r\in [1,\infty ]\), and let \(\varphi (t)\) satisfy (1.5). Then, for any \(f\in \textbf{M}^{p,r}_\varphi (X)\), we have that
Proof of Lemma 3.3
The conclusion will follow by way of the Fefferman–Stein inequality and duality. Indeed, there is a constant \(C=C(X)>0\) such that, for every \(f\in \textbf{M}^{p,r}_\varphi (X)\), and every function \(g\in L^1_{\textrm{loc}}(X)\), we have (see e.g. [27])
Thanks to duality and Corollary 3.2, we obtain
Then, we obtain Lemma 3.3. \(\square \)
Remark 3.4
By duality, and Fefferman–Stein’s inequality, the conclusion of Lemma 3.3 also holds for \(\textbf{B}^{p,r}_{\varphi }(X)\) in place of \(\textbf{M}^{p,r}_\varphi (X)\).
Thanks to Lemmas 3.1 and 3.3, we prove the \(\textbf{M}^{p,r}_\varphi \)-bound for linear Calderón–Zygmund operators as follows.
Lemma 3.5
Let \(p\in (1,\infty )\), \(r\in [1,\infty ]\), and let \(\varphi (t)\) satisfy (1.5). Let T be a linear Calderón–Zygmund operator. Then, T maps \(\textbf{M}^{p,r}_\varphi (X) \rightarrow \textbf{M}^{p,r}_\varphi (X)\). Furthermore, we have
Proof of Lemma 3.5
Let us recall the following pointwise estimate (see e.g. [15]).
For any \(q>1\), there exists a constant \(C>0\) such that
Thanks to Lemma 3.3, for any \(q\in (1,p)\) we obtain
This yields the proof of Lemma 3.5. \(\square \)
Remark 3.6
By duality, (3.5) also holds for \(\textbf{B}^{p',r'}_{\varphi }(X)\) in place of \(\textbf{M}^{p,r}_\varphi (X)\).
Our next result is an inequality of Minkowski type in \(\textbf{M}^{p,r}_\varphi (X)\), used several times in the following.
Lemma 3.7
Same hypotheses as in Lemma 3.5. Then, we have
Proof of Lemma 3.7
Applying Theorem 1.8 and Proposition 2.4 yields
Hence, we get Lemma 3.7. \(\square \)
4 Hardy Factorization in Morrey–Lorentz Spaces
Proof of Theorem 1.13
The proof follows by way of the following lemmas.
The first lemma is a fundamental result of \(\textbf{H}^1(X)\) (see e.g. [25, Lemma 4.3]).
Lemma 4.1
Let \(x_0, y_0 \in X\) be such that \(d(x_0, y_0)= M t\), for some \(t>0\), and \(M>10 A_0\). If
then there is a positive constant C independent of \(x_0, y_0, t, M\), such that
Next, we have the following result.
Lemma 4.2
If \(f\in \textbf{H}^1(X)\) can be written as
then, there exist \(\{g_k\}_{k\ge 1}, \{h_k\}_{k\ge 1}\subset L^\infty _c(X)\) such that
and
where \(M>0\) is sufficiently large.
Furthermore, we have
Proof of Lemma 4.2
Let a be an atom, supported in \(B(x_0,t)\subset X\), such that
Let \(M\ge 10\) be a real number, which will be determined later, and let \(y_0\in X\) be such that \(d(x_0,y_0)= M t\).
Now, we set
It is clear that these functions are in \(L^\infty _c(X)\). In addition, since T is homogeneous, then we have
Thanks to Proposition 2.3, we obtain
and
Combining (1.3), (4.4), and (4.5) yields
where \(C>0\) only depends on p.
Next, we show that
Put
Then, we have
We first consider \(\textbf{J}_1(x)\). Since \(\textrm{supp}(a)\subset B(x_0, t)\), then \(\textbf{J}_1(x)=0\) if \(x\notin B(x_0, t)\). And, for any \(x\in B(x_0, t)\) we use the smoothness of K in (1.8) and the homogeneity of T in order to obtain
Note that the last inequality follows from the doubling property in (1.2).
For \(\textbf{J}_2(x)\), by using the cancellation of a(x), and the fact \(\textrm{supp}(a)\subset B(x_0,t)\), we get that
Analogously to the proof of \(\textbf{J}_1\), we also obtain
Combining (4.8), (4.9), and (4.10) yields
Thus, we obtain (4.7) by applying Lemma 4.1 to F(x).
Next, we apply (4.7) to \(a =a_{k}\), for \(k\ge 1\). Then, there exist functions \(\{ g_k\}_{k\ge 1}, \{h_k\}_{k\ge 1} \subset L^{\infty }_c(X)\), such that
With this inequality noted, we get
provided that M is large enough.
This yields the proof of Lemma 4.2. \(\square \)
Now we can suppose that \(f\in \textbf{H}^1(X)\) can be written
with \(\{\lambda _{k,1}\}_{k\ge 1}\in l^1\), and \(\{a_{k,1}\}_{k\ge 1}\) are atoms.
Thanks to Lemma 4.2 and (4.6), there exist functions \(\{g_{k,1}\}, \{h_{k,1}\}\subset L^\infty _c(X)\) such that
Put
Since \(f_1\in \textbf{H}^1(X)\), then we can decompose
where \(\{\lambda _{k,2}\}_{k\ge 1}\in l^1\), and \(\{a_{k,2}\}_{k\ge 1}\) are atoms.
By (4.13), and by applying Lemma 4.2 to \(f_1\), there exist \(\{g_{k,2}\}_{k\ge 1}, \{h_{k,2}\}_{k\ge 1} \subset L^{\infty }_c(X)\), such that
Similarly, we can apply the above argument to
By induction, we can construct sequence \(\{ \lambda _{k,j}\} \in l^1\), and functions \(\{g_{k,j} \}, \{h_{k,j}\} \subset L^\infty _c(X)\), such that
which yields the desired result when \(N\rightarrow \infty \).
Thus, we complete the proof of Theorem 1.13. \(\square \)
Proof of Corollary 1.15
To obtain the upper bound of [b, T], we recall the following result (see e.g. [15, Lemma 1]).
Lemma 4.3
Let \(b\in \textrm{BMO}(X)\). Then, for any \(1<q<p\), there exists a positive constant C such that
By Lemmas 3.1, 3.3, 3.5, and 4.3, we obtain
Hence, we get the desired result.
Now, we prove the lower bound of [b, T]. To obtain the result, we utilize the Hardy factorization in Theorem 1.13, and the duality between \(\textrm{BMO}(X)\) and \(\textbf{H}^1(X)\).
As a matter of fact, \(\textbf{H}^1(X) \cap L^\infty _c(X)\) is dense in \(\textbf{H}^1(X)\).
Next, for every \(L>0\), let us put
For every \(f\in \textbf{H}^1(X)\cap L^\infty _c(X) \), it follows from Theorem 1.13 that there exist sequences \(\{ \lambda _{k,j} \} \in l^1\) and functions \(g_{k,j}, h_{k,j} \in L^\infty _c(X)\), such that
Furthermore, we have
Now, since \(b_L \rightarrow b\) in \(L^1_{\textrm{loc}}(X)\) as \(L\rightarrow \infty \), and \(f\in \textbf{H}^1(X)\cap L^\infty _c(X)\), then we have
where we denote \(\langle f, g\rangle = \int f(x) g(x) \, d\mu (x)\).
Thanks to the facts \(g_{k,j} T^*(h_{k,j}) - h_{k,j} T(g_{k,j})\in \textbf{H}^1(X)\) and \(\textrm{supp}(g_{k,j} T^*(h_{k,j}) - h_{k,j} T(g_{k,j}))\Subset X\), then we get
By Proposition 2.4, since [b, T] maps \(\textbf{B}^{p',r'}_{\varphi }(X) \rightarrow \textbf{B}^{p',r'}_{\varphi }(X)\) (see Corollary 3.2), then we obtain
Therefore,
This ends the proof of Corollary 1.15. \(\square \)
5 Compactness Characterization of [b, T] in \(\textbf{M}^{p,r}_\varphi (X)\)
In the last section, we study the compactness of [b, T] in \(\textbf{M}^{p,r}_\varphi (X)\). Then, we point out a compactness criterion in \(\textbf{M}^{p,r}_\varphi (X)\).
Lemma 5.1
Let \(p\in (1, \infty )\), \(r\in [1,\infty ]\), and let \(\varphi (t)\) satisfy (1.5). Assume that \((X,d,\mu )\) is a locally compact space such that (1.15) holds, and the set \(\mathcal {G}\) in \(\textbf{M}^{p,r}_\varphi (X)\) satisfies the following conditions:
Then, \(\mathcal {G}\) is strongly precompact set in \(\textbf{M}^{p,r}_\varphi (X)\).
Proof of Lemma 5.1
We can assume without loss of generality that \(0\in X\).
For any \(\tau >0\), let us define
Fix \(\tau >0\), we first claim that the set \(\{ \overline{f}_\tau : f\in \mathcal {G}\}\) is a precompact set in \(\mathcal {C}\left( \overline{B_R}\right) \). Thanks to the Ascoli–Arzelà theorem, it is enough to show that \(\{ \overline{f}_\tau : f\in \mathcal {G}\}\) is bounded and equicontinuous in \(\mathcal {C}\left( \overline{B_R}\right) \).
Indeed, we have from Hölder’s inequality that
uniformly in \(f\in \mathcal {G}\).
Concerning the equicontinuity, we have
By ii), we conclude that \(\{\overline{f}_\tau : f\in \mathcal {G}\}\) is equicontinuous in \(\mathcal {C}\left( \overline{B_R}\right) \), so the above claim follows.
Next, we show that
uniformly in \(f\in \mathcal {G}\).
Indeed, applying Minkowski’s inequality yields
This implies that
With this inequality noted, (5.3) follows from ii).
Now, we prove that \(\{\overline{f}_\tau :f\in \mathcal {G}\}\) is relatively compact in \(\textbf{M}^{p,r}_\varphi (X)\).
By iii), for any \(0< \varepsilon < 1\), there exist \(R_\varepsilon > 0\) such that for every \(f\in \mathcal {G}\)
Since \(\{ \overline{f_\tau }: f\in \mathcal {G}\}\) is strongly precompact in \( \mathcal {C}\left( \overline{B_R}\right) \), then for every \(\varepsilon >0\), there exist \(f^1, f^2,\dots , f^m\) in \(\mathcal {G}\), with \(m=m(\varepsilon )\in \mathbb {N}\) such that \(\{\overline{f^1_\tau },\overline{f^2_\tau },\dots ,\overline{f^m_\tau }\}\) is a finite \(\varepsilon \varphi (R_\varepsilon )\)-net in \(\{\overline{f_\tau }: f\in \mathcal {G}\}\) with respect to the norm of \( \mathcal {C}\left( \overline{B_R}\right) \).
As a result, for any \(f\in \mathcal {G}\), there exists \(j\in \{1,\dots ,m\}\) such that
Next, we prove that \(\{\overline{f^1_\tau },\overline{f^2_\tau },\dots ,\overline{f^m_\tau }\}\) is a finite \(\varepsilon \)-net of \(\{\overline{f_\tau }: f\in \mathcal {G}\}\) with respect to the norm of \(\textbf{M}^{p,r}_\varphi (X)\). It is equivalent to show that
where \(\tau >0\) is small enough.
In fact, we write
We first estimate \( (\overline{f_\tau } - \overline{f^j_\tau })\textbf{1}_{B_{R_\varepsilon }}\) in the \(\textbf{M}^{p,r}_\varphi \)-norm. Then, for any ball B(z, t) in X, we have
If \(t\ge R_\varepsilon \) then it follows from (5.7) and the monotonicity of \(\mu \left( B(z,t)\right) ^\frac{1}{p}\varphi (t)\) that
Otherwise, we use the monotonicity of \(\varphi (t)\) to obtain
By combining the two cases, we get
Concerning the term \((\overline{f_\tau } - \overline{f^j_\tau })\textbf{1}_{B^c_{R_\varepsilon }}\), by (5.3) and (5.4), we get
as \(\tau >0\) is small enough.
Thus, (5.6) follows from (5.8) and (5.9).
As a result, \(\{ \overline{f_\tau }:f\in \mathcal {G}\}\) is relatively compact in \(\textbf{M}^{p,r}_\varphi (X)\).
It suffices to show that \(\mathcal {G}\) is relatively compact in \(\textbf{M}^{p,r}_\varphi (X)\). Let \(\{f^k\}_{k\ge 1}\subset \mathcal {G}\). Since \(\{\overline{f_\tau }: f\in \mathcal {G} \}\) is strongly compact in \(\textbf{M}^{p,r}_\varphi (X)\), then there is a subsequence of \(\{f^k\}_{k\ge 1}\) (still denoted as \(\{f^k\}_{k\ge 1}\)) such that \(\overline{f^k_\tau }\) converges in \(\textbf{M}^{p,r}_\varphi (X)\) as \(k\rightarrow \infty \).
Then, it follows from (5.3) that
Therefore, \(\{f^k\}_{k\ge 1}\) is a Cauchy sequence in \(\textbf{M}^{p,r}_\varphi (X)\). Since \(\textbf{M}^{p,r}_\varphi (X)\) is complete, then \(\{f^k\}_{k\ge 1}\) converges to a function in \(\textbf{M}^{p,r}_\varphi (X)\).
This puts an end to the proof of Lemma 5.1. \(\square \)
Now, we are ready to prove Theorem 1.17.
Proof of Theorem 1.17
a) Necessity: Assume that \(b\in \textrm{CMO}(X)\). Let \(\mathcal {G}\) be a bounded set in \(\textbf{M}^{p,r}_\varphi (X)\). It is enough to show that \([b,T](\mathcal {G})\) is relatively compact in \(\textbf{M}^{p,r}_\varphi (X)\).
Indeed, since \(b\in \textrm{CMO}(X)\), then for every \(\varepsilon >0\) there exists a function \(b_\varepsilon \in \mathcal {C}^\infty _c(X) \) such that
By the triangle inequality and Corollary 1.15, we have for every \(f\in \mathcal {G}\)
With this inequality noted, it suffices to show that \([b_\varepsilon ,T](\mathcal {G})\) is relatively compact in \(\textbf{M}^{p,r}_\varphi (X)\) for a given \(\varepsilon >0\) small enough.
Since \(\mathcal {G}\) is a bounded set in \(\textbf{M}^{p,r}_\varphi (X)\), and by Theorem 1.13, then it is clear that \([b_\varepsilon ,T](\mathcal {G})\) satisfies (i).
Next, we show that \([b_\varepsilon ,T](\mathcal {G})\) also satisfies (ii). Indeed, suppose that \(\textrm{supp} (b_\varepsilon )\subset B_{R_\varepsilon }\), for some \(R_\varepsilon >10\). Then, for any \(f\in \mathcal {G}\), and for \(x\in B^c_R\), with \(R>10 A_0 R_\varepsilon \), we observe that \(d(x,y)\approx d(x,0)\) for any \(y\in B_{R_\varepsilon }\).
Thus, for any \(x\in B^c_R\) we have
For every ball \(B_t=B(x_0,t)\) in X, by (5.10) and Minkowski’s inequality, we obtain
uniformly in \(f\in \mathcal {G}\).
This implies that
Thus, \(\left\| [b_\varepsilon ,T](f) \textbf{1}_{B^c_R} \right\| _{\textbf{M}^{p,r}_\varphi } \rightarrow 0\) when \(R\rightarrow \infty \) uniformly in \(f\in \mathcal {G}\). In other words, \([b_\varepsilon ,T](\mathcal {G})\) verifies (iii).
It remains to prove the equicontinuity of \([b_\varepsilon , T]\). In fact, we show that for every \(\delta >0\), if d(z, 0) is sufficiently small (merely depending on \(\delta \)), then
uniformly in \(f\in \mathcal {G}\), where the constant \(C>0\) is independent of \(f, \delta ,d(z,0)\).
To obtain the desired result, we recall the maximal operator of T, defined by
where \(T_\tau \), the truncated operator of T, is
For convenience, we recall here Cotlar’s inequality (see [39, Lemma 6.1]). That is for all \(l>0\),
Now, for any \(x\in X\) we express
We first estimate \(\textbf{I}_1\).
Since \(b_\varepsilon \) is uniformly continuous on X, then we deduce from the last inequality that
as \(d(z,0)\rightarrow 0\).
Applying Cotlar’s inequality yields
For \(\textbf{I}_2\), we use the smoothness of kernel K, (1.14), and the doubling property of \(\mu \) in order to get
where \(D_k=B\left( x, 2^k\delta ^{-1} d(z,0)\right) \), \(k\ge 0\).
Then, we get from the last inequality that
Next, we estimate \(\textbf{I}_3\). For any \(k\ge 0\), let us set \(B_{k}=B\left( x,\delta ^{-1} 2^{-k}d(z,0)\right) \). Then, it follows from the size condition of K that
provided that \(d(z,0)<\delta ^2\).
With the last inequality noted, it follows from the \(\textbf{M}^{p,r}_\varphi \)-bound of operator \(\mathcal {M}\) that
Finally, we treat \(\textbf{I}_4\). Since \({\text {supp}}(b_\varepsilon )\subset B(0,R_\varepsilon )\), then it is sufficient to consider \(x\in B(0, 2 R_\varepsilon )\) when \(z\rightarrow 0\).
Thanks to the quasi-triangle inequality (1.1), we get
when \(z\rightarrow 0\), for all \(d(x,y)<\delta \).
Then,
By arguing as in \(\textbf{I}_3\), we also obtain
Combining (5.15), (5.16), (5.17), and (5.18) yields
uniformly in \(f\in \mathcal {G}\). Therefore, [b, T] satisfies ii).
Thanks to Lemma 5.1, we conclude that [b, T] is a compact operator on \(\textbf{M}^{p,r}_\varphi (X)\).
b) Sufficiency: Suppose that T is homogeneous, and [b, T] is a compact operator on \(\textbf{M}^{p,r}_\varphi (X)\). Thanks to Corollary 1.15, we have that \(b\in \textrm{BMO}(X)\).
Next, we show that \(b\in \textrm{CMO}(X)\). To obtain the result, we need a characterization of a function in \( \textrm{CMO}(X)\) (see, e.g., [40]).
Lemma 5.2
A function \(b\in \textrm{CMO}(X)\) if and only if b satisfies the following three conditions.
(i) \(\displaystyle \lim _{R\rightarrow \infty } \sup _{B_l,\, l>R} \frac{1}{\mu (B_l)}\int _{B_l} |b(z)-b_{B_l}| \,d\mu (z) = 0\),
(ii) \(\displaystyle \lim _{R\rightarrow \infty } \sup _{ \{ B_l, B_l \subset B(0,R)^c \}} \frac{1}{\mu (B_l)} \int _{B_l} |b(z)-b_{B_l}| \,d\mu (z) = 0\),
(iii) \(\displaystyle \lim _{\delta \rightarrow 0} \sup _{B_l, \,l<\delta } \frac{1}{\mu (B_l)} \int _{B_l} |b(z)-b_{B_l}| \,d \mu (z) = 0\).
We also need the following result for technical reasons.
Lemma 5.3
There exists a positive constant \(M\ge 10 A_0\), such that for any ball \(B(x_0, t)\) in X, there is a ball \(B(y_0,t)\), \(d(x_0,y_0)=M t\); and for any \(x\in B(x_0,t)\), \(T\left( \textbf{1}_{B(y_0,t)}\right) (x)\) does not change sign and
Proof of Lemma 5.3
Thanks to the smoothness of K, we have
If \( T(\textbf{1}_{B(y_0,t)})(x_0)>0\), then it follows from the homogeneity of T, the triangle inequality, and (5.20) that
provided that M is large enough.
By the same argument, we also obtain the conclusion if \( T(\textbf{1}_{B(y_0,R_0)})(x_0)<0\).
This puts an end to the proof of Lemma 5.3. \(\square \)
Now, we demonstrate that \(b\in \textrm{CMO}(X)\). Seeking a contradiction, we assume that \(b\not \in \textrm{CMO}(X)\). Therefore, b violates (i), (ii), and (iii) in Lemma 5.2. We consider these cases orderly.
Case 1. Suppose that b violates (i). Then, there exists a sequence of balls \(\left\{ B_k = B(x_k, R_k) \right\} _{k\ge 1} \) such that \(R_k\rightarrow \infty \) as \(k\rightarrow \infty \), and
Since \(R_k\rightarrow \infty \), we can choose a subsequence of \(\{R_k\}_{k\ge 1}\) (still denoted by \( \{R_k \}_{ k\ge 1}\)) such that
for some constant \(C>10\).
For technical reason, we denote \(m_b(\Omega )\), by the median value of function b on a bounded set \(\Omega \subset \mathbb {R}^n\) (possibly non-unique) such that
Next, for any \(k\ge 1\), let \(y_k \in X\) be such that \(d(x_k,y_k)= M R_k\), \(M>10A_0\), and put
and
also
Note that \(F_{k,1} \cap F_{l,1}=\emptyset \) whenever \(j\not =k\). Moreover, we have from the definition of the median value that
Similarly, we also obtain
Furthermore, we have from the definition of the median value
Next, it follows from (5.21) and the triangle inequality that
This implies that there exists a subsequence with respect to k such that either
or
for any \(k\ge 1\). Thus, one can assume without loss of generality that (5.27) occurs.
For any \(k\ge 1\), applying Lemma 5.3 and (5.23) yields
In addition, \(T\left( \textbf{1}_{F_{k,1}}\right) (x)\) is a constant sign in \(B_k\). Then, it follows from (5.25), (5.27), and Lemma 5.3 that
where \(\phi _k (x) = \varphi (R_k) \textbf{1}_{F_{k,1}}(x)\), for \(k\ge 1\).
Applying Hölder’s inequality in (5.29) yields
Since [b, T] maps \(\textbf{M}^{p,r}_\varphi (X) \rightarrow \textbf{M}^{p,r}_\varphi (X)\) continuously, then we deduce from the last inequality that
Next, thanks to (2.5) and the definition of \(F_{k,1}\), we get
Combining (5.30) and (5.31) yields
Thanks to the compactness of [b, T] on \(\textbf{M}^{p,r}_\varphi (X)\), there exists a subsequence of \(\left\{ [b,T] (\phi _{k}) \right\} _{k\ge 1}\) (still denoted as \(\left\{ [b,T] (\phi _{k}) \right\} _{k\ge 1}\)) such that
as \(k\rightarrow \infty \).
By (5.32), we also obtain
Next, for any \(q\in (p,\infty )\), since \(b\in \textrm{BMO}(X)\), and T is a linear of Calderón–Zygmund type, then [b, T] maps \(L^{q}(X) \rightarrow L^{q}(X)\) continuously for \(q\in (1,\infty )\).
As a result, we obtain
Since \(R_k\rightarrow \infty \), then we have \(2^{\gamma (k)} \le R_k < 2^{\gamma (k)+1}\), with \(\gamma (k)= [\log _2 R_k]\), and [l] denotes by the integer part of a real number l. Thanks to (1.5) and the monotonicity of \(\varphi \), we achieve
By inserting this fact into (5.35), we obtain
To this end, we only take q large enough such that \(2^\frac{n}{q} D<1\). Then, the right hand side of (5.36) tends to 0 as \(k\rightarrow \infty \) since (1.15),.
This implies that \(\left\| [b, T](\phi _{k}) \right\| _{L^{q}} \rightarrow 0\). Thus, \(\Phi =0\) a.e. in X. This contradicts (5.32).
Similarly, we also obtain a contradiction if (5.28) holds true. In summary, b cannot violate (i).
Case 2. Assume that b violates (ii). The proof of this case is similar to the one of Case 1. Thus, we leave its detail to the reader.
Case 3. The proof of this case is most like that of Case 1 by considering \(\delta _k\) in place of \(R_k\), with \(\delta _k\rightarrow 0\). Since we want to repeat the above proof for \(\delta _k\) in place of \(R_k\), then it is necessary to make some changes as follows:
Since \(\delta _k\rightarrow 0\), then for every \(C>10\), there is a subsequence of \(\{\delta _k\}_{k\ge 1}\) (still denoted as \(\{\delta _k\}_{k\ge 1}\)) such that \(\delta _{k+1}\le \frac{1}{C}\delta _k\).
Furthermore, we need to redefine \(F_{k,1}\) (resp. \(F_{k,2}\)):
By the definition of the median value, it is not difficult to verify that \(\mu (F_{k,1})\approx \mu (\tilde{B}_{k})\), and \(\mu (F_{k,2}) \approx \mu (\tilde{B}_{k})\) for \(k\ge 1\). This enable us to repeat the proof of Case 1 in order to get (5.33) and (5.34).
Next, for \(q\in (1,p)\) we repeat the proof of (5.35) to obtain
Since \(\varphi (t)\mu (B_t)^\frac{1}{p}\) is nondecreasing and \(\delta _k\rightarrow 0\), then we observe that
as \(k\rightarrow \infty \).
Again, we get that \([b, T](\phi _{k})\rightarrow 0\) in \(L^q(X)\), when \(k\rightarrow \infty \). This contradicts (5.34). Therefore, b must satisfy (iii).
From the above cases, we conclude that \(b\in \textrm{CMO}(X)\). Hence, we complete the proof of Theorem 1.17. \(\square \)
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Acknowledgements
The authors would like to thank the referees for their valuable comments which were very helpful for improving the manuscript. N. A. Dao is funded by University of Economics Ho Chi Minh City (UEH), Vietnam. T. T. Dung is funded by Ho Chi Minh City University of Education under the CS.2021.19.03TƉ project. X. T. Duong was supported by Australian Research Council through the grant ARC DP190100970.
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Dung, T.T., Dao, N.A., Duong, X.T. et al. Commutators on Spaces of Homogeneous Type in Generalized Block Spaces. J Geom Anal 34, 209 (2024). https://doi.org/10.1007/s12220-024-01662-1
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DOI: https://doi.org/10.1007/s12220-024-01662-1
Keywords
- Block space
- Hardy factorization
- Commutators
- Singular integral operators of Calderón–Zygmund type
- Space of homogeneous type