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p-adic singular integrals and their commutators in generalized Morrey spaces
Journal of Inequalities and Applications volume 2019, Article number: 65 (2019)
Abstract
For a prime number p, let \(\mathbb{Q}_{p}\) be the field of p-adic numbers. In this paper, we establish the boundedness of a class of p-adic singular integral operators on the p-adic generalized Morrey spaces. We also consider the corresponding boundedness for the commutators generalized by the p-adic singular integral operators and p-adic Lipschitz functions or p-adic generalized Campanato functions.
1 Introduction
Let p be a prime number, and let \(x\in \mathbb{Q}\). Then the non-Archimedean p-adic norm \(|x|_{p}\) is defined as follows: if \(x=0\), then \(|0|_{p}=0\); if \(x\neq 0\) is an arbitrary rational number with unique representation \(x=p^{\gamma }\frac{m}{n}\), where m, n are not divisible by p, and \(\gamma =\gamma (x)\in \mathbb{Z}\), then \(|x|_{p}=p^{-\gamma }\). This norm has the following properties: \(|xy|_{p}=|x|_{p}|y|_{p}\), \(|x+y|_{p}\leq \max \{|x|_{p}, |y|_{p}\}\), and \(|x|_{p}=0\) if and only if \(x=0\). Moreover, when \(|x|_{p}\neq |y|_{p}\), we have \(|x+y|_{p}=\max \{|x|_{p}, |y|_{p}\}\). Let \(\mathbb{Q}_{p}\) be the field of p-adic numbers defined as the completion of the field of rational numbers \(\mathbb{Q}\) with respect to the non-Archimedean p-adic norm \(|\cdot |_{p}\). For \(\gamma \in \mathbb{Z}\), we denote the ball \(B_{\gamma }(a)\) with center at \(a\in \mathbb{Q}_{p}\) and radius \(p^{\gamma }\) and its boundary \(S_{\gamma }(a)\) by
respectively. It is easy to see that
For \(n\in \mathbb{N}\), the space \(\mathbb{Q}_{p}^{n}=\mathbb{Q}_{p} \times \cdots \times \mathbb{Q}_{p}\) consists of all points \({x}=(x_{1},\ldots , x_{n})\) where \(x_{i}\in \mathbb{Q}_{p}\), \(i=1,\dots ,n\), \(n\geq 1\). The p-adic norm of \(\mathbb{Q}_{p}^{n}\) is defined by
Thus it is easy to see that \(|{x}|_{p}\) is a non-Archimedean norm on \(\mathbb{Q}_{p}^{n}\). The balls \(B_{\gamma }({a})\) and the sphere \(S_{\gamma }({a})\) in \(\mathbb{Q}_{p}^{n}\) for \(\gamma \in \mathbb{Z}\) are defined similarly to the case \(n=1\).
Since \(\mathbb{Q}_{p}^{n}\) is a locally compact commutative group under addition, by the standard analysis there exists the Haar measure dx on the additive group \(\mathbb{Q}_{p}^{n}\) normalized by \(\int _{B_{0}}dx=|B _{0}|_{H}=1\), where \(|E|_{H}\) denotes the Haar measure of a measurable set \(E\subset \mathbb{Q}_{p}^{n}\). Then by a simple calculation the Haar measures of any balls and spheres can be obtained. From the integral theory it is easy to see that \(|B_{\gamma }({a})|_{H}=p^{n\gamma }\) and \(|S_{\gamma }({a})|_{H}=p^{n\gamma }(1-p^{-n})\) for any \({a}\in \mathbb{Q}_{p}^{n}\). For a more complete introduction to the p-adic analysis, we refer to [1,2,3,4,5,6,7,8] and the references therein.
The p-adic numbers have been applied in the string theory, turbulence theory, statistical mechanics, quantum mechanics, and so forth (see [1, 9, 10] for detail). In the past few years, there is an increasing interest in the study of harmonic analysis on p-adic field (see [5,6,7,8] for detail).
Let \(\varOmega \in L^{\infty }(\mathbb{Q}_{p}^{n})\) be such that \(\varOmega (p^{j}x)=\varOmega (x)\) for all \(j\in \mathbb{Z}\) and \(\int _{|x|_{p}=1}\varOmega (x)\,dx=0\). Then the p-adic singular integral operators defined by Taibleson [5] are as follows:
The p-adic singular integral operator T is defined as the limit of \(T_{k}\) as k goes to −∞.
Moreover, let \(\overrightarrow{b}=(b_{1},b_{2},\ldots,b_{m})\), where \(b_{i}\in L_{\mathrm{loc}}{(\mathbb{Q}_{p}^{n})}\) for \(1\leq i\leq m\). Then the higher commutator generated by b⃗ and \(T_{k}\) can be defined as
and the commutator generated by \(\overrightarrow{b}=(b_{1},b_{2},\ldots,b _{m})\) and the p-adic singular integral operator T is defined as the limit of \(T_{k}^{\vec{b}} \) as k goes to −∞.
Under some conditions, the authors in [5, 11] showed that \(T_{k}\) were of type \((q, q)\) for \(1 < q < \infty \) and of weak type \((1,1)\) on local fields. Wu et al. [12] established the boundedness of \(T_{k}\) on p-adic central Morrey spaces. Furthermore, the λ-central BMO estimates for commutators of these singular integral operators on p-adic central Morrey spaces were obtained in [12]. Moreover, in the p-adic linear space \(\mathbb{Q}_{p} ^{n}\), Volosivets [13] gave sufficient conditions for the boundedness of the maximal function and Riesz potential in p-adic generalized Morrey spaces. Mo et al. [14] established the boundedness of the commutators generated by the p-adic Riesz potential and p-adic generalized Campanato functions in p-adic generalized Morrey spaces.
Motivated by the works of [12,13,14], we consider the boundedness of \(T_{k}\) on the p-adic generalized Morrey type spaces, as well as the boundedness of the commutators generated by \(T_{k}\) and p-adic generalized Campanato functions.
Throughout this paper, the letter C will be used to denote constants varying from line to line. The relation \(A\lesssim B\) means that \(A\leq CB\) with some positive constant C independent of appropriate quantities.
2 Some notation and lemmas
Definition 2.1
([13])
Let \(1\leq q<\infty \), and let \(\omega (x)\) be a nonnegative measurable function in \(\mathbb{Q}_{p} ^{n} \). AÂ function \(f\in L^{q}_{\mathrm{loc}}(\mathbb{Q}_{p}^{n} )\) is said to belong to the generalized Morrey space \(GM_{q,\omega }(\mathbb{Q}_{p} ^{n} )\) if
where \(\omega (B_{\gamma }(a))=\int _{B_{\gamma }(a)}\omega (x)\,dx\).
Let \(\lambda \in \mathbb{R}\). If \(\omega (B_{\gamma }(a))=|B_{\gamma }(a)|^{\lambda }\), then \(GM_{q,\omega }(\mathbb{Q}_{p}^{n} )\) is the classical Morrey space \(M_{q,\lambda }(\mathbb{Q}_{p}^{n} )\). About the generalized Morrey space, see [15], and for the classical Morrey spaces, see [16] and so on.
Moreover, let \(\lambda \in \mathbb{R}\) and \(1\leq q<\infty \). The p-adic central Morrey space \(CM_{q,\lambda }(\mathbb{Q}^{n}_{p})\) (see [8]) is defined by
Definition 2.2
([17])
For \(0<\beta <1\), the the p-adic Lipschitz space \(\varLambda _{\beta }(\mathbb{Q}^{n}_{p})\) is defined as the set of all functions \(f: \mathbb{Q}_{p}^{n}\mapsto \mathbb{C}\) such that
Definition 2.3
([13])
Let B be a ball in \(\mathbb{Q} _{p}^{n}\), \(1\leq q<\infty \), and let \(\omega (x)\) be a nonnegative measurable function in \(\mathbb{Q}_{p}^{n} \). AÂ function \(f\in L^{q} _{\mathrm{loc}}(\mathbb{Q}_{p}^{n} )\) is said to belong to the generalized Campanato space \(GC_{q,\omega }(\mathbb{Q}_{p}^{n} )\) if
where \(f_{B_{\gamma }(a)}=\frac{1}{|B_{\gamma }(a)|_{H}} \int _{B_{\gamma }(a)}f(x)\,dx\) and \(\omega (B_{\gamma }(a))= \int _{B_{\gamma }(a)}\omega (x)\,dx\).
The classical Campanato spaces can be found in [18, 19], and so on. The important particular case of \(GC_{q,\omega }( \mathbb{Q}_{p}^{n} )\) is \(BMO_{q,\lambda }(\mathbb{Q}_{p}^{n} )\), where \(1< q<\infty \) and \(0<\lambda <1/n\). The central BMO space \(CBMO_{q, \lambda }(\mathbb{Q}_{p}^{n} )\) is defined by
Lemma 2.1
([14])
Let \(1\leq q<\infty \), and let ω be a nonnegative measurable function. Let \(b\in GC_{q, \omega }(\mathbb{Q}_{p}^{n} )\). Then
for \(j,k\in \mathbb{Z}\) and any fixed \(a\in \mathbb{Q}_{p}^{n}\).
Thus, for \(j>k\), from Lemma 2.1 we deduce that
Lemma 2.2
([5])
Let \(\varOmega \in L^{\infty }( \mathbb{Q}_{p}^{n})\) be such that \(\varOmega (p^{j}x)=\varOmega (x)\) for all \(j\in \mathbb{Z}\) and \(\int _{|x|_{p}=1}\varOmega (x)\,dx=0\). If
then for \(1 < p <\infty \), there is a constant \(C > 0\) such that
for \(k\in \mathbb{Z}\), where C is independent of f and \(k \in {\mathbb{Z}}\).
Furthermore, \(T(f) = \lim_{k\rightarrow -\infty }T_{k}(f)\) exists in the \(L^{p}\) norm, and
Moreover, on the p-adic field, the Riesz potential \(I_{\alpha }^{p}\) is defined by
where \(\varGamma _{n}(\alpha )=(1-p^{\alpha -n})/(1-p^{-\alpha })\) for \(\alpha \in \mathbb{C}\), \(\alpha \neq 0\).
Lemma 2.3
([14])
Let α be a complex number with \(0< \operatorname{Re}\alpha <n\), and let \(1< r<\infty \), \(1< q< n/ \operatorname{Re}\alpha \), and \(0<1/r=1/q- \operatorname{Re}\alpha /n\). Suppose that both ω and ν are nonnegative measurable functions such that
for any \(a\in \mathbb{Q}^{n}_{p}\) and \(\gamma \in \mathbb{Z}\). Then the Riesz potential \(I^{\alpha }_{p}\) is bounded from \(GM_{q,\nu }\) to \(GM_{r,\omega }\).
3 Main results
In this section, we state the main results of the paper.
Theorem 3.1
Let \(1< q<\infty \), and let \(\varOmega (p ^{j}x)=\varOmega (x)\) for all \(j\in \mathbb{Z}\), \(\int _{|x|_{p}=1}\varOmega (x)\,dx=0\), and
Suppose that both ω and ν are nonnegative measurable functions such that
for any \(\gamma \in \mathbb{Z}\) and \(a\in \mathbb{Q}^{n}_{p}\). Then the singular integral operators \(T_{k}\) are bounded from \(GM_{q,\nu }\) to \(GM_{q,\omega }\) for all \(k\in \mathbb{Z}\). Moreover, \(T(f)=\lim_{k\rightarrow -\infty }T_{k}(f)\) exists in \(GM_{q,\omega }\), and the operator T is bounded from \(GM_{q,\nu }\) to \(GM_{q,\omega }\).
Corollary 3.1
Let \(1< q<\infty \), \(\lambda <0\), and let \(\varOmega \in L^{\infty }(\mathbb{Q}_{p}^{n})\) be such that \(\varOmega (p ^{j}x)=\varOmega (x)\) for all \(j\in \mathbb{Z}\), \(\int _{|x|_{p}=1}\varOmega (x)\,dx=0\), and
Then the operators \(T_{k}\) and T are bounded on the space \(M_{q,\lambda }\) for all \(k\in \mathbb{Z}\).
In fact, for \(\lambda <0\), taking \(\omega (B)=\nu (B)=|B|_{H}^{\lambda }\) in Theorem 3.1, we obtain Corollary 3.1. If the Morrey space \(M_{q,\lambda }(\mathbb{Q}_{p}^{n} )\) is replaced by the central Morrey space \(CM_{ q,\lambda }(\mathbb{Q}_{p}^{n} )\) in Corollary 3.1, then the conclusion is that of Theorem 4.1 in [12].
Theorem 3.2
Let \(\varOmega \in L^{\infty }(\mathbb{Q}_{p} ^{n})\) be such that \(\varOmega (p^{j}x)=\varOmega (x)\) for all \(j\in \mathbb{Z}\), \(\int _{|x|_{p}=1}\varOmega (x)\,dx=0\), and
Let \(0<\beta _{i}<1\) for \(i=1,2,\dots ,m\) be such that \(0<\beta =\sum_{i=1}^{m}\beta _{i}<n\), and let \(1< r<\infty \) and \(1< q< n/\beta \) be such that \(1/r=1/q-\beta /n\). Suppose that \(b_{i}\in {\varLambda _{\beta _{i}}}\), \(i=1,2, \dots ,m\), and both ω and ν are nonnegative measurable functions such that
for any \(\gamma \in \mathbb{Z}\) and \(a\in \mathbb{Q}^{n}_{p}\). Then the commutators \(T_{k}^{\vec{b}}\) are bounded from \(GM_{q,\nu }\) to \(GM_{r,\omega }\) for all \(k\in \mathbb{Z}\). Moreover, the commutator \(T^{\vec{b}}(f)=\lim_{k\rightarrow -\infty }T_{k}^{\vec{b}}(f)\) exists in the space of \(GM_{q,\omega }\), and \(T^{\vec{b}}\) is bounded from \(GM_{q,\nu }\) to \(GM_{q,\omega }\).
Theorem 3.3
Let \(\varOmega \in L^{\infty }(\mathbb{Q} _{p}^{n})\) be such that \(\varOmega (p^{j}x)=\varOmega (x)\) for all \(j\in \mathbb{Z}\), \(\int _{|x|_{p}=1}\varOmega (x)\,dx=0\), and
Let \(1< q,r,q_{1},\dots, q_{m}<\infty \) be such that \(1/r=1/q+1/q_{1}+1/q _{2}+\cdots +1/q_{m}\). Suppose that ω, ν, and \(\nu _{i}\) (\(i=1,2,\dots ,m\)) are nonnegative measurable functions. Suppose that \(b_{i}\in GC_{q_{i},\nu _{i}}(\mathbb{Q}_{p}^{n} )\), \(i=1,2,\dots ,m\), and the functions ω, ν, and \(\nu _{i}\) (\(i=1,2,\dots ,m\)) satisfy the following conditions:
-
(i)
\(\prod_{i=1}^{m}\nu _{i}(B_{\gamma }(a))\nu (B_{\gamma }(a))/ \omega (B_{\gamma }(a))=C<\infty \),
-
(ii)
\(\sum_{j=\gamma +1}^{\infty }\prod_{i=1}^{m}\nu _{i}(B _{j}(a))(j+1-\gamma )^{m}\nu (B_{j}(a))/\omega (B_{\gamma }(a))=C< \infty \)
for any \(\gamma \in \mathbb{Z}\) and \(a\in \mathbb{Q}^{n}_{p}\). Then the commutators \(T_{k}^{\vec{b}}\) are bounded from \(GM_{q,\nu }\) to \(GM_{r,\omega }\) for all \(k\in \mathbb{Z}\). The commutator \(T^{ \vec{b}}=\lim_{k\rightarrow -\infty }T_{k}^{\vec{b}}\) exists in the space of \(GM_{q,\omega }\), and \(T^{\vec{b}}\) is bounded from \(GM_{q,\nu }\) to \(GM_{q,\omega }\).
Corollary 3.2
Let \(\varOmega \in L^{\infty }( \mathbb{Q}_{p}^{n})\) be such that \(\varOmega (p^{j}x)=\varOmega (x)\) for all \(j\in \mathbb{Z}\), \(\int _{|x|_{p}=1}\varOmega (x)\,dx=0\), and
Let \(1< q,r,q_{1},\dots, q_{m}<\infty \) be such that \(1/r=1/q+1/q_{1}+1/q _{2}+\cdots +1/q_{m}\). Let \(0\leq \lambda _{1},\dots ,\lambda _{m}<1/n\), \(\lambda <-\sum_{i=1}^{m}\lambda _{i}\), and \(\tilde{\lambda }= \sum_{i=1}^{m}\lambda _{i}+\lambda \). If \(b_{i}\in BMO_{q_{i}, \lambda _{i}}(\mathbb{Q}_{p}^{n} )\), then the commutators \(T_{k}^{ \vec{b}}\) and \(T^{\vec{b}}\) are bounded from \(M_{q,\lambda }\) to \(M_{r,\tilde{\lambda }}\) for all \(k\in \mathbb{Z}\).
Moreover, let \(1< r,q,q_{1}<\infty \) be such that \(1/r=1/q+1/q_{1}\). Let \(0\leq \lambda _{1}<1/n\), \(\lambda <-\lambda _{1}\), and \(\tilde{\lambda }=\lambda _{1}+\lambda \). If \(b\in CBMO_{q_{1},\lambda _{1}}(\mathbb{Q}_{p}^{n} )\), then from Corollary 3.1 it follows that the commutators \(T_{k}^{b}=[T_{k}, b]\) and \(T^{b}=[T, b]\) are bounded from \(CM_{q,\lambda }\) to \(CM_{r,\tilde{\lambda }}\) for all \(k\in \mathbb{Z}\). These results are those of Theorem 4.2 in [12].
4 Proof of Theorems 3.1–3.3
Let us first give the proof of Theorem 3.1.
For any fixed \(\gamma \in \mathbb{Z}\) and \(a\in \mathbb{Q}^{n}_{p}\), it is easy to see that
where \(B^{c}_{\gamma }(a)\) is the complement to \(B_{\gamma }(a)\) in \(\mathbb{Q}^{n}_{p}\).
Using Lemma 2.2 and (3.1), it follows that
For II, let us first estimate \(|T_{k}(f\chi _{B^{c}_{\gamma }(a)})(x)|\).
Since \(x\in B_{\gamma }(a)\) and \(\varOmega \in L^{\infty }(\mathbb{Q} _{p}^{n})\), we have
Thus from (3.1) and (4.3) it follows that
Combining the estimates of (4.1), (4.2), and (4.4), we have
which means that \(T_{k}\) is bounded from \(GM_{q,\nu }\) to \(GM_{q, \omega }\) for all \(k\in \mathbb{Z}\).
Moreover, from Lemma 2.2 and the definition of \(GM_{q,\omega }( \mathbb{Q}_{p}^{n} )\) it is obvious that \(T(f)=\lim_{k\rightarrow -\infty }T_{k}(f)\) exists in \(GM_{q,\omega }\) and the operator T is bounded from \(GM_{q,\nu }\) to \(GM_{q,\omega }\).
Proof of Theorem 3.2
For any \(x\in \mathbb{Q}^{n}_{p}\), since \(\varOmega \in L^{\infty }( \mathbb{Q}_{p}^{n})\) and \(b_{i}\in {\varLambda _{\beta _{i}}}\), \(i=1,2,\dots ,m\), it is easy to see that
Thus from Lemma 2.3 it is obvious that the commutators \(T_{k}^{ \vec{b}}\) are bounded from \(GM_{q,\nu }\) to \(GM_{r,\omega }\) for all \(k\in \mathbb{Z}\).
Moreover, from the definition of \(GM_{q,\omega }(\mathbb{Q}_{p}^{n} )\) it is obvious that \(T^{\vec{b}}(f)=\lim_{k\rightarrow -\infty }T_{k}^{\vec{b}}(f)\) exists in the space of \(GM_{q,\omega }\), and the commutator \(T^{\vec{b}}\) is bounded from \(GM_{q,\nu }\) to \(GM_{q, \omega }\). □
Proof of Theorem 3.3
Without loss of generality, we need only to show that the conclusion holds for \(m=2\).
For any fixed \(\gamma \in \mathbb{Z}\) and \(a\in \mathbb{Q}^{n}_{p}\), we write \(f^{0}=f\chi _{B_{\gamma }(a)}\) and \(f^{\infty }=f \chi _{B^{c}_{\gamma }(a)}\). Then
We further estimate every part.
Since \(1/r=1/q+1/q_{1}+1/q_{2}\), from Hölder’s inequality, Lemma 2.2, and (i) it follows that
Let \(1/\bar{q}=1/q+1/q_{2}\). Then \(1/r=1/q_{1}+1/\bar{q}\). Thus, from Hölder’s inequality, Lemma 2.2, and (i) we obtain
Similarly,
For \(E_{4}\), from Lemma 2.2, Hölder’s inequality, and (i) we obtain
To estimate \(E_{5}\), we first need to consider \(|T_{k}(f^{\infty })(x)|\). In fact, by (4.3) it is easy to see that
Therefore from Hölder’s inequality, (4.6), and (ii) we get
It is similar to estimate (4.3) for \(x\in B_{\gamma }(a)\). By \(\varOmega \in L^{\infty }(\mathbb{Q}_{p}^{n})\) and (2.2) we can deduce that
Let \(1/\bar{q}=1/q+1/q_{2}\). Then \(1/r=1/q_{1}+1/\bar{q}\). Thus from Hölder’s inequality, (4.7), and (ii) it follows that
Similarly estimating \(E_{6}\), we obtain
Moreover, since \(\varOmega \in L^{\infty }(\mathbb{Q}_{p}^{n})\), by (2.2) it is easy to see that
Therefore from (4.8) and (ii) we get that
Combining (4.5) and the estimates of \(E_{1},E_{2},\dots , E_{8}\), we have
which means that the commutator \(T_{k}^{(b_{1},b_{2})}\) is bounded from \(GM_{q,\nu }\) to \(GM_{r,\omega }\).
Moreover, by Lemma 2.2 and the definition of \(GM_{q,\omega }( \mathbb{Q}_{p}^{n} )\) it is obvious that the commutator \(T^{\vec{b}}(f)= \lim_{k\rightarrow -\infty }T_{k}^{\vec{b}}(f)\) exists in the space of \(GM_{q,\omega }\), and \(T^{\vec{b}}\) is bounded from \(GM_{q,\nu }\) to \(GM_{q,\omega }\).
Therefore the proof of Theorem 3.3 is complete. □
5 Conclusion
In this paper, we established the boundedness of a class of p-adic singular integral operators on the p-adic generalized Morrey spaces. We also considered the corresponding boundedness for the commutators generalized by the p-adic singular integral operators and p-adic Lipschitz functions or p-adic generalized Campanato functions.
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Mo, H.X., Han, Z., Yang, L. et al. p-adic singular integrals and their commutators in generalized Morrey spaces. J Inequal Appl 2019, 65 (2019). https://doi.org/10.1186/s13660-019-2009-7
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DOI: https://doi.org/10.1186/s13660-019-2009-7