Spherical maximal functions, variation and oscillation inequalities on Herz spaces

Abstract In this work, the boundedness of the spherical maximal function, the mapping properties of the fractional spherical maximal functions, the variation and oscillation inequalities of Riesz transforms on Herz spaces have been established.


Introduction
In this paper, the mapping properties of the spherical maximal function and the spherical fractional maximal functions on Herz spaces are obtained. The variation and oscillation inequalities of Riesz transforms on Herz spaces are also established.
The Herz space is a generalization of the Lebesgue space. The Herz spaces have several applications on the studies of partial differential equations, Fourier series and Fourier transform (Ragusa, 2009(Ragusa, , 2012Weisz, 2008;Zhou & Cao, 2014). The mapping properties of a number of important operators from harmonic analysis such as the Hardy-Littlewood maximal function, the fractional integral operator and singular integral operators had been extended to Herz spaces (Li & Yang, 1996;Lu & Yang, 1996;Lu et al., 2008).
In (Ho, 2018e), the Young's inequality and the Hausdorff-Young inequality had also been extended to the Herz spaces by using the real interpolation method. This motivates us to use real interpolation to further study the mapping properties of some other operators on Herz spaces. In this paper, the mapping properties of the spherical maximal function, the fractional maximal functions, the variation and oscillation operators for Riesz transform are studied. They are all sublinear operators. In order to apply the real interpolation to sublinear operators, the function spaces are required to be a normed Riesz space. Since the Herz spaces are normed Riesz spaces, the estimates on the K-functional are applicable to Herz spaces and, consequently, the real interpolation yields the main results. This paper is organized as follows. The definitions of Herz spaces, the real interpolation and the normed Riesz spaces are presented in Section 2. The interpolation of the quasilinear operators on normed Riesz spaces is presented in Section 3. The main results for the spherical maximal function, the fractional spherical maximal functions, the variation and oscillation inequalities of Riesz transforms are obtained in Section 4.

Definitions and preliminaries
In this section, the definition of Herz spaces, some notions and notations from the theory of real interpolation and the Riesz spaces are recalled.
Let M 0 be the set of Lebesgue measurable function on R n . The following definitions are stated for Herz spaces. For any k 2 Z, let B k ¼ fx 2 R n : jxj 2 k g.
Definition 2.1. Let a 2 R; 0<p<1 and 0<q<1. The homogeneous Herz space consists of those Lebesgue measurable function f which satisfies where B k is the ball in R n with center at origin and radius 2 k . Similarly, the inhomogeneous Herz space consists of those Lebesgue measurable function f which satisfies

<1:
When 0<p r<1; 0<q<1 and The reader is referred to (Lu et al., 2008) for the properties of Herz spaces.
Let a 2 R and 0<q<1. The power weighted Lebesgue space L a q consists of those Lebesgue measurable function f satisfying For any weight function x : R n ! ð0; 1Þ, write It is easy to see that _ K a;q q ¼ L a q and L a q ¼ L q jÁj aq . The definition of the well known real interpolation method is recalled in the following. A pair of Banach spaces ðX 0 ; X 1 Þ is said to be an interpolation couple if there exists a linear Hausdorff space Z such that X 0 ; ! Z and X 1 ; ! Z.
Let ðX 0 ; X 1 Þ be an interpolation couple. For any f 2 X 0 þX 1 , the K-functional is defined as where the infimum is taken over all f ¼ f 0 þf 1 for which f i 2 X i , i ¼ 0, 1.

<1:
For the details about real interpolation, the reader is referred to (Triebel, 1978, Section 1.2). The above interpolation method had been generalized in (Ho, 2016(Ho, , 2018a(Ho, , 2018c(Ho, , 2018d to study the Fourier transform, the k-plane transform and the maximal estimate of the solution of some partial differential equations. In order to present the real interpolation of quasilinear operators, some notions and notations from vector lattices and Riesz spaces are recalled (Zaanen, 1997).
Let R be a partially ordered set and T be a nonempty subset of R. The element x 2 R is called the upper bound of T if t x for all t 2 T . If x is a upper bound of T such that x y where y is any other upper bound of T , x is the supremum of T . The definition of lower bound and infimum are analogous.
Let R be a partially ordered set. If every subset consisting of two elements has a supremum and an infimum, then R is a lattice.
For any lattice R, the supremum and the infimum of x; y 2 R are defined as xÚy and xÙy, respectively.
The real vector space R is a Riesz space if there is a partial order such that R is a lattice with respect to and (Zaanen, 1997, Theorem 5.1).
The following Riesz decomposition theorem for Riesz spaces is given in (Zaanen, 1997, Theorem 6.4).
Let R be a Riesz space associated with the partial order . A Riesz space R is said to be a normed Riesz space if R is equipped with a norm jj Á jj R satisfying jf j jgj ) jjf jj R jjgjj R : (2.2) For any a 2 R; 1 q<1; jf j jgj gives therefore, L a q is a normed Riesz space.

Interpolation of quasilinear operators
In this section, a folklore fact on the real interpolation of quasilinear operators (Sagher, 1971(Sagher, , 1972 is presented. It states that whenever the target spaces Y 0 , Y 1 are normed Riesz spaces, then the real interpolation of quasilinear operator is valid. Even though this is a folklore fact, for completeness, the details of the estimate on K-functional and the corresponding interpolation theorem are presented.
Since function spaces on R n are considered, for the rest of this paper, the ordering f g; f ; g 2 M 0 , is defined as f ðxÞ gðxÞ a.e. on R n . Therefore, the vector space of Lebesgue measurable functions endowed with the ordering is a Riesz space, see Definition 2.6.
As Y 0 and Y 1 are normed Riesz spaces, the boundedness of T guarantees that for some C 0 >0. Consequently, for any t 2 ð0; 1Þ By taking infimum over f ¼ gþh with g 2 X 0 and h 2 X 1 on both sides of the above inequalities, the above inequalities yield Therefore, when 0<p<1, The reader is referred to (Ho, 2018d) for the interpolation of sublinear operators by general interpolation functors.

Main results
In this section, the interpolation of quasilinear operators from the previous section is used to establish the boundedness of the spherical maximal function, the fractional spherical maximal function, the oscillation operator and the q-variation operator of the Riesz transform.

Spherical maximal function
The boundedness of the spherical maximal function on Herz spaces is established in this section. As an application of this result, some estimates of the weak solution for wave equation on Herz spaces are obtained. Let dl be the normalized surface measure on the unit sphere on R n . For any locally integrable function f, the spherical maximal function of f is defined by Francia, 1985, p.571). (Duandikoetvea & Vega, 1996;Garc ıa-Cuerva & Rubio de Francia, 1985, Corollary 7.9;Gunawan, 1998) give the following power weighted norm inequalities for the spherical maximal function. Let n ! 3; 1<q<1 and 1Àn a<qðnÀ1ÞÀn. There exists a constant C > 0 such that Obviously, M is a sublinear operator on L a 0 q þL a 1 q when 1<q<1 and 1Àn a 0 ; a 1 <qðnÀ1ÞÀn. The boundedness of the spherical maximal function on Herz spaces is presented and established in the following theorem.

Let
n ! 3; 1<q<1; 0<p<1 and 1Àn<a<qðnÀ1ÞÀn. There is a constant C > 0 such that Select 0<h<1 and 1Àn<a 0 <a 1 <qðnÀ1ÞÀn so that a ¼ ð1ÀhÞa 0 þha 1 . In view of Theorem 4.1, M is bounded on L a 0 =q q and L a 1 =q q , respectively. Consequently, Theorem 3.3 yields The above result gives estimates on the weak solution of the wave equation on Herz spaces because it is well known that the spherical maximal function can provide some estimates for the weak solution of the wave equation, see (Stein, 1976).
The above result is used to study the following classical initial-value problem For any f 2 L q ðR 3 Þ; 3 2 <q<1, the weak solution of the above initial value problem is given by the Kirchhoff's formula (Evan, 2010) The above formula gives ju x; t ð Þj CtMf x ð Þ; 8x 2 R 3 ; t > 0: Therefore, Theorem 4.2 yields the following estimate of the weak solution for the wave equation on Herz spaces. Let 3 2 <q<1; 0<p<1 and 1Àn<a<qðnÀ1ÞÀn. There exists a constant C > 0 such that Some further generalizations of the study of maximal function on Herz spaces are presented. Let / 2 L 1 ðR n Þ be a radical function satisfying Zo's condition ð jxj>2jyj sup d>0 j/ d xÀy ð Þ À/ d x ð Þjdx C; y 2 R n : Define Let 0<p<1; 1<q<1; Àn<a<nðqÀ1Þ and / 2 L 1 ðR n Þ be a radical function satisfying Zo's condition. Then, there is a constant C > 0 such that Select 0<h<1 and Àn<a 0 <a 1 <nðqÀ1Þ so that a ¼ ð1ÀhÞa 0 þha 1 . According to (Garc ıa-Cuerva & Rubio de Francia, 1985, Corollary 7.7), M / is bounded on L a 0 =q q and L a 1 =q q , respectively. Therefore, our result follows from (3.1).

Fractional spherical maximal functions
Let a>0 and dl t be the normalized surface measure on the sphere Bð0; tÞ ¼ fx 2 R n : jxj tg in R n . For any locally integrable function f, the fractional spherical maximal function of f is defined as For the mapping properties of M a , the reader may consult (Gunawan, 1998;Oberlin, 1989).
Since the Muckenhoupt weight function is involved to present the weighted norm inequality for M a , the definition of the Muckenhoupt weight function is recalled from (Grafakos, 2009, Chapter 9). Let B denote the collection of balls in R n .
For 1<p<1, a locally integrable function x : R n ! ½0; 1Þ is said to be an A p weight if where p 0 ¼ p pÀ1 . A locally integrable function x : R n ! ½0; 1Þ is said to be an A 1 weight if Moreover, A 1 to be the union of A p for all p ! 1. That is, The following is the weighted norm inequality for the fractional spherical maximal functions. then there is a constant C > 0 such that The reader is referred to (Cowling, Garc ıa-Cuerva & Gunawan, 2002, Theorem 4.4) for the proof of the preceding result.
The mapping properties for the fractional spherical maximal operators on Herz spaces are established in the following. Let 0<p<1; n nÀ1 <r<q<ðnÀ1Þr; a ¼ n r À n q and max 0; 1À q n È É <c<1À q rðnÀ1Þ . If Àn<a<n ðsÀ1Þ where then there is a constant C > 0 such that In view of (Grafakos, 2009, p.286) and the condition Àn<a<nðsÀ1Þ; j Á j a 2 A s . Select 0<h<1 and Àn<a 0 <a 1 <nðsÀ1Þ such that a ¼ ð1ÀhÞa 0 þha 1 . The above result shows that weighted norm inequality for Muckenhoupt weighted Lebesgue spaces can be used to generate mapping properties for quasilinear operators on Herz spaces. This idea is employed in the following section to study variation and oscillation inequalities for Riesz transform on Herz spaces.

Variation and oscillation inequalities
For any e>0, let R j;e f x ð Þ ¼ C n ð jxÀyj>e x j Ày j jx À yj n f y ð Þ dy; j ¼ 1; . . . ; n; hence any given ft i g 1 i¼1 with t i # 0, the oscillation operator of the Riesz transform R j , j ¼ 1 Á Á Á ; n, is given by sup t iþ1 e iþ1 <e i t i jR j;e iþ1 f À R j;e i f j 2 ! 1=2 ; and the q-variation operator is defined as where q>0 and the supremum is taken over all sequence fe i g 1 i¼1 satisfying e i # 0. The weighted norm inequalities for the oscillation operators and the q-variation operators for the Riesz transforms are established in (Ma, Torrea & Xu, 2017;Zhang & Wu, 2017).
Let q>2. For every 1 j n; 1<p<1 and x 2 A p , there exists a constant C > 0 such that The reader is referred to (Gillespie & Torres, 2004;Ma et al., 2017;Zhang & Wu, 2017) for the proofs of the above results.
The above inequalities yield the variation and oscillation inequalities for Riesz transforms on Herz spaces.