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# Fractional Calculus and Applied Analysis

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Volume 21, Issue 3

# Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces

Michael Ruzhansky
/ Durvudkhan Suragan
• Department of Mathematics, School of Science and Technology, Nazarbayev University, 53 Kabanbay Batyr Ave, Astana 010000, Kazakhstan
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• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Nurgissa Yessirkegenov
• Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
• Institute of Mathematics and Mathematical Modelling, 125 Pushkin str., Almaty 050010, Kazakhstan
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Published Online: 2018-07-12 | DOI: https://doi.org/10.1515/fca-2018-0032

## Abstract

We analyze local (central) Morrey spaces, generalized local (central) Morrey spaces and Campanato spaces on homogeneous groups. The boundedness of the Hardy-Littlewood maximal operator, Bessel-Riesz operators, generalized Bessel-Riesz operators and generalized fractional integral operators in generalized local (central) Morrey spaces on homogeneous groups is shown. Moreover, we prove the boundedness of the modified version of the generalized fractional integral operator and Olsen type inequalities in Campanato spaces and generalized local (central) Morrey spaces on homogeneous groups, respectively. Our results extend results known in the isotropic Euclidean settings, however, some of them are new already in the standard Euclidean cases.

MSC 2010: Primary 22E30; Secondary 43A80

## 1 Introduction

Consider the following Bessel-Riesz operators

$Iα,γf(x)=∫RnKα,γ(x−y)f(y)dy=∫Rn|x−y|α−n(1+|x−y|)γf(y)dy,$(1.1)

where f$\begin{array}{}{L}_{loc}^{p}\end{array}$ (ℝn), p ≥ 1, γ ≥ 0 and 0 < α < n. Here, Iα,γ and Kα,γ are called Bessel-Riesz operator and Bessel-Riesz kernel, respectively.

The boundedness of the fractional integral operators Iα,0 on Lebesgue spaces was shown by Hardy and Littlewood in [24], [25] and Sobolev in [43]. In the case of ℝn, the Hardy-Littlewood maximal operator, the generalized fractional integral operators, which are a generalized form of the Riesz potential Iα,0 = Iα, Bessel-Riesz operators and Olsen type inequalities are widely analysed on Lebesgue spaces, local (central) Morrey spaces and generalized local (central) Morrey spaces (see e.g. [1], [10], [29], [13], [11], [28], [30], [31], [23], [42], [26] and [27], as well as [6] for a recent survey). For some of their functional analytic properties, see also [7], [8] and references therein. We also refer [17], [18], [16], [5], [37] for analysis in local (central) Morrey spaces and generalized local (central) Morrey spaces, and [4], [3] in anisotropic local Morrey-type spaces.

In this paper we are interested in the boundedness of the Hardy-Littlewood maximal operator, Bessel-Riesz operators, generalized Bessel-Riesz operators, generalized fractional integral operators and Olsen type inequalities in generalized local (central) Morrey spaces on homogeneous Lie groups. The obtained results give new statements already in the Euclidean setting of ℝn when we are working with anisotropic differential structure. Furthermore, even in the isotropic situation in ℝn, one novelty of all the obtained results is also in the arbitrariness of the choice of any homogeneous quasi-norm, and some estimates are also new in the usual isotropic structure of ℝn with the Euclidean norm, which we will be indicating at relevant places.

Thus, we could have worked directly in ℝn with anisotropic structure, but since the methods work equally well in the setting of Folland and Stein’s homogeneous groups, we formulate all the results in such (greater) generality. In particular, it follows the general strategy initiated by their work, of distilling results of harmonic analysis depending only on the group and dilation structures: in this respect the present paper shows that the harmonic analysis on local (central) Morrey spaces largely falls into this category.

In turn, this continues the research direction initiated in [36] devoted to Hardy and other functional inequalities in the setting of Folland and Stein’s [15] homogeneous groups. We also refer to recent papers [32], [33], [34], [35], [38], [39], [40] and [41] for discussions related to different functional inequalities with special as well as arbitrary homogeneous quasi-norms in different settings. Local (central) Morrey spaces for non-Euclidean distances find their applications in many problems, see e.g. [20, 21] and [22].

We also refer to the recent survey paper [2] for some inequalities in fractional calculus that are used in differential or integral equations, and refer to [9] for the boundedness of the Riesz fractional integration operator from a generalized Morrey space Lp,ϕ(ℝn) to a certain Orlicz-Morrey space LΦ,ϕ(ℝn).

For the convenience of the reader let us now shortly recapture the main results of this paper.

For the definitions of the spaces appearing in the formulations below, see (3.1) for local (central) Morrey spaces LMp,q(𝔾), (3.2) for generalized local (central) Morrey spaces LMp,ϕ(𝔾), and (7.1) for generalized Camponato spaces 𝓛𝓜p,ϕ(𝔾), as well as (3.4) for the Hardy-Littlewood maximal operator M, (2.4) for Bessel-Riesz operators Iα,γ, (5.1) for generalized Bessel-Riesz operators Iρ,γ, and (6.1) for generalized fractional integral operators Tρ. Both Iρ,γ and Tρ generalise the Riesz transform and the Bessel-Riesz transform in different directions.

Thus, in this paper we show that for a homogeneous group 𝔾 of homogeneous dimension Q and any homogeneous quasi-norm |·| we have the following properties:

• If 0 < α < Q and γ > 0, then Kα,γLp1(𝔾) for $\begin{array}{}\frac{Q}{Q+\gamma -\alpha }\end{array}$ < p1 < $\begin{array}{}\frac{Q}{Q-\alpha },\end{array}$ and ∥Kα,γLp1(𝔾)$\begin{array}{}{\left(\sum _{k\in \mathbb{Z}}\frac{\left({2}^{k}R{\right)}^{\left(\alpha -Q\right){p}_{1}+Q}}{\left(1+{2}^{k}R{\right)}^{\gamma {p}_{1}}}\right)}^{\frac{1}{{p}_{1}}}\end{array}$ for any R > 0, where $\begin{array}{}{K}_{\alpha ,\gamma }:=\frac{|x{|}^{\alpha -n}}{\left(1+|x|{\right)}^{\gamma }}.\end{array}$

• For any fLMp,ϕ(𝔾) and 1 < p < ∞, we have

$∥Mf∥LMp,ϕ(G)≤Cp∥f∥LMp,ϕ(G),$

where generalized local (central) Morrey space LMp,ϕ(𝔾) and Hardy-Littlewood maximal operator Mf are defined in (3.2) and (3.4), respectively.

• Let γ > 0 and 0 < α < Q. If ϕ(r) ≤ Crβ for every r > 0, β < −α, 1 < p < ∞, and $\begin{array}{}\frac{Q}{Q+\gamma -\alpha }\end{array}$ < p1 < $\begin{array}{}\frac{Q}{Q-\alpha },\end{array}$ then for all fLMp,ϕ(𝔾) we have

$∥Iα,γf∥LMq,ψ(G)≤Cp,ϕ,Q∥Kα,γ∥Lp1(G)∥f∥LMp,ϕ(G),$

where $\begin{array}{}q=\frac{\beta {p}_{1}^{{}^{\prime }}p}{\beta {p}_{1}^{{}^{\prime }}+Q}\end{array}$ and ψ(r) = (ϕ(r))p/q. The Bessel-Riesz operator Iα,γ on a homogenous group is defined in (2.4).

• Let γ > 0 and 0 < α < Q. If ϕ(r) ≤ Crβ for every r > 0, β < −α, $\begin{array}{}\frac{Q}{Q+\gamma -\alpha }\end{array}$ < p2p1 < $\begin{array}{}\frac{Q}{Q-\alpha }\end{array}$ and p2 ≥ 1, then for all fLMp,ϕ(𝔾) we have

$∥Iα,γf∥LMq,ψ(G)≤Cp,ϕ,Q∥Kα,γ∥LMp2,p1(G)∥f∥LMp,ϕ(G),$

where 1 < p < ∞, $\begin{array}{}q=\frac{\beta {p}_{1}^{{}^{\prime }}p}{\beta {p}_{1}^{{}^{\prime }}+Q}\end{array}$ , ψ(r) = (ϕ(r))p/q.

• Let ω : ℝ+ → ℝ+ satisfy the doubling condition and assume that ω(r) ≤ Crα for every r > 0, so that Kα,γLMp2,ω(𝔾) for $\begin{array}{}\frac{Q}{Q+\gamma -\alpha }\end{array}$ < p2 < $\begin{array}{}\frac{Q}{Q-\alpha }\end{array}$ and p2 ≥ 1, where 0 < α < Q and γ > 0. If ϕ(r) ≤ Crβ for every r > 0, where β < −α < −Qβ, then for all fLMp,ϕ(𝔾) we have

$∥Iα,γf∥LMq,ψ(G)≤Cp,ϕ,Q∥Kα,γ∥LMp2,ω(G)∥f∥LMp,ϕ(G),$

where 1 < p < ∞, $\begin{array}{}q=\frac{\beta p}{\beta +Q-\alpha }\end{array}$ and ψ(r) = (ϕ(r))p/q.

• Let γ > 0 and let ρ and ϕ satisfy the doubling condition (3.3). Let 1 < p < q < ∞. Let ϕ be surjective and satisfy $\begin{array}{}{\int }_{r}^{\mathrm{\infty }}\frac{\varphi \left(t{\right)}^{p}}{t}dt\end{array}$C1(ϕ(r))p, and

$ϕ(r)∫0rρ(t)tγ−Q+1dt+∫r∞ρ(t)ϕ(t)tγ−Q+1dt≤C2(ϕ(r))p/q,$

for all r > 0. Then we have

$∥Iρ,γf∥LMq,ϕp/q(G)≤Cp,q,ϕ,Q∥f∥LMp,ϕ(G),$

where the generalized Bessel-Riesz operator Iρ,γ is defined in (5.1). This result is new already in the standard setting of ℝn.

• Let ρ and ϕ satisfy the doubling condition (3.3). Let γ > 0, and assume that ϕ is surjective and satisfies (5.3)-(5.4). Then for 1 < p < p2 < ∞ we have

$∥WIρ,γf∥LMp,ϕ(G)≤Cp,ϕ,Q∥W∥LMp2,ϕp/p2(G)∥f∥LMp,ϕ(G),$

provided that WLMp2,ϕp/p2(𝔾). This result is new even in the Euclidean cases.

• Let ρ and ϕ satisfy the doubling condition (3.3). Let 1 < p < q < ∞. Let ϕ be surjective and satisfy $\begin{array}{}{\int }_{r}^{\mathrm{\infty }}\frac{\varphi \left(t{\right)}^{p}}{t}dt\end{array}$C1(ϕ(r))p, and

$ϕ(r)∫0rρ(t)tdt+∫r∞ρ(t)ϕ(t)tdt≤C2(ϕ(r))p/q,$

for all r > 0. Then we have

$∥Tρf∥LMq,ϕp/q(G)≤Cp,q,ϕ,Q∥f∥LMp,ϕ(G),$

where the generalized fractional integral operator Tρ is defined in (6.1).

• Let ρ and ϕ satisfy the doubling condition (3.3). Let ϕ be surjective and satisfy (6.3)-(6.4). Then for 1 < p < p2 < ∞ we have

$∥WTρf∥LMp,ϕ(G)≤Cp,ϕ,Q∥W∥LMp2,ϕp/p2(G)∥f∥LMp,ϕ(G),$

provided that WLMp2,ϕp/p2(𝔾).

• Let ω : ℝ+ → ℝ+ satisfy the doubling condition and assume that ω(r) ≤ Crα for every r > 0, so that Kα,γLMp2,ω(𝔾) for $\begin{array}{}\frac{Q}{Q+\gamma -\alpha }\end{array}$ < p2 < $\begin{array}{}\frac{Q}{Q-\alpha }\end{array}$ and p2 ≥ 1, where 0 < α < Q, 1 < p < ∞, $\begin{array}{}q=\frac{\beta p}{\beta +Q-\alpha }\end{array}$ and γ > 0. If ϕ(r) ≤ Crβ for every r > 0, where β < −α < −Qβ, then we have

$∥WIα,γf∥LMp,ϕ(G)≤Cp,ϕ,Q∥W∥LMp2,ϕp/p2(G)∥f∥LMp,ϕ(G),$

provided that WLMp2,ϕp/p2(𝔾), where $\begin{array}{}\frac{1}{{p}_{2}}=\frac{1}{p}-\frac{1}{q}.\end{array}$ . This result is new already in the Euclidean setting of ℝn.

• Let ρ satisfy (6.2), (3.3), (7.3), (7.4), and let ϕ satisfy the doubling condition (3.3) and $\begin{array}{}{\int }_{1}^{\mathrm{\infty }}\frac{\varphi \left(t\right)}{t}dt\end{array}$ < ∞. If

$∫r∞ϕ(t)tdt∫0rρ(t)tdt+r∫r∞ρ(t)ϕ(t)t2dt≤C3ψ(r)$

for all r > 0, then we have

$∥T~ρf∥LMp,ψ(G)≤Cp,ϕ,Q∥f∥LMp,ϕ(G),1

where the generalized local (central) Campanato space 𝓛𝓜p,ψ(𝔾) and operator ρ are defined in (7.1) and (7.2), respectively.

This paper is structured as follows. In Section 2 we briefly recall the concepts of homogeneous groups and fix the notation. The boundedness of the Hardy-Littlewood maximal operator and Bessel-Riesz operators in generalized local (central) Morrey spaces on homogeneous groups is proved in Section 3 and in Section 4, respectively. In Section 5 we prove the boundedness of the generalized Bessel-Riesz operators and Olsen type inequality for these operators in generalized local (central) Morrey spaces on homogeneous groups. The boundedness of the generalized fractional integral operators and Olsen type inequality for these operators in generalized local (central) Morrey spaces on homogeneous groups are proved in Section 6. Finally, in Section 7 we investigate the boundedness of the modified version of the generalized fractional integral operator in Campanato spaces on homogeneous groups.

## 2 Preliminaries

A connected simply connected Lie group 𝔾 is called a homogeneous group if its Lie algebra 𝔤 is equipped with a family of dilations:

$Dλ=Exp(Alnλ)=∑k=0∞1k!(ln(λ)A)k,$

where A is a diagonalisable positive linear operator on 𝔤, and each Dλ is a morphism of 𝔤, that is, ∀X, Y ∈ 𝔤, λ > 0,; [DλX, DλY] = Dλ[X, Y].

The exponential mapping exp𝔾 : 𝔤 → 𝔾 is a global diffeomorphism and gives the dilation structure, which is denoted by Dλx or just by λx, on 𝔾.

Then we have

$|Dλ(S)|=λQ|S|and∫Gf(λx)dx=λ−Q∫Gf(x)dx,$(2.1)

where dx is the Haar measure on 𝔾, |S| is the volume of a measurable set S ⊂ 𝔾 and Q : = Tr A is the homogeneous dimension of 𝔾. Recall that the Haar measure on a homogeneous group 𝔾 is the standard Lebesgue measure for ℝn (see e.g. [14, Proposition 1.6.6]).

Let |·| be a homogeneous quasi-norm on 𝔾. We will denote the quasi-ball centred at x ∈ 𝔾 with radius R > 0 by

$B(x,R):={y∈G:|x−1y|

and we will also use the notation

$Bc(x,R):={y∈G:|x−1y|≥R}.$

The proof of the following important polar decomposition on homogeneous Lie groups was given by Folland and Stein [15], which can be also found in [14, Section 3.1.7]: there is a (unique) positive Borel measure σ on the unit sphere

$S:={x∈G:|x|=1},$(2.2)

so that for any fL1(𝔾), one has

$∫Gf(x)dx=∫0∞∫Sf(ry)rQ−1dσ(y)dr.$(2.3)

Now, for any f$\begin{array}{}{L}_{loc}^{p}\end{array}$ (𝔾), p ≥ 1 and γ ≥ 0, 0 < α < Q, we shall define the Bessel-Riesz operators on homogeneous groups by

$Iα,γf(x):=∫GKα,γ(xy−1)f(y)dy=∫G|xy−1|α−Q(1+|xy−1|)γf(y)dy,$(2.4)

where |·| is any homogeneous quasi-norm. Here, Kα,γ is the Bessel-Riesz kernel. Hereafter, C, Ci, Cp, Cp,ϕ,Q and Cp,q,ϕ,Q are positive constants, which are not necessarily the same from line to line.

Let us recall the following result, which will be used in the sequel.

#### Lemma 2.1

([26]). If b > a > 0, then $\begin{array}{}\sum _{k\in \mathbb{Z}}\frac{\left({u}^{k}R{\right)}^{a}}{\left(1+{u}^{k}R{\right)}^{b}}<\mathrm{\infty },\end{array}$ for every u > 1 and R > 0.

We now calculate the Lp-norms of the Bessel-Riesz kernel.

#### Theorem 2.1

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let |·| be a homogeneous quasi-norm. Let $\begin{array}{}{K}_{\alpha ,\gamma }\left(x\right)=\frac{|x{|}^{\alpha -Q}}{\left(1+|x|{\right)}^{\gamma }}.\end{array}$ If 0 < α < Q and γ > 0, then Kα,γLp1(𝔾) and

$∥Kα,γ∥Lp1(G)∼∑k∈Z(2kR)(α−Q)p1+Q(1+2kR)γp11p1,$

for any R > 0 and $\begin{array}{}\frac{Q}{Q+\gamma -\alpha }\end{array}$ < p1 < $\begin{array}{}\frac{Q}{Q-\alpha }\end{array}$ .

#### Remark 2.1

We note that this result was proved in [26, Theorem 3] in the Abelian case 𝔾 = (ℝn, +) and Q = n with the standard Euclidean distance $\begin{array}{}|x|=\sqrt{{x}_{1}^{2}+{x}_{2}^{2}+...+{x}_{n}^{2}}.\end{array}$

#### Proof of Theorem 2.1

Introducing polar coordinates (r, y) = $\begin{array}{}\left(|x|,\frac{x}{\mid x\mid }\right)\end{array}$ ∈ (0, ∞) × 𝔖 on 𝔾, where 𝔖 is the sphere as in (2.2), and using (2.3) for any R > 0, we have

$∫G|Kα,γ(x)|p1dx=∫G|x|(α−Q)p1(1+|x|)γp1dx=∫0∞∫Sr(α−Q)p1+Q−1(1+r)γp1dσ(y)dr=|σ|∑k∈Z∫2kR≤r<2k+1Rr(α−Q)p1+Q−1(1+r)γp1dr,$

where |σ| is the Q − 1 dimensional surface measure of the unit sphere.

Then it follows that

$∫G|Kα,γ(x)|p1dx≤|σ|∑k∈Z1(1+2kR)γp1∫2kR≤r<2k+1Rr(α−Q)p1+Q−1dr=|σ|(2(α−Q)p1+Q−1)(α−Q)p1+Q∑k∈Z(2kR)(α−Q)p1+Q(1+2kR)γp1.$

On the other hand, we obtain

$∫G|Kα,γ(x)|p1dx≥|σ|2γp1∑k∈Z1(1+2kR)γp1∫2kR≤r<2k+1Rr(α−Q)p1+Q−1dr=|σ|(2(α−Q)p1+Q−1)2γp1((α−Q)p1+Q)∑k∈Z(2kR)(α−Q)p1+Q(1+2kR)γp1.$

Therefore, for every R > 0 we arrive at

$∫G|Kα,γ(x)|p1dx∼∑k∈Z(2kR)(α−Q)p1+Q(1+2kR)γp1.$

For p1 ∈ ( $\begin{array}{}\frac{Q}{Q+\gamma -\alpha }\end{array}$ , $\begin{array}{}\frac{Q}{Q-\alpha }\end{array}$ ) using Lemma 2.1 with u = 2, a = (αQ)p1 + Q, b = γp1, we obtain $\begin{array}{}\sum _{k\in \mathbb{Z}}\frac{\left({2}^{k}R{\right)}^{\left(\alpha -Q\right){p}_{1}+Q}}{\left(1+{2}^{k}R{\right)}^{\gamma {p}_{1}}}<\mathrm{\infty }\end{array}$ which implies Kα,γLp1(𝔾). □

The following is well-known on homogeneous groups, see e.g. [14, Proposition 1.5.2].

#### Proposition 2.1

(Young’s inequality). Let 𝔾 be a homogeneous group. Suppose 1 ≤ p, q, p1 ≤ ∞ and $\begin{array}{}\frac{1}{q}+1=\frac{1}{p}+\frac{1}{{p}_{1}}\end{array}$ . If fLp(𝔾) and gLp1(𝔾), then

$∥g∗f∥Lq(G)≤∥f∥Lp(G)∥g∥Lp1(G).$

In view of Proposition 2.1 and taking into account the definition of Bessel-Riesz operator (2.4), we immediately get:

#### Corollary 2.1

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let |·| be a homogeneous quasi-norm. Then for 0 < α < Q and γ > 0 we have

$∥Iα,γf∥Lq(G)≤∥Kα,γ∥Lp1(G)∥f∥Lp(G)$

for every fLp(𝔾), where 1 ≤ p, q, p1 ≤ ∞, $\begin{array}{}\frac{1}{q}+1=\frac{1}{p}+\frac{1}{{p}_{1}}\end{array}$ and $\begin{array}{}\frac{Q}{Q+\gamma -\alpha }\end{array}$ < p1 < $\begin{array}{}\frac{Q}{Q-\alpha }\end{array}$.

Corollary 2.1 shows that the Iα,γ is bounded from Lp(𝔾) to Lq(𝔾) and

$∥Iα,γ∥Lp(G)→Lq(G)≤∥Kα,γ∥Lp1(G).$

## 3 The boundedness of Hardy-Littlewood maximal operator in generalized local (central) Morrey spaces

In this section we define local (central) Morrey and generalized local (central) Morrey spaces on homogeneous groups. Then we prove that the Hardy-Littlewood maximal operator is bounded in these spaces. Note that in the isotropic Abelian case the result was obtained by Nakai [29].

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let us define the local (central) Morrey spaces LMp,q(𝔾) by

$LMp,q(G):={f∈Llocp(G):∥f∥LMp,q(G)<∞},1≤p≤q,$(3.1)

where ∥fLMp,q(𝔾) := supr>0rQ(1/q−1/p) (∫B(0,r)|f(x)|pdx)1/p. Next, for a function ϕ : ℝ+ → ℝ+ and 1 ≤ p < ∞, we define the generalized local (central) Morrey space LMp,ϕ(𝔾) by

$LMp,ϕ(G):={f∈Llocp(G):∥f∥LMp,ϕ(G)<∞},$(3.2)

where ∥fLMp,ϕ(𝔾) := supr>0 $\begin{array}{}\frac{1}{\varphi \left(r\right)}{\left(\frac{1}{{r}^{Q}}{\int }_{B\left(0,r\right)}|f\left(x\right){|}^{p}dx\right)}^{1/p}\end{array}$ . Here we assume that ϕ is nonincreasing and tQ/pϕ(t) is nondecreasing, so that ϕ satisfies the doubling condition, i.e. there exists a constant C1 > 0 such that

$12≤rs≤2⟹1C1≤ϕ(r)ϕ(s)≤C1.$(3.3)

Now, for every f$\begin{array}{}{L}_{loc}^{p}\end{array}$ (𝔾), we define the Hardy-Littlewood maximal operator M by

$Mf(x):=supr>01|B(x,r)|∫B(x,r)|f(y)|dy,$(3.4)

where |B(x, r)| denotes the Haar measure of the (quasi-)ball B = B(x, r).

Using the definition of local Morrey spaces (3.1), one can readily obtain:

#### Lemma 3.1

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let |·| be a homogeneous quasi-norm. Then

$∥Kα,γ∥LMp2,p1(G)≤∥Kα,γ∥LMp1,p1(G)=∥Kα,γ∥Lp1(G),$(3.5)

where 1 ≤ p2p1 and $\begin{array}{}\frac{Q}{Q+\gamma -\alpha }\end{array}$ < p1 < $\begin{array}{}\frac{Q}{Q-\alpha }\end{array}$.

We now prove the boundedness of the Hardy-Littlewood maximal operator on generalized local Morrey spaces.

#### Theorem 3.1

Let 𝔾 be a homogeneous group. For any fLMp,ϕ(𝔾) and 1 < p < ∞, we have

$∥Mf∥LMp,ϕ(G)≤Cp∥f∥LMp,ϕ(G).$(3.6)

#### Remark 3.1

We note that this result was proved on stratified groups (or homogeneous Carnot groups) in [19, Corollary 3.2]. Here, Theorem 3.1 holds on general homogeneous groups.

#### Proof of Theorem 3.1

By the definition of the norm of the generalized local (central) Morrey space (3.2), we have

$∥f∥LMp,ϕ(G)=supr>01ϕ(r)1rQ∫B(0,r)|f(x)|pdx1/p.$

This implies that

$∫B(0,r)|f(x)|pdx1/p≤ϕ(r)rQp∥f∥LMp,ϕ(G),$(3.7)

for any r > 0.

On the other hand, using Corollary 2.5 (b) from Folland and Stein [15] we have

$∫G|M(fχB(0,r))(x)|pdx1/p≤Cp∫G|f(x)χB(0,r)|pdx1/p=Cp∫B(0,r)|f(x)|pdx1/p,$

which implies

$∫B(0,r)|Mf(x)|pdx1/p≤Cp∫B(0,r)|f(x)|pdx1/p.$(3.8)

Combining (3.7) and (3.8) we get for all r > 0

$1ϕ(r)1rQ∫B(0,r)|Mf(x)|pdx1/p≤Cp∥f∥LMp,ϕ(G),$

which implies ∥MfLMp,ϕ(𝔾)CpfLMp,ϕ(𝔾), completing the proof. □

## 4 Inequalities for Bessel-Riesz operators on generalized local Morrey spaces

In this section, we prove the boundedness of the Bessel-Riesz operators on the generalized local (central) Morrey space (3.2). In the Abelian case 𝔾 = (ℝn, +) and Q = n with the standard Euclidean distance |x| = $\begin{array}{}\sqrt{{x}_{1}^{2}+{x}_{2}^{2}+...+{x}_{n}^{2}}\end{array}$ the results of this section were obtained in [27].

#### Theorem 4.1

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let |·| be a homogeneous quasi-norm. Let γ > 0 and 0 < α < Q. If ϕ(r) ≤ Crβ for every r > 0, β < −α, 1 < p < ∞, and $\begin{array}{}\frac{Q}{Q+\gamma -\alpha }\end{array}$ < p1 < $\begin{array}{}\frac{Q}{Q-\alpha }\end{array}$ , then for all fLMp,ϕ(𝔾) we have

$∥Iα,γf∥LMq,ψ(G)≤Cp,ϕ,Q∥Kα,γ∥Lp1(G)∥f∥LMp,ϕ(G),$(4.1)

where $\begin{array}{}q=\frac{\beta {p}_{1}^{{}^{\prime }}p}{\beta {p}_{1}^{{}^{\prime }}+Q}\end{array}$ and ψ(r) = (ϕ(r))p/q.

#### Proof of Theorem 4.1

For every fLMp,ϕ(𝔾), we write Iα,γf(x) in the form Iα,γf(x) := I1(x) + I2(x), where $\begin{array}{}{I}_{1}\left(x\right):={\int }_{B\left(x,R\right)}\frac{|x{y}^{-1}{|}^{\alpha -Q}f\left(y\right)}{\left(1+|x{y}^{-1}|{\right)}^{\gamma }}dy\end{array}$ and $\begin{array}{}{I}_{2}\left(x\right):={\int }_{{B}^{c}\left(x,R\right)}\frac{|x{y}^{-1}{|}^{\alpha -Q}f\left(y\right)}{\left(1+|x{y}^{-1}|{\right)}^{\gamma }}dy\end{array}$ , for some R > 0.

By using dyadic decomposition for I1, we obtain

$|I1(x)|≤∑k=−∞−1∫2kR≤|xy−1|<2k+1R|xy−1|α−Q|f(y)|(1+|xy−1|)γdy≤∑k=−∞−1(2kR)α−Q(1+2kR)γ∫2kR≤|xy−1|<2k+1R|f(y)|dy≤CMf(x)∑k=−∞−1(2kR)α−Q+Q/p1(2kR)Q/p1′(1+2kR)γ.$

From this using Hölder’s inequality for $\begin{array}{}\frac{1}{{p}_{1}}+\frac{1}{{p}_{1}^{{}^{\prime }}}=1,\end{array}$ we get

$|I1(x)|≤CMf(x)∑k=−∞−1(2kR)(α−Q)p1+Q(1+2kR)γp11/p1∑k=−∞−1(2kR)Q1/p1′.$

Since

$∑k=−∞−1(2kR)(α−Q)p1+Q(1+2kR)γp11/p1≤∑k∈Z(2kR)(α−Q)p1+Q(1+2kR)γp11/p1∼∥Kα,γ∥Lp1(G),$(4.2)

we arrive at

$|I1(x)|≤C∥Kα,γ∥Lp1(G)Mf(x)RQ/p1′.$(4.3)

For the second term I2, by using Hölder’s inequality for $\begin{array}{}\frac{1}{p}+\frac{1}{{p}^{{}^{\prime }}}=1\end{array}$ we obtain that

$|I2(x)|≤∑k=0∞(2kR)α−Q(1+2kR)γ∫2kR≤|xy−1|<2k+1R|f(y)|dy≤∑k=0∞(2kR)α−Q(1+2kR)γ×∫2kR≤|xy−1|<2k+1Rdy1/p′∫2kR≤|xy−1|<2k+1R|f(y)|pdy1/p,$

that is,

$|I2(x)|≤∑k=0∞(2kR)α−Q(1+2kR)γ∫2kR2k+1R∫SrQ−1dσ(y)dr1/p′×∫2kR≤|xy−1|<2k+1R|f(y)|pdy1/p≤C∑k=0∞(2kR)α−Q(1+2kR)γ(2kR)Q/p′∫2kR≤|xy−1|<2k+1R|f(y)|pdy1/p.$

Taking into account ϕ(r) ≤ Crβ, one obtains from above that

$|I2(x)|≤C∥f∥LMp,ϕ(G)∑k=0∞(2kR)α−Q+Q/p1(1+2kR)γϕ(2kR)(2kR)Q/p1′≤C∥f∥LMp,ϕ(G)∑k=0∞(2kR)α−Q+Q/p1(1+2kR)γ(2kR)β+Q/p1′.$

Applying Hölder’s inequality again, we get

$|I2(x)|≤C∥f∥LMp,ϕ(G)∑k=0∞(2kR)(α−Q)p1+Q(1+2kR)γp11/p1∑k=0∞(2kR)βp1′+Q1/p1′.$

From the conditions p1 < $\begin{array}{}\frac{Q}{Q-\alpha }\end{array}$ and β < −α, we have $\begin{array}{}\beta {p}_{1}^{{}^{\prime }}\end{array}$ + Q < 0. By Theorem 2.1, we also have

$∑k=0∞(2kR)(α−Q)p1+Q(1+2kR)γp11/p1≤∑k∈Z(2kR)(α−Q)p1+Q(1+2kR)γp11/p1∼∥Kα,γ∥Lp1(G).$

Using these, we arrive at

$|I2(x)|≤C∥Kα,γ∥Lp1(G)∥f∥LMp,ϕ(G)RQ/p1′+β.$(4.4)

Summing up the estimates (4.3) and (4.4), we obtain

$|Iα,γf(x)|≤C∥Kα,γ∥Lp1(G)Mf(x)RQ/p1′+∥f∥LMp,ϕ(G)RQ/p1′+β.$

Assuming that f is not identically 0 and that Mf is finite everywhere, we can choose R > 0 such that $\begin{array}{}{R}^{\beta }=\frac{Mf\left(x\right)}{{\parallel f\parallel }_{L{M}^{p,\varphi }\left(\mathbb{G}\right)}},\end{array}$ that is,

$|Iα,γf(x)|≤C∥Kα,γ∥Lp1(G)∥f∥LMp,ϕ(G)−Qβp1′(Mf(x))1+Qβp1′,$

for every x ∈ 𝔾. Setting $\begin{array}{}q=\frac{\beta {p}_{1}^{{}^{\prime }}p}{\beta {p}_{1}^{{}^{\prime }}+Q}\end{array}$ , for any r > 0 we get

$∫|x|

Then we divide both sides by (ϕ(r))p/qrQ/q to get

$∫|x|

where ψ(r) = (ϕ(r))p/q. Now by taking the supremum over r > 0, we obtain that

$∥Iα,γf∥LMq,ψ(G)≤C∥Kα,γ∥Lp1(G)∥f∥LMp,ϕ(G)1−p/q∥Mf∥LMp,ϕ(G)p/q,$

which gives (4.1), after applying estimate (3.6). □

Lemma 3.1 gives the property that the Bessel-Riesz kernel belongs to local Morrey spaces, which will be used in the following theorem.

#### Theorem 4.2

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let |·| be a homogeneous quasi-norm. Let γ > 0 and 0 < α < Q. If ϕ(r) ≤ Crβ for every r > 0, β < −α, $\begin{array}{}\frac{Q}{Q+\gamma -\alpha }\end{array}$ < p2p1 < $\begin{array}{}\frac{Q}{Q-\alpha }\end{array}$ and p2 ≥ 1, then for all fLMp,ϕ(𝔾) we have

$∥Iα,γf∥LMq,ψ(G)≤Cp,ϕ,Q∥Kα,γ∥LMp2,p1(G)∥f∥LMp,ϕ(G),$(4.5)

where 1 < p < ∞, $\begin{array}{}q=\frac{\beta {p}_{1}^{{}^{\prime }}p}{\beta {p}_{1}^{{}^{\prime }}+Q}\end{array}$ and ψ(r) = (ϕ(r))p/q.

#### Proof of Theorem 4.2

Similarly to the proof of Theorem 4.1, we write Iα,γf(x) := I1(x) + I2(x), where

$I1(x):=∫B(x,R)|xy−1|α−Q(1+|xy−1|)γf(y)dy$

and

$I2(x):=∫Bc(x,R)|xy−1|α−Q(1+|xy−1|)γf(y)dy$

for R > 0. As before, we estimate the first term I1 by using the dyadic decomposition:

$|I1(x)|≤∑k=−∞−1∫2kR≤|xy−1|<2k+1R|xy−1|α−Q|f(y)|(1+|xy−1|)γdy≤∑k=−∞−1(2kR)α−Q(1+2kR)γ∫2kR≤|xy−1|<2k+1R|f(y)|dy≤CMf(x)∑k=−∞−1(2kR)α−Q+Q/p2(2kR)Q/p2′(1+2kR)γ,$

where 1 ≤ p2p1. From this using Hölder’s inequality for $\begin{array}{}\frac{1}{{p}_{2}}+\frac{1}{{p}_{2}^{{}^{\prime }}}=1,\end{array}$ we get

$|I1(x)|≤CMf(x)∑k=−∞−1(2kR)(α−Q)p2+Q(1+2kR)γp21/p2∑k=−∞−1(2kR)Q1/p2′.$

By virtue of (4.2), we have

$|I1(x)|≤C2Mf(x)∫0<|x|(4.6)

Now for I2 by using Hölder’s inequality for $\begin{array}{}\frac{1}{p}+\frac{1}{{p}^{{}^{\prime }}}=1,\end{array}$ we have

$|I2(x)|≤∑k=0∞(2kR)α−Q(1+2kR)γ(2kR)Q/p′∫2kR≤|xy−1|<2k+1R|f(y)|pdy1/p,$

that is,

$|I2(x)|≤C∥f∥LMp,ϕ(G)∑k=0∞(2kR)αϕ(2kR)(1+2kR)γ∫2kR≤|xy−1|<2k+1Rdy1/p2(2kR)Q/p2≤C∥f∥LMp,ϕ(G)∑k=0∞ϕ(2kR)(2kR)Q/p1′∫2kR≤|xy−1|<2k+1RKα,γp2(xy−1)dy1/p2(2kR)Q/p2−Q/p1,$

where we have used the following inequality

$∫2kR≤|xy−1|<2k+1RKα,γp2(xy−1)dy1/p2∼(2kR)(α−Q)+Q/p2(1+2kR)γ≥C(2kR)(α−Q)(1+2kR)γ∫2kR≤|xy−1|<2k+1Rdy1/p2.$(4.7)

Since we have ϕ(r) ≤ Crβ and

$∫2kR≤|xy−1|<2k+1RKα,γp2(xy−1)dy1/p2(2kR)Q/p2−Q/p1≲∥Kα,γ∥LMp2,p1(G)$

for every k = 0, 1, 2, …, we get

$|I2(x)|≤C∥Kα,γ∥LMp2,p1(G)∥f∥LMp,ϕ(G)∑k=0∞(2kR)β+Q/p1′.$

Taking into account $\begin{array}{}\beta +Q/{p}_{1}^{{}^{\prime }}<0,\end{array}$ we have

$|I2(x)|≤C∥Kα,γ∥LMp2,p1(G)∥f∥LMp,ϕ(G)Rβ+Q/p1′.$(4.8)

Summing up the estimates (4.6) and (4.8), we obtain

$|Iα,γf(x)|≤C∥Kα,γ∥LMp2,p1(G)(Mf(x)RQ/p1′+∥f∥LMp,ϕ(G)Rβ+Q/p1′).$

Assuming that f is not identically 0 and that Mf is finite everywhere, we can choose R > 0 such that $\begin{array}{}{R}^{\beta }=\frac{Mf\left(x\right)}{{\parallel f\parallel }_{L{M}^{p,\varphi }\left(\mathbb{G}\right)}},\end{array}$ which yields

$|Iα,γf(x)|≤C∥Kα,γ∥LMp2,p1(G)∥f∥LMp,ϕ(G)−Qβp1′(Mf(x))1+Qβp1′.$

Now by putting $\begin{array}{}q=\frac{\beta {p}_{1}^{{}^{\prime }}p}{\beta {p}_{1}^{{}^{\prime }}+Q}\end{array}$ , for any r > 0 we obtain

$∫|x|

Then we divide both sides by (ϕ(r))p/qrQ/q to get

$∫|x|

where ψ(r) = (ϕ(r))p/q. Taking the supremum over r > 0 and then using (3.6), we obtain the following desired result

$∥Iα,γf∥LMq,ψ(G)≤C∥Kα,γ∥LMp2,p1(G)∥f∥LMp,ϕ(G)1−p/q∥Mf∥LMp,ϕ(G)p/q≤Cp,ϕ,Q∥Kα,γ∥LMp2,p1(G)∥f∥LMp,ϕ(G),$

completing the proof. □

By Lemma 3.1, we note that Theorem 4.2 implies Theorem 4.1:

$∥Iα,γf∥LMq,ψ(G)≤C∥Kα,γ∥LMp2,p1(G)∥f∥LMp,ϕ(G)≤C∥Kα,γ∥Lp1(G)∥f∥LMp,ϕ(G).$

In order to improve our results, we present the following lemma, which states that the kernel Kα,γ belongs to the generalized local Morrey space LMp2,ω(𝔾) for some p2 ≥ 1 and some function ω.

#### Lemma 4.1

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let γ > 0, p2 ≥ 1 and $\begin{array}{}Q-\frac{Q}{{p}_{2}}\end{array}$ < α < Q. If ω : ℝ+ → ℝ+ with ω(r) ≥ CrαQ for every r > 0, then Kα,γLMp2,ω(𝔾).

#### Proof of Lemma 4.1

Here, it is sufficient to evaluate the following integral around zero

$∫|x|≤RKα,γp2(x)dx=∫|x|≤R|x|(α−Q)p2(1+|x|)γp2dx≤|σ|∫0

By dividing both sides of this inequality by ωp2(R)RQ and taking $\begin{array}{}{p}_{2}^{th}\end{array}$ -root, we obtain

$∫|x|≤RKα,γp2(x)dx1/p2ω(R)RQ/p2≤C1/p2.$

Then, we take the supremum over R > 0 to get

$supR>0∫|x|≤RKα,γp2(x)dx1/p2ω(R)RQ/p2<∞,$

which implies Kα,γLMp2,ω(𝔾). □

#### Theorem 4.3

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let |·| be a homogeneous quasi-norm. Let ω : ℝ+ → ℝ+ satisfy the doubling condition and assume that ω(r) ≤ Crα for every r > 0, so that Kα,γLMp2,ω(𝔾) for $\begin{array}{}\frac{Q}{Q+\gamma -\alpha }\end{array}$ < p2 < $\begin{array}{}\frac{Q}{Q-\alpha },\end{array}$ and p2 ≥ 1, where 0 < α < Q and γ > 0. If ϕ(r) ≤ Crβ for every r > 0, where β < −α < −Qβ, then for all fLMp,ϕ(𝔾) we have

$∥Iα,γf∥LMq,ψ(G)≤Cp,ϕ,Q∥Kα,γ∥LMp2,ω(G)∥f∥LMp,ϕ(G),$(4.9)

where 1 < p < ∞, $\begin{array}{}q=\frac{\beta p}{\beta +Q-\alpha }\end{array}$ and ψ(r) = (ϕ(r))p/q.

#### Proof of Theorem 4.3

As in the proof of Theorem 4.1, we write

$Iα,γf(x):=I1(x)+I2(x),$

where $\begin{array}{}{I}_{1}\left(x\right):={\int }_{B\left(x,R\right)}\frac{|x{y}^{-1}{|}^{\alpha -Q}f\left(y\right)}{\left(1+|x{y}^{-1}|{\right)}^{\gamma }}dy\end{array}$ and $\begin{array}{}{I}_{2}\left(x\right):={\int }_{{B}^{c}\left(x,R\right)}\frac{|x{y}^{-1}{|}^{\alpha -Q}f\left(y\right)}{\left(1+|x{y}^{-1}|{\right)}^{\gamma }}dy\end{array}$ , R > 0.

First, we estimate I1 by using the dyadic decomposition

$|I1(x)|≤∑k=−∞−1∫2kR≤|xy−1|<2k+1R|xy−1|α−Q|f(y)|(1+|xy−1|)γdy≤∑k=−∞−1(2kR)α−Q(1+2kR)γ∫2kR≤|xy−1|<2k+1R|f(y)|dy≤CMf(x)∑k=−∞−1(2kR)α−Q+Q/p2(2kR)Q/p2′(1+2kR)γ.$

From this using Hölder’s inequality for $\begin{array}{}\frac{1}{{p}_{2}}+\frac{1}{{p}_{2}^{{}^{\prime }}}=1,\end{array}$ we get

$|I1(x)|≤CMf(x)∑k=−∞−1(2kR)(α−Q)p2+Q(1+2kR)γp21/p2∑k=−∞−1(2kR)Q1/p2′.$

By (4.2) we have

$|I1(x)|≤CMf(x)∫0<|x|

and using ω(r) ≤ Crα, we arrive at

$|I1(x)|≤C∥Kα,γ∥LMp2,ω(G)Mf(x)RQ−α.$(4.10)

Now let us estimate the second term I2:

$|I2(x)|≤∑k=0∞(2kR)α−Q(1+2kR)γ∫2kR≤|xy−1|<2k+1R|f(y)|dy≤C∑k=0∞(2kR)α−Q(1+2kR)γ(2kR)Q/p′∫2kR≤|xy−1|<2k+1R|f(y)|pdy1/p≤C∥f∥LMp,ϕ(G)∑k=0∞(2kR)αϕ(2kR)(1+2kR)γ∫2kR≤|xy−1|<2k+1Rdy1/p2(2kR)Q/p2,$

where we have used that (∫2kR≤|xy−1|<2k+1R dy)1/p2 ∼ (2kR)Q/p2. Using (4.7) we obtain

$|I2(x)|≤C∥f∥LMp,ϕ(G)∑k=0∞(2kR)αϕ(2kR)(2kR)α−Q×∫2kR≤|xy−1|<2k+1RKα,γp2(xy−1)dy1/p2(2kR)Q/p2.$

Taking into account that ϕ(r) ≤ Crβ and ω(r) ≤ Crα for every r > 0, we have

$|I2(x)|≤C∥f∥LMp,ϕ(G)∑k=0∞(2kR)Q−α+β∫2kR≤|xy−1|<2k+1RKα,γp2(xy−1)dy1/p2ω(2kR)(2kR)Q/p2.$

Since we have

$∫2kR≤|xy−1|<2k+1RKα,γp2(xy−1)dy1/p2ω(2kR)(2kR)Q/p2≲∥Kα,γ∥LMp2,ω(G)$

for every k = 0, 1, 2, …, it follows that

$|I2(x)|≤C∥Kα,γ∥LMp2,ω(G)∥f∥LMp,ϕ(G)∑k=0∞(2kR)Q−α+β,$

and since Qα + β < 0, it implies that

$|I2(x)|≤C∥Kα,γ∥LMp2,ω(G)∥f∥LMp,ϕ(G)RQ−α+β.$(4.11)

Summing up the estimates (4.10) and (4.11), we have

$|Iα,γf(x)|≤C∥Kα,γ∥LMp2,ω(G)(Mf(x)RQ−α+∥f∥LMp,ϕ(G)RQ−α+β).$

Assuming that f is not identically 0 and that Mf is finite everywhere, we can choose R > 0 such that $\begin{array}{}{R}^{\beta }=\frac{Mf\left(x\right)}{\parallel f{\parallel }_{L{M}^{p,\varphi }\left(\mathbb{G}\right)}},\end{array}$ that is

$|Iα,γf(x)|≤C∥Kα,γ∥LMp2,ω(G)∥f∥LMp,ϕ(G)(α−Q)/β(Mf(x))1+(Q−α)/β.$

Now by putting $\begin{array}{}q=\frac{\beta p}{\beta +Q-\alpha },\end{array}$ for any r > 0 we get

$∫|x|

Then we divide both sides by (ϕ(r))p/qrQ/q to get

$∫|x|

where ψ(r) = (ϕ(r))p/q. Finally, taking the supremum over r > 0 and using (3.6), we obtain the desired result

$∥Iα,γf∥LMq,ψ(G)≤C∥Kα,γ∥LMp2,ω(G)∥f∥LMp,ϕ(G)1−p/q∥Mf∥LMp,ϕ(G)p/q≤Cp,ϕ,Q∥Kα,γ∥LMp2,ω(G)∥f∥LMp,ϕ(G),$

completing the proof. □

#### Remark 4.1

We can make the following comparison between the obtained estimates, similarly to the Euclidean case [27, Section 3], namely, that also in the case of general homogeneous groups, Theorem 4.3 gives the best estimate among the three. Indeed, if we take ω(R) := (1 + RQ/q1)RQ/p1 for some q1 > p1, then ∥Kα,γLMp2,ω(𝔾) ≤ ∥Kα,γLMp2,p1(𝔾). By Theorem 4.3 and Lemma 3.1 we obtain

$∥Iα,γf∥LMq,ψ(G)≤C∥Kα,γ∥LMp2,ω(G)∥f∥LMp,ϕ(G)≤C∥Kα,γ∥LMp2,p1(G)∥f∥LMp,ϕ(G)≤C∥Kα,γ∥Lp1(G)∥f∥LMp,ϕ(G).$

## 5 Inequalities for generalized Bessel-Riesz operator in generalized local Morrey spaces

In this section, we prove the boundedness of the generalized Bessel-Riesz operator Iρ͠,γ and establish Olsen type inequality for this operator in generalized local Morrey spaces on homogeneous groups.

We define the generalized Bessel-Riesz operator Iρ͠,γ by

$Iρ~,γf(x):=∫Gρ~(|xy−1|)(1+|xy−1|)γf(y)dy,$(5.1)

where γ ≥ 0, ρ͠ : ℝ+ → ℝ+, ρ͠ satisfies the doubling condition (3.3) and the condition

$∫01ρ~(t)tγ−Q+1dt<∞.$(5.2)

For ρ͠(t) = tαQ, γ < α < Q, we have the Bessel-Riesz kernel

$Iρ~,γ=Iα,γ=|xy−1|α−Q(1+|xy−1|)γ.$

#### Theorem 5.1

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let |·| be a homogeneous quasi-norm and let γ > 0. Let ρ͠ and ϕ satisfy the doubling condition (3.3). Let ϕ be surjective and for some 1 < p < q < ∞ satisfy

$∫r∞(ϕ(t))ptdt≤C1(ϕ(r))p,$(5.3)

and

$ϕ(r)∫0rρ~(t)tγ−Q+1dt+∫r∞ρ~(t)ϕ(t)tγ−Q+1dt≤C2(ϕ(r))p/q,$(5.4)

for all r > 0. Then we have

$∥Iρ~,γf∥LMq,ϕp/q(G)≤Cp,q,ϕ,Q∥f∥LMp,ϕ(G).$(5.5)

#### Proof of Theorem 5.1

We write Iρ͠,γf(x) = I1,ρ͠(x) + I2,ρ͠(x), where $\begin{array}{}{I}_{1,\stackrel{~}{\rho \phantom{\rule{thinmathspace}{0ex}}}}\left(x\right):={\int }_{B\left(x,R\right)}\frac{\stackrel{~}{\rho \phantom{\rule{thinmathspace}{0ex}}}\left(|x{y}^{-1}|\right)f\left(y\right)}{\left(1+|x{y}^{-1}|{\right)}^{\gamma }}dy\end{array}$ and $\begin{array}{}{I}_{2,\stackrel{~}{\rho \phantom{\rule{thinmathspace}{0ex}}}}\left(x\right):={\int }_{{B}^{c}\left(x,R\right)}\frac{\stackrel{~}{\rho \phantom{\rule{thinmathspace}{0ex}}}\left(|x{y}^{-1}|\right)f\left(y\right)}{\left(1+|x{y}^{-1}|{\right)}^{\gamma }}dy\end{array}$ for every R > 0. For I1,ρ͠(x), we have

$|I1,ρ~(x)|≤∫|xy−1|

By virtue of (3.3), we get

$|I1,ρ~(x)|≤C∑k=−∞−1ρ~(2kR)(2kR)γ∫|xy−1|<2k+1R|f(y)|dy≤CMf(x)∑k=−∞−1ρ~(2kR)(2kR)γ−Q≤CMf(x)∑k=−∞−1∫2kR2k+1Rρ~(t)tγ−Q+1dt=CMf(x)∫0Rρ~(t)tγ−Q+1dt,$

where we have used the fact that

$∫2kR2k+1Rρ~(t)tγ−Q+1dt≥Cρ~(2kR)(2kR)γ−Q+12kR≥Cρ~(2kR)(2kR)γ−Q.$(5.6)

Now, using (5.4), we obtain

$|I1,ρ~(x)|≤CMf(x)(ϕ(R))(p−q)/q.$(5.7)

For I2,ρ͠(x), applying (3.3) we have

$|I2,ρ~(x)|≤∫|xy−1|≥Rρ~(|xy−1|)(1+|xy−1|)γ|f(y)|dy≤∫|xy−1|≥Rρ~(|xy−1|)|xy−1|γ|f(y)|dy=∑k=0∞∫2kR≤|xy−1|<2k+1Rρ~(|xy−1|)|xy−1|γ|f(y)|dy≤C∑k=0∞ρ~(2kR)(2kR)γ∫|xy−1|<2k+1R|f(y)|dy.$

From this using the Hölder inequality, we obtain

$|I2,ρ~(x)|≤C∑k=0∞ρ~(2kR)(2kR)γ∫|xy−1|<2k+1Rdy1−1p∫|xy−1|<2k+1R|f(y)|dy1p≤C∑k=0∞ρ~(2kR)(2kR)γ−Q+Qp∫|xy−1|<2k+1R|f(y)|dy1p.$

Using the definition (3.2), one gets

$|I2,ρ~(x)|≤C∥f∥LMp,ϕ(G)∑k=0∞ρ~(2k+1R)ϕ(2k+1R)(2kR)γ−Q≤C∥f∥LMp,ϕ(G)∑k=0∞∫2kR2k+1Rρ~(t)ϕ(t)tγ−Q+1dt=C∥f∥LMp,ϕ(G)∫R∞ρ~(t)ϕ(t)tγ−Q+1dt,$

where we have used the fact that

$∫2kR2k+1Rρ~(t)ϕ(t)tγ−Q+1dt≥Cρ~(2k+1R)ϕ(2k+1R)(2k+1R)γ−Q+12kR≥Cρ~(2k+1R)ϕ(2k+1R)(2kR)γ−Q.$

Now, using (5.4), we obtain

$|I2,ρ~(x)|≤C∥f∥LMp,ϕ(G)(ϕ(R))p/q.$(5.8)

Summing the two estimates (5.7) and (5.8), we arrive at

$|Iρ~,γf(x)|≤C(Mf(x)(ϕ(R))(p−q)/q+∥f∥LMp,ϕ(G)(ϕ(R))p/q).$

Assuming that f is not identically 0 and that Mf is finite everywhere and then using the fact that ϕ is surjective, we can choose R > 0 such that $\begin{array}{}\varphi \left(R\right)=Mf\left(x\right)\phantom{\rule{thinmathspace}{0ex}}{\parallel f\parallel }_{L{M}^{p,\varphi }\left(\mathbb{G}\right)}^{-1},\end{array}$ Thus, for every x ∈ 𝔾, we have

$|Iρ~,γf(x)|≤C(Mf(x))pq∥f∥LMp,ϕ(G)q−pq.$

It follows that

$∫B(0,r)|Iρ~,γf(x)|q1/q≤C∫B(0,r)|Mf(x)|p1/q∥f∥LMp,ϕ(G)q−pq,$

then we divide both sides by (ϕ(r))p/qrQ/q to get

$1(ϕ(r))p/q1rQ∫B(0,r)|Iρ~,γf(x)|q1/q≤C1(ϕ(r))p/q1rQ∫B(0,r)|Mf(x)|p1/q∥f∥LMp,ϕ(G)q−pq.$

Taking the supremum over r > 0 and using the boundedness of the maximal operator M on LMp,ϕ(𝔾) from (3.6), we obtain

$∥Iρ~,γf∥LMq,ϕp/q(G)≤Cp,q,ϕ,Q∥f∥LMp,ϕ(G).$

This completes the proof. □

Now let show the Olsen type inequalities for the generalized Bessel-Riesz operator Iρ,γ.

#### Theorem 5.2

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let |·| be a homogeneous quasi-norm and let γ > 0. Let ρ͠ and ϕ satisfy the doubling condition (3.3). Let ϕ be surjective and satisfy (5.3)-(5.4). Then for 1 < p < p2 < ∞ we have

$∥WIρ~,γf∥LMp,ϕ(G)≤Cp,ϕ,Q∥W∥LMp2,ϕp/p2(G)∥f∥LMp,ϕ(G),$(5.9)

provided that WLMp2,ϕp/p2(𝔾).

#### Proof of Theorem 5.2

By using Hölder’s inequality, we have

$1rQ∫B(0,r)|W(x)Iρ~,γf(x)|pdx≤1rQ∫B(0,r)|W(x)|p2dxp/p2×1rQ∫B(0,r)|Iρ~,γf(x)|pp2p2−pdxp2−pp2.$

Now let us take the p-th roots and then divide both sides by ϕ(r) to obtain

$1ϕ(r)1rQ∫B(0,r)|W(x)Iρ~,γf(x)|pdx1/p≤1(ϕ(r))p/p21rQ∫B(0,r)|W(x)|p2dx1/p2×1(ϕ(r))p2−pp21rQ∫B(0,r)|Iρ~,γf(x)|pp2p2−pdxp2−ppp2.$

By taking the supremum over r > 0 and using the inequality (5.5), we get

$∥WIρ~,γf∥LMp,ϕ(G)≤Cp,ϕ,Q∥W∥LMp2,ϕp/p2(G)∥Iρ~,γf∥Lpp2p2−p,ϕp2−pp2(G).$

Taking into account that 1 < p < $\begin{array}{}\frac{p{p}_{2}}{{p}_{2}-p}\end{array}$ < ∞ and putting q = $\begin{array}{}\frac{p{p}_{2}}{{p}_{2}-p}\end{array}$ in (5.5), we obtain (5.9). □

## 6 Generalized fractional integral operators in generalized local Morrey spaces

In this section, we prove the boundedness of the generalized fractional integral operators and establish Olsen type inequality in generalized local Morrey spaces on homogeneous groups.

We define the generalized fractional integral operator Tρ by

$Tρf(x):=∫Gρ(|xy−1|)|xy−1|Qf(y)dy,$(6.1)

where ρ : ℝ+ → ℝ+ satisfies the doubling condition (3.3) and the condition

$∫01ρ(t)tdt<∞.$(6.2)

As in the Abelian case, for ρ(t) = tα, 0 < α < Q, we have the Riesz transform

$Tρf(x)=Iαf(x)=∫G1|xy−1|Q−αf(y)dy.$

#### Theorem 6.1

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let |·| be a homogeneous quasi-norm. Let ρ and ϕ satisfy the doubling condition (3.3). Let ϕ be also surjective and satisfy, for some 1 < p < q < ∞, the inequalities

$∫r∞ϕ(t)ptdt≤C1(ϕ(r))p,$(6.3)

and

$ϕ(r)∫0rρ(t)tdt+∫r∞ρ(t)ϕ(t)tdt≤C2(ϕ(r))p/q,$(6.4)

for all r > 0. Then we have

$∥Tρf∥LMq,ϕp/q(G)≤Cp,q,ϕ,Q∥f∥LMp,ϕ(G).$(6.5)

#### Proof of Theorem 6.1

For every R > 0, let us write Tρf(x) in the form

$Tρf(x)=T1(x)+T2(x),$

where $\begin{array}{}{T}_{1}\left(x\right):={\int }_{B\left(x,R\right)}\frac{\rho \left(|x{y}^{-1}|\right)}{\left(|x{y}^{-1}|{\right)}^{Q}}f\left(y\right)dy\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{T}_{2}\left(x\right):={\int }_{{B}^{c}\left(x,R\right)}\frac{\rho \left(|x{y}^{-1}|\right)}{\left(|x{y}^{-1}|{\right)}^{Q}}f\left(y\right)dy.\end{array}$ For T1(x), we have

$|T1(x)|≤∫|xy−1|

By view of (3.3), we get

$|T1(x)|≤C∑k=−∞−1ρ(2kR)(2kR)Q∫|xy−1|<2k+1R|f(y)|dy≤CMf(x)∑k=−∞−1ρ(2kR)≤CMf(x)∑k=−∞−1∫2kR2k+1Rρ(t)tdt=CMf(x)∫0Rρ(t)tdt.$

Here we have used the fact that

$∫2kR2k+1Rρ(t)tdt≥Cρ(2kR)∫2kR2k+1R1tdt=Cρ(2kR)ln⁡2.$(6.6)

Now, using (6.4), we obtain

$|T1(x)|≤CMf(x)(ϕ(R))(p−q)/q.$(6.7)

For T2(x), using (3.3) we have

$|T2(x)|≤∫|xy−1|≥Rρ(|xy−1|)|xy−1|Q|f(y)|dy=∑k=0∞∫2kR≤|xy−1|<2k+1Rρ(|xy−1|)|xy−1|Q|f(y)|dy≤C∑k=0∞ρ(2kR)(2kR)Q∫|xy−1|<2k+1R|f(y)|dy.$

From this using Hölder’s inequality, we obtain

$|T2(x)|≤C∑k=0∞ρ(2kR)(2kR)Q∫|xy−1|<2k+1Rdy1−1/p∫|xy−1|<2k+1R|f(y)|dy1/p≤C∑k=0∞ρ(2kR)(2kR)Q/p∫|xy−1|<2k+1R|f(y)|dy1/p≤C∥f∥LMp,ϕ(G)∑k=0∞ρ(2k+1R)ϕ(2k+1R)≤C∥f∥LMp,ϕ(G)∑k=0∞∫2kR2k+1Rρ(t)ϕ(t)t=C∥f∥LMp,ϕ(G)∫R∞ρ(t)ϕ(t)t,$

where we have used the fact that

$∫2kR2k+1Rρ(t)ϕ(t)tdt≥Cρ(2k+1R)ϕ(2k+1R)∫2kR2k+1R1tdt=Cρ(2k+1R)ϕ(2k+1R)ln⁡2.$(6.8)

Now, in view of (6.4), we obtain

$|T2(x)|≤C∥f∥LMp,ϕ(G)(ϕ(R))p/q.$(6.9)

Summing the two estimates (6.7) and (6.9), we arrive at

$|Tρf(x)|≤C(Mf(x)(ϕ(R))(p−q)/q+∥f∥LMp,ϕ(G)(ϕ(R))p/q).$

Assuming that f is not identically 0 and that Mf is finite everywhere and then using the fact that ϕ is surjective, we can choose R > 0 such that $\begin{array}{}\varphi \left(R\right)=Mf\left(x\right)\cdot {\parallel f\parallel }_{L{M}^{p,\varphi }\left(\mathbb{G}\right)}^{-1}.\end{array}$ Thus, for every x ∈ 𝔾, we have

$|Tρf(x)|≤C(Mf(x))pq∥f∥LMp,ϕ(G)q−pq.$

It follows that

$∫B(0,r)|Tρf(x)|q1/q≤C∫B(0,r)|Mf(x)|p1/q∥f∥LMp,ϕ(G)q−pq,$

then we divide both sides by (ϕ(r))p/qrQ/q to get

$1(ϕ(r))p/q1rQ∫B(0,r)|Tρf(x)|q1/q≤C1(ϕ(r))p/q1rQ∫B(0,r)|Mf(x)|p1/q∥f∥LMp,ϕ(G)q−pq.$

Taking the supremum over r > 0 and using the boundedness of the maximal operator M on LMp,ϕ(𝔾) (3.6), we obtain

$∥Tρf∥LMq,ϕp/q(G)≤Cp,q,ϕ,Q∥f∥LMp,ϕ(G).$

The proof is complete. □

Now let us turn to the Olsen type inequalities for the generalized fractional integral operator Tρ and Bessel-Riesz operator Iα,γ.

#### Theorem 6.2

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let |·| be a homogeneous quasi-norm. Let ρ and ϕ satisfy the doubling condition (3.3). Let ϕ be also surjective and satisfy (6.3)-(6.4). Then we have

$∥WTρf∥LMp,ϕ(G)≤Cp,ϕ,Q∥W∥LMp2,ϕp/p2(G)∥f∥LMp,ϕ(G),1(6.10)

provided that WLMp2,ϕp/p2(𝔾).

#### Proof of Theorem 6.2

By using Hölder’s inequality, we have

$1rQ∫B(0,r)|W(x)Tρf(x)|pdx≤1rQ∫B(0,r)|W(x)|p2dxp/p21rQ∫B(0,r)|Tρf(x)|pp2p2−pdxp2−pp2.$

Now let us take the p-th roots and then divide both sides by ϕ(r) to obtain

$1ϕ(r)1rQ∫B(0,r)|W(x)Tρf(x)|pdx1/p≤1(ϕ(r))p/p21rQ∫B(0,r)|W(x)|p2dx1/p2×1(ϕ(r))p2−pp21rQ∫B(0,r)|Tρf(x)|pp2p2−pdxp2−ppp2.$

By taking the supremum over r > 0 and using the inequality (6.5), we get

$∥WTρf∥LMp,ϕ(G)≤Cp,ϕ,Q∥W∥LMp2,ϕp/p2(G)∥Tρf∥Lpp2p2−p,ϕp2−pp2(G).$

Taking into account that 1 < p < $\begin{array}{}\frac{p{p}_{2}}{{p}_{2}-p}\end{array}$ < ∞ and putting q = $\begin{array}{}\frac{p{p}_{2}}{{p}_{2}-p}\end{array}$ in (6.5), we obtain (6.10). □

#### Theorem 6.3

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let |·| be a homogeneous quasi-norm. Let ω : ℝ+ → ℝ+ satisfy the doubling condition and assume that ω(r) ≤ Crα for every r > 0, so that Kα,γLMp2,ω(𝔾) for $\begin{array}{}\frac{Q}{Q+\gamma -\alpha }\end{array}$ < p2 < $\begin{array}{}\frac{Q}{Q-\alpha },\end{array}$ and p2 ≥ 1, where 0 < α < Q, 1 < p < ∞, $\begin{array}{}q=\frac{\beta p}{\beta +Q-\alpha }\end{array}$ and γ > 0. If ϕ(r) ≤ Crβ for every r > 0, where β < −α < −Qβ, then we have

$∥WIα,γf∥LMp,ϕ(G)≤Cp,ϕ,Q∥W∥LMp2,ϕp/p2(G)∥f∥LMp,ϕ(G),$(6.11)

provided that WLMp2,ϕp/p2(𝔾), where $\begin{array}{}\frac{1}{{p}_{2}}=\frac{1}{p}-\frac{1}{q}.\end{array}$

#### Proof of Theorem 6.3

As in Theorem 6.2, by using Hölder’s inequality for $\begin{array}{}\frac{p}{{p}_{2}}+\frac{p}{q}=1,\end{array}$ we have

$1rQ∫B(0,r)|W(x)Iα,γf(x)|pdx≤1rQ∫B(0,r)|W(x)|p2dxp/p21rQ∫B(0,r)|Iα,γf(x)|qdxp/q.$

Now we take the p-th roots and then divide both sides by ϕ(r) to get

$1ϕ(r)1rQ∫B(0,r)|W(x)Iα,γf(x)|pdx1/p≤1(ϕ(r))p/p21rQ∫B(0,r)|W(x)|p2dx1/p2×1(ϕ(r))p/q1rQ∫B(0,r)|Iα,γf(x)|qdx1/q.$

By taking the supremum over r > 0, we have

$∥WIα,γf∥LMp,ϕ(G)≤C∥W∥LMp2,ϕp/p2(G)∥Iα,γf∥LMq,ϕp/q(G),$

which implies (6.11) in view of Theorem 4.3 after putting ψ(r) = (ϕ(r))p/q. □

## 7 Inequalities for the modified version of generalized fractional integral operator in Campanato spaces

In this section, we prove the boundedness of the modified version of the operator Tρ in Campanato spaces on homogeneous groups.

We define the generalized local (central) Campanato space by

$LMp,ϕ(G):={f∈Llocp(G):∥f∥LMp,ϕ(G)<∞},$(7.1)

where

$∥f∥LMp,ϕ(G):=supr>01ϕ(r)1rQ∫B(0,r)|f(x)−fB|pdx1/p,$

with $\begin{array}{}{f}_{B}={f}_{B\left(0,r\right)}:=\frac{1}{{r}^{Q}}{\int }_{B\left(0,r\right)}f\left(y\right)dy,\end{array}$ and we assume that $\begin{array}{}\frac{\varphi \left(r\right)}{r}\end{array}$ is nonincreasing.

Next, for the function ρ : ℝ+ → ℝ+, we define the modified version of the generalized fractional integral operator Tρ by

$Tρ~f(x):=∫Gρ(|xy−1|)|xy−1|Q−ρ(|y|)(1−χB(0,1)(y))|y|Qf(y)dy,$(7.2)

where B(0, 1) := {x ∈ 𝔾: |x| < 1} and χB(0,1) is the characteristic function of B(0, 1). In this definition, we assume that ρ satisfies (6.2), (3.3) and the following conditions:

$∫r∞ρ(t)t2dt≤C1ρ(r)rforallr>0;$(7.3)

$12≤rs≤2⇒ρ(r)rQ−ρ(s)sQ≤C2|r−s|ρ(s)sQ+1.$(7.4)

For instance, the function ρ(r) = rα satisfies (6.2), (3.3) and (7.4) for 0 < α < Q, and also satisfies (7.3) for 0 < α < 1.

#### Theorem 7.1

Let 𝔾 be a homogeneous group of homogeneous dimension Q. Let |·| be a homogeneous quasi-norm. Let ρ satisfy (6.2), (3.3), (7.3), (7.4), and let ϕ satisfy the doubling condition (3.3) and $\begin{array}{}{\int }_{1}^{\mathrm{\infty }}\frac{\varphi \left(t\right)}{t}dt\end{array}$ < ∞. If

$∫r∞ϕ(t)tdt∫0rρ(t)tdt+r∫r∞ρ(t)ϕ(t)t2dt≤C3ψ(r)forallr>0,$(7.5)

then we have

$∥T~ρf∥LMp,ψ(G)≤Cp,ϕ,Q∥f∥LMp,ϕ(G),1(7.6)

#### Proof of Theorem 7.1

For every xB(0, r) and f ∈ 𝓛𝓜p,ϕ(𝔾), let us write ρf in the following form:

$T~ρf(x)=T~B(0,r)(x)+CB(0,r)1+CB(0,r)2=T~B(0,r)1(x)+T~B(0,r)2(x)+CB(0,r)1+CB(0,r)2,$

where

$T~B(0,r)(x):=∫G(f(y)−fB(0,2r))ρ(|xy−1|)|xy−1|Q−ρ(|y|)(1−χB(0,2r)(y))|y|Qdy,CB(0,r)1:=∫G(f(y)−fB(0,2r))×ρ(|y|)(1−χB(0,2r)(y))|y|Q−ρ(|y|)(1−χB(0,1)(y))|y|Qdy,CB(0,r)2:=∫GfB(0,2r)ρ(|xy−1|)|xy−1|Q−ρ(|y|)(1−χB(0,1)(y))|y|Qdy,T~B(0,r)1(x):=∫B(0,2r)(f(y)−fB(0,2r))ρ(|xy−1|)|xy−1|Qdy,T~B(0,r)2(x):=∫Bc(0,2r)(f(y)−fB(0,2r))ρ(|xy−1|)|xy−1|Q−ρ(|y|)|y|Qdy.$

Since

$ρ(|y|)(1−χB(0,2r)(y))|y|Q−ρ(|y|)(1−χB(0,1)(y))|y|Q≤0,|y|

$\begin{array}{}{C}_{B\left(0,r\right)}^{1}\end{array}$ is finite.

Now let us show that $\begin{array}{}{C}_{B\left(0,r\right)}^{2}\end{array}$ is finite. For this it is enough to prove that the following integral is finite:

$∫Gρ(|xy−1|)|xy−1|Q−ρ(|y|)(1−χB(0,1)(y))|y|Qdy=∫Gρ(|xy−1|)|xy−1|Q−ρ(|y|)|y|Qdy+∫B(0,1)ρ(|y|)|y|Qdy.$

Let us denote $\begin{array}{}A:={\int }_{\mathbb{G}}\left(\frac{\rho \left(|x{y}^{-1}|\right)}{|x{y}^{-1}{|}^{Q}}-\frac{\rho \left(|y|\right)}{|y{|}^{Q}}\right)dy.\end{array}$ For large R > 0, we write A in the form A = A1 + A2 + A3, where

$A1:=∫B(x,R)ρ(|xy−1|)|xy−1|Qdy−∫B(0,R)ρ(|y|)|y|Qdy,A2:=∫B(x,R+r)∖B(x,R)ρ(|xy−1|)|xy−1|Qdy−∫B(x,R+r)∖B(0,R)ρ(|y|)|y|Qdy,A3:=∫Bc(x,R+r)ρ(|xy−1|)|xy−1|Q−ρ(|y|)|y|Qdy.$

Since we have $\begin{array}{}{\int }_{0}^{1}\frac{\rho \left(t\right)}{t}dt<+\mathrm{\infty },\end{array}$ it implies that $\begin{array}{}\frac{\rho \left(|x{y}^{-1}|\right)}{|x{y}^{-1}{|}^{Q}},\frac{\rho \left(|y|\right)}{|y{|}^{Q}}\in {L}_{loc}^{1}\left(\mathbb{G}\right),\end{array}$ and hence A1 = 0. By (7.4) we have

$A3≤∫Bc(x,R+r)ρ(|xy−1|)|xy−1|Q−ρ(|y|)|y|Qdy≤C∫Bc(x,R+r)||xy−1|−|y||ρ(|xy−1|)|xy−1|Q+1dy.$

By using the triangle inequality (see e.g. [14, Theorem 3.1.39, p.113]) and symmetric property of homogeneous quasi-norms, we get

$A3≤C||x|+|y−1|−|y||∫R+r+∞∫Sρ(t)tQ+1tQ−1dσ(y)dt≤C|σ|r∫R+r+∞ρ(t)t2dt.$

The inequality (7.3) implies that the last integral is finite and |A3| → 0 as R → +∞. For A2, we have

$|A2|≤∫B(x,R+r)∖B(x,R−r)ρ(|xy−1|)|xy−1|Q+ρ(|y|)|y|Qdy∼((R+r)Q−(R−r)Q)ρ(R)RQ≤Crρ(R)R,$

and taking into account the conditions (3.3) and (7.3), we obtain

$|A2|≤Crρ(R)R→0asR→+∞.$

Since A → 0 as R → +∞, we have A = 0 and hence

$∫Gρ(|xy−1|)|xy−1|Q−ρ(|y|)(1−χB(0,1)(y))|y|Qdy=∫B(0,1)ρ(|y|)|y|Qdy<∞,$

which implies that $\begin{array}{}{C}_{B\left(0,r\right)}^{2}\end{array}$ is finite.

Now before estimating $\begin{array}{}{\stackrel{~}{T}}_{B\left(0,r\right)}^{1},\end{array}$ let us denote := (ffB(0,2r))χB(0,2r) and $\begin{array}{}\stackrel{~}{\varphi }\left(r\right):={\int }_{r}^{\mathrm{\infty }}\frac{\varphi \left(t\right)}{t}dt.\end{array}$ Then, we have

$|T~B(0,r)1(x)|≤∫B(0,2r)|f~(y)|ρ(|xy−1|)|xy−1|Qdy=∑k=−∞0∫2kr≤|xy−1|<2k+1rρ(|xy−1|)|xy−1|Q|f~(y)|dy.$

By using (3.3) and (6.6), we get

$|T~B(0,r)1(x)|≤C∑k=−∞0ρ(2kr)(2kr)Q∫|xy−1|<2k+1r|f~(y)|dy≤CMf~(x)∑k=−∞0ρ(2kr)≤CMf~(x)∑k=−∞0ρ(2k−1r)≤CMf~(x)∑k=−∞0∫2k−1r2krρ(t)tdt=CMf~(x)∫0rρ(t)tdt.$

Now using (7.5), we have $\begin{array}{}|{\stackrel{~}{T}}_{B\left(0,r\right)}^{1}\left(x\right)|\le C\frac{\psi \left(r\right)}{\stackrel{~}{\varphi }\left(r\right)}M\stackrel{~}{f}\left(x\right).\end{array}$ It follows that

$1ψ(r)1rQ∫B(0,r)|T~B(0,r)1(x)|pdx1/p≤C1ϕ~(r)rQ/p∫B(0,r)|Mf~(x)|pdx1/p≤C1ϕ~(r)rQ/p∥f~∥Lp(G),$

where we have used (3.8).

By the Minkowski inequality, we have

$1ϕ~(r)rQ/p∥f~∥Lp(G)=1ϕ~(r)rQ/p∥(f−fB(0,2r))χB(0,2r)∥Lp(G)≤C1ϕ~(r)rQ/p(∥(f−σ(f))χB(0,2r)∥Lp(G)+(2r)Q/p|fB(0,2r)−σ(f)|),$

where $\begin{array}{}\sigma \left(f\right)=\underset{r\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{f}_{B\left(0,r\right)}.\end{array}$

Moreover, we obtain the following inequalities exactly in the same way as in the Abelian case (see [13], Section 6)

$∥f−σ(f)∥LMp,ϕ~(G)≤C1∥f∥LMp,ϕ(G),$(7.7)

and

$|fB(0,r)−σ(f)|≤C2∥f∥LMp,ϕ(G)ϕ~(r).$(7.8)

Finally, using these inequalities we get our estimate for $\begin{array}{}{\stackrel{~}{T}}_{B\left(0,r\right)}^{1}\end{array}$ as

$T~B(0,r)1(x)|≤C∥f∥LMp,ϕ(G).$(7.9)

Now let us estimate $\begin{array}{}{\stackrel{~}{T}}_{B\left(0,r\right)}^{2}\end{array}$ . By (3.3) and (7.4), we have

$|T~B(0,r)2(x)|≤∫Bc(0,2r)|f(y)−fB(0,2r)|ρ(|xy−1|)|xy−1|Q−ρ(|y|)|y|Qdy≤C||xy−1|−|y||∫|y|≥2r|f(y)−fB(0,2r)|ρ(|y|)|y|Q+1dy.$

By using the triangle inequality (see e.g. [14, Theorem 3.1.39, p.113]) and symmetric property of homogeneous quasi-norms, we get

$|T~B(0,r)2(x)|≤C||x|+|y−1|−|y||∫|y|≥2r|f(y)−fB(0,2r)|ρ(|y|)|y|Q+1dy≤C|x|∫|y|≥2r|f(y)−fB(0,2r)|ρ(|y|)|y|Q+1dy=C|x|∑k=2∞∫2k−1r≤|y|<2krρ(|y|)|f(y)−fB(0,2r)||y|Q+1dy.$

By using (3.3) and Hölder’s inequality, we obtain

$|T~B(0,r)2(x)|≤C|x|∑k=2∞ρ(2kr)(2kr)Q+1∫|y|<2kr|f(y)−fB(0,2r)|dy≤C|x|∑k=2∞ρ(2kr)2kr1(2kr)Q∫|y|<2kr|f(y)−fB(0,2r)|pdy1/p.$

As in the Abelian case ([13]), we have

$1(2kr)Q∫B(0,2kr)|f(y)−fB(0,2r)|pdy1/p≤C∥f∥LMp,ϕ(G)∫2r2k+1rϕ(s)sds,$

for every k ≥ 2. The inequality (6.6) implies that

$∫2kr2k+1rρ(t)t2dt≥12k+1r∫2kr2k+1rρ(t)tdt≥Cρ(2kr)2kr.$

By using the last two inequalities, we get

$|T~B(0,r)2(x)|≤C|x|∥f∥LMp,ϕ(G)∑k=2∞ρ(2kr)2kr∫2r2k+1rϕ(s)sds≤C|x|∥f∥LMp,ϕ(G)∑k=2∞∫2kr2k+1rρ(t)t2∫2rtϕ(s)sdsdt≤C|x|∥f∥LMp,ϕ(G)∫2r∞ρ(t)t2∫2rtϕ(s)sdsdt=C|x|∥f∥LMp,ϕ(G)∫2r∞∫s∞ρ(t)t2dtϕ(s)sds.$

Using (7.3) and then (7.5), it implies that

$|T~B(0,r)2(x)|≤Cr∥f∥LMp,ϕ(G)∫2r∞ρ(s)ϕ(s)s2ds≤Cψ(r)∥f∥LMp,ϕ(G).$

This follows that

$1ψ(r)1rQ∫B(0,r)|T~B(0,r)2(x)|pdx1/p≤C∥f∥LMp,ϕ(G).$(7.10)

Summing the estimates (7.9) and (7.10), we obtain (7.6). □

## Acknowledgements

The first author was supported in parts by the EPSRC grant EP/R003025/1 and by the Leverhulme Grant RPG-2017-151. The second author was partially supported by the MESRK grant AP05130981 and NU SPG. The third author was supported by the MESRK grant AP05133271.

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## About the article

Received: 2017-04-17

Published Online: 2018-07-12

Published in Print: 2018-06-26

Citation Information: Fractional Calculus and Applied Analysis, Volume 21, Issue 3, Pages 577–612, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454,

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© 2018 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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