# Weighted Hardy and Potential Operators in Morrey Spaces

Authors: Natasha Samko
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We study the weightedp→q-boundedness of Hardy-type operators
in Morrey spaces ℒp,λ(ℝn) (or ℒp,λ(ℝ+1) in the one-dimensional case) for
a class of almost monotonic weights. The obtained results are applied to
a similar weighted p→q-boundedness of the Riesz potential operator.
The conditions on weights, both for the Hardy and potential operators, are necessary and sufficient in the case of power weights. In the case
of more general weights, we provide separately necessary and sufficient conditions in terms of Matuszewska-Orlicz indices of weights.


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## Body

## 1. Introduction

The well-known Morrey spacesℒp,λ introduced in [1] in relation to the study of partial differential equations and presented in various books, see [2–4], were widely investigated during the last decades, including the study of classical operators of harmonic analysis—maximal, singular, and potential operators—in these spaces; we refer for instance to papers [5–23], where Morrey spaces on metric measure spaces may be also found. Surprisingly, weighted estimates of these classical operators, in fact, were not studied. Just recently, in [24] we proved weighted p→p-estimates in Morrey spaces for Hardy operators on ℝ+1 and one-dimensional singular operators (on ℝ1 or on Carleson curves in the complex plane).In this paper we develop an approach which allows us both to obtain weightedp→q-estimations of Hardy-type operators and potential operators and extend them to the multidimensional case, for the Hardy operators (related to integration over balls). Note that, in contrast with the case of Lebesgue spaces, Hardy inequalities in Morrey norms admit the value p=1 for p; see Theorem 4.3. The progress in comparison with [24] is based on the pointwise estimation of the Hardy operators we present in Sections 3.2 and 4.1. Such an estimation is helpless in the case of Lebesgue spaces (λ=0), but proved to be effective in the case of Morrey spaces (0<λ<n). Roughly speaking, it is based on the simple fact that(1.1)|x|-n/p∉Lp(λ=0),but|x|(λ-n)/p∈Lp,λ(λ>0).The admitted weightsφ(|x-x0|) are generated by functions φ(r) from the Bary-Stechkin-type class; they may be characterized as weights continuous and positive for r∈(0,∞), with possible decay or growth at r=0 and r=∞, which become almost increasing or almost decreasing after the multiplication by some power. Such weights are oscillating between two powers at the origin and infinity (with different exponents for the origin and infinity).We also note that the obtained estimates show that the Hardy operators (with admitted weights) act boundedly not only in local and global Morrey spaces (see definitions in Section3.1), but also from a larger local Morrey space into a more narrow global Morrey space (see Theorems 4.3 and 4.4).The paper is organized as follows. In Section2 we give necessary preliminaries on some classes of weight functions. In Section 3 we prove some statements on embedding of Morrey spaces ℒp,λ(Ω) into some weighted Lp(Ω,ϱ)-spaces. In Section 4 we prove theorems on the weighted p→q-boundedness of Hardy operators in Morrey spaces. Finally, in Section 5 we apply the results of Section 4 to a similar weighted boundedness of potential operators. The conditions on weights, both for the Hardy and potential operators are necessary and sufficient in the case of power weights. In the case of more general weights, we provide separately necessary and sufficient conditions in terms of Matuszewska-Orlicz indices of weights.The main results are given in Theorems4.3, 4.4, 4.5, and 5.3 and Corollary 5.4.


## 2. Preliminaries on Weight Functions

### 2.1. Zygmund-Bary-Stechkin (ZBS) Classes and Matuszewska-Orlicz (MO) Type Indices

#### 2.1.1. On Classes of the TypeW0andW∞

In the sequel, a nonnegative functionf on [0,ℓ],0<ℓ≤∞, is called almost increasing (almost decreasing) if there exists a constant C(≥1) such that f(x)≤Cf(y) for all x≤y (x≥y, resp.). Equivalently, a function f is almost increasing (almost decreasing) if it is equivalent to an increasing (decreasing, resp.) function g, that is, c1f(x)≤g(x)≤c2f(x),c1>0,c2>0.Definition 2.1.
Let0<ℓ<∞.(1)
ByW0=W0([0,ℓ]) one denotes the class of functions φ continuous and positive on (0,ℓ] such that there exists the finite limit limx→0φ(x), and φ(x) is almost increasing on(0,ℓ);(2)
byW̃0=W̃0([0,ℓ]) one denotes the class of functions φ on [0,ℓ] such that xaφ(x)∈W0 for some a=a(φ)∈ℝ1.Definition 2.2.
Let0<ℓ<∞.(1)
ByW∞=W∞([ℓ,∞]) one denotes the class of functions φ continuous and positive on [ℓ,∞) which have the finite limit limx→∞φ(x), and φ(x) is almost increasing on [ℓ,∞);(2)
byW̃∞=W̃∞([ℓ,∞)) one denotes the class of functions φ∈W∞ such xaφ(x)∈W∞ for some a=a(φ)∈ℝ1.


#### 2.1.2. ZBS Classes and MO Indices of Weights at the Origin

In this subsection we assume thatℓ<∞.Definition 2.3.
One says that a functionφ∈W0 belongs to the Zygmund class ℤβ, β∈ℝ1, if
(2.1)∫0xφ(t)t1+βdt≤cφ(x)xβ,x∈(0,l),
and to the Zygmund class ℤγ, γ∈ℝ1, if
(2.2)∫xlφ(t)t1+γdt≤cφ(x)xγ,x∈(0,l).
One also denotes
(2.3)Φγβ∶=Zβ⋂Zγ,
the latter class being also known as Bary-Stechkin-Zygmund class [25].It is known that the property of a function is to be almost increasing or almost decreasing after the multiplication (division) by a power function is closely related to the notion of the so called Matuszewska-Orlicz indices. We refer to [26, 27] [28, page 20], [29–32] for the properties of the indices of such a type. For a function φ∈W̃0, the numbers(2.4)m(φ)=sup⁡0<x<1ln⁡(limsup⁡h→0⁡(φ(hx)/φ(h)))ln⁡⁡x=lim⁡x→0ln⁡(limsup⁡h→0⁡(φ(hx)/φ(h)))ln⁡⁡x,M(φ)=sup⁡x>1ln⁡(limsup⁡h→0⁡(φ(hx)/φ(h)))ln⁡⁡x=lim⁡x→∞ln⁡(limsup⁡h→0⁡(φ(hx)/φ(h)))ln⁡⁡x.
are known as the Matuszewska-Orlicz type lower and upper indices of the function φ(r). Note that in this definition φ(x)needs not to be an N-function: only its behaviour at the origin is of importance. Observe that 0≤m(φ)≤M(φ)≤∞forφ∈W0 and -∞<m(φ)≤M(φ)≤∞forφ∈W̃0, and the formulas are valid(2.5)m[xaφ(x)]=a+m(φ),M[xaφ(x)]=a+M(φ),a∈R1,(2.6)m([φ(x)]a)=am(φ),M([φ(x)]a)=aM(φ),a≥0,(2.7)m(1φ)=-M(φ),M(1φ)=-m(φ),(2.8)m(uv)≥m(u)+m(v),M(uv)≤M(u)+M(v)
for φ,u,v∈W̃0.The following statement is known see [26, Theorems 3.1,  3.2, and 3.5]. (In the formulation of Theorem  2.4 in [26], it was supposed that β≥0,γ>0, and φ∈W0. It is evidently true also for φ∈W̃0 and all β,γ∈ℝ1, in view of formulas (2.5).)Theorem 2.4.
Letφ∈W̃0 and β,γ∈ℝ1. Then
(2.9)φ∈Zβ⟺m(φ)>β,φ∈Zγ⟺M(φ)<γ.
Besides this,
(2.10)m(φ)=sup⁡{δ>0:φ(x)xδisalmostincreasing},(2.11)M(φ)=inf⁡{λ>0:φ(x)xλisalmostdecreasing},
and for φ∈Φγβ the inequalities
(2.12)c1xM(φ)+ɛ≤φ(x)≤c2xm(φ)-ɛ
hold with an arbitrarily small ɛ>0 and c1=c1(ɛ),c2=c2(ɛ).


#### 2.1.3. ZBS Classes and MO Indices of Weights at Infinity

Following [14, Section4.1] and [29, Section 2.2], we introduce the following definitions.Definition 2.5.
Let-∞<α<β<∞. One puts Ψαβ:=ℤ̂β∩ℤ̂α, where ℤ̂β is the class of functions φ∈W̃∞ satisfying the condition
(2.13)∫x∞(xt)βφ(t)dtt≤cφ(x),x∈(l,∞),
and ℤ̂α is the class of functions φ∈W([ℓ,∞)) satisfying the condition
(2.14)∫lx(xt)αφ(t)dtt≤cφ(x),x∈(l,∞),
where c=c(φ)>0 does not depend on x∈[ℓ,∞).The indicesm∞(φ) and M∞(φ) responsible for the behavior of functions φ∈Ψαβ([ℓ,∞)) at infinity are introduced in the way similar to (2.4):(2.15)m∞(φ)=sup⁡x>1ln⁡⁡[lim⁡inf⁡h→∞⁡(φ(xh)/φ(h))]ln⁡x,M∞(φ)=inf⁡x>1ln⁡⁡[limsup⁡h→∞⁡(φ(xh)/φ(h))]ln⁡x.Properties of functions in the classΨαβ([ℓ,∞)) are easily derived from those of functions in Φβα([0,ℓ]) because of the following equivalence:(2.16)φ∈Ψαβ([l,∞))⟺φ*∈Φ-α-β([0,l*]),
where φ*(t)=φ(1/t) and ℓ*=1/ℓ∙. Direct calculation shows that(2.17)m∞(φ)=-M(φ*),M∞(φ)=-m(φ*),φ*(t):=φ(1t).Making use of (2.16) and (2.17), one can easily reformulate properties of functions of the class Φαβ near the origin, given in Theorem 2.4 for the case of the corresponding behavior at infinity of functions of the class Ψαβ and obtain that(2.18)c1tm∞(φ)-ε≤φ(t)≤c2tM∞(φ)+ε,t≥l,φ∈W̃∞,(2.19)m∞(φ)=sup⁡{μ∈R1:t-μφ(t)isalmostincreasingon⁡[l,∞)},(2.20)M∞(φ)=inf⁡{ν∈R1:t-νφ(t)isalmostdecreasingon⁡[l,∞)}.We say that a functionφ continuous in (0,∞) is in the class W̃0,∞(ℝ+1) if its restriction to (0,1) belongs to W̃0([0,1]) and its restriction to (1,∞) belongs to W̃∞([1,∞]). For functions in W̃0,∞(ℝ+1), the notation(2.21)Zβ0,β∞(R+1)=Zβ0([0,1])∩Zβ∞([1,∞)),Zγ0,γ∞(R+1)=Zγ0([0,1])∩Zγ∞([1,∞))
has an obvious meaning. In the case, where the indices coincide: β0=β∞:=β, we will simply write ℤβ(ℝ+1) and similarly for ℤγ(ℝ+1). We also denote(2.22)Φαβ(R+1):=Zβ(R+1)∩Zγ(R+1).Making use of Theorem2.4 for Φαβ([0,1]) and relations (2.17), we easily arrive at the following statement.Lemma 2.6.
Letφ∈W̃0(ℝ+1). Then
(2.23)φ∈Zβ0,β∞(R+1)⟺m(φ)>β0,m∞(φ)>β∞,φ∈Zγ0,γ∞(R+1)⟺M(φ)<γ0,M∞(φ)<γ∞.


### 2.2. On ClassesV±μ

Note that we slightly changed the notation of the class introduced in the following definition, in comparison with its notation in [32].Definition 2.7.
Let0<μ≤1. By V±μ, One denotes the classes of functions φ nonnegative on [0,ℓ],0<ℓ≤∞, defined by the following conditions:
(2.24)V+μ:|φ(x)-φ(y)||x-y|μ≤Cφ(x+)x+μ,(2.25)V-μ:|φ(x)-φ(y)||x-y|μ≤Cφ(x-)x+μ,
where x,y∈(0,ℓ] and x+=max(x,y),x-=min(x,y).Lemma 2.8.
Functionsφ∈V+μ are almost increasing on [0,ℓ], and functions φ∈V-μ are almost decreasing on [0,ℓ].Proof.
Letφ∈V+μ and y≤x. By (2.24) we have |φ(x)-φ(y)|≤Cφ(x)(1-y/x)μ≤Cφ(x). Then φ(y)≤|φ(x)-φ(y)|+φ(x)≤(C+1)φ(x). The case φ∈V-μ is similarly treated.Corollary 2.9.
Functionsφ∈V+μ have non-negative indices 0≤m(φ)≤M(φ), and functions φ∈V-μ have non-positive indices m(φ)≤M(φ)≤0, the same being also valid with respect to the indices m∞(φ),M∞(φ) in the case ℓ=∞.Note that(2.26)V+μ⊂V+ν,V-μ⊂V-ν,0<ν<μ≤1,
the classes V±μ being trivial for μ>1. We also have(2.27)xγ∈⋃μ∈[0,1]V+μ⟺γ≥0,xγ∈⋃μ∈[0,1]V-μ⟺γ≤0,
which follows from the fact that xγ∈V+1⇔γ≥0 and xγ∈V-1⇔γ≤0 (see [24, Subsection 2.3, Remark 2.8]) and property (2.26).An example of a function which is inV+μ with some μ>0, but does not belong to the total intersection ⋂μ∈[0,1]V+μ is given by(2.28)φ(x)=axγ+b|x-x0|β∈⋂μ∈[0,β]V+μ,
where x0>0 and γ≥0,0<β<1,a>0,b>0.The following lemmas (see [24], Lemmas 2.10 and 2.11) show that conditions (2.24) and (2.25) are fulfilled with μ=1 not only for power functions but also for an essentially larger class of functions (which in particular may oscillate between two power functions with different exponents). Note that the information about this class in Lemmas 2.10 and 2.11 is given in terms of increasing or decreasing functions, without the word “almost”.Lemma 2.10.
Letφ∈W. Then(i)
φ∈V+1 in the case φ is increasing and the function φ(x)/xν  is decreasing for some ν≥0;(ii)
φ∈V-1 in the case φ(x) is decreasing and there exists a number μ≥0 such that xμφ(x) is increasing.Lemma 2.11.
Letφ∈W∩C1((0,ℓ]). If there exist ɛ>0 and ν≥0 such that 0≤φ′(x)/φ(x)≤ν/x for 0<x≤ɛ, then φ∈V+1. If there exist ɛ>0 and μ≥0 such that -μ/x≤φ′(x)/φ(x)≤0 for 0<x≤ɛ, then φ∈V-1.


## 2.1. Zygmund-Bary-Stechkin (ZBS) Classes and Matuszewska-Orlicz (MO) Type Indices

### 2.1.1. On Classes of the TypeW0andW∞

In the sequel, a nonnegative functionf on [0,ℓ],0<ℓ≤∞, is called almost increasing (almost decreasing) if there exists a constant C(≥1) such that f(x)≤Cf(y) for all x≤y (x≥y, resp.). Equivalently, a function f is almost increasing (almost decreasing) if it is equivalent to an increasing (decreasing, resp.) function g, that is, c1f(x)≤g(x)≤c2f(x),c1>0,c2>0.Definition 2.1.
Let0<ℓ<∞.(1)
ByW0=W0([0,ℓ]) one denotes the class of functions φ continuous and positive on (0,ℓ] such that there exists the finite limit limx→0φ(x), and φ(x) is almost increasing on(0,ℓ);(2)
byW̃0=W̃0([0,ℓ]) one denotes the class of functions φ on [0,ℓ] such that xaφ(x)∈W0 for some a=a(φ)∈ℝ1.Definition 2.2.
Let0<ℓ<∞.(1)
ByW∞=W∞([ℓ,∞]) one denotes the class of functions φ continuous and positive on [ℓ,∞) which have the finite limit limx→∞φ(x), and φ(x) is almost increasing on [ℓ,∞);(2)
byW̃∞=W̃∞([ℓ,∞)) one denotes the class of functions φ∈W∞ such xaφ(x)∈W∞ for some a=a(φ)∈ℝ1.


### 2.1.2. ZBS Classes and MO Indices of Weights at the Origin

In this subsection we assume thatℓ<∞.Definition 2.3.
One says that a functionφ∈W0 belongs to the Zygmund class ℤβ, β∈ℝ1, if
(2.1)∫0xφ(t)t1+βdt≤cφ(x)xβ,x∈(0,l),
and to the Zygmund class ℤγ, γ∈ℝ1, if
(2.2)∫xlφ(t)t1+γdt≤cφ(x)xγ,x∈(0,l).
One also denotes
(2.3)Φγβ∶=Zβ⋂Zγ,
the latter class being also known as Bary-Stechkin-Zygmund class [25].It is known that the property of a function is to be almost increasing or almost decreasing after the multiplication (division) by a power function is closely related to the notion of the so called Matuszewska-Orlicz indices. We refer to [26, 27] [28, page 20], [29–32] for the properties of the indices of such a type. For a function φ∈W̃0, the numbers(2.4)m(φ)=sup⁡0<x<1ln⁡(limsup⁡h→0⁡(φ(hx)/φ(h)))ln⁡⁡x=lim⁡x→0ln⁡(limsup⁡h→0⁡(φ(hx)/φ(h)))ln⁡⁡x,M(φ)=sup⁡x>1ln⁡(limsup⁡h→0⁡(φ(hx)/φ(h)))ln⁡⁡x=lim⁡x→∞ln⁡(limsup⁡h→0⁡(φ(hx)/φ(h)))ln⁡⁡x.
are known as the Matuszewska-Orlicz type lower and upper indices of the function φ(r). Note that in this definition φ(x)needs not to be an N-function: only its behaviour at the origin is of importance. Observe that 0≤m(φ)≤M(φ)≤∞forφ∈W0 and -∞<m(φ)≤M(φ)≤∞forφ∈W̃0, and the formulas are valid(2.5)m[xaφ(x)]=a+m(φ),M[xaφ(x)]=a+M(φ),a∈R1,(2.6)m([φ(x)]a)=am(φ),M([φ(x)]a)=aM(φ),a≥0,(2.7)m(1φ)=-M(φ),M(1φ)=-m(φ),(2.8)m(uv)≥m(u)+m(v),M(uv)≤M(u)+M(v)
for φ,u,v∈W̃0.The following statement is known see [26, Theorems 3.1,  3.2, and 3.5]. (In the formulation of Theorem  2.4 in [26], it was supposed that β≥0,γ>0, and φ∈W0. It is evidently true also for φ∈W̃0 and all β,γ∈ℝ1, in view of formulas (2.5).)Theorem 2.4.
Letφ∈W̃0 and β,γ∈ℝ1. Then
(2.9)φ∈Zβ⟺m(φ)>β,φ∈Zγ⟺M(φ)<γ.
Besides this,
(2.10)m(φ)=sup⁡{δ>0:φ(x)xδisalmostincreasing},(2.11)M(φ)=inf⁡{λ>0:φ(x)xλisalmostdecreasing},
and for φ∈Φγβ the inequalities
(2.12)c1xM(φ)+ɛ≤φ(x)≤c2xm(φ)-ɛ
hold with an arbitrarily small ɛ>0 and c1=c1(ɛ),c2=c2(ɛ).


### 2.1.3. ZBS Classes and MO Indices of Weights at Infinity

Following [14, Section4.1] and [29, Section 2.2], we introduce the following definitions.Definition 2.5.
Let-∞<α<β<∞. One puts Ψαβ:=ℤ̂β∩ℤ̂α, where ℤ̂β is the class of functions φ∈W̃∞ satisfying the condition
(2.13)∫x∞(xt)βφ(t)dtt≤cφ(x),x∈(l,∞),
and ℤ̂α is the class of functions φ∈W([ℓ,∞)) satisfying the condition
(2.14)∫lx(xt)αφ(t)dtt≤cφ(x),x∈(l,∞),
where c=c(φ)>0 does not depend on x∈[ℓ,∞).The indicesm∞(φ) and M∞(φ) responsible for the behavior of functions φ∈Ψαβ([ℓ,∞)) at infinity are introduced in the way similar to (2.4):(2.15)m∞(φ)=sup⁡x>1ln⁡⁡[lim⁡inf⁡h→∞⁡(φ(xh)/φ(h))]ln⁡x,M∞(φ)=inf⁡x>1ln⁡⁡[limsup⁡h→∞⁡(φ(xh)/φ(h))]ln⁡x.Properties of functions in the classΨαβ([ℓ,∞)) are easily derived from those of functions in Φβα([0,ℓ]) because of the following equivalence:(2.16)φ∈Ψαβ([l,∞))⟺φ*∈Φ-α-β([0,l*]),
where φ*(t)=φ(1/t) and ℓ*=1/ℓ∙. Direct calculation shows that(2.17)m∞(φ)=-M(φ*),M∞(φ)=-m(φ*),φ*(t):=φ(1t).Making use of (2.16) and (2.17), one can easily reformulate properties of functions of the class Φαβ near the origin, given in Theorem 2.4 for the case of the corresponding behavior at infinity of functions of the class Ψαβ and obtain that(2.18)c1tm∞(φ)-ε≤φ(t)≤c2tM∞(φ)+ε,t≥l,φ∈W̃∞,(2.19)m∞(φ)=sup⁡{μ∈R1:t-μφ(t)isalmostincreasingon⁡[l,∞)},(2.20)M∞(φ)=inf⁡{ν∈R1:t-νφ(t)isalmostdecreasingon⁡[l,∞)}.We say that a functionφ continuous in (0,∞) is in the class W̃0,∞(ℝ+1) if its restriction to (0,1) belongs to W̃0([0,1]) and its restriction to (1,∞) belongs to W̃∞([1,∞]). For functions in W̃0,∞(ℝ+1), the notation(2.21)Zβ0,β∞(R+1)=Zβ0([0,1])∩Zβ∞([1,∞)),Zγ0,γ∞(R+1)=Zγ0([0,1])∩Zγ∞([1,∞))
has an obvious meaning. In the case, where the indices coincide: β0=β∞:=β, we will simply write ℤβ(ℝ+1) and similarly for ℤγ(ℝ+1). We also denote(2.22)Φαβ(R+1):=Zβ(R+1)∩Zγ(R+1).Making use of Theorem2.4 for Φαβ([0,1]) and relations (2.17), we easily arrive at the following statement.Lemma 2.6.
Letφ∈W̃0(ℝ+1). Then
(2.23)φ∈Zβ0,β∞(R+1)⟺m(φ)>β0,m∞(φ)>β∞,φ∈Zγ0,γ∞(R+1)⟺M(φ)<γ0,M∞(φ)<γ∞.


## 2.1.1. On Classes of the TypeW0andW∞

In the sequel, a nonnegative functionf on [0,ℓ],0<ℓ≤∞, is called almost increasing (almost decreasing) if there exists a constant C(≥1) such that f(x)≤Cf(y) for all x≤y (x≥y, resp.). Equivalently, a function f is almost increasing (almost decreasing) if it is equivalent to an increasing (decreasing, resp.) function g, that is, c1f(x)≤g(x)≤c2f(x),c1>0,c2>0.Definition 2.1.
Let0<ℓ<∞.(1)
ByW0=W0([0,ℓ]) one denotes the class of functions φ continuous and positive on (0,ℓ] such that there exists the finite limit limx→0φ(x), and φ(x) is almost increasing on(0,ℓ);(2)
byW̃0=W̃0([0,ℓ]) one denotes the class of functions φ on [0,ℓ] such that xaφ(x)∈W0 for some a=a(φ)∈ℝ1.Definition 2.2.
Let0<ℓ<∞.(1)
ByW∞=W∞([ℓ,∞]) one denotes the class of functions φ continuous and positive on [ℓ,∞) which have the finite limit limx→∞φ(x), and φ(x) is almost increasing on [ℓ,∞);(2)
byW̃∞=W̃∞([ℓ,∞)) one denotes the class of functions φ∈W∞ such xaφ(x)∈W∞ for some a=a(φ)∈ℝ1.


## 2.1.2. ZBS Classes and MO Indices of Weights at the Origin

In this subsection we assume thatℓ<∞.Definition 2.3.
One says that a functionφ∈W0 belongs to the Zygmund class ℤβ, β∈ℝ1, if
(2.1)∫0xφ(t)t1+βdt≤cφ(x)xβ,x∈(0,l),
and to the Zygmund class ℤγ, γ∈ℝ1, if
(2.2)∫xlφ(t)t1+γdt≤cφ(x)xγ,x∈(0,l).
One also denotes
(2.3)Φγβ∶=Zβ⋂Zγ,
the latter class being also known as Bary-Stechkin-Zygmund class [25].It is known that the property of a function is to be almost increasing or almost decreasing after the multiplication (division) by a power function is closely related to the notion of the so called Matuszewska-Orlicz indices. We refer to [26, 27] [28, page 20], [29–32] for the properties of the indices of such a type. For a function φ∈W̃0, the numbers(2.4)m(φ)=sup⁡0<x<1ln⁡(limsup⁡h→0⁡(φ(hx)/φ(h)))ln⁡⁡x=lim⁡x→0ln⁡(limsup⁡h→0⁡(φ(hx)/φ(h)))ln⁡⁡x,M(φ)=sup⁡x>1ln⁡(limsup⁡h→0⁡(φ(hx)/φ(h)))ln⁡⁡x=lim⁡x→∞ln⁡(limsup⁡h→0⁡(φ(hx)/φ(h)))ln⁡⁡x.
are known as the Matuszewska-Orlicz type lower and upper indices of the function φ(r). Note that in this definition φ(x)needs not to be an N-function: only its behaviour at the origin is of importance. Observe that 0≤m(φ)≤M(φ)≤∞forφ∈W0 and -∞<m(φ)≤M(φ)≤∞forφ∈W̃0, and the formulas are valid(2.5)m[xaφ(x)]=a+m(φ),M[xaφ(x)]=a+M(φ),a∈R1,(2.6)m([φ(x)]a)=am(φ),M([φ(x)]a)=aM(φ),a≥0,(2.7)m(1φ)=-M(φ),M(1φ)=-m(φ),(2.8)m(uv)≥m(u)+m(v),M(uv)≤M(u)+M(v)
for φ,u,v∈W̃0.The following statement is known see [26, Theorems 3.1,  3.2, and 3.5]. (In the formulation of Theorem  2.4 in [26], it was supposed that β≥0,γ>0, and φ∈W0. It is evidently true also for φ∈W̃0 and all β,γ∈ℝ1, in view of formulas (2.5).)Theorem 2.4.
Letφ∈W̃0 and β,γ∈ℝ1. Then
(2.9)φ∈Zβ⟺m(φ)>β,φ∈Zγ⟺M(φ)<γ.
Besides this,
(2.10)m(φ)=sup⁡{δ>0:φ(x)xδisalmostincreasing},(2.11)M(φ)=inf⁡{λ>0:φ(x)xλisalmostdecreasing},
and for φ∈Φγβ the inequalities
(2.12)c1xM(φ)+ɛ≤φ(x)≤c2xm(φ)-ɛ
hold with an arbitrarily small ɛ>0 and c1=c1(ɛ),c2=c2(ɛ).


## 2.1.3. ZBS Classes and MO Indices of Weights at Infinity

Following [14, Section4.1] and [29, Section 2.2], we introduce the following definitions.Definition 2.5.
Let-∞<α<β<∞. One puts Ψαβ:=ℤ̂β∩ℤ̂α, where ℤ̂β is the class of functions φ∈W̃∞ satisfying the condition
(2.13)∫x∞(xt)βφ(t)dtt≤cφ(x),x∈(l,∞),
and ℤ̂α is the class of functions φ∈W([ℓ,∞)) satisfying the condition
(2.14)∫lx(xt)αφ(t)dtt≤cφ(x),x∈(l,∞),
where c=c(φ)>0 does not depend on x∈[ℓ,∞).The indicesm∞(φ) and M∞(φ) responsible for the behavior of functions φ∈Ψαβ([ℓ,∞)) at infinity are introduced in the way similar to (2.4):(2.15)m∞(φ)=sup⁡x>1ln⁡⁡[lim⁡inf⁡h→∞⁡(φ(xh)/φ(h))]ln⁡x,M∞(φ)=inf⁡x>1ln⁡⁡[limsup⁡h→∞⁡(φ(xh)/φ(h))]ln⁡x.Properties of functions in the classΨαβ([ℓ,∞)) are easily derived from those of functions in Φβα([0,ℓ]) because of the following equivalence:(2.16)φ∈Ψαβ([l,∞))⟺φ*∈Φ-α-β([0,l*]),
where φ*(t)=φ(1/t) and ℓ*=1/ℓ∙. Direct calculation shows that(2.17)m∞(φ)=-M(φ*),M∞(φ)=-m(φ*),φ*(t):=φ(1t).Making use of (2.16) and (2.17), one can easily reformulate properties of functions of the class Φαβ near the origin, given in Theorem 2.4 for the case of the corresponding behavior at infinity of functions of the class Ψαβ and obtain that(2.18)c1tm∞(φ)-ε≤φ(t)≤c2tM∞(φ)+ε,t≥l,φ∈W̃∞,(2.19)m∞(φ)=sup⁡{μ∈R1:t-μφ(t)isalmostincreasingon⁡[l,∞)},(2.20)M∞(φ)=inf⁡{ν∈R1:t-νφ(t)isalmostdecreasingon⁡[l,∞)}.We say that a functionφ continuous in (0,∞) is in the class W̃0,∞(ℝ+1) if its restriction to (0,1) belongs to W̃0([0,1]) and its restriction to (1,∞) belongs to W̃∞([1,∞]). For functions in W̃0,∞(ℝ+1), the notation(2.21)Zβ0,β∞(R+1)=Zβ0([0,1])∩Zβ∞([1,∞)),Zγ0,γ∞(R+1)=Zγ0([0,1])∩Zγ∞([1,∞))
has an obvious meaning. In the case, where the indices coincide: β0=β∞:=β, we will simply write ℤβ(ℝ+1) and similarly for ℤγ(ℝ+1). We also denote(2.22)Φαβ(R+1):=Zβ(R+1)∩Zγ(R+1).Making use of Theorem2.4 for Φαβ([0,1]) and relations (2.17), we easily arrive at the following statement.Lemma 2.6.
Letφ∈W̃0(ℝ+1). Then
(2.23)φ∈Zβ0,β∞(R+1)⟺m(φ)>β0,m∞(φ)>β∞,φ∈Zγ0,γ∞(R+1)⟺M(φ)<γ0,M∞(φ)<γ∞.


## 2.2. On ClassesV±μ

Note that we slightly changed the notation of the class introduced in the following definition, in comparison with its notation in [32].Definition 2.7.
Let0<μ≤1. By V±μ, One denotes the classes of functions φ nonnegative on [0,ℓ],0<ℓ≤∞, defined by the following conditions:
(2.24)V+μ:|φ(x)-φ(y)||x-y|μ≤Cφ(x+)x+μ,(2.25)V-μ:|φ(x)-φ(y)||x-y|μ≤Cφ(x-)x+μ,
where x,y∈(0,ℓ] and x+=max(x,y),x-=min(x,y).Lemma 2.8.
Functionsφ∈V+μ are almost increasing on [0,ℓ], and functions φ∈V-μ are almost decreasing on [0,ℓ].Proof.
Letφ∈V+μ and y≤x. By (2.24) we have |φ(x)-φ(y)|≤Cφ(x)(1-y/x)μ≤Cφ(x). Then φ(y)≤|φ(x)-φ(y)|+φ(x)≤(C+1)φ(x). The case φ∈V-μ is similarly treated.Corollary 2.9.
Functionsφ∈V+μ have non-negative indices 0≤m(φ)≤M(φ), and functions φ∈V-μ have non-positive indices m(φ)≤M(φ)≤0, the same being also valid with respect to the indices m∞(φ),M∞(φ) in the case ℓ=∞.Note that(2.26)V+μ⊂V+ν,V-μ⊂V-ν,0<ν<μ≤1,
the classes V±μ being trivial for μ>1. We also have(2.27)xγ∈⋃μ∈[0,1]V+μ⟺γ≥0,xγ∈⋃μ∈[0,1]V-μ⟺γ≤0,
which follows from the fact that xγ∈V+1⇔γ≥0 and xγ∈V-1⇔γ≤0 (see [24, Subsection 2.3, Remark 2.8]) and property (2.26).An example of a function which is inV+μ with some μ>0, but does not belong to the total intersection ⋂μ∈[0,1]V+μ is given by(2.28)φ(x)=axγ+b|x-x0|β∈⋂μ∈[0,β]V+μ,
where x0>0 and γ≥0,0<β<1,a>0,b>0.The following lemmas (see [24], Lemmas 2.10 and 2.11) show that conditions (2.24) and (2.25) are fulfilled with μ=1 not only for power functions but also for an essentially larger class of functions (which in particular may oscillate between two power functions with different exponents). Note that the information about this class in Lemmas 2.10 and 2.11 is given in terms of increasing or decreasing functions, without the word “almost”.Lemma 2.10.
Letφ∈W. Then(i)
φ∈V+1 in the case φ is increasing and the function φ(x)/xν  is decreasing for some ν≥0;(ii)
φ∈V-1 in the case φ(x) is decreasing and there exists a number μ≥0 such that xμφ(x) is increasing.Lemma 2.11.
Letφ∈W∩C1((0,ℓ]). If there exist ɛ>0 and ν≥0 such that 0≤φ′(x)/φ(x)≤ν/x for 0<x≤ɛ, then φ∈V+1. If there exist ɛ>0 and μ≥0 such that -μ/x≤φ′(x)/φ(x)≤0 for 0<x≤ɛ, then φ∈V-1.


## 3. On Weighted Integrability of Functions in Morrey Spaces

### 3.1. Definitions and Belongness of Some Functions to Morrey Spaces

LetΩ be an open set in ℝn.Definition 3.1.
The Morrey spacesℒp,λ(Ω),ℒlocp,λ(Ω)1≤p<∞,0≤λ<n, are defined as the space of functions f∈Llocp(Ω) such that
(3.1)‖f‖p,λ=sup⁡x∈Ω,r>0(1rλ∫B̃(x,r)|f(y)|pdy)1/p,‖f‖p,λ;loc⁡=sup⁡r>0(1rλ∫B̃(0,r)|f(y)|pdy)1/p,
respectively, where B̃(x,r)=B(x,r)∩Ω.Obviously,(3.2)Lp,λ(Ω)⊂L0p,λloc⁡(Ω).The spacesℒp,λ(Ω),ℒlocp,λ(Ω) are known under the names of global and local Morrey spaces; see for instance, [9, 10].The weighted Morrey space is defined as(3.3)Lp,γ(Ω,ω)={f:ωf∈Lp,γ(Ω)}.Remark 3.2.
As is well known, the spaceℒp,λ(Ω) as defined above is not necessarily embedded into Lp(Ω), in the case when Ω is unbounded. A typical counterexample in the case Ω=ℝn is
(3.4)f(x)=|x|(λ-n)/p∈Lp,λ(Rn).
Indeed, we have
(3.5)‖f‖p,λ=sup⁡x,r(1rλ∫B(x,r)|y|λ-ndy)1/p
which is bounded (when |x|≥2r, take into account that |y|≥r, and when |x|≤2r, make use of the inclusion B(x,r)⊂B(0,3r)).Lemma 3.3.
Letℓ=diamΩ<∞, u∈W̃0(0,ℓ), and x0∈Ω¯. The condition
(3.6)m(u)>λ-np
is sufficient for the function f(x)=u(|x-x0|) to belong to ℒp,λ(Ω),0≤λ<n. In the case u(t)=tγ, the inclusion |x-x0|γ∈ℒp,λ(Ω) with 0<λ<n  holds if γ≥(λ-n)/p, the latter condition being necessary, when x0 is an inner point of Ω or n=1 and Ω=(a,b),-∞≤a<b≤∞.Proof.
We have(3.7)‖f‖p,λ=sup⁡x∈Ω,r>0(1rλ∫B̃(x,r)up(|y-x0|)dy)1/p≤sup⁡x∈Ωx0,r>0(1rλ∫B(x,r)up(|y|)dy)1/p,
where Ωx0={x:x+x0∈Ω}. Then
(3.8)‖f‖p,λ≤Csup⁡x∈Ωx0,r>0(1rλ∫B(x,r)|y|pm(u)-pεdy)1/p,
by (2.12). If m(u)>0, we choose 0<ɛ<m(u) and then the right-hand side of the last inequality is bounded. So let m(u)≤0. We distinguish the cases (1) |x|≥2r and (2) |x|≤2r. In the case (1), |y|≥|x|-|x-y|≥r. Therefore,
(3.9)‖f‖p,λ≤Csup⁡x∈Ω,r>0(rpm(u)-pɛrλ∫B(x,r)dy)1/p=Csup⁡rr>0m(u)+(n-λ)/p-ε
which is bounded under the choice ɛ<m(u)+(n-λ)/p. In the case (2), we observe that B(x,r)⊂B(0,3r) and then the same estimate ∥f∥p,λ≤Csupr>0rm(u)+(n-λ)/p-ε follows.
In the caseu(t)=tγ, the proof of the “if” part follows the same lines as above with ɛ=0. To prove the “only if” part, it suffices to observe that
(3.10)‖f‖p,λ≥sup⁡0<r<δ(1rλ∫B(x0,r)|y-x0|pγdy)1/p≥Csup⁡0<r<δrγ+(n-λ)/p.Corollary 3.4.
Ifu∈W̃0(0,ℓ) and there exists an a<(n-λ)/p such that tau(t) is almost increasing, then u(|x-x0|)∈ℒp,λ,x0∈Ω¯.
To derive this corollary from Lemma3.3, it suffices to refer to formula (2.10).


### 3.2. Some Weighted Estimates of Functions in Morrey Spaces

Lemma 3.5.
Let1≤p<∞,0<s≤p,0≤λ<n, and v∈W̃0([0,ℓ]),0<ℓ≤∞. Then
(3.11)(∫|z|<|y||f(z)|sv(|z|)dz)1/s≤cA(|y|)‖f‖p,λ;loc⁡,0<|y|≤l,
where C>0 does not depend on y and f and
(3.12)A(r)=(∫0rtn-1-((n-λ)/p)sdtv(t))1/s
under the assumption that the last integral converges.Proof.
We have(3.13)∫|z|<|y||f(z)|sv(|z|)dz=∑k=0∞∫Bk(y)|f(z)|sv(|z|)dz,
where Bk(y)={z:2-k-1|y|<|z|<2-k|y|}. Making use of the fact that there exists a β such that tβv(t) is almost increasing, we observe that
(3.14)1v(|z|)≤Cv(2-k-1|y|).
Applying this in (3.13) and making use of the Hölder inequality with the exponent p/s≥1, we obtain
(3.15)∫|z|<|y||f(z)|sv(|z|)dz≤C∑k=0∞(2-k-1|y|)n(1-s/p)v(2-k-1|y|)(∫Bk(y)|f(z)|pdz)s/p.
Hence,
(3.16)∫|z|<|y||f(z)|sv(|z|)dz≤C∑k=0∞(2-k-1|y|)n-(n-λ)s/pv(2-k-1|y|)‖f‖p,λ;loc⁡s.
It remains to prove that
(3.17)∑k=0∞(2-k-1|y|)n-(n-λ)s/pv(2-k-1|y|)≤C[A(|y|)]s.
We have
(3.18)∫0|y|tn-1-((n-λ)/p)sdtv(t)=∑k=0∞∫2-k-1|y|2-k|y|tn-1-((n-λ)/p)sdtv(t).
Making use of the fact that tβv(t) is almost increasing with some β, we easily obtain that
(3.19)∫0|y|tn-1-((n-λ)/p)sdtv(t)≥C∑k=0∞(2-k|y|)n-((n-λ)/p)sv(2-k|y|)≥C∑k=0∞(2-k-1|y|)n-((n-λ)/p)sv(2-k-1|y|),
which proves (3.17).Corollary 3.6.
Let1≤p<∞,0<s≤p,0≤λ<n, and a<n/s-(n-λ)/p. Then
(3.20)(∫|t|<|y|(|f(t)||t|a)sdt)1/s≤c|y|n/s-(n-λ)/p-a‖f‖p,λ;loc⁡,0<|y|≤l≤∞.Lemma 3.7.
Let1≤p<∞,0≤s≤p,0≤λ<n, and v∈W̃0(ℝ+1). Then
(3.21)(∫|z|>|y|v(|z|)|f(z)|sdz)1/s≤cB(|y|)‖f‖p,λ;loc⁡,y≠0,
where C>0 does not depend on y and f and
(3.22)B(y)=(∫|y|∞tn-1-((n-λ)/p)sv(t)dt)1/s.Proof.
The proof is similar to that of Lemma3.5. We have
(3.23)∫|z|>|y|v(z)|f(z)|sdz=∑k=0∞∫Bk(y)v(z)|f(z)|sdz,
where Bk(y)={z:2k|y|<|z|<2k+1|y|}. Since there exists a β∈ℝ1 such that tβv(t) is almost increasing, we obtain
(3.24)∫Bk(y)v(|z|)|f(z)|sdz≤C∑k=0∞v(2k+1|y|)∫Bk(y)|f(z)|sdz,
where C may depend on β, but does not depend on y and f. Applying the Hölder inequality with the exponent p/s, we get
(3.25)∫|z|>|y|v(|z|)|f(z)|sdz≤C∑k=0∞v(2k+1|y|)(2k|y|)n(1-s/p)(∫Bk(y)|f(z)|pdz)s/p≤C∑k=0∞v(2k+1|y|)(2k|y|)n-((n-λ)/p)s‖f‖p,λ;loc⁡.
It remains to prove that
(3.26)∑k=0∞v(2k+1|y|)(2k|y|)n-((n-λ)/p)s≤C∫|y|∞tn-1-((n-λ)/p)sv(t)dt.
We have
(3.27)∫|y|∞tn-1-((n-λ)/p)sv(t)dt=∑k=0∞∫2k|y|2k+1|y|tn-1-((n-λ)/p)sv(t)dt≥C∑k=0∞v(2k|y|)∫2k|y|2k+1|y|tn-1-((n-λ)/p)sdt=C∑k=0∞v(2k|y|)(2k|y|)n-((n-λ)/p)s≥C∑k=0∞v(2k+1|y|)(2k|y|)n-((n-λ)/p)s,
which completes the proof.Remark 3.8.
The analysis of the proof shows that estimate (3.21) remains in force, if the assumption v∈W̃0(ℝ+1) is replaced by the condition that 1/v∈W̃0(ℝ+1) and v satisfies the doubling condition v(2t)≤cv(t).Corollary 3.9.
Let1≤p<∞,0<s≤p, and b<(n-λ)/p-n/s. Then
(3.28)(∫|z|>|y|(|z|b|f(z)|)sdz)1/s≤cyb+n/s-(n-λ)/p‖f‖p,λ;loc⁡,y≠0.


## 3.1. Definitions and Belongness of Some Functions to Morrey Spaces

LetΩ be an open set in ℝn.Definition 3.1.
The Morrey spacesℒp,λ(Ω),ℒlocp,λ(Ω)1≤p<∞,0≤λ<n, are defined as the space of functions f∈Llocp(Ω) such that
(3.1)‖f‖p,λ=sup⁡x∈Ω,r>0(1rλ∫B̃(x,r)|f(y)|pdy)1/p,‖f‖p,λ;loc⁡=sup⁡r>0(1rλ∫B̃(0,r)|f(y)|pdy)1/p,
respectively, where B̃(x,r)=B(x,r)∩Ω.Obviously,(3.2)Lp,λ(Ω)⊂L0p,λloc⁡(Ω).The spacesℒp,λ(Ω),ℒlocp,λ(Ω) are known under the names of global and local Morrey spaces; see for instance, [9, 10].The weighted Morrey space is defined as(3.3)Lp,γ(Ω,ω)={f:ωf∈Lp,γ(Ω)}.Remark 3.2.
As is well known, the spaceℒp,λ(Ω) as defined above is not necessarily embedded into Lp(Ω), in the case when Ω is unbounded. A typical counterexample in the case Ω=ℝn is
(3.4)f(x)=|x|(λ-n)/p∈Lp,λ(Rn).
Indeed, we have
(3.5)‖f‖p,λ=sup⁡x,r(1rλ∫B(x,r)|y|λ-ndy)1/p
which is bounded (when |x|≥2r, take into account that |y|≥r, and when |x|≤2r, make use of the inclusion B(x,r)⊂B(0,3r)).Lemma 3.3.
Letℓ=diamΩ<∞, u∈W̃0(0,ℓ), and x0∈Ω¯. The condition
(3.6)m(u)>λ-np
is sufficient for the function f(x)=u(|x-x0|) to belong to ℒp,λ(Ω),0≤λ<n. In the case u(t)=tγ, the inclusion |x-x0|γ∈ℒp,λ(Ω) with 0<λ<n  holds if γ≥(λ-n)/p, the latter condition being necessary, when x0 is an inner point of Ω or n=1 and Ω=(a,b),-∞≤a<b≤∞.Proof.
We have(3.7)‖f‖p,λ=sup⁡x∈Ω,r>0(1rλ∫B̃(x,r)up(|y-x0|)dy)1/p≤sup⁡x∈Ωx0,r>0(1rλ∫B(x,r)up(|y|)dy)1/p,
where Ωx0={x:x+x0∈Ω}. Then
(3.8)‖f‖p,λ≤Csup⁡x∈Ωx0,r>0(1rλ∫B(x,r)|y|pm(u)-pεdy)1/p,
by (2.12). If m(u)>0, we choose 0<ɛ<m(u) and then the right-hand side of the last inequality is bounded. So let m(u)≤0. We distinguish the cases (1) |x|≥2r and (2) |x|≤2r. In the case (1), |y|≥|x|-|x-y|≥r. Therefore,
(3.9)‖f‖p,λ≤Csup⁡x∈Ω,r>0(rpm(u)-pɛrλ∫B(x,r)dy)1/p=Csup⁡rr>0m(u)+(n-λ)/p-ε
which is bounded under the choice ɛ<m(u)+(n-λ)/p. In the case (2), we observe that B(x,r)⊂B(0,3r) and then the same estimate ∥f∥p,λ≤Csupr>0rm(u)+(n-λ)/p-ε follows.
In the caseu(t)=tγ, the proof of the “if” part follows the same lines as above with ɛ=0. To prove the “only if” part, it suffices to observe that
(3.10)‖f‖p,λ≥sup⁡0<r<δ(1rλ∫B(x0,r)|y-x0|pγdy)1/p≥Csup⁡0<r<δrγ+(n-λ)/p.Corollary 3.4.
Ifu∈W̃0(0,ℓ) and there exists an a<(n-λ)/p such that tau(t) is almost increasing, then u(|x-x0|)∈ℒp,λ,x0∈Ω¯.
To derive this corollary from Lemma3.3, it suffices to refer to formula (2.10).


## 3.2. Some Weighted Estimates of Functions in Morrey Spaces

Lemma 3.5.
Let1≤p<∞,0<s≤p,0≤λ<n, and v∈W̃0([0,ℓ]),0<ℓ≤∞. Then
(3.11)(∫|z|<|y||f(z)|sv(|z|)dz)1/s≤cA(|y|)‖f‖p,λ;loc⁡,0<|y|≤l,
where C>0 does not depend on y and f and
(3.12)A(r)=(∫0rtn-1-((n-λ)/p)sdtv(t))1/s
under the assumption that the last integral converges.Proof.
We have(3.13)∫|z|<|y||f(z)|sv(|z|)dz=∑k=0∞∫Bk(y)|f(z)|sv(|z|)dz,
where Bk(y)={z:2-k-1|y|<|z|<2-k|y|}. Making use of the fact that there exists a β such that tβv(t) is almost increasing, we observe that
(3.14)1v(|z|)≤Cv(2-k-1|y|).
Applying this in (3.13) and making use of the Hölder inequality with the exponent p/s≥1, we obtain
(3.15)∫|z|<|y||f(z)|sv(|z|)dz≤C∑k=0∞(2-k-1|y|)n(1-s/p)v(2-k-1|y|)(∫Bk(y)|f(z)|pdz)s/p.
Hence,
(3.16)∫|z|<|y||f(z)|sv(|z|)dz≤C∑k=0∞(2-k-1|y|)n-(n-λ)s/pv(2-k-1|y|)‖f‖p,λ;loc⁡s.
It remains to prove that
(3.17)∑k=0∞(2-k-1|y|)n-(n-λ)s/pv(2-k-1|y|)≤C[A(|y|)]s.
We have
(3.18)∫0|y|tn-1-((n-λ)/p)sdtv(t)=∑k=0∞∫2-k-1|y|2-k|y|tn-1-((n-λ)/p)sdtv(t).
Making use of the fact that tβv(t) is almost increasing with some β, we easily obtain that
(3.19)∫0|y|tn-1-((n-λ)/p)sdtv(t)≥C∑k=0∞(2-k|y|)n-((n-λ)/p)sv(2-k|y|)≥C∑k=0∞(2-k-1|y|)n-((n-λ)/p)sv(2-k-1|y|),
which proves (3.17).Corollary 3.6.
Let1≤p<∞,0<s≤p,0≤λ<n, and a<n/s-(n-λ)/p. Then
(3.20)(∫|t|<|y|(|f(t)||t|a)sdt)1/s≤c|y|n/s-(n-λ)/p-a‖f‖p,λ;loc⁡,0<|y|≤l≤∞.Lemma 3.7.
Let1≤p<∞,0≤s≤p,0≤λ<n, and v∈W̃0(ℝ+1). Then
(3.21)(∫|z|>|y|v(|z|)|f(z)|sdz)1/s≤cB(|y|)‖f‖p,λ;loc⁡,y≠0,
where C>0 does not depend on y and f and
(3.22)B(y)=(∫|y|∞tn-1-((n-λ)/p)sv(t)dt)1/s.Proof.
The proof is similar to that of Lemma3.5. We have
(3.23)∫|z|>|y|v(z)|f(z)|sdz=∑k=0∞∫Bk(y)v(z)|f(z)|sdz,
where Bk(y)={z:2k|y|<|z|<2k+1|y|}. Since there exists a β∈ℝ1 such that tβv(t) is almost increasing, we obtain
(3.24)∫Bk(y)v(|z|)|f(z)|sdz≤C∑k=0∞v(2k+1|y|)∫Bk(y)|f(z)|sdz,
where C may depend on β, but does not depend on y and f. Applying the Hölder inequality with the exponent p/s, we get
(3.25)∫|z|>|y|v(|z|)|f(z)|sdz≤C∑k=0∞v(2k+1|y|)(2k|y|)n(1-s/p)(∫Bk(y)|f(z)|pdz)s/p≤C∑k=0∞v(2k+1|y|)(2k|y|)n-((n-λ)/p)s‖f‖p,λ;loc⁡.
It remains to prove that
(3.26)∑k=0∞v(2k+1|y|)(2k|y|)n-((n-λ)/p)s≤C∫|y|∞tn-1-((n-λ)/p)sv(t)dt.
We have
(3.27)∫|y|∞tn-1-((n-λ)/p)sv(t)dt=∑k=0∞∫2k|y|2k+1|y|tn-1-((n-λ)/p)sv(t)dt≥C∑k=0∞v(2k|y|)∫2k|y|2k+1|y|tn-1-((n-λ)/p)sdt=C∑k=0∞v(2k|y|)(2k|y|)n-((n-λ)/p)s≥C∑k=0∞v(2k+1|y|)(2k|y|)n-((n-λ)/p)s,
which completes the proof.Remark 3.8.
The analysis of the proof shows that estimate (3.21) remains in force, if the assumption v∈W̃0(ℝ+1) is replaced by the condition that 1/v∈W̃0(ℝ+1) and v satisfies the doubling condition v(2t)≤cv(t).Corollary 3.9.
Let1≤p<∞,0<s≤p, and b<(n-λ)/p-n/s. Then
(3.28)(∫|z|>|y|(|z|b|f(z)|)sdz)1/s≤cyb+n/s-(n-λ)/p‖f‖p,λ;loc⁡,y≠0.


## 4. On Weighted Hardy Operators in Morrey Spaces

### 4.1. Pointwise Estimations

We consider the generalized Hardy operators(4.1)Hφαf(x)=|x|α-nφ(|x|)∫|y|<|x|f(t)dtφ(|t|),Hφαf(x)=|x|αφ(|x|)∫Rnf(t)dt|t|nφ(|t|).In the sequelℝn with n=1 may be read either as ℝ1 or ℝ+1 with the operators interpreted as(4.2)Hφαf(x):=xα-1φ(x)∫0xf(y)φ(y)dy,Hφαf(x):=xαφ(x)∫x∞f(y)φ(y)ydy,x>0.In the caseφ(t) is a power function, we also use the notation(4.3)H(γ)αf(x):=|x|γ+α-n∫|y|<|x|f(y)|y|γdy,H(γ)νf(x):=|x|γ+α∫|y|>|x|f(y)|y|γ+ndy
and their one-dimensional versions(4.4)H(γ)αf(x):=xγ+α-1∫0xf(y)yγdy,H(γ)αf(x):=xγ+α∫x∞f(y)yγ+1dy,x>0
adjusted for the half-axis ℝ+1.Lemma 4.1.
Let1≤p<∞ and 0<λ<n.(I)
Letφ∈W̃0. Then the Hardy operator Hφα is defined on the space ℒp,λ(ℝn) or  on the space ℒlocp,λ(ℝn), if and only if(4.5)∫0+tn-1-(n-λ)/pφ(t)dt<∞,and in this case(4.6)|Hφα(x)|≤C|x|α-nφ(|x|)∫0|x|tn-1-(n-λ)/pφ(t)dt‖f‖p,λ;loc⁡.(II)
Let1/φ∈W̃0 or φ∈W̃0 and φ(2t)≤Cφ(t). Then the Hardy operator ℋφα is defined on the space ℒp,λ(ℝn) or on the space ℒlocp,λ(ℝn), if and only if(4.7)∫ε∞tn-1-(n-λ)/pφ(t)dt<∞for everyε>0 and in this case(4.8)|Hφα(x)|≤C|x|αφ(|x|)∫|x|∞tn-1-(n-λ)/pφ(t)dt‖f‖p,λ;loc⁡.Proof.
(I)The “If” Part. The sufficiency of condition (4.5) and estimate (4.6) follow from (3.12) under the choice s=1 and v(t)=φ(t).
The “Only If” Part. We choose a function f(x) equal to |x|(λ-n)/p in a neighborhood of the origin and zero beyond this neighborhood. Then f∈ℒp,λ by Lemma 3.3. For this function f, the existence of the integral Hφαf is equivalent to condition (4.5).
(II)The “If” part. The sufficiency of condition (4.7) and estimate (4.8) follow from (3.21) under the choice s=1 and v(t)=1/tnφ(t).
The “Only If” Part. We choose a function f(x) equal to x(λ-n)/p in a neighborhood of infinity and zero beyond this neighborhood. Then f∈ℒp,λ by Remark 3.2. For this function f, the existence of the integral ℋφαf is nothing else but condition (4.7).Corollary 4.2.
s
(I) The Hardy operatorH(γ)α is defined on the space ℒp,λ(ℝn) or on the space ℒlocp,λ(ℝn), if and only if γ<n/p′+λ/p, and in this case
(4.9)|H(γ)αf(x)|≤C|x|α-(n-λ)/p‖f‖p,λ;loc⁡. (II) The Hardy operator ℋ(γ)α is defined on the space ℒp,λ(ℝn) or on the space ℒlocp,λ(ℝn), if and only if γ>λ-n/p, and in this case
(4.10)|H(γ)αf(x)|≤C|x|α-(n-λ)/p‖f‖p,λ;loc⁡.


### 4.2. Weightedp→q-Estimates for Hardy Operators in Morrey Spaces

The statements of Theorem4.3 are well known in the case of Lebesgue space λ=0 when 1<p<n/α; see, for instance, [33, p. 6, 54]. As can be seen from the results below, inequalities for the Hardy operators in Morrey spaces admit the case p=1 when λ>0.


#### 4.2.1. The Case of Power Weights

Theorem 4.3.
Let0<λ<n,0<α<n-λ, and 1≤p<(n-λ)/α. The operator H(γ)α (ℋ(γ)α, resp.) is bounded from ℒp,λ(ℝn) or ℒlocp,λ(ℝn) to ℒq,λ(ℝn), where 1/q=1/p-α/(n-λ), if and only if γ<n/p′+λ/p(γ>(λ-n)/p, resp.).Proof.
The “only if” part follows from Corollary4.2 and the “if part” from (4.9) and (4.10), since |x|α-(n-λ)/p=|x|(n-λ)/p∈ℒq,λ(ℝn); see Remark 3.2.


#### 4.2.2. The Case of General Weights

We first deal with Hardy operators on a ballB(0,ℓ),0<ℓ<∞ of a finite radius ℓ.Theorem 4.4.
Let0<λ<n,0<α<n-λ, and 1≤p<(1-λ)/α and φ∈W̃0. Then the weighted Hardy operators Hφα and ℋφα are bounded from ℒp,λ(B(0,ℓ)) or ℒlocp,λ(B(0,ℓ)) to  ℒq,λ(B(0,ℓ)), where 1/q=1/p-α/(n-λ), if
(4.11)φ∈Zλ/p+n/p′,φ∈Z(λ-n)/p,
respectively, or, equivalently,
(4.12)M(φ)<λp+np′fortheoperatorHφα,(4.13)m(φ)>λ-npfortheoperatorHφα.
The conditions
(4.14)m(φ)≤λp+np′,M(φ)≥λp-np
are necessary for the boundedness of the operators Hφα and ℋφα, respectively.Proof.
By (2.10) and (2.11), the function φ(t)/tm(φ)-ε is almost increasing, while φ(t)/tM(φ)+ε  is almost decreasing for every ɛ>0. Consequently,
(4.15)C1rm(φ)-εtm(φ)-ε≤φ(r)φ(t)≤C2rM(φ)+εtM(φ)+ε
for 0<t≤r and then
(4.16)C1|x|m(φ)-ε+α-n∫B(0,|x|)f(y)dt|y|m(φ)-ε≤Hφαf(x)≤C2|x|M(φ)+ε+α-n∫B(0,|x|)f(y)dy|y|M(φ)+ε
supposing that f(y)≥0. From the right-hand side inequality in (4.16) and Theorem 4.3, we obtain that the operator Hφα is bounded if M(φ)+ɛ<λ/p+1/p′, which is satisfied under the choice of ε>0 sufficiently small, the latter being possible by (4.12). It remains to recall that condition (4.12) is equivalent to the assumption φ∈ℤλ/p+1/p′ by Theorem 2.4. The necessity of the condition m(φ)≤λ/p+n/p′ follows from the left-hand side inequality in (4.16). The case of the operator ℋφ is similarly treated.In the case of the whole space (ℓ=∞), we admit that the weight φ(|x|) may have an “oscillation between power functions” different at the origin and infinity. Correspondingly, the behavior at the origin and infinity is characterized by different indices m(φ),M(φ) and m∞(φ),M∞(φ), as described in Section 2.1.3.Theorem 4.5.
Let0<λ<n,0<α<n-λ, and 1≤p<(1-λ)/α and φ∈W̃0,∞(ℝ+1). Then the weighted Hardy operators Hφα and ℋφα are bounded from ℒp,λ(ℝn) or ℒlocp,λ(ℝn) to ℒq,λ(ℝn),1/q=1/p-α/(n-λ), if
(4.17)φ∈Zλ/p+n/p′(R+1)φ∈Z(λ-n)/p(R+1),
respectively, or, equivalently,
(4.18)max⁡(M(φ),M∞(φ))<λp+np′fortheoperatorHφα,min⁡(m(φ),m∞(φ))>λ-npfortheoperatorHφα.
The conditions
(4.19)max⁡(m(φ),m∞(φ))≤λp+np′,min⁡(M(φ),M∞(φ))≥λp-np
are necessary for the boundedness of the operators Hφα and ℋφα, respectively.Proof.
The restriction ofHφαf(x) to B(0,1) is covered by Theorem 4.4, so that it suffices to estimate ∥Hφαf∥ℒp,λ(ℝn∖B(0,1)). For |x|>1 we have
(4.20)Hφαf(x)=C(f)|x|α-nφ(|x|)+|x|α-n∫1<|y|<|x|φ(|x|)φ(|y|)f(y)dy,
where C(f)=∫B(0,1)f(y)/φ(|y|)dy. By Lemma 3.5 we have |C(f)|≤c∥f∥p,λ;loc∫01tn-1-(1-λ)/pdt/φ(t), where the integral converges since φ(t)≥CtM(φ)+ɛ with an arbitrarily small ɛ>0 and (n-λ)/p+M(φ)<1. Then
(4.21)|C(f)xα-nφ(x)|≤cxα-n+M∞(φ)+ε‖f‖p,λ;loc⁡
by (2.18). Here xα-n+M∞(φ)+ε∈ℒq,λ(1,∞), since α-1+M∞(φ)+ε<(λ-n)/q for sufficiently small ε; see Remark 3.2.
To deal with the second term in (4.20), it suffices to observe that for 1≤|y|≤|x|<∞ we have inequality (4.15) with m(φ),M(φ) replaced by m∞(φ),M∞(φ) and then the proof follows the same lines as in Theorem (4.4) after formula (4.16).
The operatorℋφα is considered in a similar way.


## 4.1. Pointwise Estimations

We consider the generalized Hardy operators(4.1)Hφαf(x)=|x|α-nφ(|x|)∫|y|<|x|f(t)dtφ(|t|),Hφαf(x)=|x|αφ(|x|)∫Rnf(t)dt|t|nφ(|t|).In the sequelℝn with n=1 may be read either as ℝ1 or ℝ+1 with the operators interpreted as(4.2)Hφαf(x):=xα-1φ(x)∫0xf(y)φ(y)dy,Hφαf(x):=xαφ(x)∫x∞f(y)φ(y)ydy,x>0.In the caseφ(t) is a power function, we also use the notation(4.3)H(γ)αf(x):=|x|γ+α-n∫|y|<|x|f(y)|y|γdy,H(γ)νf(x):=|x|γ+α∫|y|>|x|f(y)|y|γ+ndy
and their one-dimensional versions(4.4)H(γ)αf(x):=xγ+α-1∫0xf(y)yγdy,H(γ)αf(x):=xγ+α∫x∞f(y)yγ+1dy,x>0
adjusted for the half-axis ℝ+1.Lemma 4.1.
Let1≤p<∞ and 0<λ<n.(I)
Letφ∈W̃0. Then the Hardy operator Hφα is defined on the space ℒp,λ(ℝn) or  on the space ℒlocp,λ(ℝn), if and only if(4.5)∫0+tn-1-(n-λ)/pφ(t)dt<∞,and in this case(4.6)|Hφα(x)|≤C|x|α-nφ(|x|)∫0|x|tn-1-(n-λ)/pφ(t)dt‖f‖p,λ;loc⁡.(II)
Let1/φ∈W̃0 or φ∈W̃0 and φ(2t)≤Cφ(t). Then the Hardy operator ℋφα is defined on the space ℒp,λ(ℝn) or on the space ℒlocp,λ(ℝn), if and only if(4.7)∫ε∞tn-1-(n-λ)/pφ(t)dt<∞for everyε>0 and in this case(4.8)|Hφα(x)|≤C|x|αφ(|x|)∫|x|∞tn-1-(n-λ)/pφ(t)dt‖f‖p,λ;loc⁡.Proof.
(I)The “If” Part. The sufficiency of condition (4.5) and estimate (4.6) follow from (3.12) under the choice s=1 and v(t)=φ(t).
The “Only If” Part. We choose a function f(x) equal to |x|(λ-n)/p in a neighborhood of the origin and zero beyond this neighborhood. Then f∈ℒp,λ by Lemma 3.3. For this function f, the existence of the integral Hφαf is equivalent to condition (4.5).
(II)The “If” part. The sufficiency of condition (4.7) and estimate (4.8) follow from (3.21) under the choice s=1 and v(t)=1/tnφ(t).
The “Only If” Part. We choose a function f(x) equal to x(λ-n)/p in a neighborhood of infinity and zero beyond this neighborhood. Then f∈ℒp,λ by Remark 3.2. For this function f, the existence of the integral ℋφαf is nothing else but condition (4.7).Corollary 4.2.
s
(I) The Hardy operatorH(γ)α is defined on the space ℒp,λ(ℝn) or on the space ℒlocp,λ(ℝn), if and only if γ<n/p′+λ/p, and in this case
(4.9)|H(γ)αf(x)|≤C|x|α-(n-λ)/p‖f‖p,λ;loc⁡. (II) The Hardy operator ℋ(γ)α is defined on the space ℒp,λ(ℝn) or on the space ℒlocp,λ(ℝn), if and only if γ>λ-n/p, and in this case
(4.10)|H(γ)αf(x)|≤C|x|α-(n-λ)/p‖f‖p,λ;loc⁡.


## 4.2. Weightedp→q-Estimates for Hardy Operators in Morrey Spaces

The statements of Theorem4.3 are well known in the case of Lebesgue space λ=0 when 1<p<n/α; see, for instance, [33, p. 6, 54]. As can be seen from the results below, inequalities for the Hardy operators in Morrey spaces admit the case p=1 when λ>0.


### 4.2.1. The Case of Power Weights

Theorem 4.3.
Let0<λ<n,0<α<n-λ, and 1≤p<(n-λ)/α. The operator H(γ)α (ℋ(γ)α, resp.) is bounded from ℒp,λ(ℝn) or ℒlocp,λ(ℝn) to ℒq,λ(ℝn), where 1/q=1/p-α/(n-λ), if and only if γ<n/p′+λ/p(γ>(λ-n)/p, resp.).Proof.
The “only if” part follows from Corollary4.2 and the “if part” from (4.9) and (4.10), since |x|α-(n-λ)/p=|x|(n-λ)/p∈ℒq,λ(ℝn); see Remark 3.2.


### 4.2.2. The Case of General Weights

We first deal with Hardy operators on a ballB(0,ℓ),0<ℓ<∞ of a finite radius ℓ.Theorem 4.4.
Let0<λ<n,0<α<n-λ, and 1≤p<(1-λ)/α and φ∈W̃0. Then the weighted Hardy operators Hφα and ℋφα are bounded from ℒp,λ(B(0,ℓ)) or ℒlocp,λ(B(0,ℓ)) to  ℒq,λ(B(0,ℓ)), where 1/q=1/p-α/(n-λ), if
(4.11)φ∈Zλ/p+n/p′,φ∈Z(λ-n)/p,
respectively, or, equivalently,
(4.12)M(φ)<λp+np′fortheoperatorHφα,(4.13)m(φ)>λ-npfortheoperatorHφα.
The conditions
(4.14)m(φ)≤λp+np′,M(φ)≥λp-np
are necessary for the boundedness of the operators Hφα and ℋφα, respectively.Proof.
By (2.10) and (2.11), the function φ(t)/tm(φ)-ε is almost increasing, while φ(t)/tM(φ)+ε  is almost decreasing for every ɛ>0. Consequently,
(4.15)C1rm(φ)-εtm(φ)-ε≤φ(r)φ(t)≤C2rM(φ)+εtM(φ)+ε
for 0<t≤r and then
(4.16)C1|x|m(φ)-ε+α-n∫B(0,|x|)f(y)dt|y|m(φ)-ε≤Hφαf(x)≤C2|x|M(φ)+ε+α-n∫B(0,|x|)f(y)dy|y|M(φ)+ε
supposing that f(y)≥0. From the right-hand side inequality in (4.16) and Theorem 4.3, we obtain that the operator Hφα is bounded if M(φ)+ɛ<λ/p+1/p′, which is satisfied under the choice of ε>0 sufficiently small, the latter being possible by (4.12). It remains to recall that condition (4.12) is equivalent to the assumption φ∈ℤλ/p+1/p′ by Theorem 2.4. The necessity of the condition m(φ)≤λ/p+n/p′ follows from the left-hand side inequality in (4.16). The case of the operator ℋφ is similarly treated.In the case of the whole space (ℓ=∞), we admit that the weight φ(|x|) may have an “oscillation between power functions” different at the origin and infinity. Correspondingly, the behavior at the origin and infinity is characterized by different indices m(φ),M(φ) and m∞(φ),M∞(φ), as described in Section 2.1.3.Theorem 4.5.
Let0<λ<n,0<α<n-λ, and 1≤p<(1-λ)/α and φ∈W̃0,∞(ℝ+1). Then the weighted Hardy operators Hφα and ℋφα are bounded from ℒp,λ(ℝn) or ℒlocp,λ(ℝn) to ℒq,λ(ℝn),1/q=1/p-α/(n-λ), if
(4.17)φ∈Zλ/p+n/p′(R+1)φ∈Z(λ-n)/p(R+1),
respectively, or, equivalently,
(4.18)max⁡(M(φ),M∞(φ))<λp+np′fortheoperatorHφα,min⁡(m(φ),m∞(φ))>λ-npfortheoperatorHφα.
The conditions
(4.19)max⁡(m(φ),m∞(φ))≤λp+np′,min⁡(M(φ),M∞(φ))≥λp-np
are necessary for the boundedness of the operators Hφα and ℋφα, respectively.Proof.
The restriction ofHφαf(x) to B(0,1) is covered by Theorem 4.4, so that it suffices to estimate ∥Hφαf∥ℒp,λ(ℝn∖B(0,1)). For |x|>1 we have
(4.20)Hφαf(x)=C(f)|x|α-nφ(|x|)+|x|α-n∫1<|y|<|x|φ(|x|)φ(|y|)f(y)dy,
where C(f)=∫B(0,1)f(y)/φ(|y|)dy. By Lemma 3.5 we have |C(f)|≤c∥f∥p,λ;loc∫01tn-1-(1-λ)/pdt/φ(t), where the integral converges since φ(t)≥CtM(φ)+ɛ with an arbitrarily small ɛ>0 and (n-λ)/p+M(φ)<1. Then
(4.21)|C(f)xα-nφ(x)|≤cxα-n+M∞(φ)+ε‖f‖p,λ;loc⁡
by (2.18). Here xα-n+M∞(φ)+ε∈ℒq,λ(1,∞), since α-1+M∞(φ)+ε<(λ-n)/q for sufficiently small ε; see Remark 3.2.
To deal with the second term in (4.20), it suffices to observe that for 1≤|y|≤|x|<∞ we have inequality (4.15) with m(φ),M(φ) replaced by m∞(φ),M∞(φ) and then the proof follows the same lines as in Theorem (4.4) after formula (4.16).
The operatorℋφα is considered in a similar way.


## 4.2.1. The Case of Power Weights

Theorem 4.3.
Let0<λ<n,0<α<n-λ, and 1≤p<(n-λ)/α. The operator H(γ)α (ℋ(γ)α, resp.) is bounded from ℒp,λ(ℝn) or ℒlocp,λ(ℝn) to ℒq,λ(ℝn), where 1/q=1/p-α/(n-λ), if and only if γ<n/p′+λ/p(γ>(λ-n)/p, resp.).Proof.
The “only if” part follows from Corollary4.2 and the “if part” from (4.9) and (4.10), since |x|α-(n-λ)/p=|x|(n-λ)/p∈ℒq,λ(ℝn); see Remark 3.2.


## 4.2.2. The Case of General Weights

We first deal with Hardy operators on a ballB(0,ℓ),0<ℓ<∞ of a finite radius ℓ.Theorem 4.4.
Let0<λ<n,0<α<n-λ, and 1≤p<(1-λ)/α and φ∈W̃0. Then the weighted Hardy operators Hφα and ℋφα are bounded from ℒp,λ(B(0,ℓ)) or ℒlocp,λ(B(0,ℓ)) to  ℒq,λ(B(0,ℓ)), where 1/q=1/p-α/(n-λ), if
(4.11)φ∈Zλ/p+n/p′,φ∈Z(λ-n)/p,
respectively, or, equivalently,
(4.12)M(φ)<λp+np′fortheoperatorHφα,(4.13)m(φ)>λ-npfortheoperatorHφα.
The conditions
(4.14)m(φ)≤λp+np′,M(φ)≥λp-np
are necessary for the boundedness of the operators Hφα and ℋφα, respectively.Proof.
By (2.10) and (2.11), the function φ(t)/tm(φ)-ε is almost increasing, while φ(t)/tM(φ)+ε  is almost decreasing for every ɛ>0. Consequently,
(4.15)C1rm(φ)-εtm(φ)-ε≤φ(r)φ(t)≤C2rM(φ)+εtM(φ)+ε
for 0<t≤r and then
(4.16)C1|x|m(φ)-ε+α-n∫B(0,|x|)f(y)dt|y|m(φ)-ε≤Hφαf(x)≤C2|x|M(φ)+ε+α-n∫B(0,|x|)f(y)dy|y|M(φ)+ε
supposing that f(y)≥0. From the right-hand side inequality in (4.16) and Theorem 4.3, we obtain that the operator Hφα is bounded if M(φ)+ɛ<λ/p+1/p′, which is satisfied under the choice of ε>0 sufficiently small, the latter being possible by (4.12). It remains to recall that condition (4.12) is equivalent to the assumption φ∈ℤλ/p+1/p′ by Theorem 2.4. The necessity of the condition m(φ)≤λ/p+n/p′ follows from the left-hand side inequality in (4.16). The case of the operator ℋφ is similarly treated.In the case of the whole space (ℓ=∞), we admit that the weight φ(|x|) may have an “oscillation between power functions” different at the origin and infinity. Correspondingly, the behavior at the origin and infinity is characterized by different indices m(φ),M(φ) and m∞(φ),M∞(φ), as described in Section 2.1.3.Theorem 4.5.
Let0<λ<n,0<α<n-λ, and 1≤p<(1-λ)/α and φ∈W̃0,∞(ℝ+1). Then the weighted Hardy operators Hφα and ℋφα are bounded from ℒp,λ(ℝn) or ℒlocp,λ(ℝn) to ℒq,λ(ℝn),1/q=1/p-α/(n-λ), if
(4.17)φ∈Zλ/p+n/p′(R+1)φ∈Z(λ-n)/p(R+1),
respectively, or, equivalently,
(4.18)max⁡(M(φ),M∞(φ))<λp+np′fortheoperatorHφα,min⁡(m(φ),m∞(φ))>λ-npfortheoperatorHφα.
The conditions
(4.19)max⁡(m(φ),m∞(φ))≤λp+np′,min⁡(M(φ),M∞(φ))≥λp-np
are necessary for the boundedness of the operators Hφα and ℋφα, respectively.Proof.
The restriction ofHφαf(x) to B(0,1) is covered by Theorem 4.4, so that it suffices to estimate ∥Hφαf∥ℒp,λ(ℝn∖B(0,1)). For |x|>1 we have
(4.20)Hφαf(x)=C(f)|x|α-nφ(|x|)+|x|α-n∫1<|y|<|x|φ(|x|)φ(|y|)f(y)dy,
where C(f)=∫B(0,1)f(y)/φ(|y|)dy. By Lemma 3.5 we have |C(f)|≤c∥f∥p,λ;loc∫01tn-1-(1-λ)/pdt/φ(t), where the integral converges since φ(t)≥CtM(φ)+ɛ with an arbitrarily small ɛ>0 and (n-λ)/p+M(φ)<1. Then
(4.21)|C(f)xα-nφ(x)|≤cxα-n+M∞(φ)+ε‖f‖p,λ;loc⁡
by (2.18). Here xα-n+M∞(φ)+ε∈ℒq,λ(1,∞), since α-1+M∞(φ)+ε<(λ-n)/q for sufficiently small ε; see Remark 3.2.
To deal with the second term in (4.20), it suffices to observe that for 1≤|y|≤|x|<∞ we have inequality (4.15) with m(φ),M(φ) replaced by m∞(φ),M∞(φ) and then the proof follows the same lines as in Theorem (4.4) after formula (4.16).
The operatorℋφα is considered in a similar way.


## 5. Application to Potential Operators

We consider the potential operator(5.1)Iαf(x):=∫Rnf(y)dy|x-y|n-α,0<α<n,
and in Theorem 5.3 show that its weighted boundedness in Morrey spaces—in the case of weights φ∈V+μ∪V-μ with μ=min{1,n-α}—is a consequence of the nonweighted boundedness due to Adams [5] and the weighted boundedness of Hardy operators provided by Theorem 4.5.The necessity of the boundedness of the Hardy operators for that of potential operators is a consequence of the following simple fact, whereX=X(ℝn) and Y=Y(ℝn) are arbitrary Banach function spaces in the sense of Luxemburg (cf., e.g., [34]).Lemma 5.1.
Letw=w(x) be any weight function. For the boundedness of the weighted potential operator  wIα(1/w) from  X to  Y, it is necessary that the Hardy operators  Hwα and  ℋwαα are bounded from  X to  Y, where  wα(x)=|x|-αw(x).The proof of the sufficiency of the obtained conditions is based on the pointwise estimate of the following lemma.Lemma 5.2.
Letw∈V-μ∪V+μ with  μ=min{1,n-α} be a weight and  f a non-negative function. Then the following pointwise estimate holds:
(5.2)wIα1wf(x)≤Iαf(x)+c{Hwαf(x)+H-ααf(x),ifw∈V+μ,Hαf(x)+Hwααf(x),ifw∈V-μ.Proof.
We have(5.3)(wIα1w-Iα)f(x)=∫RnK(x,y)f(y)dy,K(x,y)=w(|x|)-w(|y|)w(|y|)|x-y|n-α.
We first consider the case n-α≤1. For w∈V-μ∪V+μ with μ=n-α in this case, by the definition of the classes V±n-α, we have
(5.4)K(x,y)≤c{|x|α-nw(|x|)w(|y|),|y|<|x|,|y|α-n,|y|>|x|,whenw∈V+n-α,K(x,y)≤c{|x|α-n,|y|<|x|,|y|α-nw(|x|)w(|y|),|y|>|x|,whenw∈V-n-α,
which yield
(5.5)|(wIα1w-Iα)f(x)|≤c{Hwαf(x)+H-ααf(x),ifw∈V+n-α,Hαf(x)+Hwααf(x),ifw∈V-n-α,
with ℋ-αα=ℋwα|w≡|x|-α and prove (5.2).
Let nown-α>1. We denote n-α=m+{n-α}, where m=[n-α] and {n-α} stands for the fractional part of n-α. Now
(5.6)w∈V-1∪V+1⊂V-{n-α}∪V+{n-α}.
The procedure is similar to the previous case; we can first manage with the fractional part{n-α}, treating w as a function in V-{n-α}∪V+{n-α} like in the previous case, and then repeat a similar procedure m times treating w as a function in V-1∪V+1.
For definiteness we consider the case wherew∈V+1; the case of w∈V-1 is similarly treated. By the definition of the class V+{n-α}, we have
(5.7)|K(x,y)|≤C|x-y|m{w(|x|)w(|y|)|x|-{n-α},|y|<|x|,|y|-{n-α},|y|>|x|
(this step should be omitted when n-α is an integer), that is,
(5.8)|K(x,y)|≤K+(x,y)+K-(x,y),
where
(5.9)K+(x,y)=Cw(|x|)w(|y|)⋅θ(|x|-|y|)|x|{n-α}|x-y|m,K-(x,y)=Cθ(|y|-|x|)|y|{n-α}|x-y|m≤C2{n-α}|x-y|n-α
and θ(t)=χℝ+1(t). We only have to take care about the kernel K+(x,y). We have
(5.10)K+(x,y)=Cw(|x|)-w(|y|)w(|y|)⋅θ(|x|-|y|)|x|{n-α}|x-y|m+Cθ(|x|-|y|)|x|{n-α}|x-y|m.
We make use of the fact that w∈V+1 and obtain
(5.11)K+(x,y)≤Cw(|x|)w(|y|)⋅θ(|x|-|y|)|x|{n-α}+1|x-y|m-1+C|x-y|n-α,
where again only the first term must be studied. We repeat the same procedure m-1 times more and finally arrive at the kernel
(5.12)w(|x|)w(|y|)⋅θ(|x|-|y|)|x|{n-α}+m=|x|α-nw(|x|)w(|y|)⋅θ(|x|-|y|),
which is the kernel of the Hardy operator Hwα.We are ready for the following statement, where notation (2.22) is used.Theorem 5.3.
Let0<α<n,  0≤λ<n, and  1<p<(n-λ)/α. (i)
Letφ∈W̃0,∞(ℝ+1)∩(V-μ∪V+μ) with μ=min(1,n-α). Then the condition(5.13)φ∈Φn/p′+λ/pα+(λ-n)/p(R+1),or equivalently(5.14)α-n-λp<min(m(φ),m∞(φ)),max⁡(M(φ),M∞(φ))<np′+λp,is sufficient for the boundedness of the potential operator (5.1) from the weighted space ℒp,λ(ℝ+1,φ) to the space ℒq,λ(ℝ+1,φ), where 1/q=1/p-α/(n-λ).(ii)
Letφ∈W̃0,∞(ℝ+1). Then the condition(5.15)α-n-λp≤min⁡(M(φ),M∞(φ)),max⁡(m(φ),m∞(φ))≤n/p′+λ/pis necessary for the boundedness of the potential operator (5.1) from ℒp,λ(ℝ+1,φ) to ℒq,λ(ℝ+1,φ).Proof.
The necessity part (ii) follows from Lemma5.1 and Theorem 4.5.
Part (i). We have to prove the boundedness of the operatorφIα1/φ from ℒp,λ(ℝ+1) to ℒq,λ(ℝ+1). Since the non-weighted ℒp,λ(ℝ+1)→ℒq,λ(ℝ+1)-boundedness of the potential operator Iα is known [5], it suffices to show the boundedness of the operator φIα1/φ-Iα. For that it remains to make use of Theorem 4.5. This completes the proof.Corollary 5.4.
Let0≤λ<n,0<α<n-λ, 1<p<(n-λ)/α, and ϱ(x)=|x-x0|γ,x0∈ℝ1. Then the potential operator (5.1) is bounded from ℒp,λ(ℝn,ϱ) into ℒq,λ(ℝn,ϱ), 1/q=1/p-α/(n-λ), if and only if
(5.16)α-np<γ-λp<np′.Remark 5.5.
As can be seen from the proof of Theorem5.3, its statement remains valid under the condition
(5.17)φ∈{Zα+(λ-n)/p,ifφ∈V+μ,Zn/p′+λ/p(R+1),ifφ∈V-μ,μ=min⁡(1,n-α),
more general than (5.13). Correspondingly, condition (5.14) may be written in a more general form:
(5.18)max⁡(M(φ),M∞(φ))<np′+λp,ifφ∈V+μ,α-n-λp<min(m(φ),m∞(φ)),ifφ∈V-μ.
(Recall that min(M(φ),M∞(φ))≥0 in the case φ∈V+μ and max(m(φ),m∞(φ))≤0 in the case φ∈V-μ; see Corollary 2.9.)



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*Source: 678171-2012-02-09.xml*