# Weighted Hardy-Type Inequalities in Variable Exponent Morrey-Type Spaces

Authors: Dag Lukkassen; Lars-Erik Persson; Stefan Samko; Peter Wall
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We study thep·→q· boundedness of weighted multidimensional
Hardy-type operators Hwα· and ℋwα· of variable order αx, with radial weight wx, from a variable exponent locally generalized
Morrey space ℒp·,φ·ℝn,w to another ℒq·,ψ·ℝn,w. The exponents are assumed
to satisfy the decay condition at
the origin and infinity. We construct certain functions, defined by p, α, and φ, the belongness of which to the
resulting space ℒq·,ψ·ℝn,w is sufficient for such a boundedness. Under additional assumptions on φ/w, this condition is also necessary. We also give
the boundedness conditions in
terms of Zygmund-type integral inequalities for the functions φ and φ/w.


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## Body

## 1. Introduction

Influenced by various applications, for instance, mechanics of the continuum medium and variational problems, in the last two decades the study of various mathematical problems in the spaces withnonstandard growth attracts the attention of researchers in various fields. This notion relates first of all to the generalized Lebesgue spaces Lp(·)(Ω), Ω⊆ℝn, known also as Lebesgue spaces with variable exponent p(x). We refer to the existing books [1–3] in the field.Thisvariable exponent boom naturally touched Morrey spaces. Morrey spaces ℒp,λ (with constant exponents) in its classical version were introduced in [4] in relation to the study of partial differential equations and presented in various books; see, for example, [5–7]; we refer also to a recent overview of Morrey spaces in [8], where various generalizations of Morrey spaces may be also found.They were widely investigated during the last decades, including the study of classical operators of harmonic analysis, maximal, singular, and potential operators on Morrey spaces, and their generalizations were studied. We refer for instance to papers [9–16] and the references therein; in particular, Hardy operators in Morrey type spaces with constant p were studied in [17–20].The Morrey spacesLp(·),λ(·)(Ω) with variable exponents λ(·) and p(·) were introduced and studied in [21–23]. Generalized Morrey spaces ℳp(·),ω(·)(Ω),  Ω⊆ℝn with variable exponents were studied in [24]; see also another version of Morrey-type spaces in [25]; we also refer to [26] for the so-called complementary Morrey spaces of variable order in the spirit of ideas of [24].In the above cited paper maximal, singular, and potential operators were studied. This paper seems to be the first one where Hardy-type integral inequalities are studied in Morrey-type spaces with variable exponents. Concerning Hardy-type inequalities and related problems and applications, we refer to the books [27, 28].The paper is organized as follows. In Section2, we give necessary preliminaries on variable exponent Lebesgue spaces. In Section 3, we define our main object-variable exponent Morrey spaces and prove important weighted estimates of functions in Morrey spaces; see Theorem 10.By means of these estimates in Section4, we prove our main statements for Hardy operators in variable exponent generalized Morrey spaces. We also consider the necessity of the obtained conditions. In Section 4.4, under some additional assumptions on φ(0,r) we obtain the boundedness conditions in a different form via Zygmund-type conditions on φ(0,r)/w(r) and provide a direct relation between φ(0,r) and ψ(0,r).In Theorem18 of this section, we specially single out the nonweighted case where we show that the Hardy inequalities in Morrey spaces are completely determined by the values p(0) and p(∞).In the appendix we recall some notions related to the Bary-Zygmund-Stechkin class and Matuszewska-Orlicz indices which sporadically are used in the paper.


## 2. Preliminaries on Variable Exponent Lebesgue Spaces

We first recall the basic definitions related to variable exponent spaces. ByΩ, we always denote an open set in ℝn, Ω⊆ℝn, and ℓ=diamΩ, 0<ℓ≤∞. Let also B(x,r)={y∈ℝn:|x-y|<r} and B~(x,r)=B(x,r)∩Ω.Letp(·) be a measurable function on Ω with values in [1,∞). We suppose that
(1)1≤p-≤p(x)≤p+<∞,
where p-:=infx∈Ωp(x)>1,  p+:=supx∈Ωp(x)<∞.We denote byLp(·)(Ω) the space of all measurable functions f(x) on Ω such that Ip(·)(f)=∫Ω‍|f(x)|p(x)dx<∞. Equipped with the norm ∥f∥p(·)=inf{η>0:Ip(·)(f/η)≤1}, this is a Banach function space. For the basics on variable exponent Lebesgue spaces, we refer to [2, 29, 30].We denote byp′(·)=p(x)/(p(x)-1), x∈Ω, the conjugate exponent of p(x). The notation ℙlog(Ω) will stand for the set of variable exponents p(·) satisfying condition (1) and the local log-condition
(2)|p(x)-p(y)|≤A-ln|x-y|,|x-y|≤12x,y∈Ω,
where A=A(p)>0 does not depend on x,y.We will use also the following decay conditions:(3)|p(x)-p(0)|≤A0|ln|x||,|x|≤12,(4)|p(x)-p∞|≤A∞|ln|x||,|x|≥2.For brevity, by𝒫0,∞(Ω) we denote the set of bounded measurable functions (not necessarily with values in [1,∞)), which satisfy the decay conditions (3) and (4).Letp satisfy the log-condition (2). The inequality
(5)∥χB(x,r)∥Lp(·)(Ω)≤Crn/p(x),
for bounded open sets Ω is known and proved in [31]. For unbounded sets, if, besides (2), the exponent p satisfies the decay condition (3), then we have
(6)∥χB(x,r)∥Lp(·)(Ω)≤Crn/pr(x),
where
(7)pr(x):={p(x),if0<r<1,p(∞),ifr≥1.
See [2, Corollary  4.5.9]. The following lemma was proved in [32], Lemma 2, and a simpler proof given in [33].Lemma 1.
Letp(x) be a bounded function. Then
(8)1c0rn/p(0)≤∥χB(0,Dr)∖B(0,r)∥p(·)≤c0rn/p(0)for0<r≤a<∞,1c∞rn/p∞≤∥χB(0,Dr)∖B(0,r)∥p(·)≤c∞rn/p∞for0<b≤r<∞,
under conditions (3) or (4), respectively, where D>1 and c0≥1 and c∞≥1 depend on D and a,b, but do not depend on r.We will use a consequence of the estimates of Lemma1 in the form
(9)∥χB(0,2r)∖B(0,r)∥p(·)≤c0rn/p*(r),r∈ℝ+,
where we denoted
(10)p*(r)={p(0),r≤1p(∞),r≥1
for brevity.We refer to the appendix for the definition of the classesW¯(ℝ+) and W_(ℝ+) used in the following lemma.Lemma 2.
Letp∈𝒫0,∞(ℝn) and 1≤p-≤p+<∞ and a function a(x,r) belongs to W¯(ℝ+)∩W_(ℝ+) with respect to r uniformly in x. Then
(11)∥a(x,r)χB(0,r)∥p(·)≤C∫0rtn/p*(t)a(x,t)dtt,r>0,
where C>0 does not depend on r and x.Proof.
LetBk(0,r):=B(0,2-kr)∖B(0,2-k-1r). We have
(12)∥a(x,r)χB(0,r)∥p(·)≤∑k=0∞∥a(x,r)χBk(0,r)∥p(·).
Since a∈W¯, we obtain
(13)∥a(x,r)χB(0,r)∥p(·)≤C∑k=0∞‍a(x,2-kr)∥χB(x,2-kr)∥p(·).
By (9), we have
(14)∥χB(0,2-kr)∥p(·)≤c(2-kr)n/p*(2-kr).
Therefore,
(15)∥a(x,r)χB(0,r)∥p(·)≤C∑k=0∞‍a(x,2-kr)(2-kr)n/p*(2-kr)≤Cln2∫0rtn/p*(t)a(x,t)dtt,
and we arrive at (11). The last passage to the integral is verified in the standard way with the use of the monotonicity properties of the function tn/p*(t)a(x,t) in t, imposed by the assumptions of the lemma on a(x,t) as
(16)∫0rtn/p*(t)a(x,t)dtt=∑k=0∞‍∫2-k-1r2-krtn/p*(t)a(x,t)dtt≥∑k=0∞‍a(x,2-kr)(2-kr)+n/p*(2-kr)∫2-k-1r2-krdtt=ln2∑k=0∞‍a(x,2-kr)(2-kr)n/p*(2-kr).The classℤβ0,β∞(ℝ+1) used in the following corollary is defined in (A.18).Corollary 3.
Leta(x,·) belong to the class ℤβ0,β∞(ℝ+1) uniformly in x, where β0=n/p(0) and β∞=n/p(∞). Then
(17)∥a(x,r)χB(0,r)∥p(·)≤Crn/p*(r)a(x,r),r>0.Proof.
The statement follows from (11) by the definition of the class ℤβ0,β∞(ℝ+1).Corollary 4.
Letp∈𝒫0,∞(ℝn) and 1≤p-≤p+<∞ and a bounded function ν(x) satisfies the conditions
(18)infx∈ℝn[n+ν(x)p(0)],infx∈ℝn[n+ν(x)p(∞)]>0.
Then
(19)∥|y|ν(x)χB(0,r)∥p(·)≤crν(x)+(n/p*(r)),r>0,
where c>0 does not depend on r and x.Proof.
The statement follows directly from (11) with a(x,r)=rν(x), since ∫0r‍tν(x)+(n/p*(t))(dt/t)=rν(x)+(n/p(0))/(ν(x)+(n/p(0))) when 0<r≤1 and ∫0r‍tν(x)+(n/p*(t))(dt/t)=1/(ν(x)+(n/p(0)))+(rν(x)+(n/p(∞))-1)/(ν(x)+(n/p(∞))) when r≥1.


## 3. Variable Exponent Morrey Spaces

### 3.1. Definitions and Some Auxiliary Results for Variable Exponent Morrey Spaces

Letφ(r) be a nonnegative function on [0,ℓ], positive on (0,ℓ]. Morrey type spaces, called also generalized Morrey spaces, with constant p,1≤p<∞ are known in two versions: global ℒp,φ(Ω) and local ℒloc;x0p,φ(Ω) (we refer, for instance, to the survey paper [8]) and are defined as the spaces of functions f∈Llocp(Ω) such that
(20)supx∈Ω,r>0∥f∥Lp(B~(x,r))φ(r)<∞,supr>0∥f∥Lp(B~(x0,r))φ(r)<∞,
respectively, where x0∈Ω.Morrey spaces with variable exponentp(x) corresponding to the classical case φ(r)=rλ/p, but with variable λ(x) as well, were introduced and studied in [21]. More general approach admitting the variable function φ(x,r) were studied in [24, 25].Following [24], we introduce the variable exponent Morrey-type space by the definition below, but note that our notation differs from that of [24].Definition 5.
Let1≤p-≤p+<∞ and let φ(x,t) be a non-negative function almost increasing in r uniformly in x∈Ω. The generalized variable exponent Morrey space ℒp(·),φ(·)(Ω) is defined by the norm
(21)∥f∥ℒp(·),φ(·)=supx∈Ω,r>01φ(x,r)∥f∥Lp(·)(B~(x,r)).
We will also refer to the spaceℒp(·),φ(·)(Ω) as global generalized variable exponent Morrey space in contrast to its local version ℒx0;locp(·),φ(·)(Ω) defined by the norm
(22)∥f∥ℒx0;locp(·),φ(·)=supr>01φ(x0,r)∥f∥Lp(·)(B~(x0,r)),
where x0∈Ω.
Forw a weight function on Ω, the weighted Morrey space ℒp,φ(Ω,w) is defined by Lp,φ(Ω,w):={f:wf∈Lp,φ(Ω)}.
By the definition of the norm in the variable exponent Lebesgue space, we the can also write that(23)∥f∥ℒp(·),φ(·)=supx∈Ω,r>0inf{λ=λ(x,r)>0:∫B~(x,r)|f(y)λφ(x,r)|p(y)dy≤1}.
From which one can see that for bounded exponents p one has
(24)f∈ℒp(·),φ(·)(Ω)⟺supx,r∫B~(x,r)|f(y)φ(x,r)|p(y)dy<∞.
The following lemma provides some minimal assumptions on the functionφ(x,r) under which the so-defined spaces contain “nice” functions.Lemma 6.
Let≤p-≤p+<∞. Under the decay condition
(25)|p(x)-p(x0)|≤A0|ln|x-x0||,|x-x0|≤12,
the assumption
(26)supr>0[min{1,r}]n/p(x0)φ(x0,r)<∞
is sufficient for bounded functions f with compact support (in the case of unbounded set Ω)  to belong to the local Morrey space ℒx0;locp(·),φ(·)(Ω). Similarly under the log-condition (2), the condition
(27)supx∈Ω,r>0[min{1,r}]n/p(x)φ(x,r)<∞
guarantees that such functions belong to the global Morrey space ℒp(·),φ(·)(Ω).Proof.
Use (19) and (5), respectively.Everywhere in the sequel we assume that the assumptions (26) and (27) for the spaces ℒx0;locp(·),φ(·)(Ω) and ℒp(·),φ(·)(Ω), respectively, are satisfied.We need the following lemma on variable exponent powers of functions in Bary-Stechkin class. For this class, Matuszewska-Orlicz indices, and all the related notation, we refer to the appendix.Lemma 7.
Let1≤p-≤p+<∞, φ∈W¯([0,ℓ])∩W_([0,ℓ]), and p satisfy the decay condition (3). Then
(28)1Cφp(0)(r)≤φp(y)(r)≤Cφp(0)(r),for|y|≤r≤ℓ,
where ℓ<∞ and C≥1 do not depend on r and y.Proof.
We have to prove that1/C≤[φ(r)]p(y)-p(0)≤C; that is,
(29)|p(y)-p(0)|·|lnφ(r)|≤C1(=lnC).
It suffices to consider the case r≤1. By Theorem A.4, the assumption φ∈W¯([0,ℓ])∩W_([0,ℓ]) implies that the function φ has finite indices m(φ) and M(φ), and (A.11) holds. Bounds in (A.11) yield the inequality |lnφ(r)|≤c1+c2|lnr| with some positive c1 and c2. Then (29) follows from the decay condition at the origin, since ln(1/r)≤ln(1/|y|).In papers [17, 18], there were given various conditions for radial type functions to belong to Morrey spaces with nonvariable characteristics. The reader can easily adjust them for the case when they are variable. We do not dwell on this, but in the next lemma we give a certain example of a function in the space ℒ0;locp(·),φ(·)(ℝn), important for our further goals.Lemma 8.
Let≤p-≤p+<∞ and p satisfy the decay condition (3) at the origin and φ(0,·)∈W¯([0,ε0])∩W_([0,ε0]) for some ε0>0, and
(30)∫0hφ(0,t)tdt≤cφ(0,h)
for small h and some c>0. Then the function f0(x)=(φ(0,|x|)/|x|n/p(x))χB(0,R)(x), where 0<R<∞, belongs to ℒ0;locp(·),φ(·)(ℝn).
If additionally we suppose thatp satisfies the decay condition (4) at infinity, φ(0,·)∈W¯([N,∞))∩W_([N,∞)) for some N>0, the inequality (30) holds also for large h,φ(0,r)≥c>0 for large r→∞ and p-=p(∞); then the same holds with R=∞.Proof.
We have to check that(31)supr∫B(0,r)|f0(y)φ(0,r)|p(y)dy=sup0<r<R∫B(0,r)|φ(0,|y|)φ(0,r)|p(y)dy|y|n<∞.
By Lemma 7, this is guaranteed by the condition
(32)supr1φp(0)(0,r)∫0rφp(0)(0,t)dtt<∞.
The latter is equivalent to the condition m(φp(0)(0,t))>0, that is, m(φ(0,t))>0, which in its turn is equivalent to (30) and consequently holds.
In the case ofR=∞, the proof follows the same lines. This time instead of (31) it suffices only to check that
(33)supr>N∫B(0,r)|φ(0,|y|)φ(0,r)|p(y)dy|y|n<∞
for some large N>0. Here φ(0,|y|)p(y)~φ(0,|y|)p(∞) by the decay condition at infinity imposed on p. Therefore,
(34)supr>N∫B(0,r)|φ(0,|y|)φ(0,r)|p(y)dy|y|n≤supr>N∫B(0,r)|φ(0,|y|)φ(0,r)|p(∞)dy|y|n,
after which the arguments are similar to those for the case R<∞.Corollary 9.
Let1≤p-≤p+<∞ and p satisfy the decay conditions (3) and (4), φ(0,·)∈W¯(ℝ+)∩W_(ℝ+), and the inequality (30) holds for all h∈ℝ+, φ(0,r)≥c>0 for large r→∞ and p-=p(∞); then
(35)φ(0,|x|)|x|n/p*(|x|)∈ℒ0;locp(·),φ(·)(ℝn).


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

Theorem 10.
Let1≤p-≤p+<∞ and p∈𝒫0,∞(ℝn). Suppose also that u∈W_(ℝ+),φ(0,·)/u(·)∈W_(ℝ+) and v∈W¯(ℝ+1),v(·)φ(0,·)∈W¯(ℝ+1). Then
(36)∫|z|<r|f(z)|u(|z|)dz≤c𝒜(r)∥f∥ℒ0;locp(·),φ(·),∫|z|>rv(|z|)|f(z)|dz≤cℬ(r)∥f∥ℒ0;locp(·),φ(·)
with
(37)𝒜(r)=∫0rt(n/p*(t))-1φ(0,t)u(t)dt,ℬ(r)=∫r∞t(n/p*′(t))-1φ(0,t)v(t)dt,
where r∈ℝ+ and c>0 do not depend on r and f.Proof.
We have(38)∫|z|<r|f(z)|u(|z|)dz=∑k=0∞‍∫Bk(y)|f(z)|u(|z|)dz,
where Bk(y)={z:2-k-1r<|z|<2-kr}. Making use of the fact that there exists a β such that tβu(t) is almost decreasing, we observe that 1/u(|z|)≤C/u(2-kr) on Bk(y). Applying this in (38) and making use of the Hölder inequality with the exponent p(x), we obtain
(39)∫|z|<r|f(z)|u(|z|)dz≤∑k=0∞‍∥χBk∥Lp′(·)(Bk)u(2-kr)·∥f∥Lp(·)(B(0,2-kr)).
By (9), we have ∥χBk∥Lp′(·)(Bk)≤C(2-kr)n/p*(2-kr), so that
(40)∫|z|<r|f(z)|u(|z|)dz≤C∑k=0∞‍(2-kr)n/p*′(2-kr)φ(0,2-kr)u(2-kr)∥f∥ℒ0;locp(·),φ(·).
It remains to prove that(41)∑k=0∞‍(2-kr)n/p*′(2-kr)φ(0,2-kr)u(2-kr)≤C𝒜(r).
We have𝒜(r)=∑k=0∞‍∫2-k-1r2-kr‍t(n/p*′(t))-1(φ(t)/u(t))dt. Since the function tn/p*′(t) is increasing for all t∈ℝ+ and the function φ(0,t)/tbu(t) is almost decreasing with some b, we obtain
(42)𝒜(r)≥C∑k=0∞‍(2-kr)n/p*′(2-kr)φ(0,2-kr)u(2-kr)≥C,
which proves (41) and completes the proof of the first inequality in (36).
For the second inequality in (36), we proceed in a similar way as
(43)∫|z|>rv(t)|f(z)|dz=∑k=0∞‍∫Bk(y)v(z)|f(z)|dz,
where Bk(y)={z:2kr<|z|<2k+1r}. Since there exists a β∈ℝ such that tβv(t) is almost increasing, we obtain
(44)∑k=0∞‍∫Bk(y)v(|z|)|f(z)|dz≤C∑k=0∞‍v(2k+1r)∫Bk(y)|f(z)|dz.
Applying the Hölder inequality with the variable exponent p(·) and taking (9) into account, we get
(45)∫|z|>rv(|z|)|f(z)|dz≤C∑k=0∞‍v(2k+1r)∥χBk(y)∥p′(·)∥fχB(0,2k+1r)∥p(·)≤C∑k=1∞‍v(2kr)(2kr)n/p*′(2kr)φ(0,2kr)·∥f∥ℒ0;locp(·),φ(·).
It remains to prove that ∑k=1∞‍v(2kr)(2kr)n/p*′(2kr)φ(2kr)≤Cℬ(r), which easily follows by the monotonicity of the involved functions as
(46)ℬ(r)=∑k=0∞∫2kr2k+1rtn/(p*′(t)-1)v(t)φ(t)dt‍≥C∑k=0∞‍v(2kr)φ(2kr)(2kr)n/p*′(2kr).


## 3.1. Definitions and Some Auxiliary Results for Variable Exponent Morrey Spaces

Letφ(r) be a nonnegative function on [0,ℓ], positive on (0,ℓ]. Morrey type spaces, called also generalized Morrey spaces, with constant p,1≤p<∞ are known in two versions: global ℒp,φ(Ω) and local ℒloc;x0p,φ(Ω) (we refer, for instance, to the survey paper [8]) and are defined as the spaces of functions f∈Llocp(Ω) such that
(20)supx∈Ω,r>0∥f∥Lp(B~(x,r))φ(r)<∞,supr>0∥f∥Lp(B~(x0,r))φ(r)<∞,
respectively, where x0∈Ω.Morrey spaces with variable exponentp(x) corresponding to the classical case φ(r)=rλ/p, but with variable λ(x) as well, were introduced and studied in [21]. More general approach admitting the variable function φ(x,r) were studied in [24, 25].Following [24], we introduce the variable exponent Morrey-type space by the definition below, but note that our notation differs from that of [24].Definition 5.
Let1≤p-≤p+<∞ and let φ(x,t) be a non-negative function almost increasing in r uniformly in x∈Ω. The generalized variable exponent Morrey space ℒp(·),φ(·)(Ω) is defined by the norm
(21)∥f∥ℒp(·),φ(·)=supx∈Ω,r>01φ(x,r)∥f∥Lp(·)(B~(x,r)).
We will also refer to the spaceℒp(·),φ(·)(Ω) as global generalized variable exponent Morrey space in contrast to its local version ℒx0;locp(·),φ(·)(Ω) defined by the norm
(22)∥f∥ℒx0;locp(·),φ(·)=supr>01φ(x0,r)∥f∥Lp(·)(B~(x0,r)),
where x0∈Ω.
Forw a weight function on Ω, the weighted Morrey space ℒp,φ(Ω,w) is defined by Lp,φ(Ω,w):={f:wf∈Lp,φ(Ω)}.
By the definition of the norm in the variable exponent Lebesgue space, we the can also write that(23)∥f∥ℒp(·),φ(·)=supx∈Ω,r>0inf{λ=λ(x,r)>0:∫B~(x,r)|f(y)λφ(x,r)|p(y)dy≤1}.
From which one can see that for bounded exponents p one has
(24)f∈ℒp(·),φ(·)(Ω)⟺supx,r∫B~(x,r)|f(y)φ(x,r)|p(y)dy<∞.
The following lemma provides some minimal assumptions on the functionφ(x,r) under which the so-defined spaces contain “nice” functions.Lemma 6.
Let≤p-≤p+<∞. Under the decay condition
(25)|p(x)-p(x0)|≤A0|ln|x-x0||,|x-x0|≤12,
the assumption
(26)supr>0[min{1,r}]n/p(x0)φ(x0,r)<∞
is sufficient for bounded functions f with compact support (in the case of unbounded set Ω)  to belong to the local Morrey space ℒx0;locp(·),φ(·)(Ω). Similarly under the log-condition (2), the condition
(27)supx∈Ω,r>0[min{1,r}]n/p(x)φ(x,r)<∞
guarantees that such functions belong to the global Morrey space ℒp(·),φ(·)(Ω).Proof.
Use (19) and (5), respectively.Everywhere in the sequel we assume that the assumptions (26) and (27) for the spaces ℒx0;locp(·),φ(·)(Ω) and ℒp(·),φ(·)(Ω), respectively, are satisfied.We need the following lemma on variable exponent powers of functions in Bary-Stechkin class. For this class, Matuszewska-Orlicz indices, and all the related notation, we refer to the appendix.Lemma 7.
Let1≤p-≤p+<∞, φ∈W¯([0,ℓ])∩W_([0,ℓ]), and p satisfy the decay condition (3). Then
(28)1Cφp(0)(r)≤φp(y)(r)≤Cφp(0)(r),for|y|≤r≤ℓ,
where ℓ<∞ and C≥1 do not depend on r and y.Proof.
We have to prove that1/C≤[φ(r)]p(y)-p(0)≤C; that is,
(29)|p(y)-p(0)|·|lnφ(r)|≤C1(=lnC).
It suffices to consider the case r≤1. By Theorem A.4, the assumption φ∈W¯([0,ℓ])∩W_([0,ℓ]) implies that the function φ has finite indices m(φ) and M(φ), and (A.11) holds. Bounds in (A.11) yield the inequality |lnφ(r)|≤c1+c2|lnr| with some positive c1 and c2. Then (29) follows from the decay condition at the origin, since ln(1/r)≤ln(1/|y|).In papers [17, 18], there were given various conditions for radial type functions to belong to Morrey spaces with nonvariable characteristics. The reader can easily adjust them for the case when they are variable. We do not dwell on this, but in the next lemma we give a certain example of a function in the space ℒ0;locp(·),φ(·)(ℝn), important for our further goals.Lemma 8.
Let≤p-≤p+<∞ and p satisfy the decay condition (3) at the origin and φ(0,·)∈W¯([0,ε0])∩W_([0,ε0]) for some ε0>0, and
(30)∫0hφ(0,t)tdt≤cφ(0,h)
for small h and some c>0. Then the function f0(x)=(φ(0,|x|)/|x|n/p(x))χB(0,R)(x), where 0<R<∞, belongs to ℒ0;locp(·),φ(·)(ℝn).
If additionally we suppose thatp satisfies the decay condition (4) at infinity, φ(0,·)∈W¯([N,∞))∩W_([N,∞)) for some N>0, the inequality (30) holds also for large h,φ(0,r)≥c>0 for large r→∞ and p-=p(∞); then the same holds with R=∞.Proof.
We have to check that(31)supr∫B(0,r)|f0(y)φ(0,r)|p(y)dy=sup0<r<R∫B(0,r)|φ(0,|y|)φ(0,r)|p(y)dy|y|n<∞.
By Lemma 7, this is guaranteed by the condition
(32)supr1φp(0)(0,r)∫0rφp(0)(0,t)dtt<∞.
The latter is equivalent to the condition m(φp(0)(0,t))>0, that is, m(φ(0,t))>0, which in its turn is equivalent to (30) and consequently holds.
In the case ofR=∞, the proof follows the same lines. This time instead of (31) it suffices only to check that
(33)supr>N∫B(0,r)|φ(0,|y|)φ(0,r)|p(y)dy|y|n<∞
for some large N>0. Here φ(0,|y|)p(y)~φ(0,|y|)p(∞) by the decay condition at infinity imposed on p. Therefore,
(34)supr>N∫B(0,r)|φ(0,|y|)φ(0,r)|p(y)dy|y|n≤supr>N∫B(0,r)|φ(0,|y|)φ(0,r)|p(∞)dy|y|n,
after which the arguments are similar to those for the case R<∞.Corollary 9.
Let1≤p-≤p+<∞ and p satisfy the decay conditions (3) and (4), φ(0,·)∈W¯(ℝ+)∩W_(ℝ+), and the inequality (30) holds for all h∈ℝ+, φ(0,r)≥c>0 for large r→∞ and p-=p(∞); then
(35)φ(0,|x|)|x|n/p*(|x|)∈ℒ0;locp(·),φ(·)(ℝn).


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

Theorem 10.
Let1≤p-≤p+<∞ and p∈𝒫0,∞(ℝn). Suppose also that u∈W_(ℝ+),φ(0,·)/u(·)∈W_(ℝ+) and v∈W¯(ℝ+1),v(·)φ(0,·)∈W¯(ℝ+1). Then
(36)∫|z|<r|f(z)|u(|z|)dz≤c𝒜(r)∥f∥ℒ0;locp(·),φ(·),∫|z|>rv(|z|)|f(z)|dz≤cℬ(r)∥f∥ℒ0;locp(·),φ(·)
with
(37)𝒜(r)=∫0rt(n/p*(t))-1φ(0,t)u(t)dt,ℬ(r)=∫r∞t(n/p*′(t))-1φ(0,t)v(t)dt,
where r∈ℝ+ and c>0 do not depend on r and f.Proof.
We have(38)∫|z|<r|f(z)|u(|z|)dz=∑k=0∞‍∫Bk(y)|f(z)|u(|z|)dz,
where Bk(y)={z:2-k-1r<|z|<2-kr}. Making use of the fact that there exists a β such that tβu(t) is almost decreasing, we observe that 1/u(|z|)≤C/u(2-kr) on Bk(y). Applying this in (38) and making use of the Hölder inequality with the exponent p(x), we obtain
(39)∫|z|<r|f(z)|u(|z|)dz≤∑k=0∞‍∥χBk∥Lp′(·)(Bk)u(2-kr)·∥f∥Lp(·)(B(0,2-kr)).
By (9), we have ∥χBk∥Lp′(·)(Bk)≤C(2-kr)n/p*(2-kr), so that
(40)∫|z|<r|f(z)|u(|z|)dz≤C∑k=0∞‍(2-kr)n/p*′(2-kr)φ(0,2-kr)u(2-kr)∥f∥ℒ0;locp(·),φ(·).
It remains to prove that(41)∑k=0∞‍(2-kr)n/p*′(2-kr)φ(0,2-kr)u(2-kr)≤C𝒜(r).
We have𝒜(r)=∑k=0∞‍∫2-k-1r2-kr‍t(n/p*′(t))-1(φ(t)/u(t))dt. Since the function tn/p*′(t) is increasing for all t∈ℝ+ and the function φ(0,t)/tbu(t) is almost decreasing with some b, we obtain
(42)𝒜(r)≥C∑k=0∞‍(2-kr)n/p*′(2-kr)φ(0,2-kr)u(2-kr)≥C,
which proves (41) and completes the proof of the first inequality in (36).
For the second inequality in (36), we proceed in a similar way as
(43)∫|z|>rv(t)|f(z)|dz=∑k=0∞‍∫Bk(y)v(z)|f(z)|dz,
where Bk(y)={z:2kr<|z|<2k+1r}. Since there exists a β∈ℝ such that tβv(t) is almost increasing, we obtain
(44)∑k=0∞‍∫Bk(y)v(|z|)|f(z)|dz≤C∑k=0∞‍v(2k+1r)∫Bk(y)|f(z)|dz.
Applying the Hölder inequality with the variable exponent p(·) and taking (9) into account, we get
(45)∫|z|>rv(|z|)|f(z)|dz≤C∑k=0∞‍v(2k+1r)∥χBk(y)∥p′(·)∥fχB(0,2k+1r)∥p(·)≤C∑k=1∞‍v(2kr)(2kr)n/p*′(2kr)φ(0,2kr)·∥f∥ℒ0;locp(·),φ(·).
It remains to prove that ∑k=1∞‍v(2kr)(2kr)n/p*′(2kr)φ(2kr)≤Cℬ(r), which easily follows by the monotonicity of the involved functions as
(46)ℬ(r)=∑k=0∞∫2kr2k+1rtn/(p*′(t)-1)v(t)φ(t)dt‍≥C∑k=0∞‍v(2kr)φ(2kr)(2kr)n/p*′(2kr).


## 4. On Weighted Hardy Operators in Generalized Morrey Spaces

### 4.1. Pointwise Estimations

We consider the following generalized Hardy operators:(47)Hwα(·)f(x)=|x|α(x)-nw(|x|)∫|y|<|x|f(y)dyw(|y|),ℋwα(·)f(x)=|x|α(x)w(|x|)∫|y|>|x|,f(y)dy|y|nw(|y|),
where α(x) is a non-negative measurable function on ℝn. In the one-dimensional case, their versions
(48)Hwα(·)f(x)=xα(x)-1w(x)∫0xf(t)dtw(t),ℋwα(·)f(x)=xα(x)w(x)∫x∞f(t)dttw(t),x>0
on the half-axis ℝ+1 may be also admitted, so that the sequel ℝn with n=1 may be read either as ℝ1 or ℝ+1.We also use the notationHα(·)=Hwα(·)|w≡1.Our next result on the boundedness of weighted Hardy operators presented in Theorem13 is prepared by our estimations in Theorem 10. It is in fact a consequence of Theorem 10. We find it useful to divide this consequence into two parts. First, in Theorem 11, we reformulate Theorem 10 in the form to emphasize that we have pointwise estimates of Hardy operatorsHwα(·) and ℋwα(·) in terms of the Morrey norm of the function f. Then as an immediate consequence of Theorem 11 we formulate Theorem 13 for global Morrey spaces.Theorem 11.
Letp∈𝒫0,∞(ℝn) and 1≤p-≤p+<∞. Let also the weight w satisfy the conditions
(49)w∈W_(ℝ+),φ(0,·)w(·)∈W_(ℝ+),
in the case of the operator Hwα(·), and the conditions
(50)1w∈W¯(ℝ+),φ(0,·)w(·)∈W¯(ℝ+).
in the case of the operator ℋwα(·). The conditions
(51)∫0εt(n/p′(0))-1φ(0,t)w(t)dt<∞,∫ε∞t-(n/p(∞))-1φ(0,t)w(t)dt<∞,
with ε>0, are sufficient for the Hardy operators Hwα(·) and ℋwα(·), respectively, to be defined on the space ℒ0;locp(·),φ(·)(ℝn). Under these conditions,
(52)|Hwα(·)f(x)|≤CKw,α(x)∥f∥ℒ0;locp(·),φ(·),|ℋwα(·)f(x)|≤C𝒦w,α(x)∥f∥ℒ0;locp(·),φ(·),
where
(53)Kw,α(x)=|x|α(x)-nw(|x|)𝒜(|x|)=|x|α(x)-nw(|x|)∫0|x|t(n/p*′(t))-1φ(0,t)w(t)dt,𝒦w,α(x)=|x|α(x)w(|x|)∫|x|∞t-(n/p*(t))-1φ(0,t)w(t)dt.Proof.
The sufficiency of the conditions in (51) and estimates in (52) follows from (36) under the choice v(t)=w(t), for the operator Hwα(·) and v(t)=1/tnw(t) for the operator ℋwα(·).


### 4.2. On the Necessity of the Conditions in (51)

Observe that the conditions in (51) are natural in the sense that they are necessary under some additional assumptions on the function φ defining the Morrey space.Lemma 12.
Letp be as in Theorem 11 and φ(0,·)∈W¯([0,ε0])∩W_([0,ε0]) for some ε0>0, and (30) holds. Then the conditions in (51) are necessary for the Hardy operators Hα(·)W and ℋα(·)W, respectively, to be defined on the space ℒ0;locp(·),φ(·)(ℝn).Proof.
Choosef0(x)=(φ(0,|x|)/|x|n/p(x))χB(0,R)(x). Then f∈ℒ0;locp(·),φ(·)(ℝn) by Lemma 8. It remains to note that the conditions in (51) are nothing else but the statement that Hwα(·)f0 and ℋwα(·)f0, respectively, exist.


### 4.3. Weightedp→q Norm Estimates for Hardy Operators

The statements of Theorem13 are well known in the case of Lebesgue space, that is, in the case φ≡1, with constant exponents, when 1<p<n/α; see for instance [27, pages 6, 54]. For the classical Morrey spaces ℒp,λ(ℝn) with constant exponents p and λ, statements of such type for Hardy operators have been obtained in [17, 19].Hardy inequalities in the variable exponent Lebesgue spaces were studied in [34–36]; see also the references therein.Note that, in contrast to variable exponent Lebesgue spaces, inequalities for the Hardy operators in Morrey spaces admit the caseinfxp(x)=1 when φ(0,0)=0 in the case of local Morrey spaces and supxφ(x,0)=0 in the case of global Morrey spaces.We suppose that the condition(54)supr>0[min{1,r}]n/p(0)φ(0,r)<∞
holds, which ensures that the space ℒ0,locp(·),φ(·)(ℝn) is nonempty by Lemma 6.Theorem 13.
Let1≤p-≤p+<∞, 1≤q-≤q+<∞ and p∈𝒫0,∞(Ω) as well as the functions φ and ψ satisfy the assumption (54). Let also the weight w satisfy the conditions in (49) in the case of the operator Hwα(·) and the conditions in (50) in the case of the operator ℋwα(·). Then the operators Hwα(·) and ℋwα(·) are bounded from ℒ0,locp(·),φ(·)(ℝn) to ℒ0,locq(·),ψ(·)(ℝn), if
(55)Kw,α∈ℒloc;0q(·),ψ(·)(ℝn),𝒦w,α∈ℒloc;0q(·),ψ(·)(ℝn),
respectively. If p and φ(0,r) satisfy the assumptions of Corollary 9, then the conditions in (55) are also necessary for the boundedness of the operators Hwα(·) and ℋwα(·).Proof.
The sufficiency of the conditions in (55) for the boundedness follows from the estimates in (52).
As regards the necessity, the requirements in (55) are nothing else but the statement that
(56)Hwα(·)(f0)∈ℒloc;0q(·),ψ(·)(ℝn),ℋwα(·)(f0)∈ℒloc;0q(·),ψ(·)(ℝn),
respectively, where
(57)f0(x)=φ(0,|x|)|x|n/p*(|x|).
The functionf0 belongs to ℒ0,locp(·),φ(·)(ℝn) by Corollary 9. Consequently, the conditions, (55) are necessary.Corollary 14.
Under the same assumptions onp,φ and w as in Theorem 13, the Hardy operators Hwα(·) and ℋwα(·) are bounded from the global Morrey space ℒp(·),φ(·)(ℝn) to the global space ℒq(·),ψ(·)(ℝn), if
(58)Kw,α∈ℒq,ψ(ℝn),𝒦w,α∈ℒq,ψ(ℝn),
respectively.Proof.
Since∥f∥ℒ0,locp(·),φ(·)(ℝn)≤∥f∥ℒp(·),φ(·), the statement immediately follows from the pointwise estimates in (52).Remark 15.
Theorem18 is specifically a “Morrey-type” statement in the sense that the case of Lebesgue spaces (the case λ=0) is not included. This, in particular, is reflected in the admission of values p(x)=1 in Theorem 18, which is impossible for Lebesgue spaces.


### 4.4. Findingψ(0,r) by a Given φ(0,r)

The main theorem of the preceding section, Theorem13 on the ℒ0,locp(·),φ(·)(ℝn)→ℒ0,locq(·),ψ(·)(ℝn) boundedness, provides a relation between the given function φ(0,r) and ψ(0,r) in an indirect form, via the conditions in (55). In the theorems below, under some additional assumptions on the functions φ(0,r) we obtain the boundedness conditions in a form of Zygmund-type integral conditions imposed on φ(0,r)/w(r) and give a direct relation between φ(0,r) and ψ(0,r).In these theorems, we use the following assumptions on the functionφ(0,t) defining the data space ℒ0,locp(·),φ(·)(ℝn):
(59)φ(0,r)∈W_(ℝ+)∩ℤβ~0,β~∞(ℝ+),
where
(60)β~0=np(0)-nq(0)-α(0),β~∞=np(∞)-nq(∞)-α(∞)
(see (A.18) for the definition of the classes ℤβ0,β∞(ℝ+)). Recall that the assumption φ(0,r)∈ℤβ~0,β~∞(ℝ+) in (59) may be equivalently rewritten in terms of the Matuszewska-Orlicz indices m(φ),m∞(φ) of the function φ(0,t) as
(61)m(φ)>β~0,m∞(φ)>β~∞.Theorem 16.
Let1≤p-≤p+<∞, 1≤q-≤q+<∞, and p,q∈𝒫0,∞(ℝn) as well as the function φ satisfy the assumption in (54) and (59) and let the weight w fulfill the conditions in (49). If
(62)φ(0,r)w(r)∈ℤβ0,β∞(ℝ+)withβ0=-np′(0),φ(0,r)w(r)∈ℤβ0,β∞(ℝ+)withβ∞=-np′(∞),
then the Hardy operator Hwα(·) is bounded from ℒ0,locp(·),φ(·)(ℝn) to ℒ0,locq(·),ψ(·)(ℝn), with
(63)ψ(0,r)=rα*(r)-(n/p*(r))+(n/q*(r))φ(0,r).Proof.
By (69), we have
(64)Kw,α(x)=|x|α(x)-nw(|x|)∫0|x|t(n/p*′(t))-1φ(0,t)w(t)dt≤C|x|α*(|x|)-(n/p*(|x|))φ(0,|x|).
To check that Kw,α∈ℒloc;0q(·),φ(·)(ℝn), by the definition in (20), we have to estimate the norm
(65)∥|x|α*(|x|)-(n/p*(|x|))φ(0,|x|)χB(0,r)∥Lq(·).
We apply Lemma 2 with a(x,r)=rα*(r)-(n/p*(r))φ(0,r), which is possible by (59) and obtain
(66)∥|x|α*(|x|)-(n/p*(|x|))φ(0,|x|)χB(0,r)∥Lq(·)≤C∫0rtα*(t)-(n/p*(t))+(n/q*(t))φ(0,t)dtt.
Then by (59) we get
(67)∥|x|α*(|x|)-(n/p*(|x|))φ(0,|x|)χB(0,r)∥Lq(·)≤Crα*(r)-(n/p*(r))+(n/q*(r))φ(0,r).
Therefore,
(68)∥|x|α*(|x|)-(n/p*(|x|))φ(0,|x|)∥ℒloc;0q(·),φ(·)(ℝn)≤Crα*(r)-(n/p*(r))+(n/q*(r))φ(0,r)ψ(0,r),
and we arrive at (63).Theorem 17.
Let1≤p-≤p+<∞, 1≤q-≤q+<∞, p,q∈𝒫0,∞(ℝn), and φ satisfy (54) and (59) and let the weight w fulfill (50). If
(69)φ(0,r)w(r)∈ℤγ0,γ∞(ℝ+)withγ0=np(0),γ∞=np(∞),
then the Hardy operator ℋwα(·) is bounded from ℒ0,locp(·),φ(·)(ℝn) to ℒ0,locq(·),ψ(·)(ℝn), where ψ(0,r) is the function (63).Proof.
By (69), we have
(70)𝒦w,α(x)=|x|α(x)w(|x|)∫|x|∞t-(n/p*(t))-1φ(0,t)w(t)dt≤C|x|α*(|x|)-(n/p*(|x|))φ(0,|x|),
after which the proof is the same as that of Theorem 16.We single out an important case of non-weighted Hardy operators(71)Hα(·)f(x)=|x|α(x)-n∫|y|<|x|f(y)dy,ℋα(·)f(x)=|x|α(x)∫|y|>|x|f(y)dy|y|n
in variable exponent Morrey spaces of classical type, that is, with the function φ(0,r)=φλ/p(0,r), defined by
(72)φλ/p(0,r)={rλ/p(0),r≤1=rλ/p*(r),rλ/p(∞),r≤1=rλ/p*(r),
where 0<λ<n and then ψ(0,r)=φλ/q(0,r)=rλ/q*(r) by (63).Theorem 18.
Letφλ/p be of form (72), 1≤p(x)≤p+<∞, let p,q,α∈𝒫0,∞(ℝn) be such that α(0)≥0, α(∞)≥0, and
(73)1q(0)=1p(0)-α(0)n-λ,1q(∞)=1p(∞)-α(∞)n-λ.
Then the Hardy operators Hα(·) and ℋα(·) are bounded from ℒ0,locp(·),φλ/p(·)(ℝn) to ℒ0,locq(·),φλ/qi(·)(ℝn), if
(74)α(0)p(0)<n-λ,α(∞)p(∞)<n-λ.
In the case of infx∈ℝnp(x)=p(∞), the conditions in (74) are also necessary.Proof.
In the sufficiency part, the theorem may be derived from Theorems16 and 17, but we find it more convenient to derive it from more general statement of Theorem 13, since the functions Kw,α and 𝒦w,α may be explicitly calculated in this case and
(75)Kw,α(x)=|x|α(x)-((n-λ)/p*(|x|)),𝒦w,α(x)=|x|α(x)+(λ/p*(|x|))-(n/p*′(|x|)).
Since α∈𝒫0,∞, we have
(76)Kw,α(x)~|x|α*(|x|)-((n-λ)/p*(|x|)),𝒦w,α(x)=|x|α*(|x|)+(λ/p*(|x|))-(n/p*′(|x|)),
where the notation α*(|x|) has the same meaning as in (10).
To check thatKw,α∈ℒloc;0q(·),φ(·)(ℝn), by the definition in (20) we have to estimate the norm ∥|x|α*(|x|)-((n-λ)/p*(|x|))χB(0,r)(x)∥q(·). To this end, we may apply Corollary 4 with ν(x)=α*(|x|)-((n-λ)/p*(|x|)) and p replaced by q. The assumptions on ν(x) of that corollary are satisfied if 0<λ<n and max{α(0),α(∞)}<n-λ, which holds under the assumptions of the theorem. Thus, by Corollary 4,
(77)∥|x|α*(|x|)-((n-λ)/p*(|x|))χB(0,r)(x)∥q(·)≤Crα*(r)-((n-λ)/p*(r))+(n/q*(r)).
Then the required condition
(78)1rλ/p*(r)∥|x|α*(|x|)-((n-λ)/p*(|x|))χB(0,r)(x)∥q(·)<∞
is guaranteed by (77) and (73).
Similarly the case of the operatorℋα(·) is treated.
The necessity of the conditions (74) becomes evident if we note that in the case under consideration they are just the same as the conditions in (55) which are necessary by Theorem 13.


## 4.1. Pointwise Estimations

We consider the following generalized Hardy operators:(47)Hwα(·)f(x)=|x|α(x)-nw(|x|)∫|y|<|x|f(y)dyw(|y|),ℋwα(·)f(x)=|x|α(x)w(|x|)∫|y|>|x|,f(y)dy|y|nw(|y|),
where α(x) is a non-negative measurable function on ℝn. In the one-dimensional case, their versions
(48)Hwα(·)f(x)=xα(x)-1w(x)∫0xf(t)dtw(t),ℋwα(·)f(x)=xα(x)w(x)∫x∞f(t)dttw(t),x>0
on the half-axis ℝ+1 may be also admitted, so that the sequel ℝn with n=1 may be read either as ℝ1 or ℝ+1.We also use the notationHα(·)=Hwα(·)|w≡1.Our next result on the boundedness of weighted Hardy operators presented in Theorem13 is prepared by our estimations in Theorem 10. It is in fact a consequence of Theorem 10. We find it useful to divide this consequence into two parts. First, in Theorem 11, we reformulate Theorem 10 in the form to emphasize that we have pointwise estimates of Hardy operatorsHwα(·) and ℋwα(·) in terms of the Morrey norm of the function f. Then as an immediate consequence of Theorem 11 we formulate Theorem 13 for global Morrey spaces.Theorem 11.
Letp∈𝒫0,∞(ℝn) and 1≤p-≤p+<∞. Let also the weight w satisfy the conditions
(49)w∈W_(ℝ+),φ(0,·)w(·)∈W_(ℝ+),
in the case of the operator Hwα(·), and the conditions
(50)1w∈W¯(ℝ+),φ(0,·)w(·)∈W¯(ℝ+).
in the case of the operator ℋwα(·). The conditions
(51)∫0εt(n/p′(0))-1φ(0,t)w(t)dt<∞,∫ε∞t-(n/p(∞))-1φ(0,t)w(t)dt<∞,
with ε>0, are sufficient for the Hardy operators Hwα(·) and ℋwα(·), respectively, to be defined on the space ℒ0;locp(·),φ(·)(ℝn). Under these conditions,
(52)|Hwα(·)f(x)|≤CKw,α(x)∥f∥ℒ0;locp(·),φ(·),|ℋwα(·)f(x)|≤C𝒦w,α(x)∥f∥ℒ0;locp(·),φ(·),
where
(53)Kw,α(x)=|x|α(x)-nw(|x|)𝒜(|x|)=|x|α(x)-nw(|x|)∫0|x|t(n/p*′(t))-1φ(0,t)w(t)dt,𝒦w,α(x)=|x|α(x)w(|x|)∫|x|∞t-(n/p*(t))-1φ(0,t)w(t)dt.Proof.
The sufficiency of the conditions in (51) and estimates in (52) follows from (36) under the choice v(t)=w(t), for the operator Hwα(·) and v(t)=1/tnw(t) for the operator ℋwα(·).


## 4.2. On the Necessity of the Conditions in (51)

Observe that the conditions in (51) are natural in the sense that they are necessary under some additional assumptions on the function φ defining the Morrey space.Lemma 12.
Letp be as in Theorem 11 and φ(0,·)∈W¯([0,ε0])∩W_([0,ε0]) for some ε0>0, and (30) holds. Then the conditions in (51) are necessary for the Hardy operators Hα(·)W and ℋα(·)W, respectively, to be defined on the space ℒ0;locp(·),φ(·)(ℝn).Proof.
Choosef0(x)=(φ(0,|x|)/|x|n/p(x))χB(0,R)(x). Then f∈ℒ0;locp(·),φ(·)(ℝn) by Lemma 8. It remains to note that the conditions in (51) are nothing else but the statement that Hwα(·)f0 and ℋwα(·)f0, respectively, exist.


## 4.3. Weightedp→q Norm Estimates for Hardy Operators

The statements of Theorem13 are well known in the case of Lebesgue space, that is, in the case φ≡1, with constant exponents, when 1<p<n/α; see for instance [27, pages 6, 54]. For the classical Morrey spaces ℒp,λ(ℝn) with constant exponents p and λ, statements of such type for Hardy operators have been obtained in [17, 19].Hardy inequalities in the variable exponent Lebesgue spaces were studied in [34–36]; see also the references therein.Note that, in contrast to variable exponent Lebesgue spaces, inequalities for the Hardy operators in Morrey spaces admit the caseinfxp(x)=1 when φ(0,0)=0 in the case of local Morrey spaces and supxφ(x,0)=0 in the case of global Morrey spaces.We suppose that the condition(54)supr>0[min{1,r}]n/p(0)φ(0,r)<∞
holds, which ensures that the space ℒ0,locp(·),φ(·)(ℝn) is nonempty by Lemma 6.Theorem 13.
Let1≤p-≤p+<∞, 1≤q-≤q+<∞ and p∈𝒫0,∞(Ω) as well as the functions φ and ψ satisfy the assumption (54). Let also the weight w satisfy the conditions in (49) in the case of the operator Hwα(·) and the conditions in (50) in the case of the operator ℋwα(·). Then the operators Hwα(·) and ℋwα(·) are bounded from ℒ0,locp(·),φ(·)(ℝn) to ℒ0,locq(·),ψ(·)(ℝn), if
(55)Kw,α∈ℒloc;0q(·),ψ(·)(ℝn),𝒦w,α∈ℒloc;0q(·),ψ(·)(ℝn),
respectively. If p and φ(0,r) satisfy the assumptions of Corollary 9, then the conditions in (55) are also necessary for the boundedness of the operators Hwα(·) and ℋwα(·).Proof.
The sufficiency of the conditions in (55) for the boundedness follows from the estimates in (52).
As regards the necessity, the requirements in (55) are nothing else but the statement that
(56)Hwα(·)(f0)∈ℒloc;0q(·),ψ(·)(ℝn),ℋwα(·)(f0)∈ℒloc;0q(·),ψ(·)(ℝn),
respectively, where
(57)f0(x)=φ(0,|x|)|x|n/p*(|x|).
The functionf0 belongs to ℒ0,locp(·),φ(·)(ℝn) by Corollary 9. Consequently, the conditions, (55) are necessary.Corollary 14.
Under the same assumptions onp,φ and w as in Theorem 13, the Hardy operators Hwα(·) and ℋwα(·) are bounded from the global Morrey space ℒp(·),φ(·)(ℝn) to the global space ℒq(·),ψ(·)(ℝn), if
(58)Kw,α∈ℒq,ψ(ℝn),𝒦w,α∈ℒq,ψ(ℝn),
respectively.Proof.
Since∥f∥ℒ0,locp(·),φ(·)(ℝn)≤∥f∥ℒp(·),φ(·), the statement immediately follows from the pointwise estimates in (52).Remark 15.
Theorem18 is specifically a “Morrey-type” statement in the sense that the case of Lebesgue spaces (the case λ=0) is not included. This, in particular, is reflected in the admission of values p(x)=1 in Theorem 18, which is impossible for Lebesgue spaces.


## 4.4. Findingψ(0,r) by a Given φ(0,r)

The main theorem of the preceding section, Theorem13 on the ℒ0,locp(·),φ(·)(ℝn)→ℒ0,locq(·),ψ(·)(ℝn) boundedness, provides a relation between the given function φ(0,r) and ψ(0,r) in an indirect form, via the conditions in (55). In the theorems below, under some additional assumptions on the functions φ(0,r) we obtain the boundedness conditions in a form of Zygmund-type integral conditions imposed on φ(0,r)/w(r) and give a direct relation between φ(0,r) and ψ(0,r).In these theorems, we use the following assumptions on the functionφ(0,t) defining the data space ℒ0,locp(·),φ(·)(ℝn):
(59)φ(0,r)∈W_(ℝ+)∩ℤβ~0,β~∞(ℝ+),
where
(60)β~0=np(0)-nq(0)-α(0),β~∞=np(∞)-nq(∞)-α(∞)
(see (A.18) for the definition of the classes ℤβ0,β∞(ℝ+)). Recall that the assumption φ(0,r)∈ℤβ~0,β~∞(ℝ+) in (59) may be equivalently rewritten in terms of the Matuszewska-Orlicz indices m(φ),m∞(φ) of the function φ(0,t) as
(61)m(φ)>β~0,m∞(φ)>β~∞.Theorem 16.
Let1≤p-≤p+<∞, 1≤q-≤q+<∞, and p,q∈𝒫0,∞(ℝn) as well as the function φ satisfy the assumption in (54) and (59) and let the weight w fulfill the conditions in (49). If
(62)φ(0,r)w(r)∈ℤβ0,β∞(ℝ+)withβ0=-np′(0),φ(0,r)w(r)∈ℤβ0,β∞(ℝ+)withβ∞=-np′(∞),
then the Hardy operator Hwα(·) is bounded from ℒ0,locp(·),φ(·)(ℝn) to ℒ0,locq(·),ψ(·)(ℝn), with
(63)ψ(0,r)=rα*(r)-(n/p*(r))+(n/q*(r))φ(0,r).Proof.
By (69), we have
(64)Kw,α(x)=|x|α(x)-nw(|x|)∫0|x|t(n/p*′(t))-1φ(0,t)w(t)dt≤C|x|α*(|x|)-(n/p*(|x|))φ(0,|x|).
To check that Kw,α∈ℒloc;0q(·),φ(·)(ℝn), by the definition in (20), we have to estimate the norm
(65)∥|x|α*(|x|)-(n/p*(|x|))φ(0,|x|)χB(0,r)∥Lq(·).
We apply Lemma 2 with a(x,r)=rα*(r)-(n/p*(r))φ(0,r), which is possible by (59) and obtain
(66)∥|x|α*(|x|)-(n/p*(|x|))φ(0,|x|)χB(0,r)∥Lq(·)≤C∫0rtα*(t)-(n/p*(t))+(n/q*(t))φ(0,t)dtt.
Then by (59) we get
(67)∥|x|α*(|x|)-(n/p*(|x|))φ(0,|x|)χB(0,r)∥Lq(·)≤Crα*(r)-(n/p*(r))+(n/q*(r))φ(0,r).
Therefore,
(68)∥|x|α*(|x|)-(n/p*(|x|))φ(0,|x|)∥ℒloc;0q(·),φ(·)(ℝn)≤Crα*(r)-(n/p*(r))+(n/q*(r))φ(0,r)ψ(0,r),
and we arrive at (63).Theorem 17.
Let1≤p-≤p+<∞, 1≤q-≤q+<∞, p,q∈𝒫0,∞(ℝn), and φ satisfy (54) and (59) and let the weight w fulfill (50). If
(69)φ(0,r)w(r)∈ℤγ0,γ∞(ℝ+)withγ0=np(0),γ∞=np(∞),
then the Hardy operator ℋwα(·) is bounded from ℒ0,locp(·),φ(·)(ℝn) to ℒ0,locq(·),ψ(·)(ℝn), where ψ(0,r) is the function (63).Proof.
By (69), we have
(70)𝒦w,α(x)=|x|α(x)w(|x|)∫|x|∞t-(n/p*(t))-1φ(0,t)w(t)dt≤C|x|α*(|x|)-(n/p*(|x|))φ(0,|x|),
after which the proof is the same as that of Theorem 16.We single out an important case of non-weighted Hardy operators(71)Hα(·)f(x)=|x|α(x)-n∫|y|<|x|f(y)dy,ℋα(·)f(x)=|x|α(x)∫|y|>|x|f(y)dy|y|n
in variable exponent Morrey spaces of classical type, that is, with the function φ(0,r)=φλ/p(0,r), defined by
(72)φλ/p(0,r)={rλ/p(0),r≤1=rλ/p*(r),rλ/p(∞),r≤1=rλ/p*(r),
where 0<λ<n and then ψ(0,r)=φλ/q(0,r)=rλ/q*(r) by (63).Theorem 18.
Letφλ/p be of form (72), 1≤p(x)≤p+<∞, let p,q,α∈𝒫0,∞(ℝn) be such that α(0)≥0, α(∞)≥0, and
(73)1q(0)=1p(0)-α(0)n-λ,1q(∞)=1p(∞)-α(∞)n-λ.
Then the Hardy operators Hα(·) and ℋα(·) are bounded from ℒ0,locp(·),φλ/p(·)(ℝn) to ℒ0,locq(·),φλ/qi(·)(ℝn), if
(74)α(0)p(0)<n-λ,α(∞)p(∞)<n-λ.
In the case of infx∈ℝnp(x)=p(∞), the conditions in (74) are also necessary.Proof.
In the sufficiency part, the theorem may be derived from Theorems16 and 17, but we find it more convenient to derive it from more general statement of Theorem 13, since the functions Kw,α and 𝒦w,α may be explicitly calculated in this case and
(75)Kw,α(x)=|x|α(x)-((n-λ)/p*(|x|)),𝒦w,α(x)=|x|α(x)+(λ/p*(|x|))-(n/p*′(|x|)).
Since α∈𝒫0,∞, we have
(76)Kw,α(x)~|x|α*(|x|)-((n-λ)/p*(|x|)),𝒦w,α(x)=|x|α*(|x|)+(λ/p*(|x|))-(n/p*′(|x|)),
where the notation α*(|x|) has the same meaning as in (10).
To check thatKw,α∈ℒloc;0q(·),φ(·)(ℝn), by the definition in (20) we have to estimate the norm ∥|x|α*(|x|)-((n-λ)/p*(|x|))χB(0,r)(x)∥q(·). To this end, we may apply Corollary 4 with ν(x)=α*(|x|)-((n-λ)/p*(|x|)) and p replaced by q. The assumptions on ν(x) of that corollary are satisfied if 0<λ<n and max{α(0),α(∞)}<n-λ, which holds under the assumptions of the theorem. Thus, by Corollary 4,
(77)∥|x|α*(|x|)-((n-λ)/p*(|x|))χB(0,r)(x)∥q(·)≤Crα*(r)-((n-λ)/p*(r))+(n/q*(r)).
Then the required condition
(78)1rλ/p*(r)∥|x|α*(|x|)-((n-λ)/p*(|x|))χB(0,r)(x)∥q(·)<∞
is guaranteed by (77) and (73).
Similarly the case of the operatorℋα(·) is treated.
The necessity of the conditions (74) becomes evident if we note that in the case under consideration they are just the same as the conditions in (55) which are necessary by Theorem 13.



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