Sobolev’s inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over non doubling measure spaces

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R E S E A R C H

Open Access

Sobolev’s inequality for Riesz potentials of

functions in generalized Morrey spaces with

variable exponent attaining the value



over

non-doubling measure spaces

Yoshihiro Sawano

1,2*

_{and Tetsu Shimomura}

3

*_{Correspondence:}

[email protected]

1_{Department of Mathematics, Kyoto}

University, Kyoto, 606-8502, Japan

2_{Current address: Department of}

Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan Full list of author information is available at the end of the article

Abstract

Our aim in this paper is to give Sobolev’s inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over

non-doubling measure spaces. The main result is oriented to the outrange of the well-known Adams theorem.

MSC: Primary 31B15; secondary 46E35; 26A33

Keywords: Riesz potentials; Sobolev’s inequality; Morrey spaces of variable exponents; non-doubling measure

1 Introduction

The boundedness of fractional integral operators on Morrey spaces is known as the Adams theorem. Recently, many endpoint results have been obtained for this theorem, and in this paper we extend them to generalized Morrey spaces with variable exponent attaining the value  over non-doubling measure spaces.

In  Morrey observed that a weaker regularity sufficed in order that the solutions in elliptic differential equations were smooth []. This observation grew up to be a useful tool for partial differential equations in general. Nowadays, his technique turned out to be a wide theory of function spaces called Morrey spaces; see also []. The (original) Morrey spaceLp,λ₍_RN_{) with }_≤_p_<_∞_{and  <}_λ_≤_N_{is a normed space whose norm is given by} f_Lp,λ≡sup_x_∈_RN_,_r_>( 

rN–λ

B(x,r)|f(y)|pdy)



p _for_f ∈_Lp

loc(RN).

We are oriented to building up a theory of a metric measure space (X,d,μ) for which the notion of dimension is not equipped withμ, wheredis a distance function andμis a Borel measure. For example, we encounter the situation where more than one dimension comes into play.

. InRconsider the setX≡Y∪Z, whereY≡ {(x, , ) :x∈R}and

Z≡ {(x,y,z) :z=x₊_y}_{. Denote by}Hs_the_s_{-dimensional Hausdorﬀ measure in}

R_for__≤_s_≤__{. Consider}_μ_≡_H_|_Y₊_H_|_Z_{. Then}_μ_{has two diﬀerent dimensions.}

. The above example is a little artiﬁcial. Consider the Cantor dust, which is given by

E≡∞_j_=Ej, whereEjis given recursively by

E≡[, ],

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Ej+=

e∈{,}

 e+

 Ej

=

 e+



x:x∈Ej,e∈ {, }

 _⊂_R_.

Then the Hausdorﬀ dimension ofEiswhile eachEjis a union of closed cubes inR.

As the second example shows, when we consider the subsets inR_{, it is natural to}

ap-proximate them with sets having diﬀerent dimension. Faced with such an inconvenience, we work on a separable metric spaceXequipped with a nonnegative Radon measureμ, where the notion of dimensions ofXandμdoes not come into play.

We assume that

μ{x}=  (.)

for allx∈X. ByB(x,r) we denote the open ball centered atxof radiusr> . We assume

 <μB(x,r)<∞ (.)

forx∈Xandr>  for simplicity. We writed(x,y) for the distance of the pointsxandy

inX. We are interested in the operator of the formf →_XK(·,y)f(y)dμ(y), where f is aμ-measurable function with a certainμ-integrability. Note that (.) is indispensable because we envisage examples in whichK:X×X→[,∞] has singularity at the diag-onal. Meanwhile, (.) does not lose any generality; it is just a matter of restrictingXto supp(μ).

LetGbe a bounded open set inX. Our argument to follow heavily depends upon its diameterdG. Letp≥,ν>  andk> . Deﬁne the Morrey normf_Lp,ν,β;k₍_G₎by

f_Lp,ν,β;k₍_G₎≡sup

x,r

rν₍_log₍_e_{+ /}_r₎₎β

μ(B(x,kr))

B(x,r)∩G

f(y)pdμ(y) /p

for μ-measurable functions f, where (x,r) runs over all elements in G×(,dG). The Morrey spaceLp,ν,β;k₍_G_{) is the set of all}_μ_{-measurable functions}_f _{for which the norm} f_Lp,ν,β;k₍_G₎is ﬁnite. Whenβ= ,Lp,ν,β;k(G) is denoted byLp,ν;k(G).

In this paper, we aim to understand how the fractional integral operators behave in gen-eralized Morrey spaces with variable exponent attaining the value  over non-doubling measure spaces. We obtain the following constant exponent case as a corollary of a more general theorem in non-doubling and variable exponent setting (see Theorem . below). The result is new even for a constant exponent case.

Theorem . Letα,β,ν be constants.Deﬁne an index p*_by 

p* =  – α

ν,and letγ > . Deﬁne

Uα,f(x)≡

G

d(x,y)α

μ(B(x, d(x,y)))f(y)dμ(y) (.)

for a positiveμ-measurable function f.Then there exists a constant C> such that



μ(B(z, r))

B(z,r)

Uα,f(x)p

*

loge+Uα,f(x)

–γ+αβp*/ν dμ(x)

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for all z∈G and <r<dG, whenever f is a nonnegativeμ-measurable function on G satisfyingfL,ν,β;₍_G₎≤.

We remark thatUα,extends naturally the fractional integral operator deﬁned by

Koki-lashvili onRN_{[]. We also remark that an extension to quasi-metric spaces was performed} in [].

Theorem . describes the missing part of the Adams inequality, which we recall now.

Proposition .(Adams, []) Let <α<n,  <p,q<∞and <λ≤N.Assume_q=_p–α_λ.

Then there exists a constant C> such thatIαfLq,λ≤Cf_Lp,λfor all f ∈Lp,λ(RN).

In the endpoint case, whenp= , two natural questions about Proposition . arise.

. Can we prove a similar result by enlargingLq,λ₍_RN₎_slightly?

. Can we prove a similar result by shrinkingLp,λ₍_RN₎_slightly?

The second question is considered in []. We aim to consider the ﬁrst question in the present paper.

It is the condition on μthat counts in the present paper; we do not postulate onμ

the so-called doubling condition. Recall that a Radon measureμis said to be doubling if there exists a constantC>  such thatμ(B(x, r))≤Cμ(B(x,r)) for allx∈supp(μ) (=X) andr> . Otherwise,μis said to be non-doubling. In connection with the r-covering lemma, the doubling condition had been a key condition in harmonic analysis. However, Nazarov, Treil and Volberg showed that the doubling condition was not necessary by using the modiﬁed maximal operator [, ]. The idea of replacingd(x,y) with d(x,y) in (.) originates from these papers. In the present paper, we show that this idea works even for Riesz potentials.

Here, we summarize the structure of the space Lp,ν;k₍_G_{) in Proposition . and} Re-mark . below. Note that the Morrey spaceLp,ν;k₍_G_{) does depend upon the parameter}

k, which is illustrated by the following proposition.

Proposition . ([]) There does exist a bounded separable metric space(X,d,μ)such that Lp,ν;₍_G₎_{and L}p,ν;₍_G₎_{do not coincide as sets}_.

About the modiﬁed Morrey norm, we have the following remarks.

Remark . Letf :X→[,∞] be aμ-measurable function andGbe a bounded open subset ofX.

. From the deﬁnition of the norms, we learnf_Lp,ν;k(G)≤ fLp,ν;k(G)for allk>k> 

andp≥.

. Ifp≥p≥,k≥andν/p=ν/p> , thenfLp,ν;k₍_G₎≤ f_Lp,ν;k₍_G₎by the Hölder inequality.

. Ifμis a doubling measure, thenf_Lp,ν;k₍_G₎andf_Lp,ν;₍_G₎are equivalent for allp≥,

k> andν> .

Our result can be readily translated into the Morrey spaceLp,λ₍_G_{), where}_Lp,λ₍_G_{) is the set}

of all functionsf such that

f_Lp,λ₍_G₎≡ sup

x∈G,r∈(,dG)



rN–λ

G∩B(x,r)

f(y)pdy

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Keeping our fundamental result, Theorem ., in mind, let us describe our results in full generality. We seek to establish Sobolev’s inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value  over non-doubling measure spaces, which will extend the results in our earlier papers [–]. In the present paper, we show that a modiﬁcation enables us to obtain boundedness results in generalized Morrey spaces with variable exponent attaining the value  over non-doubling measure spaces even when

inf

x∈Xp(x) = .

We consider variable exponentsp(·) andq(·) onXsuch that

(P) ≤p–≡infx∈Xp(x)≤supx∈Xp(x)≡p+<∞;

(P) |p(x) –p(y)| ≤C/log(e+d(x,y)–₎_whenever_x_∈_X_and_y_∈_X_;

(Q) –∞<q–≡infx∈Xq(x)≤supx∈Xq(x)≡q+<∞;

(Q) |q(x) –q(y)| ≤C/log(e+ (log(e+d(x,y)–₎₎₎_whenever_x_∈_X_and_y_∈_X_.

In general, ifp(·) satisﬁes (P) (resp.q(·) satisﬁes (Q)), thenp(·) (resp.q(·)) is said to satisfy the log-Hölder (resp. loglog-Hölder) condition. Observe that these conditions are much weaker than Lipschitz continuity.

Next, we redeﬁne the Riesz potential. Recall that we are working on a bounded subset

GofX. For a boundedμ-measurable functionα(·) :X→(,∞) and a constantκ> , we deﬁne the Riesz potential of (variable) orderα(·) for a nonnegativeμ-measurable function

f onGby

Uα(·),κf(x)≡

G

d(x,y)α(x)_f₍_y₎

μ(B(x,κd(x,y)))dμ(y).

Here and in what follows, we tacitly assume thatf =  outsideG. Observe that this natu-rally extends the Riesz potential operator

Uαf(x)≡

RN f(y) |x–y|N–αdy

when (X,| · |,μ) is theN-dimensional Euclidean space andμ=dx. We also assume

α–≡inf

x∈Xα(x) >  (.)

forα(·) appearing in the deﬁnition of the operatorUα(·),κ.

Now, we are going to formulate our results in full generality. First of all, we set

(x,r) =p(·),q(·)(x,r)≡rp(x)

log(c+r)q(x) (x∈X,r> ); (.) here we assume

() p(·),q(·)(x,·)is convex on[,∞)for everyx∈X.

Note that () holds for somec≥eif and only if there is a positive constantKsuch that

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(see [, Theorem .]). Observe from () that the functiont →t–₍_x_,_t_{) is}

nondecreas-ing on (,∞) for ﬁxedx∈X.

Now, we redefine and extend the definition of Morrey norms adapted to the func-tion. Letk>  be a fixed parameter and letGbe a bounded subset ofX. Let us de-note bydG the diameter ofG. For boundedμ-measurable functionsν:X→(,∞) and

β:X→(–∞,∞), we introduce the familyL,ν,β;k₍_G_{) of all}_μ_{-measurable functions}_f _on

Gsuch that for someλ∈(,∞),

sup x∈G,<r<dG

rν(x)₍_log₍_c_{+ /}_r₎₎β(x)

μ(B(x,kr))

G∩B(x,r)

y,f(y)/λdμ(y)≤. (.)

Denote byfL,ν,β;k₍_G₎the smallest value ofλsatisfying (.). Note thatνandβneed not

be continuous.

The spaceL,ν,β;k₍_G_{) is a further generalization of generalized Morrey spaces with} non-doubling measures onRN, which are taken up in [, ]. Nowadays, generalized Morrey spaces are not generalization for its own sake. Note that generalized Morrey spaces occur naturally when we consider the limiting case as Proposition . below shows.

Proposition . ([, Theorem .]) Let <p<∞and <λ<N. Then there exists a

positive constant Cp,λsuch that

B

f(x)dx≤Cp,λ|B|

 +|B|–p_log

e+  |B|

( – )λ/p_f Lp,λ

holds for all f ∈Lp,λ₍_RN₎_with_{( –} ₎λ/p_f _∈_Lp,λ₍_RN₎_{and for all balls B}_.

In view of the integral kernel of ( – )–α/_{(see []) and the Adams theorem, we have}

( – )–α/_:_Lp,λ_RN_→_Lq,λ_RN _(.)

is bounded as long as

 <p,q<∞,  <λ≤N, 

q=



p– α λ.

However, ifα=λ_p, the numberqnot being ﬁnite, the boundedness assertion (.) is no longer true. Hence, Proposition . can be considered as a substitute of (.). Proposi-tion . shows that the price to pay is the factorlog(e+_|_Q_|). We refer to [] for a coun-terexample showing that (.) is no longer true forα=λ_p.

Remark that ifX=RN_{, the parameter}_k_{is not essential as long as}_k_{>  as Proposition .} below shows.

Proposition .([]) Let k,k> and X=RN be the Euclidean space.Suppose that G

is a bounded open set.Assume in addition thatνandβsatisfy the log-Hölder continuity

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What is diﬀerent from the classical setting endowed with Lebesgue measure is that we need to take an attentive care of the parametersk appearing in (.). For example, un-like the doubling measure spaces, we need delicate geometric observations (see (.) for example).

Among many other function spaces, Morrey spaces with variable exponents are worth investigating because our result Theorem . shows that the smoothing eﬀect that the fractional integral operators enjoy is local. It is believed thatpin the deﬁnition ofLp,ν;k₍_G₎ measures the local integrability, whileνseems to control the global integrability.

Here we make a historical remark about the research of this ﬁeld. The caseμ=dxof Theorem . is covered in [, Theorem .] and [, Corollary .]. Roughly speaking, we seek to discuss the boundedness of the operatorUα(·),:f →Uα(·),f from the Morrey

spaceL,ν,β;₍_G_{) to another Morrey space}_L,ν,β;₍_G_{) with suitable}₍_x_,_r_{), which extends}

the results in [–]. Whenp–> , the maximal functions play a crucial tool by Hedberg’s

trick (see []). In view of [, , , ], the modiﬁed maximal operator is bounded and this was useful for the case whenp–> . In casep–= , our strategy is to give an estimate

ofUα(·),f by the use of another Riesz-type potentials of order , which plays a role of the

maximal functions. For related results, see [–].

Finally, we explain some notations used in the present paper. The functionχEdenotes the characteristic function ofE. Throughout the present paper, letCdenote various con-stants independent of the variables in question. In analogy withp±andq±, we deﬁneα±,

β±andν±.

2 Sobolev’s inequality in the casep–= 1

Recall that α : X →(,∞), ν :X →(,∞) and β : X →(–∞,∞) are bounded μ -measurable functions andα–> . Throughout this section, we additionally assume that

inf x∈X

/p(x) –α(x)/ν(x)> . (.)

Note thatν(x)≥α(x)p(x) forx∈X. Thus, fromp–≥ and (.), we have

ν–≥α–> . (.)

Our ﬁrst aim is to give the following Morrey version of Sobolev-type inequality for Riesz potentials

Uα(·),f(x)≡

G

μ(B(x, d(x,y)))dμ(y). (.)

We consider the Sobolev exponent

/p*(x)≡/p(x) –α(x)/ν(x) (x∈X) (.)

and the new modular function

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Theorem . Assume that p(·)satisﬁes(P)with p–= , (P)and(.).Then, for each

ε> ,there exists a constant C> such that



μ(B(z, r))

B(z,r)

x,Uα(·),f(x)

logc+Uα(·),f(x)

–(+ε)

dx

≤Cr–ν(z)log(c+ /r)–β(z)–ε

for all z∈G and <r<dG, whenever f is a nonnegativeμ-measurable function on G satisfyingfL,ν,β;₍_G₎≤.

Remark . In Theorem ., it is known that we cannot takeε=  (see [, Remark .] and O’Neil [, Theorem .]).

Here we outline the proof of Theorem .. Forε>  andx∈X, and setting

ρε(r)≡



μ(B(x, r))

log(c+ /r)–ε–, (.)

we consider the logarithmic potential (of order )

Jεf(x) =

G

ρε

d(x,y)f(y)p(y)logc+f(y)q(y)dμ(y). (.)

Decompose

Uα(·),f(x) =

B(x,δ)

μ(B(x, d(x,y)))dμ(y) +

G\B(x,δ)

μ(B(x, d(x,y)))dμ(y)

=I(δ) +I(δ), say. (.)

Following the Hedberg trick [], we plan to controlI(δ) byJεf(x) not by maximal

func-tions (Lemma .). Next, we estimateI(δ) by the use of Young’s inequality (Lemma .).

Finally, an optimization (see (.) below), together with the boundedness property ofJε

(Lemma .), yields the desired estimateUα(·),f(x) in terms ofJεf(x).

Lemma . For <δ<dG,x∈G and a nonnegativeμ-measurable function f on G,set

I(δ)≡

B(x,δ)

μ(B(x, d(x,y)))dμ(y).

Letε> be ﬁxed.Then there exists a constant C> such that

I(δ)≤C

δα(x)–ν(x)/p(x)logc+δ––(q(x)+β(x))/p(x)

+δα(x)+(p(x)–)ν(x)/p(x)logc+δ–β(x)–(q(x)+β(x))/p(x)+(+ε)Jεf(x)

,

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Proof Recall thatμdoes not charge a pointx(see (.) above) and thatα–>  by virtue of

(.). We thus have

B(x,δ)

d(x,y)α(x)

=

∞

j=

B(x,–j+_δ₎_\_B₍_x_,–j_δ₎

d(x,y)α(x)

≤

∞

j=

B(x,–j+_δ₎

(–j+_δ₎α(x)

μ(B(x, –j+_δ₎₎dμ(y)

≤

∞

j=

–j+δα(x)=   – –α(x)δ

α(x)_≤ 

 – –α–δ

α(x)₌_C_δα(x)_.

Consequently,

B(x,δ)

d(x,y)α(x)

μ(B(x, d(x,y)))dμ(y)≤Cδ

α(x)_. _(.)

From (), observe that the functiont →t–₍_x_,_t_{) is nondecreasing on (,}_∞_{) for ﬁxed}

x∈X. With this in mind, fork> , which we specify in (.), we decompose and estimate the integral deﬁningI(δ) according to the level set{|f|>k}:

I(δ)≤k

B(x,δ)

d(x,y)α(x)

+

B(x,δ)

d(x,y)α(x)

μ(B(x, d(x,y)))f(y)

f(y)

k

p(y)–_log

(c+f(y)) log(c+k)

q(y)

dμ(y).

The ﬁrst term is estimated by (.):

k

B(x,δ)

d(x,y)α(x)

μ(B(x, d(x,y)))dμ(y)≤Ckδ

α(x)_.

Recall also thatρεis deﬁned by (.). Inserting them, we have

I(δ)≤C

kδα(x)+

B(x,δ)

d(x,y)α(x)

μ(B(x, d(x,y)))g(y)



k

p(y)–

 log(c+k)

q(y)

dμ(y)

≤C

kδα(x)+δα(x)logc+δ–+ε

×

B(x,δ)

ρε

d(x,y)g(y)k–p(y)

 log(c+k)

q(y)

dμ(y) ,

whereg(y) =f(y)p(y)₍_log₍_c₊_f₍_y₎₎₎q(y)_{. We set}

k≡k(x) =δ–ν(x)/p(x)logc+δ––(q(x)+β(x))/p(x). (.) Fory∈B(x,δ), from (P) and the boundedness oflogp,q,βandν, we deduce

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so that

k–p(y)≤Ck–p(x). (.)

Similarly, by (Q) we have

log(c+k)–q(y)≤Clog(c+k)–q(x). (.)

Consequently, it follows from (.) and (.) that

I(δ)≤C

kδα(x)

+δα(x)logc+δ–+εk–p(x)

 log(c+k)

q(x)

B(x,δ)

ρε

d(x,y)g(y)dμ(y) .

Recall now thatJεf(x) is given by (.). Observe also that

log(c+k) =logc+δ–ν(x)/p(x)logc+δ––(q(x)+β(x))/p(x)∼logc+δ–,

sinceν–> ,p–≥ andp,q,βare all bounded. Thus,

I(δ)≤C

+δα(x)logc+δ–+ε–q(x)δν(x)/p(x)logc+δ–(q(x)+β(x))/p(x)p(x)–Jεf(x)

≤Cδα(x)–ν(x)/p(x)logc+δ––(q(x)+β(x))/p(x)

+δα(x)+(p(x)–)ν(x)/p(x)logc+δ–β(x)–(q(x)+β(x))/p(x)+(+ε)Jεf(x)

.

Now, the result follows.

In our earlier work, we obtained the following lemma, whose proof is simple; we do not recall the proof.

The quantityI(δ) will be taken care of by using the next lemma.

Lemma .([, Lemma .]) Let f be a nonnegative μ-measurable function on G such

that

fL,ν,β;₍_G₎≤. (.)

Then

I(δ)≡

G\B(x,δ)

≤Cδα(x)–ν(x)/p(x)logc+δ––(q(x)+β(x))/p(x) (.)

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What remains to show for the proof of Theorem . is to give a boundedness property on Morrey spaces forJεf(x).

Lemma . There exists a constant C> such that



μ(B(z, r))

B(z,r)

Jεf(x)dμ(x)≤Cr–ν(z)

log(c+ /r)–β(z)–ε (.)

∀z∈G,  <r<dGand nonnegativeμ-measurable functions f satisfyingfL,ν,β;₍_G₎≤. Proof Forz∈Gand  <r<dG, write

Jεf(x) =

B(z,r)

ρε

d(x,y)g(y)dμ(y) +

G\B(z,r)

ρε

d(x,y)g(y)dμ(y)

=J(x) +J(x), (.)

whereg(y) =f(y)p(y)₍_log₍_c₊_f₍_y₎₎₎q(y)_{. Then we have}

B(z,r)

J(x)dμ(x) =

B(z,r)

B(z,r)

ρε

d(x,y)dμ(x)

g(y)dμ(y)

≤

B(z,r)

B(y,r)

ρε

d(x,y)dμ(x)

g(y)dμ(y)

by the Fubini theorem and a geometric observation. Since we are assuming (.), we can use a routine dyadic decomposition to obtain

B(z,r)

J(x)dμ(x)≤

B(z,r)

_∞

j=

B(y,–j+_r₎_\_B₍_y_,–j+_r₎

ρε

d(x,y)dμ(x)

g(y)dμ(y).

We now insert the deﬁnition ofρε(see (.) above) and we have

B(z,r)

J(x)dμ(x)

≤

B(z,r)

_∞

j=

B(y,–j+_r₎_\_B₍_y_,–j+_r₎

(log(c+ /(–j+r)))–ε–

μ(B(x, –j+_r₎₎ dμ(x)

g(y)dμ(y).

A geometric observation shows that μ(B(x, –j+_r₎₎_≥_μ₍_B₍_y_{, }–j+_r_{)) provided}_d₍_x_,_y₎_≤

–j+_r_{. Thus, we have}

B(z,r)

J(x)dμ(x)≤

B(z,r)

_∞

j=

logc+ /–j+r–ε–

g(y)dμ(y).

Now, we invoke the assumption thatε>  and thatr<dG. From these conditions, (.) and the deﬁnition ofg, we deduce

B(z,r)

J(x)dμ(x)≤C

log(c+ /r)–ε

B(z,r)

g(y)dμ(y)

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so that



μ(B(z, r))

B(z,r)

J(x)dμ(x)≤Cr–ν(z)

log(c+ /r)–β(z)–ε. (.)

Next, we claim thatx∈B(z,r) andy∈/B(z, r) imply that



d(x,y)≤d(y,z)≤d(x,y) (.)

and that

Bx, d(x,y)⊃Bz, d(z,y). (.)

Here we check only (.). Whenw∈B(z, d(z,y)), we have

d(w,x)≤d(z,x) +d(w,z)≤d(z,x) + d(z,y)≤ 

d(y,z) + d(z,y) = 

d(z,y)≤d(y,x).

We use (.), (.) and (.) forJ(x) to obtain

J(x) =

G\B(z,r)

ρε

d(x,y)g(y)dμ(y)

≤C

G\B(z,r)

(log(c+ /d(z,y)))–ε–

μ(B(x, d(z,y))) g(y)dμ(y)

=C

∞

j=

B(z,j+_r₎_\_B₍_z_,j_r₎

(log(c+ /d(z,y)))–ε–

μ(B(z, d(z,y))) g(y)dμ(y).

Consequently, from (.), (.) and the assumption thatβis a bounded function, we ob-tain

J(x)≤C

∞

j=

(log(c+ /(j+r)))–ε–

μ(B(z, j+_r₎₎

B(z,j+_r₎g(y)d

μ(y)

≤C

∞

j=

j+r–ν(z)logc+ /j+r–β(z)–ε–.

Recall thatν–>  and thatβis bounded. Therefore, it follows that

J(x)≤C

dG

r

t–ν(z)log(c+ /t)–β(z)–εdt

t ≤Cr

–ν(z)_log₍_c_{+ /}_r₎–β(z)–ε

forx∈B(z,r). Hence, we see that



μ(B(z, r))

B(z,r)

J(x)dμ(x)≤Cr–ν(z)

log(c+ /r)–β(z)–ε. (.)

From (.), (.) and (.), we conclude (.).

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Proof of Theorem. We may assume thatf≥. Forδ> , write

Uα(·),f(x) =I(δ) +I(δ)

according to (.). In view of Lemma ., we ﬁnd

I(δ)≤C

+δα(x)+(p(x)–)ν(x)/p(x)logc+δ–β(x)–(q(x)+β(x))/p(x)+(+ε)J

withJ=Jεf(x). Moreover, Lemma . yields

I(δ)≤Cδα(x)–ν(x)/p(x)

logc+δ––(q(x)+β(x))/p(x), (.) so that

Uα(·),f(x)≤C

+δα(x)+(p(x)–)ν(x)/p(x)logc+δ–β(x)–(q(x)+β(x))/p(x)+(+ε)J. Now, letting

δ≡mindG,J–/ν(x)

log(c+J)–(β(x)+(+ε))/ν(x), (.) we obtain

Uα(·),f(x)≤C

 +J/p*(x)log(c+J)–α(x)β(x)/ν(x)–q(x)/p(x)+(+ε)/p∗(x). By Lemma ., we obtain



μ(B(z, r))

B(z,r)

x,Uα(·),f(x)

–(+ε)

dμ(x)

≤C 

μ(B(z, r))

B(z,r)

( +J)dμ(x)≤Cr–ν(z)_log₍_c_{+ /}_r₎–β(z)–ε

forz∈Gand  <r<dG, which completes the proof of Theorem ..

Remark . [, Example .] shows that Theorem . is best possible to exponents ap-pearing in the Morrey condition.

3 Sobolev’s inequality in the casep–= 1andq–> 0

Letp–=  andq–> . In this section, we assume that there exists a constantq>  such that

sp(x)–log(c+s)q(x)–q≤Ctp(x)–log(c+t)q(x)–q, (.) whenever  <s<t<∞andx∈X. A direct consequence of (.) is that

sp(x)–log(c+s)q(x)–ρ≤Ctp(x)–log(c+t)q(x)–ρ (.) for  <ρ≤q. Letp*and be as in (.) and (.), respectively. Under this assumption,

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Theorem . Let p–= and q–> .Deﬁneby(.).Assume that p(·),q(·),α(·)andν(·)

satisfy(.)and(.).Then there exists a constant C> such that



μ(B(z, r))

B(z,r)

x,Uα(·),f(x)

–

dμ(x)

≤Cr–ν(z)log(c+ /r)–β(z)

for all z∈G and <r<dG, whenever f is a nonnegativeμ-measurable function on G

satisfyingf_L,ν,β;₍_G₎≤.

Forε>  andx∈X, we let

ρ–ε(r)≡



μ(B(x, r))

log(c+ /r)ε– (.)

as before.

Note that this deﬁnition extends naturally (.). For a nonnegativeμ-measurable func-tionf onG, we deﬁne the (non-linear) logarithmic potential

Lεf(x)≡

{y∈G:d(x,y)–ε_<_f₍_y₎_} ρ–ε

d(x,y)logc+f(y)–ε

×f(y)p(y)logc+f(y)q(y)dμ(y). (.)

To prove Theorem ., we modify Lemmas . and . in the following manner, respec-tively.

Lemma . Let <ε≤q/and deﬁne

F(δ)≡

{y∈B(x,δ):d(x,y)–ε_<_f₍_y₎_}

d(x,y)α(x)

μ(B(x, d(x,y)))

log(c+f(y)) log(c+ /d(x,y))

ε

f(y)dμ(y)

for <δ<dGand a nonnegativeμ-measurable function f on G.Then there exists a

con-stant C> such that

F(δ)≤Cδα(x)–ν(x)/p(x)logc+δ––(q(x)+β(x))/p(x)

+δα(x)+(p(x)–)ν(x)/p(x)logc+δ–β(x)–(q(x)+β(x))/p(x)+Lεf(x)

.

Proof We abbreviate{y∈B(x,δ) :d(x,y)–ε_<_f₍_y₎_}_to_E_{. For a constant}_k_{> , which will be}

speciﬁed in (.), we let

E_k≡y∈B(x,δ) :d(x,y)–ε_<_f₍_y₎_≤_k_, _E

k≡E\Ek. OnE

k, using (.) and the dyadic decomposition, we estimatef very crudely to obtain

E_k

d(x,y)α(x)

μ(B(x, d(x,y)))

ε

f(y)dμ(y)

≤klog(c+k)ε

B(x,δ)

d(x,y)α(x)

μ(B(x, d(x,y)))

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=klog(c+k)ε

∞

j=

B(x,–j+_δ₎_\_B₍_x_,–j_δ₎

d(x,y)α(x)

μ(B(x, d(x,y)))

logc+ /d(x,y)–εdμ(y)

≤klog(c+k)ε

∞

j=

B(x,–j+_δ₎

(–j+_δ₎α(x)

μ(B(x, –j+_δ₎₎

logc+ /–j+δ–εdμ(y)

≤klog(c+k)ε

∞

j=

–j+δα(x)logc+ /–j+δ–ε.

If we pass to a continuous variable and keep in mind thatα–> , then we obtain

E k

d(x,y)α(x)

μ(B(x, d(x,y)))

ε

f(y)dμ(y)

≤Cklog(c+k)ε δ



tα(x)–log(c+ /t)–εdt

≤Cklog(c+k)εδα(x)–α–/logc+δ––ε

δ



tα–/–_dt

=Cklog(c+k)εδα(x)logc+δ––ε. In summary, we obtained

E_k

d(x,y)α(x)

μ(B(x, d(x,y)))

ε

f(y)dμ(y)

≤Cklog(c+k)εδα(x)logc+δ––ε. (.)

OnE

k, we use (.) andε≤q/.

E_k

d(x,y)α(x)

μ(B(x, d(x,y)))

ε

f(y)dμ(y)

≤C

E k

d(x,y)α(x)

μ(B(x, d(x,y)))

ε f(y)

×

f(y)

k

p(y)–_log₍_c₊_f₍_y₎₎

log(c+k)

q(y)–ε dμ(y)

=C

E_k

d(x,y)α(x)₍_log₍_c_{+ /}_d₍_x_,_y₎₎₎–ε₍_log₍_c₊_f₍_y₎₎₎–ε

μ(B(x, d(x,y))) g(y)

k–p(y)

(log(c+k))q(y)–εdμ(y)

≤Cδα(x)logc+δ––ε

×

E

ρ–ε

d(x,y)logc+f(y)–εg(y) k

–p(y)

(log(c+k))q(y)–εdμ(y)

for allx∈G, whereg(y) =f(y)p(y)(log(c+f(y)))q(y). Hence, from (.), we deduce

F(δ)≤C

klog(c+k)εδα(x)logc+δ––ε+δα(x)logc+δ––ε

×

E

ρ–ε

d(x,y)logc+f(y)–εg(y) k

–p(y)

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We set

k≡δ–ν(x)/p(x)logc+δ––(q(x)+β(x))/p(x). (.) Then we have fory∈B(x,δ),

k–p(y)≤Ck–p(x)

and

log(c+k)–q(y)≤Clog(c+k)–q(x)

as we did in (.) and (.). Consequently, it follows that

F(δ)≤Cδα(x)–ν(x)/p(x)logc+δ––(q(x)+β(x))/p(x)

+δα(x)+(p(x)–)ν(x)/p(x)logc+δ–β(x)–(q(x)+β(x))/p(x)+Lεf(x)

.

Now the result follows.

The next lemma is a counterpart for Lemma ..

Lemma . Let f be a nonnegativeμ-measurable function satisfyingf_L,ν,β;₍_G₎≤and deﬁne Lεf(x)by(.).Ifis given by(.),then there exists a constant C> independent of z∈G,  <r<dGand f such that



μ(B(z, r))

B(z,r)

Lεf(x)dμ(x)≤Cr–ν(z)

log(c+ /r)–β(z). (.)

Proof Letf be a nonnegativeμ-measurable function on Gsatisfying f_L,ν,β;₍_G₎ ≤.

Write

Lεf(x) =

{y∈B(z,r):d(x,y)–ε_<_f₍_y₎_} ρ–ε

d(x,y)logc+f(y)–εg(y)dμ(y)

+

{y∈G\B(z,r):d(x,y)–ε_<_f₍_y₎_} ρ–ε

d(x,y)logc+f(y)–εg(y)dμ(y)

=L(x) +L(x), (.)

whereg(y)≡f(y)p(y)₍_log₍_c₊_f₍_y₎₎₎q(y)_{. Let us estimate}

B(z,r)

L(x)dμ(x)

=

B(z,r)

{x∈G:d(x,y)–ε_<_f₍_y₎_} ρ–ε

d(x,y)dμ(x)logc+f(y)–εg(y)dμ(y).

For eachy∈G, we decompose

x∈G:d(x,y)–ε<f(y)=

∞

j=

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Then, using (.), (.) and a crude estimate of the integrand, we obtain

B(z,r)

L(x)dμ(x)

≤C

B(z,r)

_∞

j=

B(y,j+_f₍_y₎–/ε₎

(log(c+ /(j_f₍_y₎–/ε₎₎₎ε–

μ(B(x, j+_f₍_y₎–/ε₎₎ dμ(x)

×logc+f(y)–εg(y)dμ(y) ≤C

B(z,r)

_∞

j=

logc+ /jf(y)–/εε–

logc+f(y)–εg(y)dμ(y).

From this and the fact thatf_L,ν,β;₍_G₎≤, we deduce

B(z,r)

L(x)dμ(x)≤C

B(z,r)

g(y)dμ(y)≤Cr–ν(z)_log₍_c_{+ /}_r₎–β(z)

μB(z, r),

so that



μ(B(z, r))

B(z,r)

L(x)dμ(x)≤Cr–ν(z)

log(c+ /r)–β(z). (.)

Note thatx∈B(z,r) andy∈/B(z, r) imply that



d(x,y)≤d(y,z)≤d(x,y)

and that

Bx, d(x,y)⊃Bz, d(z,y).

ForL, we insert (.) directly to obtain a pointwise estimate forx∈B(z,r):

L(x)≤C

G\B(z,r)

(log(c+ /d(x,y)))–

μ(B(x, d(x,y))) g(y)dμ(y)

≤C

G\B(z,r)

(log(c+ /d(z,y)))–

μ(B(z, d(z,y))) g(y)dμ(y).

As before, we decomposeG\B(z, r) dyadically to estimateL(x) from above. The result

is

L(x)≤C

∞

j=

B(z,j+_r₎_\_B₍_z_,j_r₎

(log(c+ /d(z,y)))–

μ(B(z, d(z,y))) g(y)dμ(y)

≤C

∞

j=

B(z,j+_r₎_\_B₍_z_,j_r₎

(log(c+ /(j+r)))–

μ(B(z, j+_r₎₎ g(y)dμ(y)

≤C

∞

j=

(log(c+ /(j+_r₎₎₎–

μ(B(z, j+_r₎₎

B(z,j+_r₎g(y)d

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From this and the fact thatfL,ν,β;₍_G₎≤, we deduce

L(x)≤C

∞

j=

j+r–ν(z)logc+ /j+r–β(z)–

≤C

dG

r

t–ν(z)_log₍_c_{+ /}_t₎–β(z)–dt

t

≤Cr–ν(z)log(c+ /r)–β(z)–

forx∈B(z,r). Hence, we see that



μ(B(z, r))

B(z,r)

L(x)dμ(x)≤Cr–ν(z)

log(c+ /r)–β(z)–. (.)

Thus, putting (.), (.) and (.) together, we obtain (.).

Proof of Theorem. We may assume thatf ≥ by considering|f|if necessary. Unlike Theorem ., we have freedom to chooseε, so that we deﬁneε≡min{α–/,q/}. Let us

obtain a pointwise estimate ofUα(·),f(x) forx∈G.

Forδ> , whereδis speciﬁed in (.) below, we decompose

Uα(·),f(x) =

B(x,δ)

+

G\B(x,δ)

=I(δ) +I(δ).

As in the proof of (.), we have

B(x,δ)

d(x,y)α(x)–ε

μ(B(x, d(x,y)))dμ(y)≤Cδ

α(x)–ε_≤ C

sinceα––ε=α––min{α–/,q/}=max{α–/,α––q/}> .

Note thatε≤q/. In view of Lemma ., we ﬁnd

I(δ)≤

B(x,δ)

d(x,y)α(x)–ε

+

{y∈B(x,δ):d(x,y)–ε_<_f₍_y₎_}

μ(B(x, d(x,y)))

(log(c+f(y))) log(c+d(x,y)–ε₎

ε

f(y)dμ(y)

≤Cδα(x)–ν(x)/p(x)logc+δ––(q(x)+β(x))/p(x)

+δα(x)+(p(x)–)ν(x)/p(x)logc+δ–β(x)–(q(x)+β(x))/p(x)+L

withL=Lεf(x). Moreover, Lemma . yields

I(δ)≤Cδα(x)–ν(x)/p(x)

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so that

Uα(·),f(x)≤C

+δα(x)+(p(x)–)ν(x)/p(x)logc+δ–β(x)–(q(x)+β(x))/p(x)+L. Now, letting

δ≡mindG,L–/ν(x)

log(c+L)–(β(x)+)/ν(x), (.) we obtain

Uα(·),f(x)≤C

 +L/p*(x)_log₍_c₊_L₎–α(x)β(x)/ν(x)–q(x)/p(x)+/p∗(x)

.

In view of Lemma ., we ﬁnd



μ(B(z, r))

B(z,r)

x,Uα(·),f(x)

–

dμ(x)

≤C 

μ(B(z, r))

B(z,r)

( +L)dμ(x)≤Cr–ν(z)_log₍_c_{+ /}_r₎–β(z)

,

which completes the proof of Theorem ..

Remark . [, Example .] shows that Theorem . is best possible to exponents ap-pearing in the Morrey condition.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All the authors participated in all the discussions and they read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan.}2_{Current address: Department of Mathematics}

and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan.

3_{Department of Mathematics Graduate School of Education, Hiroshima University, Higashi-Hiroshima, 739-8524, Japan.}

Acknowledgements

The authors are thankful to the anonymous referees for their valuable comments about the history of fractional integral operators on non-doubling measure spaces.

Received: 18 June 2012 Accepted: 12 November 2012 Published: 7 January 2013 References

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