DOI 10.4171/RLM/736
Functional Analysis — On maximal and potential operators with rough kernels in variable exponent spaces, by Humberto Rafeiro and Stefan Samko, com-municated on 12 February 2016.
Ab str act. — In the framework of variable exponent Lebesgue and Morrey spaces we prove some boundedness results for operators with rough kernels, such as the maximal operator, fractional maximal operator, sharp maximal operators and fractional operators. The approach is based on some pointwise estimates.
Key wo rds : Rough operators, variable exponent Lebesgue spaces, variable exponent Morrey spaces
Mat hemat ics S ubject Cla ss i fica tion: 42B25, 46E30
1. Introduction
The main operators of harmonic analysis, maximal, singular and of potential type, with the so called rough kernel have been widely studied, see e.g. [5, 6, 7, 8, 9, 12, 16, 17, 23, 24, 25, 26, 29, 34, 39] and references therein. The study of operators with rough kernels was based on the usage of the rotation method, which goes back to [15].
The theory of variable exponent spaces has received a thrust in recent years, due mainly to some applications, for example in the modeling of electro-rheological fluids [4, 3, 37] as well as thermo-electro-rheological fluids [11], in the study of image processing [1, 2, 13, 14, 18, 19, 41] and in di¤erential equations with non-standard growth [28, 33]. For details on variable Lebesgue spaces one can refer to [21, 22, 31, 32] and the references therein.
We want to study the boundedness of some rough operators in the framework of variable exponent Lebesgue and Morrey spaces. Since the classical proof is based upon the rotation method which is not well suited for the case of variable exponents we obtained some pointwise estimates in Section 3 to circumvent this problem.
Operator with rough kernel have already been considered in the variable exponent setting in [20], where their study was based on the extrapolation theory. Our approach is based upon some pointwise estimates and do not use extrap-olation theorems and allows, in particular, to consider potential operators and fractional maximal functions of variable order aðxÞ.
The paper is arranged as follows: In Section 2 we give necessary preliminaries on variable exponent Lebesgue and Morrey spaces. In Section 3 we provide
pointwise estimates for maximal and potential operators with rough kernels. Sections 4 and 5 contain the main results on the boundedness of such operators.
2. Variable exponent spaces
2.1. Variable Lebesgue spaces
We refer to the books [21, 22], but recall some basics we need on variable exponent Lebesgue spaces. Let U J Rn be an open set and pðÞ be a real-valued measurable function on U with values in [1, lÞ. We suppose that
1 a pa pðxÞ a pþ< l;
ð1Þ
where p :¼ ess infx A UpðxÞ, pþ :¼ ess supx A UpðxÞ. As usual, by p0ðxÞ ¼ pðxÞ=
ðpðxÞ 1Þ we indicate the conjugate exponent of pðxÞ and we have the relations ðp0Þ
þ¼ ðpÞ0andðp0Þ¼ ðpþÞ0. By LpðÞðUÞ we denote the space of real-valued
measurable functions f on U such that
IpðÞð f Þ :¼
Z
U
j f ðxÞjpðxÞdx < l:
Equipped with the norm
k f kpðÞ ¼ inf h > 0 : IpðÞ f h a1 ;
this is a Banach function space. The variable Lebesgue exponent norm has the following property k flk pðÞ¼ k f k l lpðÞ; ð2Þ for l b 1
p. In the subsequent sections, we will also need the relation between the modular IpðÞð f Þ and the norm.
Le mma2.1. For every f a LpðÞðUÞ, the inequalities
k f kpþ
LpðÞðUÞa IpðÞð f Þ a k f kLppðÞðUÞ; if k f kLpðÞðUÞa1; ð3Þ
k f kp
LpðÞðUÞa IpðÞð f Þ a k f kLpþpðÞðUÞ; if k f kLpðÞðUÞb1; ð4Þ
are valid.
In the sequel we use the well known log-condition
jpðxÞ pðyÞj a A
lnjx yj; jx yj a 1
2; x; y a U; ð5Þ
where A¼ AðpÞ > 0 does not depend on x, y. In the case U is unbounded, we also use the decay condition: there exists a number pl að1; lÞ; such that
jpðxÞ plj a
A lnðe þ jxjÞ: ð6Þ
In the sequel we use the following notation. For an open set U J Rn; by PlogðUÞ we denote the set of exponents pðxÞ with 1 < pa pþ < l, satisfying
the log and decay conditions (the latter required if U is unbounded).
By PðUÞ we denote the set of exponents p with 1 < pa pþ < lsuch that
the maximal operator is bounded in the space LpðÞðUÞ: The following fact is known, see [38] or [31, Theorem 2.62].
Lem ma 2.2. Let U be a bounded open set in Rn and p a PlogðUÞ. In the case
supx A UaðxÞpðxÞ < n, then
wUnBðx; rÞ jx jnaðxÞ p0ðÞ a CraðxÞ n pðxÞ; ð7Þ
where the constant C does not depend on x.
2.2. Variable exponent Morrey spaces
For more about Morrey spaces see [27, 35, 36]. Let l be a measurable function on U with values in ½0; n. We define the variable Morrey space LpðÞ; lðÞðUÞ as the set of all real-valued measurable functions on U such that
IpðÞ; lðÞð f Þ :¼ sup x A U ; r>0 rlðxÞ Z Uðx; rÞ j f ðyÞjpð yÞdy < l;
where Uðx; rÞ ¼ U B Bðx; rÞ. The norm in the space LpðÞ; lðÞðUÞ can be intro-duced in two forms, namely
k f k1¼ inf h > 0 : IpðÞ; lðÞ f h a1 and k f k2:¼ sup x A U; r>0 kr lðxÞ pðÞf w Uðx; rÞkpðÞ; ð8Þ
which are equal, see [10]. We take
k f kLpðÞ; lðÞðUÞ¼ k f k1 or k f kLpðÞ; lðÞðUÞ¼ k f k2
whichever is more convenient. We also have the following important property (see [10, 30]).
Le mma2.3. For every f a LpðÞ; lðÞðUÞ, the inequalities
k f kpþ
LpðÞ; lðÞðUÞa IpðÞ; lðÞð f Þ a k f kLppðÞ; lðÞðUÞ; if k f kLpðÞ; lðÞðUÞa1; ð9Þ
k f kp
LpðÞ; lðÞðUÞa IpðÞ; lðÞð f Þ a k f kLpþpðÞ; lðÞðUÞ; if k f kLpðÞ; lðÞðUÞb1; ð10Þ
are valid.
Similarly to property (2), from (8) we have
k fskLpðÞ; lðÞðUÞ¼ sup x A U; r>0 kr lðxÞ spðÞf w Uðx; rÞk s LspðÞðUÞ ð11Þ ¼ k f kLsspðÞ; lðÞðUÞ; ð12Þ whenever s b 1=p.
The following theorem is known, see [10].
Theorem 2.4. Let U be an open bounded set in Rn, 1 < pa pðxÞ a pþ < l,
0 a lðxÞ a lþ < n and p aPlogðUÞ. Then the maximal operator
MfðxÞ ¼ sup r>0 1 rn Z Uðx; rÞ j f ðyÞj dy
is bounded in the space LpðÞ; lðÞðUÞ.
3. Pointwise estimates of maximal and fractional rough operators
To deal with the boundedness problem of rough operators, we will use some pointwise estimates related to the maximal rough operator, the fractional maxi-mal rough operator and the fractional rough operator.
3.1. The case of the maximal operator
Let us consider the maximal operator with rough kernel
MWfðxÞ :¼ sup r>0 1 rn Z j yj<r
jWðyÞ f ðx yÞj dy; ð13Þ
in the case W¼ 1, we simply write MWf ¼ Mf . It is known that the operator MW
is bounded in LpðRn
Þ, 1 < p < l, if W a L1ðSn1Þ and W is a homogeneous
function of degree 0, i.e.
WðlxÞ ¼ WðxÞ ð14Þ
Lem ma 3.1. Let f a Ls0
locðR
nÞ, W satisfy (14) and W a LsðSn1Þ with s > 1. Then
MWfðxÞ a kWkLsðSn1Þ n1s ðMðj f js0ÞðxÞÞs 01: ð15Þ Proo f. We have 1 rn Z j yj<r jWðyÞ f ðx yÞj dy ¼ 1 rn Z r 0 %n1d% Z Sn1 jWðsÞ f ðx %sÞj ds akWkL sðSn1Þ rn Z r 0 %n1s d% %n1 Z Sn1 j f ðx %sÞjs0ds 1 s 0 akWkL sðSn1Þ n1srn n s Z r 0 %n1d% Z Sn1 j f ðx %sÞjs0ds 1 s 0 ¼kWkLsðSn1Þ n1s 1 rn Z j yj<r j f ðx yÞjs0dy 1 s 0
where we used (14), the Ho¨lder inequality on Sn1and on the intervalð0; rÞ. The
inequality (15) now follows. r
3.2. The case of fractional maximal operator
The fractional maximal operator MW; a is defined as
MW; afðxÞ :¼ sup r>0 1 rna Z j yj<r
jWðyÞj j f ðx yÞj dy: ð16Þ
Lem ma 3.2. Let f a LpðÞðUÞ, 0 < a < n, 1 < pa pþ < l and q be defined
pointwise by 1=qðxÞ ¼ 1=pðxÞ a=n. Then we have the following pointwise estimate
MW; afðxÞ ð17Þ a½MjWj n naðj f ðÞj pðÞ qðÞ n naÞðxÞ1 a n Z U j f ðyÞjpð yÞdy a n : ð18Þ
Proo f. Since pð yÞ
qð yÞþ apð yÞ n ¼ 1, we have 1 rna Z jx yj<r jWðx yÞj j f ðyÞj pð yÞ qð yÞj f ðyÞj apð yÞ n dy a 1 rn Z jx yj<r
jWðx yÞjnan j f ðyÞj n na pð yÞ qð yÞdy 1a=n ½IpðÞð f Þa=n a½MjWj n naðj f ðÞj pðÞ qðÞ n
naÞðxÞ1a=n½I
pðÞð f Þa=n
3.3. The case of sharp maximal operator
The sharp maximal operator, also known as the Fe¤erman–Stein operator, is a very well-known operator in harmonic analysis. We now introduce the rough sharp maximal operator as
MWafðxÞ ¼ fa WðxÞ ¼ sup r>0 1 rn Z j yj<r jWðyÞj j f ðx yÞ fBðx; rÞj dy
where fB is the integral average, namely
fB:¼ 1 jBj Z B fðyÞ dy:
Le mma3.3. Let W satisfy (14) and W a L1ðSn1Þ. Then
j fa WðxÞj a MWfðxÞ þ 1 nMfðxÞkWkL1ðSn1Þ: ð19Þ Pr oof. We have 1 rn Z j yj<r jWðyÞj j f ðx yÞ fBðx; rÞj dy a 1 rn Z j yj<r jWðyÞj j f ðx yÞj dy þ 1 rn Z j yj<r jWðyÞj j fBðx; rÞj dy ¼ I þ II : It is immediate that I a MWfðxÞ: ð20Þ We now estimate II : II a MfðxÞ 1 rn Z j yj<r jWðyÞj dy ð21Þ ¼ Mf ðxÞ 1 rn Z r 0 %n1d% Z Sn1 jWðsÞj ds ¼1 nMfðxÞkWkL1ðSn1Þ
where the first inequality follows from the fact that fBðx; rÞa MfðxÞ. Taking into
3.4. The case of fractional operator
In a similar fashion to the maximal operator with rough kernel (13) we can define the fractional operator with rough kernel
IWaðxÞfðxÞ :¼ Z
Rn
Wðx yÞ
jx yjnaðxÞ fðyÞ dy: ð22Þ
Lem ma 3.4. Let f a Ls0
locðR
nÞ, 1 < s < l, W satisfy (14), W a LsðSn1Þ and
aðxÞ > 0 almost everywhere. Then
jIWaðxÞð f wBðx; rÞÞðxÞj a 1 aðxÞ1s kWkLsðSn1Þr aðxÞ s ðIaðxÞðj f js 0 wBðx; rÞÞðxÞÞs 01; ð23Þ
at all points x a Rnsuch that aðxÞ > 0.
Proo f. We have Z Bðx; rÞ Wðx yÞ jx yjnaðxÞ fðyÞ dy ¼ Z r 0 %aðxÞ1d% Z Sn1 WðsÞ f ðx %sÞ ds akWkLsðSn1Þ Z r 0 %aðxÞ1s d% %aðxÞ1 Z Sn1 j f ðx %sÞjs0ds 1 s 0 akWkL sðSn1Þ aðxÞ1s raðxÞs Z r 0 %aðxÞ1d% Z Sn1 j f ðx %sÞjs0ds 1 s 0 ¼kWkLsðSn1Þ aðxÞ1s raðxÞs Z Bðx; rÞ j f js0ðyÞ jx yjnaðxÞdy 1 s 0 ¼ CkWkLsðSn1Þr aðxÞ s ðIaðxÞðj f js 0 wBðx; rÞÞÞs 01;
where we used (14), the Ho¨lder inequality on Sn1and on the intervalð0; rÞ. The
inequality (23) now follows. r
Cor ollar y 3.5. Let W a LsðSn1Þ, 1 < s < l, infx A UaðxÞ > 0 and
f a Lsloc0 ðRnÞ. Then
jIWaðxÞð f wBðx; rÞÞðxÞj a CkWkLsðSn1ÞraðxÞðMðj f js 0
ÞðxÞÞs 01: ð24Þ
Pr oof. The proof follows from the estimate (23) and the well-known pointwise
estimate
IaðxÞð f wBðx; rÞÞ a CraðxÞMfðxÞ
see, e.g. [31, Theorem 2.55] for the case of variable a. r
Le mma3.6. Let W a LsðSn1Þ, 1 < s < l, a a LlðUÞ. Then
jIWaðxÞð f wRnnBðx; rÞÞðxÞj a CkWk LsðSn1Þr aðxÞ s bðxÞ s 0ðIaðxÞþbðxÞðj f js 0 wRnnBðx; rÞÞÞ 1 s 0 ð25Þ
where b is an arbitrary function chosen so as
inf x A U bðxÞ aðxÞ s 1 >0 ð26Þ and C¼ Cða; s; bÞ. Pr oof. We have Z jx yj>r Wðx yÞ jx yjnaðxÞ fðyÞ dy ¼ Z l r %aðxÞ1d% Z Sn1 WðsÞ f ðx %sÞ ds akWkLsðSn1Þ Z l r %aðxÞ1s bðxÞ s 0 d% %aðxÞþbðxÞ1 Z Sn1j f ðx %sÞj s0 ds 1 s 0 ;
where b > 0 will be chosen later. Hence Z jx yj>r Wðx yÞ jx yjnaðxÞ fðyÞ dy a CkWkLsðSn1Þr aðxÞ s bðxÞ s 0 Z l r %aðxÞþbðxÞ1d% Z Sn1 j f ðx %sÞjs0ds 1 s 0
under the choice bðxÞ >aðxÞs1. Consequently, Z jx yj>r Wðx yÞ jx yjnaðxÞ fðyÞ dy a CkWkLsðSn1Þr aðxÞ s bðxÞ s 0 Z jx yj>r j f js0ðyÞ jx yjnaðxÞbðxÞ dy 1 s 0 ;
Lem ma 3.7. Let U be a bounded open set, p a PlogðUÞ,
sup
x A U
pðxÞaðxÞ < n; ð27Þ
infx A UaðxÞ > 0, W a LsðSn1Þ, f a LpðÞðUÞ and 1 < s < l. Then
jIWaðxÞð f wUnBðx; rÞÞðxÞj a CkWkLsðSn1ÞraðxÞ n pðxÞk f k LpðÞðUÞ: ð28Þ Proo f. By (25) we have jIWaðxÞð f wUnBðx; rÞÞðxÞj ð29Þ a CkWkLsðSn1Þr aðxÞ s bðxÞ s 0 Z jx yj>r j f js0ðyÞ jx yjnaðxÞbðxÞdy 1 s 0 :
We apply the Ho¨lder inequality with the variable exponent pðxÞs0 and get
jIWaðxÞð f wUnBðx; rÞÞðxÞj a CkWkLsðSn1Þr aðxÞ s bðxÞ s 0k j f js 0 k 1 s 0 pðÞ s 0 wUnBðx; rÞ jx jnaðxÞbðxÞ 1 s 0 pðÞ s 0 0 :
By the property (2) and the estimate (7), we then obtain
jIaðxÞ
Wð f wUnBðx; rÞÞðxÞj a CkWkLsðSn1ÞraðxÞ n pðxÞk f k
pðÞ;
with (7) applicable provided that
sup
x A U
½ pðxÞaðxÞ þ pðxÞbðxÞ < ns0: ð30Þ
Since we also have a restriction
inf
x A U½bðxÞ aðxÞ=ðs 1Þ > 0;
ð31Þ
it is not hard to see that the choice of bðxÞ satisfying both (30) and (31) is possible
by the assumption (27). r
Lem ma 3.8. Let U be a bounded open set, W a LsðSn1Þ, 1 < s < l,
infx A UaðxÞ > 0, p a PlogðUÞ and supx A U½lðxÞ þ aðxÞpðxÞ < n. Then
jIWaðxÞð f wUnBðx; rÞÞðxÞj a CkWkLsðSn1Þr
aðxÞnlðxÞ pðxÞ ð32Þ
Pr oof. From (25) we have Z jx yj>r Wðx yÞ fðyÞ jx yjnaðxÞdy akWkLsðSn1Þr aðxÞ s bðxÞ s 0 Z jx yj>r j f js0ðyÞ jx yjnaðxÞbðxÞdy |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} J 1 s 0
when inf bðxÞ aðxÞs1 >0. We now estimate J. By a dyadic decomposition and using the log-condition on the exponent p we obtain
J a CX l j¼1 Z 2jr<jx yj<2jþ1r ð2jrÞ lðxÞs 0
pð yÞj f js0ðyÞjx yjaðxÞþbðxÞnþ lðxÞs 0 pðxÞ dy ð33Þ a CX l j¼1 kð2jrÞlðxÞs 0pðÞj f js0ðÞk L pðÞ s 0ðBðx; 2jþ1rÞÞ k jx jaðxÞþbðxÞnþ lðxÞs 0 pðxÞk L pðÞ s 0 0 ðUnBðx; 2jrÞÞ : The term kð2jrÞlðxÞs 0pðÞj f js0ðÞk L pðÞ s 0ðBðx; 2jþ1rÞÞ
can be estimated by the modular, namely IpðÞ s 0 ðð2jrÞ lðxÞs 0 pðÞj f js0ðÞÞ a Cð2jþ1rÞlðxÞ Z Bðx; 2jþ1rÞj f ðyÞj pð yÞdy a CIpðÞ; lðÞð f Þ a C
uniformly, by the hypothesis IpðÞ; lðÞð f Þ a 1. Regarding the other term in (33) we
obtain k jx jaðxÞþbðxÞnþ lðxÞs 0 pðxÞk L pðÞ s 0 0 ðUnBðx; 2jrÞÞ að2jrÞaðxÞþbðxÞ s 0ðnlðxÞÞ pðxÞ whenaðxÞ þ bðxÞ þlðxÞspðxÞ0 pðxÞs0 < n.
Gathering all the above estimates, we obtain Z jx yj>r Wðx yÞ f ðyÞ jx yjnaðxÞ dy a CkWkLsðSn1Þr aðxÞnlðxÞpðxÞ Xl j¼1 ð2jÞaðxÞþbðxÞs 0ðnlðxÞÞpðxÞ 1 s 0 ;
whenever aðxÞ þ bðxÞ þlðxÞspðxÞ0 pðxÞs0 < n, which also implies the convergence of the series. Since infx A U
bðxÞ aðxÞs1 >0 and supx A U½lðxÞ þ aðxÞpðxÞ < n there
4. Theorems for the variable exponent Lebesgue spaces
A classical result regarding rough maximal operator states that if W satisfies (14) and W a L1ðSn1Þ that M
W is of typeðp; pÞ for 1 < p a l. The classical proof is
based upon the rotation method which is not well suited for the case of variable exponents.
We use the pointwise estimates obtained in Section 3 to obtain boundedness results regarding rough operators in the framework of variable exponent spaces.
The orem 4.1. Let W satisfy (14), W a LsðSn1Þ, s b ðp0Þ
þ, p
s0 aPðUÞ. Then the operator MWis bounded in the space LpðÞðUÞ.
Proo f. By (15) we have
kMWfkLpðÞðUÞa CkðMðj f j s0
ÞÞs 01k LpðÞðUÞ: By the property (2) we then get
kMWfkLpðÞðUÞa CkMðj f js 0 Þk 1 s 0 L pðÞ s 0ðUÞ a Ck j f js0k 1 s 0 L pðÞ s 0 ðUÞ ¼ Ck f kLpðÞðUÞ
where the second inequality comes from the fact thatsp0 aPðUÞ. r
We can now show the boundedness of the fractional maximal rough operator, namely we have the following:
The orem 4.2. Let W satisfy (14), W a LnasnðSn1Þ, s b 1a
n
q 0 þ and
ð1a=nÞq
s0 a PðUÞ. Then the operator MW; a isðLpðÞðUÞ ! LqðÞðUÞÞ-bounded.
Proo f. Let f a LpðÞðUÞ such that IpðÞð f Þ ¼ 1. Then
kMW; afkLqðÞðUÞak½MjWjnan ðj f ðÞj pðÞ qðÞ n naÞ1a=nk LqðÞðUÞ ¼ kMjWj n naðj f ðÞj pðÞ qðÞ n naÞk1a=n Lð1a=nÞqðÞðUÞ ak j f ðÞj pðÞ qðÞ n nak1a=n Lð1a=nÞqðÞðUÞ ¼ k j f ðÞjqðÞpðÞk LqðÞðUÞ a1
where the first inequality comes from the pointwise inequality (18), the second one from the Theorem 4.1 and the last one from the fact that IqðÞðj f j
pðÞ qðÞÞ ¼
IpðÞð f Þ ¼ 1 and the definition of variable exponent Lebesgue norm. The general case follows from the homogeneity of the fractional maximal rough operators. r
Theorem 4.3. Let W satisfy (14), W a LsðSn1Þ, s b ðp0Þ
þ, p
s0 aPðUÞ. Then the operator MaW is bounded in the space LpðÞðUÞ.
Pr oof. The proof follows from the pointwise estimate (19)
jMa
WfðxÞj a MWfðxÞ þ Mf ðxÞ
kWkL1ðSn1Þ
n ;
the fact that M is bounded whenever p a PðUÞ (this is valid under the assump-tionsp0 a PðUÞ, see [21, Theorem 4.37 or Theorem 3.38]) and Theorem 4.1. r
We now want to show the validity of a Sobolev type theorem for the rough fractional integral operator IWaðxÞ, namely.
Theorem 4.4. Let U be a bounded open set, W a LsðSn1Þ, 1 < s < l,
infx A UaðxÞ > 0, supx A UaðxÞpðxÞ < n, s b ðp0Þþand p a PlogðUÞ. Then the rough
fractional operator IWaðxÞisðLpðÞðUÞ ! LqðÞðUÞÞ-bounded, where 1 qðxÞ¼
1 pðxÞ
aðxÞ n .
Pr oof. For arbitrary fixed r > 0, from Corollary 3.5 and Lemma 3.7 we have
the following pointwise inequality
jIWaðxÞð f ÞðxÞj a CkWkLsðSn1ÞðraðxÞðMðj f js 0 wBðx; rÞÞðxÞÞs 01 þ raðxÞ n pðxÞk f k pðÞÞ: Taking r¼ k f kpðÞ ðMj f js0Þ1=s0 pðxÞ n we obtain jIWaðxÞð f ÞðxÞj a CkWkLsðSn1Þ½ðMj f js 0 Þ1=s01aðxÞ pðxÞn k f k aðxÞ pðxÞ n pðÞ : ð34Þ
To show the boundedness of IWaðxÞ in the variable exponent Lebesgue space, from (4), it is enough to show that
IqðÞðIWaðxÞfÞ a C
for all functions f in the unit sphere, i.e.k f kLpðÞðUÞ¼ 1, and the general result follows from homogeneity. From the pointwise inequality (34) it is enough to show the boundedness of
IqðÞð½ðMj f js 0 Þ1=s01aðxÞ pðxÞn Þ ¼ I pðÞððMj f js 0 Þ1=s0Þ: ð35Þ
To bound the right-hand side in (35) it is enough, from (4), to bound the norm, i.e. to bound
kðMj f js0Þ1=s0kLpðÞðUÞ:
From (2) and the fact that s >ðp0Þ
þ implies that p=s0 >1, we get
kðMj f js0Þ1=s0kLpðÞðUÞ¼ kðMj f j s0 Þk1=s0 L pðÞ s 0ðUÞ ak j f js0k1=s0 L pðÞ s 0ðUÞ ¼ k f kLpðÞðUÞ
which is bounded sincek f kLpðÞðUÞ¼ 1. r
Cor ollar y 4.5. Let U be a bounded open set, W a LsðSn1Þ, 1 < s < l,
infx A UaðxÞ > 0, supx A UaðxÞpðxÞ < n, s b ðp0Þþ and p a P log
ðUÞ. Then the rough fractional operator MW; aðxÞ is ðLpðÞðUÞ ! LqðÞðUÞÞ-bounded, where 1
qðxÞ¼ 1 pðxÞ aðxÞ n and MW; aðxÞfðxÞ :¼ sup r>0 1 rnaðxÞ Z j yj<r
jWðyÞj j f ðx yÞj dy:
Proo f. Use Theorem 4.4 and the inequality
MW; aðxÞf a CnIWaðxÞf ;
which follows from the definitions of the operators. r
5. Boundedness results in variable exponent Morrey spaces
We can now state the boundedness results for rough operators in the framework of variable exponent Morrey spaces.
The orem 5.1. Let U be an open bounded set in Rn, 0 a lðxÞ a lþ< n,
s bðp0Þ
þ, W a LsðSn1Þ, 1 < s < l, p a PlogðUÞ. Then the rough maximal
operator MWis bounded in the space LpðÞ; lðÞðUÞ.
Proo f. By (15) we have
kMWfkLpðÞ; lðÞðUÞa CkðMj f j s0
Þs 01k
LpðÞ; lðÞðUÞ: By the property (12) we then get
kMWfkLpðÞ; lðÞðUÞa CkMj f js 0 k 1 s 0 L pðÞ s 0;lðÞðUÞ a Ck j f js0k 1 s 0 L pðÞ s 0;lðÞðUÞ ¼ Ck f kLpðÞ; lðÞðUÞ
where we used Theorem 2.4 in the second inequality. r
Theorem 5.2. Let U be a bounded open set, W satisfy (14), W a LnasnðSn1Þ,
s b 1a n
q 0 þ and ð1a=nÞqs0 aPlogðUÞ. Then the rough fractional maximal operator MW; a isðLpðÞ; lðÞðUÞ ! LqðÞ; lðÞðUÞÞ-bounded.
Pr oof. Analyzing the proof of inequality (18) we obtain a similar result for the
case of variable exponent Morrey spaces, namely
MW; afðxÞ a C½MjWj n naðj f ðÞj pðÞ qðÞ n naðxÞÞ1an½I pðÞ; lðÞð f Þ a n;
where now C depends on the diameter of U . The rest of the proof now follows
closely the same lines as the proof of Theorem 4.2. r
Theorem 5.3. Let W satisfy (14), W a LsðSn1Þ, s b ðp0Þ
þ, p a P log
ðUÞ. Then the operator MWais bounded in the space LpðÞ; lðÞðUÞ.
Pr oof. As in the case of Theorem 4.3, the proof follows from the pointwise
inequality (19). r
Theorem 5.4. Let U be a bounded open set, W a LsðSn1Þ, 1 < s < l,
infx A UaðxÞ > 0, supx A U½lðxÞ þ aðxÞpðxÞ < n, s b ðp0Þþ and p a PlogðUÞ. Then
the rough fractional operator IWaðxÞ is ðLpðÞ; lðÞðUÞ ! LqðÞ; lðÞðUÞÞ-bounded,
where 1 qðxÞ¼ 1 pðxÞ aðxÞ nlðxÞ.
Pr oof. Gathering (24) and (32), we obtain the pointwise estimate
jIWaðxÞð f ÞðxÞj a CkWkLsðSn1ÞðraðxÞðMj f js 0 Þs 01 þ raðxÞ nlðxÞ pðxÞÞ: Taking r¼ ½ðMj f js0Þs 01 pðxÞ nlðxÞ we arrive at jIWaðxÞð f ÞðxÞj a CkWkLsðSn1Þ½ðMj f js 0 Þs 01 pðxÞ qðxÞ: ð36Þ By (36) we obtain IqðÞ; lðÞðIWaðxÞð f ÞÞ a IpðÞ; lðÞððMj f js 0 Þs 01Þ;
Ac know ledg ments . H. Rafeiro was partially supported by Pontificia Universidad Javeriana under the research project ‘‘Study of rough operators in non-standard function spaces’’, ID PPT: 7117. The research of S. Samko was supported by the grant No. 15-01-02732 of Russian Fund of Basic Research. We want to thank the anonymous referee for useful comments and suggestions, which lead us to the Theorem 5.4.
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Received 15 August 2015,
and in revised form 25 January 2016.
Humberto Rafeiro Pontificia Universidad Javeriana Departamento de Matema´ticas, Facultad de Ciencias Cra. 7 No. 43-82 Bogota´, Colombia silva-h@javeriana.edu.co Stefan Samko Universidade do Algarve Department of Mathematics, Faculty of Sciences 8005-139 Faro, Portugal ssamko@ualg.pt
