R E S E A R C H
Open Access
The
L
p
r
×
L
p
r
× · · · ×
L
p
k
r
k
boundedness of
rough multilinear fractional integral
operators in the Lorentz spaces
Vagif S Guliyev
1,2, Ismail Ekincioglu
3*and Sh A Nazirova
4*Correspondence: [email protected] 3Department of Mathematics, Dumlupınar University, Kütahya, Turkey
Full list of author information is available at the end of the article
Abstract
In this paper, we prove the O’Neil inequality for thek-linear convolution operator in the Lorentz spaces. As an application, we obtain the necessary and sufficient conditions on the parameters for the boundedness of thek-sublinear fractional maximal operatorM,α(f) and thek-linear fractional integral operatorI,α(f) with rough kernels from the spacesLp1r1×Lp2r2× · · · ×LpkrktoLqs, where
n/(n+
α
)≤p<q<∞, 0 <r≤s<∞,pis the harmonic mean ofp1,p2,. . .,pk> 1 andris the harmonic mean ofr1,r2,. . .,rk> 0.
MSC: Primary 42B20; 42B25; 42B35; secondary 47G10
Keywords: O’Neil inequality;k-linear convolution; rearrangement estimate; k-sublinear fractional maximal function;k-linear fractional integral; harmonic mean; Lorentz space
1 Introduction
Fractional maximal and fractional integral operators are two important operators in har-monic analysis and partial differential equations. Multilinear maximal operator and mul-tilinear fractional integral operator and related topics have been areas of research of many mathematicians such as Coifman and Grafakos [], Grafakos [, ], Grafakos and Kalton [], Kenig and Stein [], Ding and Lu [], Guliyev and Nazirova [, ], Ragusa [] and others.
Letk≥ be an integer andθj(j= , , . . . ,k) be fixed, distinct and nonzero real numbers,
and let f = (f, . . . ,fk). Thek-linear convolution operator f⊗gis defined by
(f⊗g)(x) =
Rnf(x–
θy)· · ·fk(x–θky)g(y)dy.
Let∈Ls(Sn–),s≥ andbe homogeneous of degree zero onRn, and let <α<n,
whereSn– is the unit sphere inRn. Thek-sublinear fractional maximal function with rough kernel is defined by
M,α(f)(x) =sup
r>
rn–α
|y|<r
(y)f(x–θy) . . .fk(x–θky)dy,
and thek-linear fractional integral with rough kernel is defined by
I,α(f)(x) =
Rn
(y)
|y|n–αf(x–θy)· · ·fk(x–θky)dy.
This paper consists of four sections. In Section , some lemmas needed to facilitate the proofs of our theorems and the O’Neil inequality for rearrangements of thek-linear convolution operator f ⊗g proved in [] are given. In Section , we prove the O’Neil inequality for thek-linear convolution operator in the Lorentz spaces. Finally, in Sec-tion , we obtain rearrangement estimates for the multilinear fracSec-tional maximal func-tion and multilinear fracfunc-tional integral with rough kernels. We prove the boundedness of the multilinear fractional maximal operator M,α and the multilinear fractional
in-tegral operator I,α with rough kernels from the spaces Lpr ×Lpr × · · · ×Lpkrk to
Lqs, n/(n+α)≤p<q<∞, <r≤s≤ ∞, wherepandr are the harmonic means of
p,p, . . . ,pk> andr,r, . . . ,rk> , respectively. We show that the conditions on the
pa-rameters ensuring the boundedness cannot be weakened.
2 Preliminaries
We need the following two generalized Hardy inequalities (see []) which are to be used in the proof of Theorem ..
We denote byM(Rn) the set of all extended real-valued measurable functions onRn.
Whenvis a non-negative measurable function on (,∞), we say thatvis a weight. We denoteW(t) =tw(τ)dτ,V(t) =tv(τ)dτandU(r,t) =tru(τ)dτ. For simplicity we sup-pose that <V(t) <∞, <W(t) <∞for allt> andV(∞) =∞,W(∞) =∞.
Lemma .[] Let <r≤s<∞and let v,w be weights.Then the inequality
∞
g(t)sw(t)dt
/s
≤C
∞
g(t)rv(t)dt
/r
(.)
holds for all non-negative and non-increasing g on(,∞)if and only if
A≡sup
t>
W/s(t)V–/r(t) <∞,
and the best constant C in(.)equals A.
Lemma .[, ] Let r,s∈(,∞)and let v,w be weights. (i)Let <r≤s<∞.Then the inequality
∞
t
t
g(τ)dτ
s
w(t)dt
/s
≤C
∞
g(t)rv(t)dt
/r
(.)
holds for all non-negative and non-increasing g on(,∞)if and only if A<∞,
A≡sup
t>
∞
t
w(τ) τs dτ
/s t
v(τ)τr
Vr(τ)dτ
/r
<∞,
(ii)Let <r≤,r≤s.Then(.)holds if and only if A<∞,
A≡sup
t>
t
∞
t
w(τ) τs dτ
/s
V–/r(t) <∞,
and the best constant C in(.)satisfies C≈A+A.
Lemma .[] Let r,s∈(,∞)and let u,v,w be weight functions. (i)Let <r≤s<∞.Then the inequality
∞
∞
t
g(τ)u(τ)dτ
s
w(t)dt
/s
≤C
∞
g(t)rv(t)dt
/r
(.)
holds for all non-negative and non-increasing g on(,∞)if and only if
A≡sup
t>
t
Us(t,τ)w(τ)dτ
/s
V–/r(t) <∞,
also
A≡sup
t>
W/s(t)
∞
t
Ur(τ,t)V–r(τ)v(τ)dτ
/r
<∞,
and the best constant C in(.)satisfies C≈A+A.
(ii)Let <r≤,r≤s.Then(.)holds if and only if A<∞and the best constant C in (.)equals A.
Lemma .[] Let r∈(,∞)and let u,v,w be weight functions. (i)Let <r<∞.Then the inequality
sup t>
∞
t
g(τ)u(τ)dτ
w(t)≤C
∞
g(t)rv(t)dt
/r
(.)
holds for all non-negative and non-increasing g on(,∞)if and only if
A≡sup
t>
w(t)
∞
t
Ur(τ,t)V–r(τ)v(τ)dτ
/r
<∞,
and the best constant C in(.)equals A.
(ii)Let <r≤and r≤s.Then(.)holds if and only if
A≡sup
t>
sup
<τ<t
U(τ,t)w(τ)V–/r(t) <∞,
and the best constant C in(.)equals A.
Lemma .[] Let r∈(,∞)and let u,v,w be weight functions. (i)Let <r<∞.Then the inequality
sup t>
t
k(t,τ)g(τ)u(τ)dτ
w(t)≤C
∞
g(t)rv(t)dt
/r
holds for all non-negative and non-increasing g on(,∞)if and only if
A≡sup
t>
w(t)
t
t
s
k(t,τ)V–(τ)dτ
r
v(s)ds
/r
<∞,
and the best constant C in(.)equals A.
(ii)Let <r≤,r≤s.Then(.)holds if and only if
A≡sup
t>
sup
τ>
Kt,min(τ,t)w(τ)V–/r(t) <∞,
and the best constant C in(.)equals A.
Letgbe a measurable function onRn. The distribution function ofgis defined by the
equality
λg(t) = x∈Rn:g(x)>t, t≥.
We shall denote byL(Rn) the class of all measurable functionsgonRn, which are finite
almost everywhere and such thatλg(t) <∞for allt> (see []). If a functiongbelongs
toL(Rn), then its non-increasing rearrangement is defined to be the functiong∗which is non-increasing on (,∞) equi-measurable with|g(x)|:
t> :g∗(t) >τ=λg(τ)
for allτ ≥. Moreover, by the Hardy-Littlewood theorem (see [], p.) and for every
f,f∈L(Rn),
Rn
f(x)f(x)dx≤
∞
f∗(t)f∗(t)dt.
Equi-measurable rearrangements of functions play an important role in various fields of mathematics. We give some of the main important properties (see, for example, []):
() if <t<t+τ, then
(g+h)∗(t+τ)≤g∗(t) +h∗(τ),
() if <p<∞, then
Rn
g(x)pdx=
∞
g∗(t)pdt,
() for anyt> and for any setE,
sup
|E|=t
E
g(x)dx=
t
g∗(τ)dτ.
We denote byWLp(Rn) the weakLpspace of all measurable functionsgwith finite norm
fWLp=sup
t>
t/pf∗(t) <∞, ≤p<∞.
Definition . If <p,q<∞, then the Lorentz spaceLpq(Rn) is the set of all classes of
measurable functionsf with the finite quasi-norm
fpq≡ fLpq=
∞
t/pf∗(t)qdt
t
/q
.
If <p≤ ∞,q=∞, thenLp∞(Rn) =WLp(Rn).
If ≤q≤porp=q=∞, then the functionalfpqis a norm (see []). Ifp=q=∞,
then the spaceL∞∞(Rn) is denoted byL∞(Rn).
In the case <p,q<∞we define
f(pq)=
∞
t/pf∗∗(t)qdt
t
/q
(with the usual modification if <p≤ ∞,q=∞) which is a norm onLpq(Rn) for <p<∞,
≤q≤ ∞orp=q=∞. If <p≤ ∞and ≤q≤ ∞, then
fpq≤ f(pq)≤pfpq
that is, the quasi-normsfpqandf(pq)are equivalent.
Lemma .[] Let f,f, . . . ,fk∈L(Rn),k≥.Then,for all x∈Rnand nonzero real
num-bersθ, . . . ,θk,
Rn
f(x–θy)f(x–θy)· · ·fk(x–θky)dy≤Cθ
∞
f∗(t)f∗(t)· · ·fk∗(t)dt, (.)
where Cθ=|θ. . .θk|–n.
Let f = (f,f, . . . ,fk) and define
f∗(t) =f∗(t)· · ·fk∗(t), f∗∗(t) =
t
t
f∗(τ)· · ·fk∗(τ)dτ, t> .
In the following, we give the O’Neil inequality for rearrangements of the multilinear convolution operator f⊗gproved in [].
Lemma .[] Let f,f, . . . ,fk,g∈L(Rn).Then,for all <t<∞,the following inequality
holds:
(f⊗g)∗∗(t)≤Cθ
tf∗∗(t)g∗∗(t) +
∞
t
f∗(s)g∗(s)ds
. (.)
Corollary .[] Let f,f, . . . ,fk∈L(Rn)and g∈WLm(Rn), <m<∞.Then
(f⊗g)∗(t)≤(f⊗g)∗∗(t)
≤CθgWLm
mt–/m
t
f∗(τ)dτ+
∞
t
τ–/mf∗(τ)dτ
Lemma .[] Let f,f, . . . ,fk,g∈L(Rn).Then for any t>
(f⊗g)∗∗(t)≤Cθ
∞
t
f∗∗(t)g∗∗(t)dt. (.)
Corollary . Let f,f, . . . ,fk∈L(Rn)and g∈WLm(Rn), <m<∞.Then
(f⊗g)∗(t)≤(f⊗g)∗∗(t)≤mCθgWLm
∞
t
τ–/mf∗∗(τ)dτ. (.)
3 O’Neil inequality for the multilinear convolutions in the Lorentz spaces In this section, we prove the O’Neil inequality for the multilinear convolutions in the Lorentz spaces. It is said thatpis the harmonic mean ofp,p, . . . ,pk> if /p= /p+/p+ · · ·+ /pk. Iffj∈Lpjrj(Rn),j= , , . . . ,k, then we say that f∈Lpr×Lpr× · · · ×Lpkrk(R
n).
Theorem .(O’Neil inequality fork-linear convolution in the Lorentz spaces) Suppose that <m<∞,g∈WLm(Rn),p and r are the harmonic means of p,p, . . . ,pk> and
r,r, . . . ,rk> ,respectively.If <p<m, <r≤s<∞or m/( +m)≤p≤, <r≤,r≤
s<∞or p=m, <r<∞,s=∞or p=m, <r≤,s=∞f∈Lpr×Lpr×· · ·×Lpkrk(Rn)
and/p– /q= /m,thenf⊗g∈Lqs(Rn)and
f⊗gqsCθK(p,q,r,s,m)
k
j=
fjpjrjgWLm,
where K(p,q,r,s,m) =κand
κ≈
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
mA+mA+A+A, if <p<m, <r≤s<∞,
mA+mA+A, if m
+m ≤p≤, <r≤,r≤s<∞
mA+mA, if p=m, <r<∞,s=∞,
mA+mA, if p=m, <r≤,s=∞
and
A=
mq s(m+q)
/s
r p
/r
, A=
r p
mq s(q–m)
/s
p r
/r
,
A=
r p
/r
mq s(q–m)
/s
, A=
m+/s
r p
/r
Bs+ ,sm/q/s,
A=
m+/rr p
q s
/s
Br+ ,rm/p–r/r, A=m
r p
/r
,
A=
m+/rBr+ ,rm/p–r/r,
A=
r p
/r
p p–r
+/r
B
r+ , r
p–r
/r
, A=
r p
/r
.
Here B(s,r) =( –τ)s–τr–dτ is the beta function.
Lpr×Lpr× · · · ×Lpkrk(R
n). By using inequality (.), we have
f⊗gqs=(f⊗g)∗(t)t/q–/sLs(,∞)
≤Cθ
∞
mt–/m
t
f∗(τ)dτ+
∞
t
τ–/mf∗(τ)dτ
s
ts/q–dt
/s
≤Cθm
∞
t
f∗(τ)dτ
s
t–s/m+s/q–dt
/s
+Cθ
∞
∞
t
τ–/mf∗(τ)dτ
s
ts/q–dt
/s
.
CaseI. Suppose that <p<m(equivalentlym<q<∞), <r≤s<∞. From Lemma ., for the validity of the inequality for <r≤s<∞
∞ t t
f∗(τ)dτ
s
ts–s/m+s/q–dt
/s
≤C
∞
t/pf∗(t)rdt
t
/r
, (.)
the necessary and sufficient condition is
A=sup
t>
W/s(t)V–/r(t) =
mq s(m+q)
/s r p /r sup t>
t/m+/q–/p<∞
⇔ /p– /q= /mandA=
mq s(m+q)
/s
r p
/r
and
A=sup
t>
∞
t
w(τ) τs dτ
/s t
v(τ)τr
Vp(τ)dτ
/r
= r
psupt>
∞
t
τ–s/m+s/q–dτ
/s t
τr/p–+r–rr/pdτ
/r
= r
p
mq s(q–m)
/s p r /r sup t>
t–/m+/q–/p<∞
⇔ /p– /q= /mandA=
r p
mq s(q–m)
/s
p r
/r
.
Note that the best constant C in (.) satisfies C ≈ A +A. Furthermore, from Lemma . for the validity of the inequality for <r≤s<∞
∞
∞
t
τ–/mf∗(τ)dτ
s
ts/q–dt
/s
≤C
∞
t/pf∗(t)rdt
t
/r
, (.)
the necessary and sufficient condition is
A=msup
t>
t
t/m–τ/msτs/q–dτ
/s t
τr/p–dτ
–/r =m r p /r sup t> t
t/m–τ/msτs/q–dτ
/s
=m+/s
r p
/r
Bs+ ,sm/q/ssup t>
t–/m+/q–/p<∞
⇔ /p– /q= /mandA=
m+/s
r p
/r
Bs+ ,sm/q/s
and
A=sup
t>
W/s(t)
∞
t
Ur(τ,t)V–r(τ)v(τ)dτ
/r =m r p q s /s sup t>
t/q
∞
t
τ/m–t/mrτ–rr/p+r/p–dτ
/r =m r p q s
/s ∞
λ/m– rλ–r/p–dλ
/r sup
t>
t/q+/m–/p
=m+/rr p
q s
/s
–λ/mrλ–r/m+r/p–dλ
/r sup
t>
t/q+/m–/p
=m+/rr p
q s
/s
( –τ)rτ–r+rm/p–dτ
/r sup
t>
t/q+/m–/p
=m+/rr p
q s
/s
Br+ ,rm/p–r/rsup t>
t/q+/m–/p<∞
⇔ /p– /q= /mandA=
m+/rr p
q s
/s
Br+ ,rm/p–r/r.
Note that the best constantCin (.) satisfiesC≈A+A.
CaseII. Letm/( +m)≤p≤, <r≤ andr≤s<∞. From Lemma ., for the validity of inequality (.), the necessary and sufficient condition isA<∞and
A=sup
t>
t
∞
t
w(τ) τs dτ
/s
V–/r(t)
= r p /r sup t> t ∞ t
τ–s/m+s/q–dτ
/s
t–/p
=
r p
/r mq
s(q–m)
/s sup
t>
t–/m+/q–/p
⇔ /p– /q= /mandA=
r p
/r
mq s(q–m)
/s
.
Note that the best constantCin (.) satisfiesC≈A+A. From Lemma ., for the validity of inequality (.), the necessary and sufficient condition isA<∞. Consequently, using inequalities (.), (.) and applying the Hölder inequality, we obtain
f⊗gqs≤Cθ
mC+C
∞
t/pf∗(t)rdt
t
/r
gWLm
=CθK(p,q,r,s,m)
∞ k j=
fj∗(t)t/pjrdt
t
/r
≤CθK(p,q,r,s,m) k j= ∞
fj∗(t)t/pjrjdt
t
/rj
gWLm
=CθK(p,q,r,s,m)
k
j=
fjpjrjgWLm.
CaseIII. Letp=m,q=s=∞, <r<∞orp=m,q=s=∞, <r≤ and f∈Lpr×
Lpr× · · · ×Lpkrk(R
n). By using inequality (.), we have
f⊗g∞=sup
t>
(f⊗g)∗(t)
≤Cθsup
t>
mt–/m
t
f∗(τ)dτ+
∞
t
τ–/mf∗(τ)dτ
gWLm
≤Cθmsup
t>
t–/m
t
f∗(τ)dτ
+sup t>
∞
t
τ–/mf∗(τ)dτ
gWLm
≤Cθm
∞
t/pf∗(t)rdt
t
gWLm.
From Lemma ., for the validity of the inequality for <r<∞
sup t>
t–/m
t
f∗(τ)dτ
≤C
∞
t/pf∗(t)rdt
t
/r
, (.)
the necessary and sufficient condition is
A=
r p /r sup t>
t–/m
t
t
s
τ–r/pdτ
r
sr/p–ds
/r
=
r p
/r r
p–rsupt>
t–/m
t
t–r/p–τ–r/prτr/p–dτ
/r = r p /r p p–r
+/r
–τ–r/psτr/p–dτ
/r sup
t>
t–/m–/p+
= r p /r p p–r
+/r
B
r+ , r
p–r
/r sup
t>
t–/m–/p+<∞
⇔ p=mandA=
r p
/r p
p–r
+/r
B
r+ , r
p–r
/r
.
From Lemma ., for the validity of the inequality for <r≤
sup t>
t–/m
t
f∗(τ)dτ
≤C
∞
t/pf∗(t)rdt
t
/r
, (.)
the necessary and sufficient condition is
A=sup
t>
sup
τ>
Kt,min(τ,t)w(τ)V–/r(t) =
r p /r sup t>
t/m–/p<∞
⇔ p=mandA=
r p
/r
From Lemma ., for the validity of the inequality for <r<∞
sup t>
∞
t
τ–/mf∗(τ)dτ
≤C
∞
t/pf∗(t)rdt
t
/r
, (.)
the necessary and sufficient condition isA
A=sup
t>
∞
t
Ur(τ,t)V–r(τ)v(τ)dτ
/r
=
∞
t
τ/m–t/mrτ–rr/p+r/p–dτ
/r
=
∞
λ/m– rλ–r/p–dλ
/r sup
t>
t/m–/p
=m
r
p
–λ/mrλ–r/m+r/p–dλ
/r sup
t>
t/m–/p
=m
r
p
( –τ)rτ–r+rm/p–dτ
/r sup
t>
t/m–/p
=m
r
p
Br+ ,rm/p–r/rsup t>
t/m–/p<∞
⇔ p=mandA=
mr p
Br+ ,rm/p–r/r.
Furthermore, from Lemma ., for the validity of the inequality for <r≤
sup t>
∞
t
τ–/mf∗(τ)dτ
≤C
∞
t/pf∗(t)rdt
t
/r
, (.)
the necessary and sufficient condition is
A=sup
t>
sup
<τ<t
U(τ,t)w(τ)V–/r(t)
=msup t>
sup
<τ<t
t/m–τ/m
t
τr/p–dτ
–/r
=m
r p
/r sup
t>
sup
<τ<t
t/m–τ/mt–/p
=m
r p
/r sup
t>
t/m–/p<∞
⇔ p=mandA=m
r p
/r
.
Thus the proof of Theorem . is completed.
Corollary .[] Suppose that <m<∞,g∈WLm(Rn)and p is the harmonic mean
/p– /q= /m,thenf⊗g∈Lq(Rn)and
f⊗gq≤CθK(p,q,m)
k
j=
fjpjgWLm,
where in the case <p=r<m,q=s
K(p,q,m) =m
m m+q
/q
+m
m q–m
/q
+m+/qBq+ ,m/q+m+/pBp+ ,pm/p–p/p,
and in the case m/( +m)≤p=r≤,m<q=s
K(p,q,m) =m
m m+q
/q
+m+ m
q–m
/q
+m+/qBq+ ,m/q.
4 TheLp1r1×Lp2r2× ··· ×Lpkrkboundedness of rough multilinear fractional
integral operators
In this section, we prove the Sobolev type theorem for the rough multilinear fractional integralI,αf.
Lemma . Let <α<n,be homogeneous of degree zero onRn,∈L
n/(n–α)(Sn–)and
g(x) = (x) |x|n–α.
Then g∈WLn/(n–α)(Rn)and
gWLn/(n–α)=n
α/n–
Ln/(n–α), (.)
where
Ln/(n–α)=
Sn–
xn/(n–α)dσx
(n–α)/n
.
Proof Note that
g∗(t) = (nt)α/n–
Ln/(n–α), g
∗∗(t) = n
αg
∗(t),
thereforeg∈WLn/(n–α)(Rn) and equality (.) is valid.
Lemma . Suppose that <α<n,∈Ls(Sn–)and s≥.Then
M,αf(x)≤I||,α
|f|(x), (.)
Proof Indeed, for allr> , we have
I||,α
|f|(x)≥
E(,r) |(y)|
|y|n–αf(x–θy) . . .fk(x–θky)dy
≥
rn–α
E(,r)
(y)f(x–θy) . . .fk(x–θky)dy,
whereE(,r) is the open ball centered at the origin of radiusr. Taking supremum over all
r> , we get (.).
By Lemmas . and ., we obtain a pointwise rearrangement estimate of the rough
k-sublinear fractional maximal integralM,αfandk-linear fractional integralI,αf.
Lemma .[] Suppose thatis homogeneous of degree zero onRnand∈L
n/(n–α)(Sn–),
<α<n.Then the following inequalities hold:
(I,αf)∗(t)≤(I,αf)∗∗(t)
≤Cθnα/n–Ln/(n–α)
n
αt
α/n–
t
f∗(τ)dτ+
∞
t
τα/n–f∗(τ)dτ
,
(M,αf)∗(t)≤(M,αf)∗∗(t)
≤Cθnα/n–Ln/(n–α)
n
αt
α/n–
t
f∗(τ)dτ+
∞
t
τα/n–f∗(τ)dτ
.
From Theorem . and Lemma ., we get the following.
Theorem . Letbe homogeneous of degree zero onRn,∈Ln/(n–α)(Sn–), <α<n,
p and r be the harmonic means of p,p, . . . ,pk> and r,r, . . . ,rk> ,respectively,and
<r≤s≤ ∞,q satisfy/q= /p–α/n.If <p<n/α, <r≤s<∞or n/(n+α)≤p≤, <r≤s<∞or p=n/α,r= ,then I,αis a bounded operator from Lpr×Lpr× · · · ×
Lpkrk(R
n)to L
qs(Rn)and
I,αfqs≤Cθnα/n–K
p,q,r,s,n/(n–α)Ln/(n–α) k
j= fjpjrj.
Corollary .[] Letbe homogeneous of degree zero onRn,∈Ln/(n–α)(Sn–), <α<n,
p be the harmonic mean of p,p, . . . ,pk> ,and q satisfy/q= /p–α/n.Then I,αis a
bounded operator from Lp×Lp× · · · ×Lpk(R
n)to L
q(Rn)for n/(n+α)≤p<n/α(
equiv-alently≤q<∞)and
I,αfq≤Cθnα/n–K
p,q,n/(n–α)Ln/(n–α) k
j= fjpj.
Corollary .[] Letbe homogeneous of degree zero onRn,∈L
n/(n–α)(Sn–), <α<n,
p be the harmonic mean of p,p, . . . ,pk> ,and q satisfy/q= /p–α/n.Then M,αis a
equiv-alently≤q≤ ∞)and
M,αfq≤Cθnα/n–K
p,q,n/(n–α)Ln/(n–α) k
j= fjpj,
when n/(n+α)≤p<n/α,and
M,αf∞≤CθLn/(n–α) k
j=
fjpj, p=n/α.
Finally, in the following theorem we obtain the necessary and sufficient conditions for the roughk-linear fractional integral operatorI,αto be bounded from the Lorentz spaces
Lpr×Lpr× · · · ×Lpkrk(R
n) toL
qs(Rn),n/(n+α)≤p<q<∞, <r≤s<∞.
Theorem . Let <α<n,be homogeneous of degree zero onRn,∈L
n/(n–α)(Sn–),
p and r be the harmonic means of p,p, . . . ,pk> and r,r, . . . ,rk> ,respectively.If <
p<n/α, <r≤s<∞or n/(n+α)≤p≤, <r≤s<∞,then the condition/p– /q=α/n is necessary and sufficient for the boundedness of I,αfrom Lpr×Lpr× · · · ×Lpkrk(Rn)
to Lqs(Rn).
Proof Sufficiency of the theorem follows from Theorem ..
Necessity.Suppose that the operatorI,αis bounded fromLpr×Lpr× · · · ×Lpkrk(R
n)
toLqs(Rn), andn/(n+α)≤p<n/α(equivalently ≤q<∞). Define ft(x) =: f(tx) fort>
andfpr=
k
j=fjpjrj. Then it can be easily shown that
ftpr= k
j=
(fj)tp
jrj=
k
j=
t–n/pjf
jpjrj=t
–n/pf pr
and
I,αft(x) =t–αI,αf(tx), I,αftqs=t–α–n/qI,αfqs.
Since the operatorI,αis bounded fromLpr×Lpr× · · · ×Lpkrk(R
n) toL
qs(Rn), we have
I,αfqs≤Cfpr,
whereCis independent of f. Then we get
I,αfqs=tα+n/qI,αftqs≤Ctα+n/qftpr=Ctα+n/q–n/pfpr.
If /p< /q+α/n, then for all f∈Lpr×Lpr× · · · ×Lpkrk(R
n) we haveI
,αfLq,s= as
t→. If /p> /q+α/n, then for all f∈Lpr×Lpr× · · · ×Lpkrk(R
n) we haveI
,αfqs=
ast→ ∞. Therefore we get /p= /q+α/n.
Corollary . [] Let <α <n,p be the harmonic mean of p,p, . . . ,pk> ,be
ho-mogeneous of degree zero on Rn and∈L
n/(n–α)(Sn–). If n/(n+α)≤p<n/α, then the
condition /p– /q =α/n is necessary and sufficient for the boundedness of I,α from
Remark . Note that the sufficiency part of Corollary . was proved in [] and in the case≡ in [], and in the case∈Ls(Sn–),s>n/(n–α) in [].
Theorem . Let <α<n,be homogeneous of degree zero onRn,∈L
n/(n–α)(Sn–),
p and r be the harmonic means of p,p, . . . ,pk> and r,r, . . . ,rk> ,respectively.If <
p<n/α, <r≤s<∞or n/(n+α)≤p≤, <r≤s<∞,then the condition/p– /q=α/n is necessary and sufficient for the boundedness of M,αfrom Lpr×Lpr× · · · ×Lpkrk(R
n)
to Lqs(Rn).
Proof Sufficiency part of the theorem follows from Theorem . and Lemma ..
Necessity.Suppose that the operatorM,αis bounded fromLpr×Lpr×· · ·×Lpkrk(R
n)
toLqs(Rn), andn/(n+α)≤p<n/α, <r≤s<∞. Then we have
M,αft(x) =t–αM,αf(tx)
and
M,αftqs=t–α–
n qM
,αfqs.
By the same argument in Theorem ., we obtain p–q=α
n.
Corollary .[] Let <α<n,p be the harmonic mean of p,p, . . . ,pk> ,be
ho-mogeneous of degree zero onRnand∈Ln/(n–α)(Sn–).If n/(n+α)≤p≤n/α,then the
condition /p– /q=α/n is necessary and sufficient for the boundedness of M,α from
Lp×Lp× · · · ×Lpk(R
n)to L q(Rn).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1Institute of Mathematics and Mechanics, Baku, Azerbaijan.2Department of Mathematics, Ahi Evran University, Kirsehir, Turkey.3Department of Mathematics, Dumlupınar University, Kütahya, Turkey.4Khazar University, Baku, Azerbaijan.
Acknowledgements
The research of V Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan, Grant EIF-2014-9(15)-46/10/1 and by the grant of Presidium of Azerbaijan National Academy of Science 2015.
Received: 5 November 2014 Accepted: 30 January 2015
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