One-weight inequalities with general weights for Riemann-Liouville transform and n-dimensional fractional integral operator in variable exponent Lebesgue spaces defined on Rn are investigated. In particular, we derive necessary and sufficient conditions governing one-weight inequalities for these operators on the cone of nonnegative decreasing functions in Lp(x) spaces.

1. Introduction

We derive necessary and sufficient conditions governing the one-weight inequality for the Riemann-Liouville operator
(1)Rαf(x)=1xα∫0xf(t)(x-t)1-αdt0<α<1
and n-dimensional fractional integral operator
(2)Iαg(x)=1|x|α∫|y|<|x|g(t)|x-t|n-αdt0<α<n,
on the cone of nonnegative decreasing function in Lp(x) spaces.

In the last two decades a considerable interest of researchers was attracted to the investigation of the mapping properties of integral operators in so-called Nakano spaces Lp(·) (see, e.g., the monographs [1, 2] and references therein). Mathematical problems related to these spaces arise in applications to mechanics of the continuum medium. For example, Ružicka [3] studied the problems in the so-called rheological and electrorheological fluids, which lead to spaces with variable exponent.

Weighted estimates for the Hardy transform
(3)(H1f)(x)=∫0xf(t)dt,x>0,
in Lp(·) spaces were derived in the papers [4] for power-type weights and in [5–9] for general weights. The Hardy inequality for nonnegative decreasing functions was studied in [10, 11]. Furthermore Hardy type inequality was studied in [12, 13] by Rafeiro and Samko in Lebesgue spaces with variable exponent.

Weighted problems for the Riemann-Liouville transform in Lp(x) spaces were explored in the papers [5, 14–16] (see also the monograph [17]).

Historically, one and two weight Hardy inequalities on the cone of nonnegative decreasing functions defined on R+ in the classical Lebesgue spaces were characterized by Arino and Muckenhoupt [18] and Sawyer [19], respectively.

It should be emphasized that the operator Iαf(x) is the weighted truncated potential. The trace inequity for this operator in the classical Lebesgue spaces was established by Sawyer [20] (see also the monograph [21], Ch.6 for related topics).

In general, the modular inequality
(*)∫01|∫0xf(t)dt|q(x)v(x)dx≤c∫01|f(t)|p(t)w(t)dt
for the Hardy operator is not valid (see [22], Corollary 2.3, for details). Namely, the following fact holds: if there exists a positive constant c such that inequality (*) is true for all f≥0, where q; p; w; and v are nonnegative measurable functions, then there exists b∈[0,1] such that w(t)>0 for almost every t<b; v(x)=0 for almost every x>b, and p(t) and q(x) take the same constant values a.e. for t∈(0;b) and x∈(0;b)∩{v≠0}.

To get the main result we use the following pointwise inequalities:
(4)c1(Tf)(x)≤(Rαf)(x)≤c2(Tf)(x),c3(Hg)(x)≤(Iαg)(x)≤c4(Hg)(x),
for nonnegative decreasing functions, where c1, c2, c3, and c4 are constants and are independent of f, g, and x, and
(5)Tf(x)=1x∫0xf(t)dt,Hg(x)=1|x|n∫|y|<|x|g(y)dy.

In the sequel by the symbol Tf≈Tg we mean that there are positive constants c1 and c2 such that c1Tf(x)≤Tg(x)≤c2Tf(x). Constants in inequalities will be mainly denoted by c or C; the symbol R+ means the interval (0,+∞).

2. Preliminaries

We say that a radial function f:Rn→R+ is decreasing if there is a decreasing function g:R+→R+ such that g(|x|)=f(x), x∈Rn. We will denote g again by f. Let p:Rn→Rn be a measurable function, satisfying the conditions p-=essinfx∈Rnp(x)>0, p+=esssupx∈Rnp(x)<∞.

Given p:Rn→R+ such that 0<p-≤p+<∞ and a nonnegative measurable function (weight) u in Rn, let us define the following local oscillation of p:
(6)φp(·),u(δ)=esssupx∈B(0,δ)∩suppup(x)-essinfx∈B(0,δ)∩suppup(x),
where B(0,δ) is the ball with center 0 and radius δ.

We observe that φp(·),u(δ) is nondecreasing and positive function such that
(7)limδ→∞φp(·),u(δ)=pu+-pu-,
where pu+ and pu- denote the essential infimum and supremum of p on the support of u, respectively.

By the similar manner (see [10]) the function ψp(·),u(η) is defined for an exponent p:R+↦R+ and weight v on R+:
(8)ψp(·),v(η)=esssupx∈(0,η)∩suppvp(x)-essinfx∈(0,η)∩suppvp(x).

Let D(R+) be the class of nonnegative decreasing functions on R+ and let DR(Rn) be the class of all nonnegative radially decreasing functions on Rn. Suppose that u is measurable a.e. positive function (weight) on Rn. We denote by Lp(x)(u,Rn) the class of all nonnegative functions on Rn for which
(9)Sp(f)=∫Rn|f(x)|p(x)u(x)dμ(x)<∞.

For essential properties of Lp(x) spaces we refer to the papers [23, 24] and the monographs [1, 2].

Under the symbol Ldecp(x)(u,R+) we mean the class of nonnegative decreasing functions on R+ from Lp(x)(u,Rn)∩DR(Rn).

Now we list the well-known results regarding one-weight inequality for the operator T. For the following statement we refer to [18].

Theorem A.

Let r be constant such that 0<r<∞. Then the inequity
(10)∫0∞v(x)(Tf(x))rdx≤c∫0∞v(x)(f(x))rdx,f∈Lr(v,R+),f↓
for a weight v holds, if and only if there exists a positive constant C such that for all s>0(11)∫s∞(sx)rv(x)dx≤C∫0sv(x)dx.

Condition (11) is called Br condition and was introduced in [18].

Theorem B (see [<xref ref-type="bibr" rid="B3">10</xref>]).

Let v be a weight on (0,∞) and p:R+→R+ such that 0<p-≤p+<∞, and assume that ψp(·),v(0+)=0. The following facts are equivalent:

there exists a positive constant c such that, for any f∈D(R+),
(12)∫0∞(Tf(x))p(x)v(x)dx≤C∫0∞(f(x))p(x)v(x)dx;

for any r,s>0,
(13)∫r∞(rsx)p(x)v(x)dx≤C∫0rv(x)sp(x)dx;

p|suppv≡p0 a.e. and v∈Bp0.

Proposition 1.

For the operators T,H,Rα, and Iα, the following relations hold:

(14)Rαf≈Tf,0<α<1,f∈D(R+);

(15)Iαg≈Hg,0<α<n,g∈DR(Rn).

Proof.

(a) Upper estimate: represent Rαf as follows:
(16)Rαf(x)=1xα∫0x/2f(t)(x-t)1-αdt+1xα∫x/2xf(t)(x-t)1-αdt=S1(x)+S2(x).
Observe that if t<x/2, then x/2<x-t. Hence
(17)S1(x)≤c1x∫0x/2f(t)dt≤cTf(x),
where the positive constant c does not depend on f and x. Using the fact that f is decreasing we find that
(18)S2(x)≤cf(x2)≤cTf(x).

Lower estimate follows immediately by using the fact that f is nonnegative and the obvious estimate x-t≤x and 0<t<x.

(b) Upper estimate: let us represent the operator Iα as follows:
(19)Iαg(x)=1|x|α∫|y|<|x|/2g(y)|x-y|n-αdy+1|x|α∫|x|/2<|y|<|x|g(y)|x-y|n-αdy≕S1′(x)+S2′(x).
Since |x|/2≤|x-y| for |y|<|x|/2 we have that
(20)S1′(x)≤c|x|n∫|y|<|x|/2g(y)dy≤cHg(x).
Taking into account the fact that f is radially decreasing on Rn we find that there is a decreasing function f:R+→R+ such that
(21)S2′(x)≤f(|x|2)·1|x|α∫|x|/2<|y|<|x||x-y|α-ndy.
Let Fx={y:|x|/2<|y|<|x|}. Then we have
(22)∫Fx|x-y|α-ndy=∫0∞|{y∈Fx:|x-y|α-n>t}|dt≤∫0|x|α-n|{y∈Fx:|x-y|α-n>t}|dt+∫|x|α-n∞|{y∈Fx:|x-y|α-n>t}|dt≕I1+I2.
It is easy to see that
(23)I1≤∫0|x|α-n|B(0,|x|)|dt=c|x|α;
while using the fact that n/(n-α)>1 we find that
(24)I2≤∫|x|α-n∞|{y∈Fx:|x-y|≤t1/(α-n)}|dt≤c∫|x|α-n∞tn/(α-n)dt=cα,n|x|α.
Finally we conclude that
(25)S2′(x)≤cf(|x|2)≤cHf(x).
Lower estimate follows immediately by using the fact that f is nonnegative and the obvious estimate |x-y|≤|x|, where 0<|y|<|x|.

We will also need the following statement.

Lemma 2.

Let r be a constant such that 0<r<∞. Then the inequality
(26)∫Rn(Hf(x))ru(x)dx≤C∫Rn(f(x))ru(x)dx,f∈Ldecr(u,Rn),
holds, if and only if there exists a positive constant C such that, for all s>0,
(27)∫|x|>s(s|x|)r|x|r(1-n)u(x)dx≤C∫|x|<s|x|r(1-n)u(x)dx.

Proof.

We will see that inequality (26) is equivalent to the inequality
(28)∫0∞u~(t)(Tf¯(t))rdt≤C∫0∞u~(t)(f¯(t))rdt,
where u~(t)=t(n-1)(1-r)u¯(t), f¯(t)=tn-1f(t), and u¯(t)=∫Sn-1u(tx¯)dσ(x¯).

Indeed, using polar coordinates in Rn we have
(29)∫Rn(Hf(x))ru(x)dx=∫Rnu(x)(1tn∫|y|<tf(y)dy)rdx=∫0∞tn-1(1tn∫|y|<tf(y)dy)r(∫Sn-1u(tx¯)dσx¯)dt=C∫0∞tn-1t-nrtr(1t∫0tτn-1f(τ)dτ)ru¯(t)dt=C∫0∞tn-1tr(1-n)u¯(t)(1t∫0tf¯(τ)dτ)rdt≤C∫0∞u~(t)(f¯(t))rdt=C∫0∞t(n-1)(1-r)t(n-1)r(f(t))rdt=C∫Rn(f(x))ru(x)dx.

Conversely taking the test function fr(x)=χB(0,r)(x)|x|1-n, r>0, in modular inequality (26), one can easily obtain inequality (27).

3. The Main Results

To formulate the main results we need to prove the following proposition.

Proposition 3.

Let u be a weight on Rn and p:Rn→R+ such that 0<p-≤p+<∞, and assume that φp(·),u(0+)=0. The following statements are equivalent:

there exists a positive constant C such that, for any f∈DR(Rn),
(30)∫Rn(Hf(x))p(x)u(x)dx≤C∫Rn(f(x))p(x)u(x)dx;

for any r,s>0,
(31)∫|x|>r(rs|x|n)p(x)u(x)dx≤C∫B(0,r)|x|(1-n)p(x)u(x)sp(x)dx;

p|suppu≡p0 a.e. and u∈Bp0.

Proof.

We use the arguments of [10]. To show that (a) implies (b) it is enough to test the modular inequality (30) for the function fr,s(x)=(1/s)χB(0,r)(x)|x|1-n, s,r>0. Indeed, it can be checked that
(32)Hfr,s(x)={1|x|ns∫|y|≤|x||y|1-ndy,if|x|≤r;1|x|ns∫|y|≤r|y|1-ndy,if|x|>r.

Further, we find that
(33)∫|x|>ru(x)(Hfr,s)p(x)dx≤∫Rnu(x)(Hfr,s)p(x)dx≤C∫Rnu(x)(1sχB(0,r)(x)|x|1-n)p(x)dx.
Therefore
(34)∫|x|>ru(x)(rs|x|n)p(x)dx≤C∫B(0,r)|x|(1-n)p(x)u(x)sp(x)dx.
To obtain (c) from (b) we are going to prove that condition (b) implies that φp(·),u(δ) is a constant function; namely, φp(·),u(δ)=pu+-pu- for all δ>0. This fact and the hypothesis on φp(·),u(δ) imply that φp(·),u(δ)≡0, and hence, due to (7),
(35)p|suppu≡pu+-pu-≡p0a.e.
Finally (31) means that u∈Bp0. Let us suppose that φp(·),u is not constant. Then one of the following conditions holds:

there exists δ>0 such that
(36)α=esssupx∈B(0,δ)∩suppup(x)<pu+<∞,
and, hence, there exists ϵ>0 such that
(37)|{|x|>δ:p(x)≥α+ϵ}∩suppu|>0,
or

there exists δ>0 such that
(38)β=essinfx∈B(0,δ)∩ suppup(x)>pu->0,
and then, for some ϵ>0,
(39)|{|x|>δ:p(x)≤β-ϵ}∩suppu|>0.

In case (i) we observe that condition (b), for r=δ, implies that
(40)∫|x|>δ(δs)p(x)u(x)|x|np(x)dx≤C∫B(0,δ)|x|(1-n)p(x)u(x)sp(x)dx.
Then using (36) we obtain, for s<min(1,δ),
(41)(δs)α+ϵ∫{|x|≥δ:p(x)≥α+ϵ}u(x)|x|np(x)dx≤Csα∫B(0,δ)u(x)|x|(1-n)p(x)dx,
which is clearly a contradiction if we let s↓0. Similarly in case (ii) let us consider the same condition (b), for r=δ, and fix now s>1. Taking into account (38) we find that
(42)1sβ-ϵ∫{|x|≥δ:p(x)≤β-ϵ}(δ|x|n)p(x)u(x)dx≤Csβ∫B(0,δ)|x|(1-n)p(x)u(x)dx,
which is a contradiction if we let s↑∞.

Finally, the fact that condition (c) implies (a) follows from [18,Theorem 1.7].

Theorem 4.

Let u be a weight on (0,∞) and p:R+→R+ such that 0<p-≤p+<∞. Assume that ψp(·),v(0+)=0. The following facts are equivalent:

there exists a positive constant C such that, for any f∈D(R+),
(43)∫R+(Rαf(x))p(x)u(x)dx≤C∫R+(f(x))p(x)u(x)dx;

condition (13) holds;

condition (c) of Theorem B is satisfied.

Proof.

Proof follows by using Theorem B and Proposition 1(a).

Theorem 5.

Let u be a weight on Rn and p:Rn→R+ such that 0<p-≤p+<∞, and assume that φp(·),u(0+)=0. The following facts are equivalent:

there exists a positive constant C such that, for any f∈DR(Rn),
(44)∫Rn(Iαf(x))p(x)u(x)dx≤C∫Rn(f(x))p(x)u(x)dx;

condition (31) holds;

condition (c) of Proposition 3 holds.

Proof.

Proof follows by using Propositions 3 and 1(b).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to Professor A. Meskhi for drawing their attention to the problem studied in this paper and helpful remarks. The authors are also grateful to the editor and anonymous reviewer for their careful review, valuable comments, and remarks to improve this paper.

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