# Generalized Fractional Integral Operators on Generalized Local Morrey Spaces

Authors: V. S. Guliyev; A. F. Ismayilova; A. Kucukaslan; A. Serbetci
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

We study the continuity properties of the generalized fractional integral operatorIρ on the generalized local Morrey spaces LMp,φ{x0} and generalized Morrey spaces Mp,φ. We find conditions on the triple (φ1,φ2,ρ) which ensure the Spanne-type boundedness of Iρ from one generalized local Morrey space LMp,φ1{x0} to another LMq,φ2{x0}, 1<p<q<∞, and from LM1,φ1{x0} to the weak space WLMq,φ2{x0}, 1<q<∞. We also find conditions on the pair (φ,ρ) which ensure the Adams-type boundedness of Iρ from Mp,φ1/p to Mq,φ1/q for 1<p<q<∞ and from M1,φ to WMq,φ1/q for 1<q<∞. In all cases the conditions for the boundedness of Iρ are given in terms of Zygmund-type integral inequalities on (φ1,φ2,ρ) and (φ,ρ), which do not assume any assumption on monotonicity of φ1(x,r), φ2(x,r), and φ(x,r) in r.


---

## Body

## 1. Introduction

The theory of boundedness of classical operators of the real analysis, such as the maximal operator, Riesz potential, and the singular integral operators, from one weighted Lebesgue space to another one is well studied by now. Along with weighted Lebesgue spaces, Morrey-type spaces also play an important role in the theory of partial differential equations. Morrey spaces were first introduced by Morrey [1] in 1938 to study local behavior properties of the solutions of second-order elliptic partial differential equations. Furthermore, there are important applications for the theory of partial differential equations related to obtaining sharp a priori estimates and studying regularity properties of solutions in Morrey spaces. Recently, they proved to be useful also for the Navier-Stokes equations [2, 3]. However no attempt has been made to extend these results by using more generalized Morrey-type spaces. For example, sharp regularity properties of strong solutions to elliptic and parabolic equations with VMO coefficients in terms of general Morrey-type spaces are a good place to start the investigation.Forx∈Rn and r>0, we denote by B(x,r) the open ball centered at x of radius r and by B∁(x,r) denote its complement. Let |B(x,r)| be the Lebesgue measure of the ball B(x,r).Letf∈L1loc⁡(Rn). The fractional maximal operator Mα and the Riesz potential Iα are defined by (1)Mαfx=sup⁡t>0Bx,t-1+α/n∫Bx,t‍fydy,0≤α<n,Iαf(x)=∫Rn‍f(y)dyx-yn-α,0<α<n.If α=0, then M≡M0 is the Hardy-Littlewood maximal operator.For a measurable functionρ:(0,∞)→(0,∞) the generalized Riesz potential Iρ is defined by (2)Iρfx=∫Rn‍ρx-yx-ynfydyfor any suitable function f on Rn. If ρ(t)≡tα, 0<α<n, then we get the Riesz potential operator Iα.The generalized fractional integral operatorIρ was initially investigated in [4–6]. Nowadays many authors have been culminating important observations about Iρ especially in connection with Morrey spaces. Nakai [6] proved the boundedness of Iρ from the generalized Morrey spaces M1,φ1 to the spaces M1,φ2 for suitable functions φ1, φ2 satisfying the doubling condition. The boundedness of Iρ from the generalized Morrey spaces Mp,φ1 to the spaces Mq,φ2 is studied by Eridani [7], Gunawan [8], Eridani et al. [9], Nakai [10], and Eridani et al. [11]. Guliyev [12] proved the Spanne- and Adams-type boundedness of Iα in the spaces Mp,φ(Rn) without any assumption on monotonicity of φ.In this study, by using the method given by Guliyev in [13] (see also [12, 14]), we prove the Spanne-type boundedness of the operator Iρ from one generalized local Morrey space LMp,φ1{x0} to another one LMq,φ2{x0}, 1<p<q<∞, and from LM1,φ1{x0} to the weak space WLMq,φ2{x0}, 1<q<∞. We also prove the Adams-type boundedness of the operator Iρ from generalized Morrey space Mp,φ1/p to another one Mq,φ1/q for 1<p<q<∞ and from M1,φ to WMq,φ1/q for 1<q<∞.ByA≲B we mean that A≤CB with some positive constant C independent of appropriate quantities. If A≲B and B≲A, we write A≈B and say that A and B are equivalent.


## 2. Generalized Local Morrey SpacesLMp,φ{x0}

We find it convenient to define the generalized Morrey spaces in the form as follows.Definition 1.
Letφ(x,r) be a positive measurable function on Rn×(0,∞) and 1≤p<∞. We denote by Mp,φ≡Mp,φ(Rn) the generalized Morrey space, the space of all functions f∈Lploc⁡(Rn) with finite quasinorm: (3)fMp,φ=sup⁡x∈Rn,r>0φx,r-1Bx,r-1/pfLp(B(x,r)).Also by WMp,φ≡WMp,φ(Rn) we denote the weak generalized Morrey space of all functions f∈WLploc⁡(Rn) for which (4)fWMp,φ=sup⁡x∈Rn,r>0φx,r-1Bx,r-1/pfWLpBx,r<∞.According to this definition, we recover the Morrey spaceMp,λ and weak Morrey space WMp,λ under the choice φ(x,r)=r(λ-n)/p: (5)Mp,λ=Mp,φφ(x,r)=r(λ-n)/p,WMp,λ=WMp,φφ(x,r)=r(λ-n)/p.Definition 2.
Letφ(x,r) be a positive measurable function on Rn×(0,∞) and 1≤p<∞. We denote by LMp,φ≡LMp,φ(Rn) the generalized local (central) Morrey space, the space of all functions f∈Lploc⁡(Rn) with finite quasinorm: (6)fLMp,φ=sup⁡r>0φ0,r-1B0,r-1/pfLp(B(0,r)).Also by WLMp,φ≡WLMp,φ(Rn) we denote the weak generalized local (central) Morrey space of all functions f∈WLploc⁡(Rn) for which (7)fWLMp,φ=sup⁡r>0φ0,r-1B0,r-1/pfWLp(B(0,r))<∞.Definition 3.
Letφ(x,r) be a positive measurable function on Rn×(0,∞) and 1≤p<∞. For any fixed x0∈Rn we denote by LMp,φ{x0}≡LMp,φ{x0}(Rn) the generalized local Morrey space, the space of all functions f∈Lploc⁡(Rn) with finite quasinorm: (8)fLMp,φ{x0}=f(x0+·)LMp,φ.Also by WLMp,φ{x0}≡WLMp,φ{x0}(Rn) we denote the weak generalized local Morrey space of all functions f∈WLploc⁡(Rn) for which (9)fWLMp,φ{x0}=f(x0+·)WLMp,φ<∞.According to this definition, we recover the local Morrey spaceLMp,λ{x0} and weak local Morrey space WLMp,λ{x0} under the choice φ(x0,r)=r(λ-n)/p: (10)LMp,λ{x0}=LMp,φ{x0}φ(x0,r)=r(λ-n)/p,WLMp,λ{x0}=WLMp,φ{x0}φ(x0,r)=r(λ-n)/p.Furthermore, we have the following embeddings: (11)Mp,φ⊂LMp,φx0,fLMp,φ{x0}≤fMp,φ,WMp,φ⊂WLMp,φx0,fWLMp,φ{x0}≤fWMp,φ.Wiener [15, 16] looked for a way to describe the behavior of a function at infinity. The conditions he considered are related to appropriate weighted Lq spaces. Beurling [17] extended this idea and defined a pair of dual Banach spaces Aq and Bq′, where 1/q+1/q′=1. To be precise, Aq is a Banach algebra with respect to the convolution, expressed as a union of certain weighted Lq spaces; the space Bq′ is expressed as the intersection of the corresponding weighted Lq′ spaces. Feichtinger [18] observed that the space Bq can be described by(12)fBq=sup⁡k≥02-kn/qfχkLq(Rn),where χ0 is the characteristic function of the unit ball {x∈Rn:|x|≤1}, χk is the characteristic function of the annulus {x∈Rn:2k-1<|x|≤2k}, k=1,2,…. By duality, the space Aq(Rn), called Beurling algebra now, can be described by(13)fAq=∑k=0∞‍2-kn/q′fχkLq(Rn).LetB˙q(Rn) and A˙q(Rn) be the homogeneous versions of Bq(Rn) and Aq(Rn) by taking k∈Z in (42) and (13) instead of k≥0 there.Ifλ<0 or λ>n, then LMp,λ{x0}(Rn)=Θ, where Θ is the set of all functions equivalent to 0 on Rn. Note that LMp,0(Rn)=Lp(Rn) and LMp,n(Rn)=B˙p(Rn): (14)iiiiiiiB˙p,μ=LMp,φφ0,r=rn+npμ,WB˙p,μ=WLMp,φφ0,r=rn+npμ,iiiiiiiiiiiiiiiiiiiiiiiiiμ∈-1p,0.In order to study the relationship between centralBMO spaces and Morrey spaces, Alvárez et al. [19] introduced λ-central bounded mean oscillation spaces and central Morrey spaces B˙p,μ(Rn)≡LMp,rn+npμ(Rn), μ∈[-1/p,0]. If μ<-1/p or μ>0, then B˙p,μ(Rn)=Θ. Note that B˙p,-1/p(Rn)=Lp(Rn) and B˙p,0(Rn)=B˙p(Rn). Also define the weak central Morrey spaces WB˙p,μ(Rn)≡WLMp,n+npμ(Rn).The classical result by Hardy-Littlewood-Sobolev states that if1<p<q<∞, then the operator Iα is bounded from Lp(Rn) to Lq(Rn) if and only if α=n1/p-1/q and for p=1<q<∞, the operator Iα is bounded from L1(Rn) to WLq(Rn) if and only if α=n1-1/q. Spanne and Adams studied boundedness of the Riesz potential in Morrey spaces. Their results can be summarized as follows.Theorem 4 (Spanne, but published by Peetre [20]).
Let0<α<n, 1<p<n/α, and 0<λ<n-αp. Moreover, let 1/p-1/q=α/n and λ/p=μ/q. Then, for p>1, the operator Iα is bounded from Mp,λ to Mq,μ and, for p=1, Iα is bounded from M1,λ to WMq,μ.Theorem 5 (Adams [21]).
Let0<α<n, 1<p<n/α, 0<λ<n-αp, and 1/p-1/q=α/(n-λ). Then, for p>1, the operator Iα is bounded from Mp,λ to Mq,λ and, for p=1, Iα is bounded from M1,λ to WMq,λ.Some authors [8, 12, 22–25] generalized Theorems 4 and 5 to generalized Morrey spaces and called them Spanne-type and Adams-type results for Iα.In [23] the following condition was imposed on φ(x,r):(15)c-1φ(x,r)≤φ(x,t)≤cφ(x,r)whenever r≤t≤2r, where c≥1 does not depend on t, r and x∈Rn, jointly with the condition(16)∫r∞‍tαpφx,tpdtt≤Crαpφx,rp,where C>0 does not depend on r and x∈Rn.In [23] the following Spanne-type result was proved for Iα on Mp,φ.Theorem 6.
Let1<p<∞, 0<α<n/p, 1/q=1/p-α/n, and φ(x,r) satisfy the conditions (15) and (16). Then the operator Iα is bounded from Mp,φ to Mq,φ.The following Spanne-type result forIα on LMp,φ{x0}, containing results obtained in [23, 26], was proved in [12, 13] (see also [14]).Theorem 7.
Letx0∈Rn, 1≤p<∞, 0<α<n/p, 1/q=1/p-α/n, and (φ1,φ2) satisfy the condition(17)∫t∞rαφ1(x0,r)drr≤Cφ2(x0,t),where C does not depend on x0 and t. Then the operator Iα is bounded from LMp,φ1{x0} to LMq,φ2{x0} for p>1 and from LM1,φ1{x0} to WLMq,φ2{x0} for p=1.From Theorem7 we get the following Spanne-type result for Iα on Mp,φ.Corollary 8.
Let1≤p<∞, 0<α<n/p, 1/q=1/p-α/n, and (φ1,φ2) satisfy the condition(18)∫t∞‍rαφ1(x,r)drr≤Cφ2(x,t),where C does not depend on x and t. Then the operator Iα is bounded from Mp,φ1 to Mq,φ2 for p>1 and from M1,φ1 to WMq,φ2 for p=1.The following Spanne-type result forIα on Mp,φ, containing results obtained in [12], was proved in [27].Theorem 9.
Let1≤p<∞, 0<α<n/p, 1/q=1/p-α/n, and (φ1,φ2) satisfy the condition(19)∫t∞‍essinf⁡r<s<∞⁡φ1(x,s)sn/qrn/q+1dr≤Cφ2(x,t),where C does not depend on x and t. Then the operator Iα is bounded from Mp,φ1 to Mq,φ2 for 1<p<q<∞ and from M1,φ1 to WMq,φ2 for 1<q<∞.


## 3. Some Weighted Inequalities

Letv be a nonnegative function on (0,∞). We denote by L∞,v(0,∞) the space of all functions g(t), t>0, with finite norm (20)gL∞,v(0,∞)=sup⁡t>0vtgtand L∞(0,∞)≡L∞,1(0,∞). Let M(0,∞) be the set of all Lebesgue-measurable functions on (0,∞) and M+(0,∞) its subset consisting of all nonnegative functions on (0,∞). We denote by M+(0,∞;↑) the cone of all functions in M+(0,∞) which are nondecreasing on (0,∞) and (21)A=φ∈M+(0,∞;↑):limt→0+φ(t)=0.The following theorem is valid.Theorem 10.
Letv1, v2 be nonnegative measurable functions satisfying 0<vjL∞(t,∞)<∞, j=1,2, for any t>0.
Then the identity operatorI is bounded from L∞,v1(0,∞) to L∞,v2(0,∞) on the cone A if and only if(22)v2v1L∞(·,∞)-1L∞(0,∞)<∞.Proof.
IfF, G are nonnegative functions on (0,∞) and F is nondecreasing, then (23)esssup⁡t∈(0,∞)F(t)G(t)=esssup⁡t∈(0,∞)F(t)esssup⁡s∈(t,∞)G(s),t∈(0,∞).Also if F, G are nonnegative functions on (0,∞) and F is nonincreasing, then (24)esssup⁡t∈(0,∞)F(t)G(t)=esssup⁡t∈(0,∞)F(t)esssup⁡s∈(0,t)G(s),t∈(0,∞).Therefore for all φ∈A(25)esssup⁡t>0v(t)φ(t)=esssup⁡t>0φ(t)S¯v(t),where (26)(S¯g)(t):=gL∞(t,∞),t∈(0,∞).
First we prove sufficiency. Assume that condition (22) holds. Then for all φ∈A(27)IφL∞,v20,∞=(S¯v1)-1S¯v1φL∞,v20,∞≤esssup⁡t>0S¯v1tφt·(S¯v1)-1L∞,v20,∞=v2v1L∞(·,∞)-1L∞(0,∞)esssup⁡t>0v1(t)φ(t)by (25).
To prove necessity assume thatI is bounded from L∞,v1(0,∞) to L∞,v2(0,∞) on the cone A; that is, (28)IφL∞,v2(0,∞)≤cφL∞,v1(0,∞),φ∈A,where c>0 is independent of φ.
We note that(S¯v1)-1χ(1/n,∞)∈A for all n∈N and take φ=(S¯v1)-1χ(1/n,∞). Observe that (29)(S¯v1)-1χ1/n,∞L∞,v10,∞≤(S¯v1)-1L∞,v10,∞=esssup⁡t>0v1(t)esssup⁡t≤τ<∞v1τ-1≤1.
Also for allτ>0 for sufficiently large n(30)(S¯v1)-1χ1/n,∞L∞,v20,∞≥esssup⁡t≤s<∞χ1/n,∞(s)(S¯v1)-1(s)L∞,v2τ,∞=(S¯1)-1v2L∞τ,∞;hence(31)S¯1-1v2L∞(τ,∞)≤cfor all τ>0 and condition (22) follows.We will use the following statement on the boundedness of the weighted Hardy operator:(32)Hwg(t):=∫t∞g(s)w(s)ds,0<t<∞,where w is a weight.The following theorem was proved in [28] (see, also [29]).Theorem 11.
Letv1, v2, and w be weights on (0,∞) and let v1(t) be bounded outside a neighborhood of the origin. The inequality(33)esssup⁡t>0v2(t)Hwg(t)≤Cesssup⁡t>0v1(t)g(t)holds for some C>0 for all nonnegative and nondecreasing g on (0,∞) if and only if(34)B:=sup⁡t>0v2(t)∫t∞w(s)dsesssup⁡s<τ<∞⁡v1(τ)<∞.Moreover, the value C=B is the best constant for (33).Remark 12.
In (33) and (34) it is assumed that 1/∞=0 and 0·∞=0.


## 4. Spanne-Type Result for the OperatorIρ in LMp,φ{x0}

We assume that(35)∫1∞‍ρ(t)tndtt<∞,so that the fractional integrals Iρf are well defined, at least for characteristic functions 1/|x|2n of complementary balls: (36)f(x)=χRn∖B(0,1)(x)x2n.In addition, we will also assume that ρ satisfies the growth condition: there exist constants C1>0 and 0<2k1<k2<∞ such that(37)sup⁡r/2<s≤3r/2ρ(s)sn≤C1∫k1rk2r‍ρ(t)tndtt,r>0.This condition is weaker than the usual doubling condition for the functionρ(t)/tn: there exists a constant C2>0 such that (38)1C2ρ(t)tn≤ρ(r)rn≤C2ρ(t)tn,whenever r and t satisfy r, t>0 and 1/2≤r/t≤2.In the sequel for the generalized fractional integral operatorIρ we always assume that ρ satisfies the conditions (37) and then denote the set of all such functions by G~0. We will write, when ρ∈G~0,(39)ρ~(r):=Crn∫r∞‍ρ(t)tndtt.Remark 13.
Typical examples ofρ(t) that we envisage are, for 0<α<n, (40)ρ(t)≡tαlog⁡et,0<t≤1,tαlog⁡(et),1≤t<∞,and, for c>0, (41)ρ(t)≡tα,0<t≤1,ece-ct2,1≤t<∞.The second one is used to control the Bessel potential (see also [30]).The following theorem was proved in [11].Theorem 14.
(1) Let1<p<q<∞. Then the operator Iρ is bounded from Lp(Rn) to Lq(Rn) if and only if there exists C>0 such that for all r>0(42)ρ(r)≤Crn/p-n/q.
(2) Let1<q<∞. Then the operator Iρ is bounded from L1(Rn) to WLq(Rn) if and only if there exists C>0 such that for all r>0(43)ρ(r)≤Crn-n/q.The following lemma is valid.Lemma 15.
Let1≤p<∞ and ρ(t) satisfy the conditions (35) and (37). If the condition (42) is fulfilled, then for p>1 the inequality (44)IρfLqBx0,r≲fLpBx0,2r+rn/q∫2r∞‍fLp(B(x0,t))ρ(t)tn/pdttholds for any ball B(x0,r) and for all f∈Lploc⁡(Rn).
If the condition (43) is fulfilled, then for p=1 the inequality(45)IρfWLqBx0,r≲fL1Bx0,2r+rn/q∫2r∞‍fL1(B(x0,t))ρ(t)tndttholds for any ball B(x0,r) and for all f∈L1loc⁡(Rn).Proof.
Let1<p<∞, 0<α<n/p, and 1/q=1/p-α/n. For arbitrary x0∈Rn, set B=B(x0,r) for the ball centered at x0 and of radius r. Write f=f1+f2 with f1=fχ2B and f2=fχ(2B)∁. Hence (46)IρfLq(B)≤Iρf1Lq(B)+Iρf2Lq(B).
Sincef1∈Lp(Rn), Iρf1∈Lq(Rn) and from condition (42) we get the boundedness of Iρ from Lp(Rn) to Lq(Rn) (see Theorem 14) and it follows that (47)Iρf1Lq(B)≤Iρf1Lq(Rn)≤Cf1Lp(Rn)=CfLp(2B),where constant C>0 is independent of f.
It is clear thatx∈B, y∈(2B)∁ implies (1/2)|x0-y|≤|x-y|≤(3/2)|x0-y|. Then from conditions (35), (37) and by Fubini’s theorem we have (48)∫(2B)∁‍ρx-yx-ynfydy≲∫(2B)∁‍fy∫k1x0-yk2x0-y‍ρttn+1dtdy≈∫2k1r∞‍∫2k1r<x0-y<t‍fydyρttn+1dt≲∫2r∞‍∫Bx0,t‍fydyρ(t)tn+1dt.Applying Hölder’s inequality, we get(49)∫(2B)∁‍ρx-yx-ynfydy≲∫2r∞‍fLp(B(x0,t))ρ(t)tn/p+1dt.
Moreover, for allp∈[1,∞), the inequality(50)Iρf2Lq(B)≲rn/q∫2r∞‍fLp(B(x0,t))ρ(t)tn/p+1dtis valid. Thus (51)IρfLq(B)≲fLp(2B)+rn/q∫2r∞‍fLp(B(x0,t))ρ(t)tn/p+1dt.
Letp=1. From the weak (1,q) boundedness of Iρ and (43) it follows that(52)Iρf1WLq(B)≤Iρf1WLq(Rn)≲f1L1(Rn)=fL1(2B).
Then from (50) and (52) we get the inequality (45).The following theorem is one of the main results of this paper.Theorem 16.
Letx0∈Rn, 1≤p<∞, and the function ρ satisfy the conditions (35), (37), and (42). Let also (φ1,φ2) satisfy the conditions(53)essinf⁡t<s<∞φ1(x0,s)sn/p≤Cφ2x0,t2tn/q,∫r∞‍essinf⁡t<s<∞φ1x0,ssn/pρ(t)tn/p+1dt≤Cφ2(x0,r),where C does not depend on x0 and r. Then the operator Iρ is bounded from LMp,φ1{x0} to LMq,φ2{x0} for p>1 and from LM1,φ1{x0} to WLMq,φ2{x0} for p=1. Moreover, for p>1,(54)IρfLMq,φ2{x0}≲fLMp,φ1{x0},and, for p=1,(55)IρfWLMq,φ2{x0}≲fLM1,φ1{x0}.Proof.
By Lemma15 and Theorems 10 and 11 we have, for p>1,(56)IρfLMq,φ2x0≲sup⁡r>0φ2x0,r-1r-n/qfLpBx0,2r+sup⁡r>0φ2x0,r-1∫r∞‍fLpBx0,tρttn/p+1dt≈sup⁡r>0φ1x0,r-1r-n/pfLpBx0,r=fLMp,φ1x0,and, for p=1,(57)IρfWLMq,φ2x0≲sup⁡r>0φ2x0,r-1r-n/qfL1Bx0,2r+sup⁡r>0φ2x0,r-1∫r∞‍fL1Bx0,tρttn+1dt≈sup⁡r>0φ1x0,r-1r-nfL1Bx0,r=fLM1,φ1{x0}.Corollary 17.
Let1≤p<∞, the function ρ satisfies the conditions (35), (37), and (42). Let also (φ1,φ2) satisfy the conditions(58)essinf⁡t<s<∞φ1x,ssn/p≤Cφ2x,t2tn/q,∫r∞‍essinf⁡t<s<∞φ1x,ssn/pρ(t)tn/p+1dt≤Cφ2x,r,where C does not depend on x and r. Then the operator Iρ is bounded from Mp,φ1 to Mq,φ2 for p>1 and from M1,φ1 to WMq,φ2 for p=1.In the caseρ(t)=tα from Theorem 16 we get new Spanne-type result on generalized local Morrey spaces.Corollary 18.
Letx0∈Rn, 0<α<n, 1≤p<q<∞, and 1/p-1/q=α/n. Let also (φ1,φ2) satisfy the condition(59)∫r∞‍(essinf⁡t<s<∞φ1(x0,s)sn/p)dttn/q+1dt≤Cφ2(x0,r),where C does not depend on r. Then the operator Iα is bounded from LMp,φ1{x0} to LMq,φ2{x0} for p>1 and from LM1,φ1{x0} to WLMq,φ2{x0} for p=1.Also in the casesρ(t)=tα and φ(x,t)=t(λ-n)/p, 0<λ<n, from Theorem 16 we get local Morrey space variant of Theorem 4.Corollary 19.
Letx0∈Rn, 0<α<n, 1<p<n/α, and 0<λ<n-αp. Moreover, let 1/p-1/q=α/n and λ/p=μ/q. Then, for p>1, the operator Iα is bounded from LMp,λ{x0} to LMq,λ{x0} and, for p=1, Iα is bounded from LM1,λ{x0} to WLMq,λ{x0}.


## 5. Adams-Type Result for the OperatorIρ in Mp,φ

The following Adams-type result was proved in [31] (see also [12]).Theorem 20.
Let1≤p<∞, 0<α<n/p, and q>p and let φ(x,t) satisfy the conditions(60)sup⁡r<t<∞φ(x,t)≤Cφ(x,r),(61)∫r∞‍tαφx,t1/pdtt≤Cr-αp/(q-p),where C does not depend on x∈Rn and r>0.
Then the operatorIα is bounded from Mp,φ1/p to Mq,φ1/q for p>1 and from M1,φ to WMq,φ1/q for p=1.The following Theorem was proved in [32].Theorem 21.
Let1≤p<∞ and (φ1,φ2) satisfies the condition(62)sup⁡r<t<∞φ1(x,t)≤Cφ2(x,r),where C does not depend on x and r. Then, for p>1, the Hardy-Littlewood maximal operator M is bounded from Mp,φ1 to Mp,φ2 and, for p=1, M is bounded from M1,φ1 to WM1,φ2.The following theorem is a main result of this paper on Adams-type estimate for generalized fractional integral operatorIρ. In the case ρ(t)=tα we get Theorem 20 from this theorem.Theorem 22.
Let1≤p<∞, q>p, and ρ(t) satisfy the conditions (37) and (42). Let also φ(x,t) satisfy the condition (60) and(63)∫r∞‍φx,t1/pρ(t)tdt≤Cρr-p/(q-p),where C does not depend on x∈Rn and r>0.
Then the operatorIρ is bounded from Mp,φ1/p to Mq,φ1/q for p>1 and from M1,φ to WMq,φ1/q for p=1.Proof.
Letx0∈Rn, 1<p<∞, 0<α<n/p, q>p, and f∈Mp,φ1/p. Write f=f1+f2, where B=B(x,r), f1=fχ2B, and f2=fχ(2B)∁. Then we have (64)Iρf(x)=Iρf1(x)+Iρf2(x).For Iρf1(y), y∈B(x,r), following Hedberg’s trick (see, e.g., [33, page 354]), we obtain(65)Iρf1y≤∫Bx,2r‍ρy-zy-znfzdz≈∑k=-∞0‍∫Bx,2k+1r∖Bx,2kr‍ρy-zy-znfzdz≲∑k=-∞0‍∫2k+1-1k1r2k+1-1k2r‍ρssn+1ds∫Bx,2k+1r‍fzdz≈Mfx∑k=-∞0‍∫2k+1-1k1r2k+1-1k2r‍ρssds=Mfx∫0k2r‍ρssds=Mfxρ~k2r≲Mf(x)ρ(r).
ForIρf2(y), y∈B(x,r), from (49) we have(66)Iρf2y≲∫B∁x,2r‍ρy-zy-znfzdz≲∫2r∞‍fLp(B(x,t))ρ(t)tn/p+1dt.Then from condition (63) and inequality (66) for all y∈B(x,r) we get(67)Iρfy≲ρrMfx+∫r∞‍fLpBx,tρttn/p+1dt≤ρ(r)Mf(x)+fMp,φ1/p∫r∞‍φx,t1/pρttdt≲ρ(r)Mf(x)+ρr-p/q-pfMp,φ1/p.
Hence choosingρ(r)=(fMp,φ1/p/Mf(x))(q-p)/q for all y∈B(x,r), we have (68)|Iρf(y)|≲Mfxp/qfMp,φ1/p1-p/q.Consequently the statement of the theorem follows in view of the boundedness of the maximal operator M in Mp,φ1/p provided by Theorem 21 in virtue of condition (60). If 1<p<q<∞, then (69)IρfMq,φ1/q=sup⁡x∈Rn,t>0φx,t-1/qt-n/qIρfLqBx,t≲fMp,φ1/p1-p/qsup⁡x∈Rn,t>0φx,t-1/qt-n/qMfLpBx,tp/q=fLMp,φ1/p1-p/qsup⁡x∈Rn,t>0φx,t-1/pt-n/pMfLpBx,tp/q=fMp,φ1/p1-p/qMfMp,φ1/pp/q≲fMp,φ1/p,and if 1<q<∞, then (70)IρfWMq,φ1/q=sup⁡x∈Rn,t>0φx,t-1/qt-n/qIρfWLqBx,t≲fM1,φ1-1/qsup⁡x∈Rn,t>0φx,t-1/qt-n/qMfWL1Bx,t1/q=fM1,φ1-1/qsup⁡x∈Rn,t>0φx,t-1t-nMfWL1Bx,t1/q=fM1,φ1-1/qMfWM1,φ1/q≲fM1,φ.
Thus the proof of the theorem is completed.



---

*Source: 594323-2015-04-22.xml*