# Operator Inequalities of Morrey Spaces Associated with Karamata Regular Variation

Authors: Jiajia Wang; Zunwei Fu; Shaoguang Shi; Ling Mi
Journal: Journal of Function Spaces
 (2017)

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## Abstract

Karamata regular variation is a basic tool in stochastic process and the boundary blow-up problems for partial differential equations (PDEs). Morrey space is closely related to study of the regularity of solutions to elliptic PDEs. The aim of this paper is trying to bring together these two areas and this paper is intended as an attempt at motivating some further research on these areas. A version of Morrey space associated with Karamata regular variation is introduced. As application, some estimates of operators, especially one-sided operators, on these spaces are considered.


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## Body

## 1. Introduction

A positive measurable functionw:R×R+→R+ is called regularly varying at infinity with index ρ, written as w∈RVρ, if, for each ξ>0,x0∈R and some ρ∈R,(1)limh→∞⁡wx0,ξhwx0,h=ξρ,where I(x0,h) is an interval whose center at x0 and radius h and w(x0,h)=∫I(x0,h)w(x)dx. In particular, when ρ=0, w is called slowly varying at infinity. w is the classical Karamata regular variation. Karamata regular variation theory was first introduced and established by Karamata in 1930. It is a basic tool in stochastic process [1, 2] and has been applied to study the boundary behavior of solutions to boundary blow-up elliptic problems and singular nonlinear Dirichlet problems; for some of this work, see [3–7] and the references given there.Ifw∈RV0, then(2)limh→∞⁡wx0,ξhwx0,h=1holds uniformly for ξ∈[C1,C2] with 0<C1<C2 [5]. And for s>0 and h→∞, the following asymptotic behavior is true:(3)∫h∞wx0,tts+1dt≃wx0,hshs+1.Karamata regular variation at 0 can also be defined by a positive measurable function w with h→∞ replaced by h→0.In [8], Nakai introduced a generalized weighted Morrey apace with the weight function w satisfying the following conditions:(4)1C≤wa,twa,h≤C,h≤t≤2h,(5)∫h∞wa,tt2dt≤Cwa,hh,where any a∈R. Inspired by Nakai, a general case of (2) and (3) can be defined as(6)1C≤wx0,ξhwx0,h≤C,where ξ∈[C1,C2] with 0<C1<C2 and(7)∫h∞wx0,tts+1dt≤Cwx0,hhs.It is of interest to know that when h→0 or h→∞ in (6) and (7), the function w can be seen as Karamata regular variation at 0 and infinity, respectively.Letw satisfy (6) and (7). Then the Morrey space associated with Karamata regular variation (K-Morrey space) can be adopted from [8] as (8)MKp,wR=f∈LlocpR:fMKp,wR≕supI⁡1wI∫Ifxpdx<∞,1≤p<∞.It is obvious that MKp,w(R) is a Banach space with norm fMKp,w(R). If w(I)≡1, then MKp,w(R)=Lp(R). If w(I)=r, then MKp,w(R)=L∞(R). And if w(I)=rλ with 0<λ<1, then MKp,w(R) is the classical Morrey space which was first introduced by Morrey [9] to investigate the local behavior of solutions to the second order elliptic PDEs.It is also worth pointing out that whens=1, ξ∈[1,2], and (6) and (7) are exactly the same as (4) and (5), MKp,w(R) is the generalized Morrey space Lp,w(R) introduced by Nakai [8] with n=1. In his well-known paper, Nakai proved the interesting result that the Hardy-Littlewood maximal operator, singular integral operator, and the Riesz potential were bounded on certain space. As stated in [8], we can also prove that w(I)=(∫Iu(x))α(0<α<1) with 1≤p≤1/α and u(x)∈Ap [10] is an example of functions satisfying (6) and (7).In this paper, we shall consider some estimates for one-sided operators on theK-Morrey space MKp,w(R). Let us first recall some basic definitions of one-sided operators. The reasons to study one-sided operators involve the requirements of ergodic theory [11]. The study of weighted theory for one-sided operators was first introduced by Sawyer [12] and many authors thereafter ([13–19]). The one-sided Hardy-Littlewood maximal operators [12] are defined by(9)M+fx=suph>0⁡1h∫xx+hfydy,M-fx=suph>0⁡1h∫x-hxfydy,which arise in the ergodic maximal function. It is well known that M+ and M- are bounded on Lp(R) spaces (1<p<∞) and bounded from L1(R) spaces to weak L1(R) spaces. Such operators are also bounded onK-Morrey spaces, which we now formulated as follows.Theorem 1.
Letw satisfy (6) and (7) with 0<s≤1. Then one has the following:
(a) For1<p<∞, there is a constant C>0 such that (10)M+fMKp,wR≤CfMKp,wR.
(b) There is a constantC>0 such that, for any t>0 and for any I⊂R, one has (11)x∈I:M+fx>t≤CwItfMK1,wR.LetT+ be an one-sided integral operator with one-sided kernel K+(x,y) supported in R-=(-∞,0) and satisfy (12)K+x,y≤Cy-x.That is,(13)T+fx=p.v.∫x∞K+x,yfydy,x∈supp⁡fc.Both the one-sided Calderón-Zygmund singular integral operator [13] and the one-sided oscillatory singular operator [17] are examples of operators T+. Our second result is as follows.Theorem 2.
Letw satisfy (6) and (7) with 0<s≤1. Then one has the following:
(a) IfT+ is bounded on Lp(R) with 1<p<∞, then there is a constant C>0 such that (14)T+fMKp,wR≤CfMKp,wR.
(b) IfT+ is bounded from L1(R) space to weak L1(R) space, then there is a constant C>0 such that for any t>0 and for any I(15)x∈I:T+fx>t≤CwItfMK1,wR.Remark 3.
Theorem2 provides a criterion for the boundedness of one-sided singular integral operators onK-Morrey spaces.By the corresponding boundedness in [13, 15, 17], both the one-sided Calderón-Zygmund singular integral operator and the one-sided oscillatory singular operator satisfy Theorem 2.In the fractional case, both the Rieman-Liouville fractional integral(16)Iα+fx=∫x∞fyy-x1-αdy,0<α<1,and the Weyl fractional integral (17)Iα-fx=∫-∞xfyx-y1-αdy,0<α<1,are examples of one-sided fractional integrals. Without loss of generality, we take Iα+ as our model in the following analysis. The last goal of this section is to show that Iα+ is also bounded onK-Morrey space, which can be stated in the following theorem.Theorem 4.
Let0<α<1, 0<s≤1, 1≤p<s/α, 1/q=1/p-α, w satisfy (6), and ∫h∞(wx0,t/ts-αp+1)dt≤C(w(x0,h)/hs-αp) with x0∈R. Then one has the following:
(a) If1<p<∞, then there is a constant C>0 such that (18)Iα+fMKq,qw/pR≤CfMKp,wR.
(b) Ifp=1, then there is a constant C>0 such that for any t>0 and for any I(19)x∈I:Iα+fx>t≤CwItqfMK1,wRq.Section2 contains the proofs of Theorems 1–4. In Section 3, we extend the main results to n-dimensional case, which cover the main results of [8]. Throughout this paper, C is a constant which may change from line to line.


## 2. Preliminaries

In this section, some lemmas are described by some methods adopted from [20].Lemma 5 (see [21]).
Letφ≥0 be measurable functions. Then one has the following:
(a) For every1<p<∞, there is a constant C>0 such that(20)∫RM+fxpφxdx≤C∫RfxpM-φxdx.
(b) There is a constantC>0 such that(21)∫x:M+fx>tφxdx≤Ct∫RfxM-φxdx.The principal significance of Lemma5 is that it allows one to obtain a version of one-sided Fefferman-Stein inequality. The following lemma will prove extremely useful in the proofs of the main results.Lemma 6.
Let0<δ≤1, 0<s≤1, w satisfy (6), and (22)∫h∞wx0,ttsδ+1dt≤Cwx0,hhsδ.Then for 1≤p<∞, there is a constant C>0 such that (23)∫fxpM-χIxδdx≤CwIfMKp,wRp.Proof.
The proof of Lemma6 has a root in [8, Lemma 1]. We adopted its proof here for the one-sided case. Let χI be the characteristic function of I=I(x0,h). Then M-χI≤1 for x∈2I. For x∈2k+1I/2kI and y∈I, the following can be shown: (24)M-χIx=suph>0⁡12kh∫x-2khxχIydy=suph>0⁡12kh∫x0x0+hχIydy≤C2-k,k=1,2,3,….This clearly forces (25)∫fxpM-χIxδdx≤C∫2IfxpM-χIxδdx+∑k=1∞∫2k+1I/2kIfxpM-χIxδdx≤C∫2Ifxpdx+∑k=1∞∫2k+1I/2kIfxp2-kδdx≤Cw2I+∑k=1∞2-kδw2k+1IfMKp,wRp≤Chδ∑k=0∞w2kI2khδfMKp,wRp.We conclude from (6) that (26)1C≤wx0,twx0,2kh≤C,2kh≤t≤2k+1h,hence that(27)∫2kh2k+1hwx0,ttsδ+1dt≥1C∫2kh2k+1hwx0,2khtsδ+1dt≥1Cwx0,2kh2khsδ,and finally that(28)∫2kh2k+1hwx0,ttsδ+1dt≤Cwx0,2kh2khsδ.Equations (27) and (28) show that w(x0,2kh)/(2kh)sδ is comparable to ∫2kh2k+1h(wx0,t/tsδ+1)dt, which derives that (29)∫fxpM-χIxδdx≤Chδ∑k=0∞2khsδ2khδw2kI2khsδfMKp,wRp≤Chδ∑k=0∞2khs-1δ∫2kh2k+1hwx0,ttsδ+1dtfMKp,wRp≤Chsδ∑k=0∞∫2kh2k+1hwx0,ttsδ+1dtfMKp,wRp≤Chsδ∫h∞wx0,ttsδ+1dtfMKp,wRp≤Cwx0,hfMKp,wRp.On account of the estimates given above, Lemma 6 is proved.Lemma 7 (see [8]).
Suppose thatφ(h):R+→R+. If there is a constant C>0 such that, for any h>0, ∫h∞(φt/t)dt≤Cφ(h), then there are constants ε>0 and C′>0 such that, for any h>0, (30)∫h∞φttεtdt≤C′φhhε.The remainder lemma of this section will be devoted to the boundedness of one-sided fractional integrals on Lebesgue spaces.Lemma 8 (see [14]).
Let0<α<1, 1≤p<s/α, 0<s≤1, and 1/q=1/p-α. Then there exists constant C>0 such that (31)Iα+fxLqR≤CfLpR,p>1;Iα+fxLq,∞R≤CfL1R.Having disposed of the above lemmas, the estimates for the one-sided operators onK-Morrey spaces can be proved in this section. The method used here was partly adopted from [8].Proof of Theorem1.
Let us first prove (a). Taking into account (20) and Lemma 6 with δ=1, the following can be shown easily(32)∫IM+fxpχIxdx≤C∫fxpM-χIxdx≤CwIfMKp,wRp.
Applying (21) and Lemma 6 with δ=1, we note that (33)x∈I:M+fx>t=∫x:M+fx>tχIxdx≤Ct∫fxM-χIxdx≤CtwIfMK1,wR.The proof of Theorem 1 is completed.Proof of Theorem2.
(a) For anyI, let f=f2I+f(2I)c≕f1+f2 to produce (34)T+fx≤T+f1x+T+f2x.We conclude from the fact that T+ is bounded on Lp(Rn) that (35)∫IT+f1xpdx≤T+f1Lpp≤Cw2IfMKp,wRp.
The task is now to deal with the termT+f2. The fact that x∈I and y∈(2I)c allows the user to estimate M-χI as(36)M-χIy=suph>0⁡1h∫y-hyχIzdz≥hy-x.This clearly forces (37)∫K+x,yf2ydy≤C∫2Icf2yy-xdy≤Ch∫2IcfyM-χIydy.Applying Lemma 7 to φ(t)=w(x0,t)/ts, it is easy to check that (38)∫h∞wx0,tts-ε+1dt≤Cwx0,hhs-ε.Let δ=(s-ε)/s. Using Hölder’s inequality, it may be concluded that (39)∫K+x,yf2ydy≤C1h∫fpM-χIyδdy1/p1h∫M-χIyp-δ/p-1dyp-1/p≕J1×J2.Repeated application of Lemma 6 enables us to write (40)J1≤Ch-1/pwI1/pfMKp,wR.For the term J2, the fact that M-χI(y)≤1,y∈2I, and M-χI(y)≤2-k for y∈(2k+1I)/(2kI) shows that (41)J2≤1h∫2Idy+∑k=1∞∫2k+1I/2kIM-χIyp-δ/p-1dy≤Ch∑k=0∞2-kp-δ/p-12k+1I≤C∑k=0∞2-kε/p-1≤C.Hence, (42)∫K+x,yf2ydy≤Ch-1/pwI1/pfMKp,wR.The proof of (a) is completed by showing that (43)∫IT+f2xpdx≤C∫I1hwIfMKp,wRpdx≤CwIfMKp,wRp.
(b) Forf and I, let f=f2I+f(2I)c≕f1+f2 to produce (44)T+fx≤T+f1x+T+f2x.Since T+ is bounded from L1 to L1,∞, the following can be proved: (45)x∈I:T+f1x>t≤CtwIfMK1,wR.Applying the same analysis as (a) and Lemma 6 with p=δ=1, (46)T+f2x≤Ch∫fyM-χIydy≤ChwIfMK1,wR.Therefore, (47)x∈I:T+f2x>t≤Ctw2IfMK1,wR.We have thus completed the proof of Theorem 2.Proof of Theorem4.
An argument similar to that of Theorem2 can be used to prove Theorem 4. (a) For any I, let f=f2I+f(2I)c≕f1+f2 to produce (48)Iα+fx≤Iα+f1x+Iα+f2x.Applying Lemma 8, we conclude that (49)∫IIα+f1xqdx≤Iα+f1Lqq≤Cf1LpRq,which implies (50)1wIq/p∫IIα+f1xqdx1/q≤CfMKp,wR.For x∈I and y∈(2I)c, the argument in (36) shows that (51)1y-x1-α≤CM-χIyI1-α.Hence, (52)Iα+f2x≤C∫2Icf2yy-x1-αdy≤Ch1-α∫2IcfyM-χIy1-αdy.Applying Lemma 7 to φ(t)=w(x0,t)/ts-αp, the following is true: (53)∫h∞wx0,tts-αp-ε+1dt≤Cwx0,hhs-αp-ε.Let δ=(s-αp-ε)/s. Hölder’s inequality can be used to obtain (54)Iα+f2x≤C1h1-α∫fpM-χIyδdy1/p1h∫M-χIyp-αp-δ/p-1dyp-1/p.The fact (55)1h∫M-χIyp-αp-δ/p-1dy≤1h∫2Idy+∑k=1∞∫2k+1I/2kIM-χIyp-αp-δ/p-1dy≤Ch∑k=1∞2-kp-αp-δ/p-12k+1I≤Cand Lemma 6 with δ=1-αp/s<1 allow us to estimate Iα+f2(x) as (56)Iα+f2x≤Ch-1/qwI1/pfMKp,wR,which produces (57)1wIq/p∫IIα+f2xqdx1/q≤CfMKp,wR.
(b) Forf and I, let f=f2I+f(2I)c≕f1+f2 produce (58)Iα+fx≤Iα+f1x+Iα+f2x.The fact that Iα+ is bounded from L1 to Lq,∞ allows us to get (59)x∈I:Iα+f1x>t≤Ctqw2IfMK1,wRq.Applying the same analysis as that of (a) and Lemma 6 with p=1, δ=1-α=1/q<1, the following can be confirmed easily: (60)Iα+f2x≤Ch1/qwIfMK1,wR.This produces the following inequality: (61)x∈I:Iα+f2x>t≤CtqwIfMK1,wRq,which is our desired result.


## 3. Boundedness of Operators onn-Dimensional K-Morrey Spaces

Since one-sided operators are defined onR, we built Theorems 1–4 in one dimension. The theorems in Section 2 gain interest if we realize that they are still hold for n-dimension. In fact, we can also defineK-Morrey space on Rn(n≥2) with 0<s≤n and consider the boundedness of Hardy-Littlewood maximal operator, singular integral operator, and the Riesz potential on these spaces applying the method in [8] with only a slight modification. Let w:Rn×R+→R+ and Ω(x0,r) be the cube whose center at x0 with edges has length r and is parallel to the coordinate axes. Then the definition of n-dimensional Morrey space associated with Karamata regular variation (K-Morrey space) can be defined by (62)MKp,wRn=f∈LlocpRn:fMKp,wRn≕supB⊂Rn⁡1wB∫Bfxpdx<∞,1≤p<∞,if w satisfies the following conditions:(63)1C≤wx0,ξrwx0,r≤C,where ξ∈[C1,C2] with 0<C1<C2 and(64)∫r∞wx0,tts+1dt≤Cwx0,rhs,0<s≤n.Theorem 9.
LetM be the Hardy-Littlewood maximal operator (65)Mfx=supΩ∋x⁡1Ω∫Ωfydy.Assume that w satisfies (63) and (64) with 0<s≤n. Then one has the following:
(a) For1<p<∞, there is a constant C>0 such that (66)MfMKp,wRn≤CfMKp,wRn.
(b) There is a constantC>0 such that for any λ>0 and for any cube Ω(67)x∈Ω:Mfx>λ≤CwΩtfMK1,wRn.Theorem 10.
LetT be a singular integral operator (68)Tfx=p.v.∫RnKx,yfydy,x∈supp⁡fc,with the kernel K(x,y) satisfying (69)Kx,y≤Cx-yn.Assume that w satisfies (63) and (64) with 0<s≤n. Then one has the following:
(a) IfT is bounded on Lp(Rn) with 1<p<∞, then there is a constant C>0 such that (70)TfMKp,wRn≤CfMKp,wRn.
(b) IfT is bounded from L1(Rn) space to weak L1(Rn) space, then there is a constant C>0 such that, for any λ>0 and for any cube Ω⊂Rn, one has (71)x∈Ω:Tfx>λ≤CwΩλfMK1,wRn.The last result is about the Riesz potentialIα(72)Iαfx=∫Rnfyy-xn-αdy,0<α<n.Theorem 11.
Let0<α<n, 0<s≤n, 1≤p<s/α, 1/q=1/p-α/n, w satisfy (63), and ∫r∞(wx0,t/ts-αp+1)dt≤C(w(x0,r)/rs-αp) with x0∈Rn. Then one has the following:
(a) If1<p<∞, then there is a constant C>0 such that (73)IαfMKq,qw/pRn≤CfMKp,wRn.
(b) Ifp=1, then there is a constant C>0 such that for any λ>0 and for any cube Ω⊂Rn(74)x∈Ω:Iαfx>λ≤CwΩλqfMK1,wRnq.Whens=n, it is of interest to know that Theorems 9–11 can be seen as an extension of that of [8] in the sense that these theorems agree with [8, Theorems 1–3], respectively.With a slight modification in the proofs of [8, Theorems 1–3], Theorems 9–11 can be obtained easily; we omit its proof here for the similarity.



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