Let L=-Δ+V be a Schrödinger operator acting on L2(Rn), n≥1, where V≢0 is a nonnegative locally integrable function on Rn. In this paper, we will first define molecules for weighted Hardy spaces HLp(w)(0<p≤1) associated with L and establish their molecular characterizations. Then, by using the atomic decomposition and molecular characterization of HLp(w), we will show that the imaginary power Liγ is bounded on HLp(w) for n/(n+1)<p≤1, and the fractional integral operator L-α/2 is bounded from HLp(w) to HLq(wq/p), where 0<α<min{n/2,1}, n/(n+1)<p≤n/(n+α), and 1/q=1/p-α/n.
1. Introduction
Let n≥1 and V be a nonnegative locally integrable function defined on Rn, not identically zero. We define the form Q by(1)Qu,v≔∫Rn∇u·∇vdx+∫RnVuvdxwith domain D(Q)=V×V, where (2)V≔u∈L2Rn:∂u∂xk∈L2Rn for k=1,2,…,n,Vu∈L2Rn.It is well known that this symmetric form is closed. Note also that it was shown by Simon [1] that this form coincides with the minimal closure of the form given by the same expression but defined on C0∞(Rn) (the space of C∞ functions with compact supports). In other words, C0∞(Rn) is a core of the form Q.
Let us denote by L the self-adjoint operator associated with Q. The domain of L is given by (3)DL≔u∈V:∃v∈L2Rn such that Qu,φ=∫Rnv·φdx,∀φ∈V.Formally, we write L=-Δ+V as a Schrödinger operator with potential V. Let {e-tL}t>0 be the semigroup of linear operators generated by -L and let pt(x,y) be their kernels. Since V is a locally integrable nonnegative function on Rn, then the Feynman-Kac formula implies that the semigroup kernels pt(x,y) associated with e-tL satisfy the estimates:(4)0≤ptx,y≤14πtn/2exp-x-y24tfor all t>0 and all x,y∈Rn.
Since the Schrödinger operator L is a self-adjoint and positive definite operator acting on L2(Rn), then L admits the following spectral decomposition: (5)L=∫0∞λdELλ,where EL(λ) denotes its spectral resolution. For any γ∈R, we can define the imaginary power Liγ associated with L by the formula(6)Liγ≔∫0∞λiγdELλ.By the functional calculus for L, we can also define the operator Liγ as follows:(7)Liγfx≔1Γ-iγ∫0∞t-iγ-1e-tLfxdt.By the spectral theory, we know that LiγL2→L2=1 for all γ∈R. Moreover, it was proved by Shen [2] that Liγ is a Calderón-Zygmund operator provided that V∈RHn/2 (reverse Hölder class). We refer the reader to [2–4] for related results concerning the imaginary powers of self-adjoint operators.
For any 0<α<n, the fractional integrals L-α/2 associated with L are defined by(8)L-α/2fx≔1Γα/2∫0∞tα/2-1e-tLfxdt.Since the kernel pt(x,y) of {e-tL}t>0 satisfies the Gaussian upper bounds (4), then it is easy to check that L-α/2fx≤C·Iα(f)(x) for all x∈Rn, where Iα denotes the classical fractional integral operator (see [5]):(9)Iαfx≔Γn-α/22απn/2Γα/2∫Rnfyx-yn-αdy.Hence, by using the Lp-Lq boundedness of Iα (see [5]), we have(10)L-α/2fLq≤CIαfLq≤CfLp,where 1<p<n/α and 1/q=1/p-α/n. For more information about the fractional integrals L-α/2 associated with some classes of operators, one can see [6–9].
In [10], Song and Yan introduced the weighted Hardy space HL1(w) associated with L in terms of the square function and established its atomic decomposition theory. Furthermore, they also showed that the Riesz transform ∇L-1/2 associated with L is bounded on Lp(w) for 1<p<2 and bounded from HL1(w) to the classical weighted Hardy space H1(w) (see [11, 12] for H1(w)).
Recently, in [13], we defined the weighted Hardy spaces HLp(w) associated with L for 0<p<1 and gave their atomic decompositions. We also obtained that ∇L-1/2 is bounded from HLp(w) to the classical weighted Hardy space Hp(w) (see also [11, 12] for Hp(w)) when n/(n+1)<p<1. In this paper, we first define molecules for the weighted Hardy spaces HLp(w) associated with L and then establish their molecular characterizations. As applications of the molecular characterization combining with the atomic decomposition of HLp(w), we will obtain some estimates of Liγ and L-α/2 on HLp(w) for n/(n+1)<p≤1. Our main results are stated as follows.
Theorem 1.
Let L=-Δ+V, n/(n+1)<p≤1, and w∈A1∩RH2/p′. Then, for any γ∈R, the imaginary power Liγ is bounded from HLp(w) to the weighted Lebesgue space Lp(w).
Theorem 2.
Let L=-Δ+V, n/(n+1)<p≤1, and w∈A1∩RH2/p′. Then, for any γ∈R, the imaginary power Liγ is bounded on HLp(w).
Theorem 3.
Suppose that L=-Δ+V. Let 0<α<n/2, n/(n+1)<p≤1, 1/q=1/p-α/n, and w∈A1∩RH2/p′. Then, the fractional integral operator L-α/2 is bounded from HLp(w) to Lq(wq/p).
Theorem 4.
Suppose that L=-Δ+V. Let 0<α<min{n/2,1}, n/(n+1)<p≤n/(n+α), 1/q=1/p-α/n, and w∈A1∩RH2/p′. Then, the fractional integral operator L-α/2 is bounded from HLp(w) to HLq(wq/p).
2. Notations and Preliminaries
Let us first recall some standard definitions and notations. The classical Ap weight theory was first introduced by Muckenhoupt in the study of weighted Lp boundedness of Hardy-Littlewood maximal functions in [14]. A weight w is a nonnegative, locally integrable function defined on Rn, and B=B(x0,rB) denotes the ball with the center x0 and radius rB. We say that w∈A1, if (11)1B∫Bwxdx≤C·essinfx∈Bwx,for every ball B⊆Rn,where C is a positive constant which is independent of B. A weight function w is said to belong to the reverse Hölder class RHs, if there exist two constants s>1 and C>0 such that the following reverse Hölder inequality holds: (12)1B∫Bwxsdx1/s≤C1B∫Bwxdx,for every ball B⊆Rn.
Given a ball B and λ>0, λB denotes the ball with the same center as B whose radius is λ times that of B. For a given weight function w and a measurable set E, we denote the Lebesgue measure of E by E and the weighted measure of E by w(E), where w(E)=∫Ew(x)dx.
We give the following results which will be often used in the sequel.
Lemma 5 (see [15]).
Let w∈A1. Then, for any ball B, there exists an absolute constant C>0 such that (13)w2B≤CwB.In general, for any λ>1, we have (14)wλB≤C·λnwB,where C does not depend on B nor on λ.
Lemma 6 (see [15]).
Let w∈A1. Then, there exists a constant C>0 such that (15)C·EB≤wEwBfor any measurable subset E of a ball B.
Given a weight function w on Rn, for 0<p<∞, we denote by Lp(w) the weighted space of all functions f satisfying (16)fLpw≔∫Rnfxpwxdx1/p<∞.In particular, when w equals a constant function, we will denote Lp(w) simply by Lp(Rn) and define (17)fLp≔∫Rnfxpdx1/p<∞.
Throughout this paper, we will use C to denote a positive constant, which is independent of the main parameters and not necessarily the same at each occurrence. By A~B, we mean that there exists a constant C>1 such that 1/C≤A/B≤C. Moreover, we denote the conjugate exponent of s>1 by s′=s/(s-1).
3. Atomic Decomposition and Molecular Characterization of Weighted Hardy Spaces
Let L=-Δ+V. For any t>0, we define Pt=e-tL and (18)Qt,k≔-tkdkPsdsks=t=tLke-tL,k=1,2,….We denote Qt,k simply by Qt when k=1. First note that Gaussian upper bounds carry over from heat kernels to their time derivatives.
Lemma 7 (see [16, 17]).
For every k=1,2,…, there exist two positive constants Ck and ck such that the kernel pt,k(x,y) of the operator Qt,k satisfies (19)pt,kx,y≤Ck4πtn/2exp-x-y2cktfor all t>0 and almost all x,y∈Rn.
Let R+n+1=Rn×(0,∞). For any x∈Rn, we set (20)Γx=y,t∈R+n+1:x-y<t.For a given function f∈L2(Rn), we consider the square function associated with Schrödinger operator L, which is defined by (see [18, 19])(21)SLfx≔∬ΓxQt2fy2dydttn+11/2,x∈Rn.Set (22)HL2Rn=RL¯=Lu∈L2Rn:u∈L2Rn¯,where R(L) stands for the range of L in L2(Rn). Given a weight function w on Rn, in [10, 13], the authors defined the weighted Hardy spaces HLp(w) for 0<p≤1 as the completion of HL2(Rn) in the norm given by the Lp(w)-quasinorm of square functions; that is, (23)fHLpw≔SLfLpw.For the Schrödinger operator L=-Δ+V, it can be shown that HL2(Rn)=L2(Rn) (see, e.g., [19]). In [10], Song and Yan characterized weighted Hardy spaces HL1(w) in terms of atoms in the following way and obtained their atomic characterizations.
Definition 8 (see [10]).
Let M∈N and p=1. A function a(x)∈L2(Rn) is called a (1,2,M)-atom with respect to w (or a w-(1,2,M)-atom) if there exist a ball B=B(x0,rB) and a function b∈D(LM) such that
a=LMb;
suppLkb⊆B, k=0,1,…,M;
rB2LkbL2(B)≤rB2MB1/2[w(B)]-1, k=0,1,…,M.
Theorem 9 (see [10]).
Let M∈N and w∈A1∩RH2. If f∈HL1(w)∩L2(Rn), then there exist a family of w-(1,2,M)-atoms {aj} and a sequence of real numbers {λj} with ∑j|λj|≤CfHL1(w) such that f can be represented in the form f(x)=∑jλjaj(x), and the sum converges in the sense of both L2(Rn)-norm and HL1(w)-norm.
Similarly, in [13], we introduced the notion of weighted atoms for HLp(w) (0<p<1) and proved their atomic characterizations.
Definition 10 (see [13]).
Let M∈N and 0<p<1. A function a(x)∈L2(Rn) is called a (p,M;L)-atom with respect to w (or a w-(p,M;L)-atom) if there exist a ball B=B(x0,rB) and a function b∈D(LM) such that
a=LMb;
suppLkb⊆B, k=0,1,…,M;
(rB2L)kbL2(B)≤rB2M|B|1/2[w(B)]-1/p, k=0,1,…,M.
Theorem 11 (see [13]).
Let M∈N and 0<p<1.
If w∈A1 and f∈HLp(w)∩L2(Rn), then there exist a family of w-(p,M;L)-atoms {aj} and a sequence of real numbers {λj} with ∑j|λj|p≤CfHLp(w)p such that f can be represented in the form f(x)=∑jλjaj(x), and the sum converges in the sense of both L2(Rn)-norm and HLp(w)-norm.
If w∈A1∩RH2/p′ and n/(n+1)<p<1, then every w-(p,M;L)-atom a is in HLp(w) and its HLp(w)-norm is uniformly bounded; that is, there exists a constant C>0 independent of a such that aHLp(w)≤C.
For every bounded Borel function F:[0,∞)→C, we define the operator F(L):L2(Rn)→L2(Rn) by the following formula: (24)FL≔∫0∞FλdELλ,where EL(λ) is the spectral decomposition of L. Therefore, the operator cos(tL) is well defined on L2(Rn). Moreover, it follows from [20] that there exists a constant c0 such that the Schwartz kernel Kcos(tL)(x,y) of cos(tL) has support contained in {(x,y)∈Rn×Rn:x-y≤c0t}. By the functional calculus for L and Fourier inversion formula, whenever F is an even bounded Borel function with F^∈L1(R), we can write F(L) in terms of cos(tL). More precisely (25)FL=2π-1∫-∞∞F^tcostLdt,which gives (26)KFLx,y=2π-1∫t≥c0-1x-yF^tKcostLx,ydt.
Lemma 12 (see [10, 19]).
Let φ∈C0∞(R) be even and suppφ⊆[-c0-1,c0-1]. Let Φ denote the Fourier transform of φ. Then, for each j=0,1,… and for all t>0, the Schwartz kernel K(t2L)jΦ(tL)(x,y) of (t2L)jΦ(tL) satisfies (27)suppKt2LjΦtL⊆x,y∈Rn×Rn:x-y≤t.
For a given real number s>0, we define (28)Fs≔ψ:C⟶C measurable,ψz≤C·zs1+z2s.Then, for any nonzero function ψ∈F(s), we can prove the following estimate (see [10, 19]):(29)∫0∞ψtLfL22dtt1/2=CfL2,where C=(∫0∞|ψ(t)|2dt/t)1/2. Inspired by [19, 21, 22], we are now going to define the weighted molecules corresponding to the weighted atoms mentioned above.
Definition 13.
Let ε>0, M∈N, and 0<p≤1. A function m(x)∈L2(Rn) is called a w-(p,M,ε;L)-molecule associated with L, if there exist a ball B=B(x0,rB) and a function b∈D(LM) such that
Clearly, for every w-(p,M;L)-atom a, it is also a w-(p,M,ε;L)-molecule for all ε>0. Then, we are able to establish the following molecular characterizations for the weighted Hardy spaces HLp(w) (0<p≤1) associated with L.
Theorem 14.
Let ε>0, M∈N, and 0<p≤1.
If f∈HLp(w)∩L2(Rn) and w∈A1, then there exist a family of w-(p,M,ε;L)-molecules {mj} and a sequence of real numbers {λj} with ∑j|λj|p≤CfHLp(w)p such that f(x)=∑jλjmj(x), and the sum converges in the sense of both L2(Rn)-norm and HLp(w)-norm.
Assume that M>n/2(1/p-1/2) and w∈A1∩RH2/p′. Then, every w-(p,M,ε;L)-molecule m is in HLp(w). Moreover, there exists a constant C>0 independent of m such that mHLp(w)≤C.
Proof.
(i) This is a straightforward consequence of Theorems 9 and 11. (ii) We will use some ideas from [19, 22]. Suppose that m is a w-(p,M,ε;L)-molecule associated with a ball B=B(x0,rB). Let φ∈C0∞(R) be even with suppφ⊆[-(2c0)-1,(2c0)-1] and let Φ denote the Fourier transform of φ. We set Ψ(x)=x2Φ(x), x∈R. By the L2-functional calculus of L, for every m∈L2(Rn), we can establish the following version of the Calderón reproducing formula:(30)mx=cΨ∫0∞t2LMΨ2tLmxdtt,where the above equality holds in the sense of L2(Rn)-norm. Set (31)U0B=2B,UjB=2j+1B∖2jB,j=1,2,….Then, we can decompose (32)Rn×0,∞=⋃j=0∞UjB×0,2jrB∪⋃j=1∞2jB×2j-1rB,2jrB.For any measurable set E in Rn, we denote χE the characteristic function of the set E. Hence, by formula (30), we are able to write (33)mx=cΨ∑j=0∞∫02jrBt2LMΨ2tLmχUjBxdtt+cΨ∑j=1∞∫2j-1rB2jrBt2LMΨ2tLmχ2jBxdtt≔∑j=0∞mj1x+∑j=1∞mj2x.Let us first consider the terms {mj(1)}j=0∞. We will show that each mj(1) is a multiple of a w-(p,M;L)-atom with a sequence of coefficients in lp. Indeed, for every j=0,1,2,…, one can write (34)mj1x=LMbjx,where (35)bjx=cΨ∫02jrBt2MΨ2tLmχUjBxdtt.By using Lemma 12, we can easily conclude that, for every k=0,1,…,M, supp(Lkbj)⊆2j+1B. In addition, by the duality argument, we get (36)2j+1rB2LkbjL22j+1B=suphL22j+1B≤1∫2j+1B2j+1rB2Lkbjxhxdx.Then, it follows from Hölder’s inequality and estimate (29) that (37)∫2j+1B2j+1rB2Lkbjxhxdx=cΨ2j+1rB2k·∫02jrB∫2j+1Bt2MLkΨtLmχUjBy·ΨtLhydydtt≤cΨ2j+1rB2k·2jrB2M-2k∫02jrB∫2j+1Bt2LkΨtL·mχUjBy·ΨtLhydydtt≤cΨ2j+1rB2M∫0∞t2LkΨtL·mχUjBL22dtt1/2∫0∞ΨtL·hχ2j+1BL22dtt1/2≤cΨ2j+1rB2M·mχUjBL2·hχ2j+1BL2≤C·2-jε2j+1rB2M·2j+1B1/2w2j+1B-1/p.Hence, (38)2j+1rB2LkbjL22j+1B≤C·2-jε2j+1rB2M2j+1B1/2w2j+1B-1/p,which implies our desired result. Next we consider the terms {mj(2)}j=1∞. For every j=1,2,…, we write (39)mj2x=cΨ∫2j-1rB2jrBt2LMΨ2tLmxdtt-cΨ∫2j-1rB2jrBt2LMΨ2tLmχ2jBcxdtt≔mj21x-mj22x.To deal with the term mj(21), we recall that m=LMb for some b∈D(LM), and then (40)mj21x=cΨ∫2j-1rB2jrBt2LMΨ2tLLMbxdtt=LMbj21x,where (41)bj21x=cΨ∫2j-1rB2jrBt2LMΨ2tLbxdtt.Since b(x)=b(x)χ2jB(x)+∑l=j∞b(x)χUl(B)(x), then we can further write (42)bj21x=b1,j21x+∑l=j∞blj21x,where (43)b1,j21x=cΨ∫2j-1rB2jrBt2LMΨ2tLbχ2jBxdtt,blj21x=cΨ∫2j-1rB2jrBt2LMΨ2tLbχUlBxdtt.By using Lemma 12 again, we have supp(Lkb1,j(21))⊆2jB and supp(Lkblj(21))⊆2l+1B for every k=0,1,…,M. Moreover, it follows from Minkowski’s integral inequality that (44)2jrB2Lkb1,j21L22jB=cΨ2jrB2k·∫2j-1rB2jrBt2MLM+kΨ2tLbχ2jBdttL22jB≤cΨ2jrB2k·∫2j-1rB2jrBt2LM+kΨ2tLbχ2jBL22jBdtt2k+1≤Cbχ2jBL22jB≤C∑l=0j-1bχUlBL22jB≤C∑l=0j-12-lεrB2M2lB1/2w2lB-1/p.When 0≤l≤j-1, then 2lB⊆2jB. Since w∈A1, then, by using Lemma 6, we can get (45)w2lBw2jB≥C·2lB2jB.Consequently, (46)2jrB2Lkb1,j21L22jB≤C·2-j2M-n1/p-1/2·2jrB2M2jB1/2w2jB-1/p∑l=0∞12lε·12ln/p-n/2≤C·2-j2M-n1/p-1/2·2jrB2M2jB1/2·w2jB-1/p.On the other hand, (47)2l+1rB2Lkblj21L22l+1B=cΨ2l+1rB2k·∫2j-1rB2jrBt2MLM+kΨ2tLbχUlBdttL22l+1B≤cΨ2l+1rB2k∫2j-1rB2jrBt2LM+kΨ2tL·bχUlBL22l+1Bdtt2k+1≤C2l+1rB2k·bχUlBL22l+1B·12jrB2k≤C·2-lε2l+1rB2M·2l+1B1/2w2l+1B-1/p.Observe that 2M>n(1/p-1/2). Thus, from the above discussions, we have already proved that each mj(21) is a multiple of a w-(p,M;L)-atom with a sequence of coefficients in lp. Finally, we estimate the terms {mj(22)}j=1∞. For every j=1,2,…, we decompose mj(22) as follows: (48)mj22x=cΨ∑l=j∞∫2j-1rB2jrBt2LMΨ2tLmχUlBxdtt=∑l=j∞LMblj22x,where (49)blj22x=cΨ∫2j-1rB2jrBt2MΨ2tLmχUlBxdtt.It follows immediately from Lemma 12 that supp(Lkblj(22))⊆2l+1B for every k=1,2,…,M and l≥j. Moreover, (50)2l+1rB2Lkblj22L22l+1B=cΨ2l+1rB2k·∫2j-1rB2jrBt2MLkΨ2tLmχUlBdttL22l+1B≤cΨ2l+1rB2k2lrB2M-2k·∫2j-1rB2jrBt2LkΨ2tLmχUlBL22l+1Bdtt≤cΨ2l+1rB2MmχUlBL22l+1B≤C·2-lε2l+1rB2M2l+1B1/2w2l+1B-1/p.Therefore, we have showed that each mj(22) is also a multiple of a w-(p,M;L)-atom with a sequence of coefficients in lp. This completes the proof of Theorem 14.
4. Proof of Theorems 1 and 2Proof of Theorem 1.
By the known result, we have that, for any γ∈R, the operator Liγ is linear and bounded on L2(Rn) (see, e.g., [2, 23]). Since HLp(w)∩L2(Rn) is dense in HLp(w), then, by Theorems 9 and 11 and a standard density argument, it is enough for us to show that, for any w-(p,M;L)-atom a, M∈N, there exists a constant C>0 independent of a such that Liγ(a)Lp(w)≤C. Let a be a w-(p,M;L)-atom associated with a ball B=B(x0,rB), aL2(B)≤B1/2wB-1/p. We write (51)LiγaLpwp=∫2BLiγaxpwxdx+∑k=1∞∫2k+1B∖2kBLiγaxpwxdx≔I1+I2.We set s=2/p>1. Note that w∈RHs′, and then it follows from Hölder’s inequality, the L2 boundedness of Liγ, and Lemma 5 that(52)I1≤∫2BLiγax2dxp/2∫2Bwxs′dx1/s′≤CaL2Bp·w2B2B1/s≤C.On the other hand, for any x∈2k+1B∖2kB, k=1,2,…, by expression (7), we can write (53)Liγax≤C∫0∞e-tLaxdtt≤C∫0rB2e-tLaxdtt+C∫rB2∞e-tLaxdtt≔I+II.For the term I, we observe that when x∈2k+1B∖2kB, y∈B, then x-y≥2k-1rB. Hence, by using Hölder’s inequality and estimate (4), we deduce(54)e-tLax≤C·t2k-1rBn+1∫Baydy≤C·t2krBn+1·aL2BB1/2≤C·wB-1/p·t2kn+1·rB.So we have (55)I≤C·12kn+1wB1/p·1rB∫0rB2dtt≤C·12kn+1wB1/p.We now turn to estimate the other term II. In this case, since there exists a function b∈D(LM) such that a=LMb and bL2(B)≤rB2M|B|1/2[w(B)]-1/p, then it follows from Hölder’s inequality and Lemma 7 that(56)e-tLax=tLMe-tLbx·1tM≤C·12k-1rBn+1∫Bbydy·1tM-1/2≤C·12krBn+1bL2BB1/2·1tM-1/2≤C·rB2M-12kn+1wB1/p·1tM-1/2.Consequently, (57)II≤C·12kn+1wB1/p·rB2M-1∫rB2∞dttM+1/2≤C·12kn+1wB1/p,where in the last inequality we have used the fact that M≥1. Therefore, by combining the above estimates for I and II, we obtain the following pointwise inequality:(58)Liγax≤C·12kn+1wB1/p,whenx∈2k+1B∖2kB.Notice that w∈A1. Substituting inequality (58) into the term I2 and using Lemma 5, then we have(59)I2≤C∑k=1∞12kpn+1wB·w2k+1B≤C∑k=1∞12kpn+1-n≤C,where the last series is convergent since p>n/(n+1). Summarizing estimates (52) and (59) derived above, we complete the proof of Theorem 1.
Proof of Theorem 2.
Since HLp(w)∩L2(Rn) is dense in HLp(w) and the operator Liγ is linear and bounded on L2(Rn), then, in view of Theorems 9, 11, and 14, it suffices to verify that, for every w-(p,2M;L)-atom a, the function m=Liγ(a) is a multiple of a w-(p,M,ε;L)-molecule for some ε>0, and the multiplicative constant is independent of a. Let a be a w-(p,2M;L)-atom with suppa⊆B=B(x0,rB). By the definition, there exists a function b∈D(L2M) such that (60)a=L2MbrB2LkbL2B≤rB4MB1/2wB-1/p,k=0,1,…,2M.We set b~=Liγ(LMb), and then m=LM(b~). Obviously, we have m(x)∈L2(Rn). Moreover, for k=0,1,…,M, we can deduce (61)rB2Lkb~L22B=1rB2MLiγrB2LM+kbL22B≤C·1rB2MrB2LM+kbL2B≤C·rB2MB1/2wB-1/p.It remains to estimate (rB2L)kb~L2(2j+1B∖2jB) for k=0,1,…,M, j=1,2,…. We write (62)rB2Lkb~x=LiγrB2LkLMbx≤C∫0rB2e-tLrB2kLM+kbxdtt+C∫rB2∞e-tLrB2kLM+kbxdtt≔I′+II′.As mentioned in the proof of Theorem 1, we know that when x∈2j+1B∖2jB, y∈B, then x-y≥2j-1rB, j=1,2,…. It follows from Hölder’s inequality and estimate (4) that(63)I′≤C∫0rB2t2j-1rBn+1rB2kLM+kbL2B1/2dtt≤C·12jrBn+11rB2M·rB4MB1/2wB-1/pB1/2·∫0rB2dtt≤C·12jn+1·rB2MwB-1/p.Since w∈A1 and B⊆2jB, j=1,2,…, then, by using Lemma 6, we can get(64)wBw2jB≥C·B2jB.Hence, (65)I′≤C·12jn+1-n/p·rB2Mw2jB-1/p,whenx∈2j+1B∖2jB.Applying Hölder’s inequality and Lemma 7, we obtain(66)II′≤C·rB2k∫rB2∞tLM+ke-tLbxdttM+k+1≤C·rB2k∫rB2∞12j-1rBn+1bL2BB1/2dttM+k+1/2≤C·12jrBn+1rB4M+2kBwB-1/p∫rB2∞dttM+k+1/2≤C·12jn+1·rB2MwB-1/p.It follows immediately from inequality (64) that (67)II′≤C·12jn+1-n/p·rB2Mw2jB-1/p,whenx∈2j+1B∖2jB.Combining the above estimates for I′ and II′, we thus obtain (68)rB2Lkb~L22j+1B∖2jB≤C·12jn+1-n/p·rB2M2jB1/2w2jB-1/p.Observe that p>n/(n+1). If we set ε=(n+1)-n/p, then we have ε>0. Therefore, we have proved that the function m=Liγ(a) is a multiple of a w-(p,M,ε;L)-molecule. This completes the proof of Theorem 2.
5. Proof of Theorems 3 and 4Proof of Theorem 3.
As in the proof of Theorem 1, assuming first that f∈HLp(w)∩L2(Rn), we are going to prove that, for every w-(p,M;L)-atom a associated with a ball B=B(x0,rB), M>(3n)/4, there exists a constant C>0 independent of a such that L-α/2(a)Lq(wq/p)≤C. We write (69)L-α/2aLqwq/pq=∫2BL-α/2axqwxq/pdx+∑k=1∞∫2k+1B∖2kBL-α/2axqwxq/pdx≔J1+J2.By our assumption 0<α<n/2, 1/q=1/p-α/n, then we are able to choose a number μ>q such that 1/μ=1/2-α/n. Set s=2/p. By a simple calculation, we can see that (70)qp·μq′=1p·qμμ-q=1p·1p-12-1=s′,1-qμ=q1q-1μ=q1p-12=qps′.Applying Hölder’s inequality with exponent ν=μ/q>1, the L2-Lμ boundedness of L-α/2 (see (10)), Lemma 5, and w∈RHs′, we get(71)J1≤∫2BL-α/2axq·μ/qdxq/μ·∫2Bwxq/p·μ/q′dx1-q/μ=∫2BL-α/2axμdxq/μ·∫2Bwxs′dxq/ps′≤CaL2Bqw2B2B1/sq/p≤C.We now turn to deal with J2. For any x∈2k+1B∖2kB, k=1,2,…, by expression (8), we can write (72)L-α/2ax≤C∫0∞e-tLaxdtt1-α/2≤C∫0rB2e-tLaxdtt1-α/2+C∫rB2∞e-tLaxdtt1-α/2≔III+IV.For the term III, it follows immediately from (54) that (73)III≤C·12kn+1wB1/p·1rB∫0rB2dtt1/2-α/2≤C·rBα2kn+1wB1/p.For the other term IV, by the previous estimate (56), we thus have (74)IV≤C·12kn+1wB1/p·rB2M-1∫rB2∞dttM+1/2-α/2≤C·rBα2kn+1wB1/p,where the last inequality holds since M>(3n)/4>1/2+α/2. Combining the above estimates for III and IV, we obtain the following pointwise inequality:(75)L-α/2ax≤C·rBα2kn+1wB1/p,whenx∈2k+1B∖2kB.Note that w∈A1. Substituting inequality (75) into the term J2, we get (76)J2≤C∑k=1∞rBα2kn+1wB1/pq∫2k+1Bwxq/pdx.Using Hölder’s inequality with exponent ν=μ/q>1, Lemma 5, and the condition w∈A1∩RHs′, we can deduce that(77)J2≤C∑k=1∞rBα2kn+1wB1/pq·2k+1Bq/μ·∫2k+1Bwxq/p·μ/q′dx1-q/μ≤C∑k=1∞rBα2kn+1wB1/pq·2k+1Bq/μ·w2k+1B2k+1B1/sq/p≤C∑k=1∞12kqn+1-n≤C,where in the last inequality we have used the fact that q>p>n/(n+1). By combining inequality (77) with (71), we obtain the desired result. Therefore, by a standard density argument, we can show that the same conclusion holds for all f∈HLp(w). This concludes the proof of Theorem 3.
Proof of Theorem 4.
As in the proof of Theorem 2, it is enough to show that, for every w-(p,2M;L)-atom a, function m=L-α/2(a) is a multiple of a w-(p,M,ε;L)-molecule for some ε>0 and the multiplicative constant is independent of a. Let a be a w-(p,2M;L)-atom with suppa⊆B=B(x0,rB) and a=L2M(b), where M>(3n)/4>max{n/2(1/p-1/2),1/2+α/2}, b∈D(L2M). Setting b~=L-α/2(LMb), then we have m=LM(b~). It is easy to check that m(x)∈L2(Rn). As before, since 0<α<n/2, then we may choose a number μ>2 such that 1/μ=1/2-α/n. For k=0,1,…,M, by using Hölder’s inequality and the L2-Lμ boundedness of L-α/2, we obtain(78)rB2Lkb~L22B≤1rB2ML-α/2rB2LM+kbLμ2B2B1/2-1/μ≤C·1rB2MrB2LM+kbL2BB1/2-1/μ≤C·rB2MB1/2+α/nwB-1/p.Noting that 1/q=1/p-α/n, then a straightforward computation yields that q/p<2/p′ whenever 0<α<n/2. By our assumption w∈RH2/p′, then we have w∈RHq/p. Consequently, (79)wq/pBp/q≤C·wBB1-p/q,which implies(80)wB-1/p≤C·B1/q-1/pwq/pB-1/q.Substituting inequality (80) into (78), we can get(81)rB2Lkb~L22B≤C·rB2MB1/2wq/pB-1/q.It remains to estimate (rB2L)kb~L2(2j+1B∖2jB) for k=0,1,…,M, j=1,2,…. We write (82)rB2Lkb~x=L-α/2rB2LkLMbx≤C∫0rB2e-tLrB2kLM+kbxdtt1-α/2+C∫rB2∞e-tLrB2kLM+kbxdtt1-α/2≔III′+IV′.For the term III′, by using the same arguments as in the proof of (63), we have (83)III′≤C·12jn+1·rB2MwB-1/p1rB∫0rB2dtt1/2-α/2≤C·12jn+1·rB2M+αwB-1/p.For the term IV′, we proceed as that of (66) and then obtain (84)IV′≤C·12jn+1·wB-1/prB4M+2k-1∫rB2∞dttM+k+1/2-α/2≤C·12jn+1·rB2M+αwB-1/p.Combining the above estimates for III′ and IV′, we can get (85)rB2Lkb~x≤C·12jn+1·rB2M+αwB-1/p,whenx∈2j+1B∖2jB.Since w∈A1, then it follows from inequality (64) that (86)rB2Lkb~x≤C·12jn+1-n/p·rB2M+αw2jB-1/p,whenx∈2j+1B∖2jB.Similar to the proof of (80), we can also show that (87)w2jB-1/p≤C·2jB1/q-1/pwq/p2jB-1/q.Hence, we finally obtain (88)rB2Lkb~x≤C·12jn+1-n/q·rB2Mwq/p2jB-1/q,whenx∈2j+1B∖2jB.Therefore,(89)rB2Lkb~L22j+1B∖2jB≤C·12jn+1-n/q·rB2M2jB1/2wq/p2jB-1/q.Observe that 1≥q>p>n/(n+1). If we set ε=(n+1)-n/q, then ε>0. Summarizing estimates (81) and (89) derived above, we conclude the proof of Theorem 4.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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