We establish the boundedness of some Schrödinger type operators on weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.
1. Introduction
In this paper, we consider the Schrödinger differential operator
(1)ℒ=-Δ+V(x)onℝn,n≥3,
where V(x) is a nonnegative potential belonging to the reverse Hölder class Bq for q≥n/2.
A nonnegative locally Lq integrable function V(x) on ℝn is said to belong to Bq(q>1) if there exists C>0 such that the reverse Hölder inequality
(2)(1|B|∫BVqdx)1/q≤C(1|B|∫BVdx)
holds for every ball B in ℝn; see [1].
For x∈ℝn, the function ρ(x) is defined by
(3)ρ(x)∶=supr>0{r:1rn-2∫B(x,r)V(y)dy≤1}.
Let p∈[1,∞), α∈(-∞,∞) and λ∈[0,1). For f∈Llocp(ℝn) and V∈Bq(q>1), we say f∈Lα,V,ωp,λ(ℝn) (weighted Morrey spaces related to the potential V) provided that
(4)∥f∥Lα,V,ωp,λ(ℝn)p=supB(x0,r)⊂ℝn(1+rρ(x0))αω(B(x0,2r))-λ×∫B(x0,r)|f(x)|pω(x)dx<∞,
where B=B(x0,r) denotes a ball with centered at x0 and radius r, and the weight functions ω∈Apρ,∞ (see Section 2). The space Lα,V,ωp,λ(ℝn) could be viewed as an extension of weighted Lebesgue spaces (i.e., when α=λ=0, ∥f∥L0,V,ωp,0(ℝn)=∥f∥Lωp(ℝn)). In particular, when α=0 or V=0, ω=1 and 0<λ<1, the space Lα,V,ωp,λ(ℝn) is the classic Morrey space Lp,λ(ℝn) (see [2]). When α=0 or V=0 and 0<λ<1, Lωp,λ(ℝn) was first introduced in [3] (see also [4]), where ω∈Ap(ℝn) (Muckenhoupt weights class). It is easy to see that Lα,V,ωp,λ(ℝn)⊂Lωp,λ(ℝn) for α>0, and Lωp,λ(ℝn)⊂Lα,V,ωp,λ(ℝn) for α<0. In addition, when ω=1, the Lα,V,ωp,λ(ℝn) has been studied in [5].
From [1, 6], we know some Schrödinger type operators, such as ∇(-Δ+V)-1∇ with V∈Bn, ∇(-Δ+V)-1/2 with V∈Bn, (-Δ+V)-1/2∇ with V∈Bn, (-Δ+V)iγ with γ∈ℝ and V∈Bn/2, and ∇2(-Δ+V)-1 with V is a nonnegative polynomial, are standard Calderón-Zygmund operators; see [7]. In particular, the kernels K of operators above all satisfy
(5)|K(x,y)|≤Ck(1+(|x-y|/ρ(x)))k1|x-y|n
for any k∈ℕ. Hence, in the rest of this paper, we always assume that T denotes the above operators.
Recently, Bongioanni et al. [8] proved Lp(ℝn)(1<p<∞) boundedness for commutators of Riesz transforms associated with Schrödinger operator with BMOρ functions which include the classic BMO function, and they [9] established the weighted boundedness for Riesz transforms, fractional integrals, and Littlewood-Paley functions associated with Schrödinger operator with weight Apρ,∞ class which includes the Muckenhoupt weight class. Very recently, the author [10, 11] established the weighted norm inequalities for some Schrödinger type operators, which include commutators of Riesz transforms, fractional integrals, and Littlewood-Paley functions with BMOρ functions; see also [12, 13].
The aim of this paper is to study the boundedness properties of some Schrödinger type operators on the weighted Morrey spaces ∥f∥Lα,V,ωp,λ(ℝn). Our main results in this paper are formulated as follows.
Theorem 1.
Suppose α∈(-∞,∞) and λ∈(0,1).
If 1<p<∞ and ω∈Apρ,∞, then
(6)∥Tf∥Lα,V,ωp,λ(ℝn)≤C∥f∥Lα,V,ωp,λ(ℝn),
where C is independent of f.
If p=1 and ω∈A1ρ,∞, then for any t>0,
(7)ω(B(x,2r))-λ(1+rρ(x))α×tω({y∈B(x,r):|Tf(y)|>t})≤C∥f∥Lα,V,ω1,λ(ℝn)
holds for all balls B, where C is independent of x, r, t, and f.
Let b∈BMOρ (see its definition in Section 2); we define the commutator of T by
(8)[b,T]f=bTf-T(bf).
Theorem 2.
Suppose b∈BMOρ, α∈(-∞,∞) and λ∈(0,1).
If 1<p<∞ and ω∈Apρ,∞, then
(9)∥[b,T]f∥Lα,V,ωp,λ(ℝn)≤C∥f∥Lα,V,ωp,λ(ℝn),
where C is independent of f.
If p=1 and ω∈A1ρ,∞, then, for any t>0,
(10)ω(B(x,2r))-λ(1+rρ(x))α×ω({y∈B(x,r):|[b,T]f(y)|>t})≤CsupB(x,r)⊂ℝnω(B(x,2r))-λ(1+rρ(x))α×∫B(x,r)|f(y)|tln(2+|f(y)|t)ω(y)dy
holds for all balls B(x,r), where C is independent of x, r, t, and f.
Next, we discuss the Littlewood-Paley g function related to Schrödinger operators defined by
(11)g(f)(x)=(∫0∞|ddte-tL(f)(x)|2tdt)1/2,
and the commutator gb of g with b∈BMO(ρ) is defined by
(12)gb(f)(x)=(∫0∞|ddte-tL((b(x)-b(·))f)(x)|2tdt)1/2.
Theorem 3.
Suppose V∈Bn/2, α∈(-∞,∞), and λ∈(0,1).
If 1<p<∞ and ω∈Apρ,∞, then
(13)∥g(f)∥Lα,V,ωp,λ(ℝn)≤C∥f∥Lα,V,ωp,λ(ℝn),
where C is independent of f.
If p=1 and ω∈A1ρ,∞, then, for any t>0,
(14)ω(B(x,2r))-λ(1+rρ(x))α×tω({y∈B(x,r):|g(f)(y)|>t})≤C∥f∥Lα,V,ω1,λ(ℝn)
holds for all balls B, where C is independent of x, r, t, and f.
Theorem 4.
Suppose V∈Bn/2, b∈BMOρ, α∈(-∞,∞), and λ∈(0,1).
If 1<p<∞ and ω∈Apρ,∞, then
(15)∥gb(f)∥Lα,V,ωp,λ(ℝn)≤C∥f∥Lα,V,ωp,λ(ℝn),
where C is independent of f.
If p=1 and ω∈A1ρ,∞, then, for any t>0,
(16)ω(B(x,2r))-λ(1+rρ(x))α×ω({y∈B(x,r):|gb(f)(y)|>t})≤CsupB(x,r)⊂ℝn(1+rρ(x))αω(B(x,2r))-λ×∫B(x,r)|f(y)|tln(2+|f(y)|t)ω(y)dy
holds for all balls B, where C is independent of x, r, t, and f.
Finally, we consider the boundedness of fractional integrals related to Schrödinger operators.
Let ℒ=-Δ+V with V∈Bq for q≥n/2 and its associated semigroup:
(17)Ttf(x)=e-tℒf(x)=∫ℝnkt(x,y)f(y)dy,f∈L2(ℝn),t>0.
The ℒ-fractional integral operator is defined by
(18)ℐβf(x)=ℒ-β/2f(x)=∫0∞e-tℒf(x)tβ/2-1dtfor0<β<n.
Theorem 5.
Suppose V∈Bn/2, α∈(-∞,∞) and 0<β<n.
If 1<p<n/β, 1/q=1/p-β/n, ν=q/p, 0<λ<1/ν, and ωq∈A1+q/p′ρ,∞, where p′=p/(p-1), then
(19)∥ℐβf∥Lα,V,ωqq,νλ(ℝn)≤C∥f∥Lα,V,ωpp,λ(ℝn),
where C is independent of f.
If p=1, q=n/(n-β), 0<λ<1, and ω∈A1ρ,∞, then, for any t>0,
(20)ω(B(x,2r))-λ(1+rρ(x))α×tω({y∈B(x,r):|ℐβf(y)|>t})1/q≤C∥f∥Lα,V,ω1/q1,λ(ℝn)
holds for all balls B(x,r), where C is independent of x,r,t, and f.
Let b∈BMOρ; we define the commutator of ℐβ by
(21)[b,ℐβ]f=bℐβf-ℐβ(bf).
Theorem 6.
Let b∈BMO, V∈Bn/2, α∈(-∞,∞), and 0<β<n.
If 1<p<n/β, 1/q=1/p-β/n, ν=q/p, 0<λ<1/ν, and ωq∈A1+q/p′ρ,∞, then
(22)∥[b,ℐβ]f∥Lα,V,ωqq,νλ(ℝn)≤C∥f∥Lα,V,ωpp,λ(ℝn),
where C is independent of f.
If p=1, q=n/(n-β), 0<λ<1, and ω∈A1ρ,∞, then, for any t>0,
(23)ω(B(x,2r))-λ(1+rρ(x))α×ω({y∈B(x,r):|[b,ℐβ]f(y)|>t})≤CsupB(x,r)⊂ℝnω(B(x,2r))-λ(1+rρ(x))α×Φ(∫B|f(y)|tln(2+|f(y)|t)Θ(ω(y))dy)
holds for all balls B=B(x,r), where Φ(t)=[tlog(2+tβ/n)]n/(n-β) and Θ(t)=t1-β/nlog(e+t-β/n), and C is independent of x,r,t, and f.
We remark that even if in the ω=1 case, Theorems 2, 3, 4, and 6 are also new; see [5].
Throughout this paper, C is a positive constant which is independent of the main parameters and not necessary the same at each occurrence.
2. Some Notation and Basic Results
We first recall some notation. Given B=B(x,r) and λ>0, we will write λB for the λ-dilate ball, which is the ball with the same center x and with radius λr. Given a Lebesgue measurable set E and a weight ω, |E| will denote the Lebesgue measure of E and ω(E)=∫Eωdx. ∥f∥Lp(ω) will denote (∫ℝn|f(y)|pω(y)dy)1/p for 0<p<∞.
Lemma 7 (see [1]).
There exists l0>0 and C0>1 such that
(24)1C0(1+|x-y|ρ(x))-l0≤ρ(y)ρ(x)≤C0(1+|x-y|ρ(x))l0/(l0+1).
In particular, ρ(x)~ρ(y) if |x-y|<Cρ(x).
In this paper, we write Ψθ(B)=(1+r/ρ(x0))θ, where θ>0, x0 and r denotes the center and radius of B, respectively.
A weight will always mean a nonnegative function which is locally integrable. As in [9], we say that a weight ω belongs to the class Apρ,θ for 1<p<∞, if there is a constant C such that for all ball B=B(x,r)(25)(1Ψθ(B)|B|∫Bω(y)dy)×(1Ψθ(B)|B|∫Bω-1/(p-1)(y)dy)p-1≤C.
We also say that a nonnegative function ω satisfies the A1ρ,θ condition if there exists a constant C for all balls B(26)MVθ(ω)(x)≤Cω(x),a.e.x∈ℝn,
where
(27)MVθf(x)=supx∈B1Ψθ(B)|B|∫B|f(y)|dy.
Since Ψθ(B)≥1, obviously, Ap⊂Apρ,θ for 1≤p<∞, where Ap denote the classical Muckenhoupt weights; see [7]. We will see that Ap⊂Apρ,θ for 1≤p<∞ in some cases. In fact, let θ>0 and 0≤γ≤θ; it is easy to check that ω(x)=(1+|x|)-(n+γ)∉A∞=⋃p≥1Ap and ω(x)dx is not a doubling measure, but ω(x)=(1+|x|)-(n+γ)∈A1ρ,θ provided that V=1 and Ψθ(B(x0,r))=(1+r)θ.
For convenience, we always assume that Ψ(B) denotes Ψθ(B), Apρ,∞=⋃θ>0Apρ,θ, and A∞ρ,∞=⋃p≥1Apρ,∞.
Lemma 8 (see [10]).
Let 0<θ<∞; then
if 1≤p1<p2<∞, then Ap1ρ,θ⊂Ap2ρ,θ.
ω∈Apρ,θ if and only if ω-1/(p-1)∈Ap′ρ,θ, where 1/p+1/p′=1.
If ω∈Apρ,θ for 1≤p<∞, then there exists a constant such that for any λ>1(28)ω(λB(x0,r))≤C(1+λrρ(x0))(l0+1)θω(B(x0,r)).
Lemma 9 (see [12]).
Let 0<θ<∞, 1≤p<∞. If ω∈Apρ,θ, then there exists positive constants δ,η, and C such that
(29)(1|B|∫Bω(y)1+δdy)1/(1+δ)≤C1|B|∫Bω(y)dy(1+rρ(x0))η
for all ball B(x0,r).
As a consequence of Lemma 9, we have the following result.
Corollary 10 (see [12]).
Let 0<θ<∞, 1≤p<∞. If ω∈Apρ,θ, then there exist positive constants q>1, η and C such that
(30)ω(E)ω(B)≤C(|E||B|)1/q(1+rρ(x0))η
for any measurable subset E of a ball B(x0,r).
Bongioanni et al. [8] introduce a new space BMOθ(ρ) defined by
(31)∥f∥BMOθ(ρ)=supB⊂ℝn1Ψθ(B)|B|∫B|f(x)-fB|dx<∞,
where fB=(1/|B|)∫Bf(y)dy and Ψθ(B)=(1+r/ρ(x0))θ and θ>0.
In particular, Bongioanni et al. [8] proved the following result for BMOθ(ρ).
Proposition 11.
Let θ>0 and 1≤s<∞. If b∈BMOθ(ρ), then
(32)(1|B|∫B|b-bB|s)1/s≤cC0θs∥b∥BMOθ(ρ)(1+rρ(x0))θ′,
for all B=B(x0,r), with x∈ℝn and r>0, where θ′=(l0+1)θ and C0 is defined in Lemma 7 and c is a constant depending only on n.
Obviously, the classical BMO is properly contained in BMOθ(ρ); for more examples, see [8]. For convenience, we let BMOρ=⋃θ>0BMOθ(ρ).
Tang [10] gave the following result, which is equivalent to Proposition 11.
Proposition 12.
Suppose that f∈BMOθ(ρ). There exist positive constants γ and C such that for any ball B=B(x0,r)(33)1|B|∫Bexp{γ∥f∥BMOθ(ρ)Ψθ′(B)|f(x)-fB|}dx≤C.
Applying Corollary 10 and Proposition 11, we can obtain the following result.
Proposition 13.
If f∈BMOθ(ρ) and ω∈Apρ,θ(p>1), then there exist positive constants c1,c2, and η such that for every ball B=B(x,r) and every λ>0, one has
(34)ω({x∈B:|f(x)-fB|>λ})≤c1ω(B)exp{-c2λ∥f∥BMOθ(ρ)Ψθ′(B)}(1+rρ(x))η,
where fB=(1/|B|)∫Bf(y)dy, Ψθ′(B)=(1+r/ρ(x0))θ′ and θ′=(l0+1)θ.
Proof.
We adapt the same argument of pages 145-146 in [7]. We first assume ∥f∥BMO(ρ)Ψθ′(B)=1. We apply Chebysheff’s inequality and Proposition 11; we obtain
(35)|{x∈B:|f(x)-fB|>λ}|≤(cC0θs)sλ-s|B|
for 0<λ<∞, 1≤s<∞. From this and by Corollary 10, there exist constants q>1 and η such that
(36)ω({x∈B:|f(x)-fB|>λ})≤(cC0θs)s/qλ-s/qω(B)(1+rρ(x))η,
for 0<λ<∞, 1≤s<∞.
If λ≥2cC0θ/q, we take s=λ/(2cC0θ/q)≥1. Then
(37)ω({x∈B:|f(x)-fB|>λ})≤(12)sω(B)(1+rρ(x))η=e-c1λω(B)(1+rρ(x))η,
where c1=(2cC0θ/q)-1ln2. However, if λ≤2cC0θ/q, then e-c1λ≥e-c12(cC0θ/q)=1/2, and
(38)ω({x∈B:|f(x)-fB|>λ})≤2e-c1λω(B)(1+rρ(x))η
in that range of λ. Altogether then, if we drop the normalization on f by replacing f by f/(∥f∥BMOθ(ρ)Ψθ′(B)), we can obtain the conclusion by taking c1=2 and c2=(2cC0θ/q)-1ln2.
From Proposition 13, it is easy to see the following.
Corollary 14.
If f∈BMOρ and ω∈A∞ρ,∞, then there exist positive constants C and η such that for every ball B=B(x,r), one has
(39)1ω(B)∫B|f(x)-fB|ω(x)dx≤(1+rρ(x))η∥f∥BMOρp,
where fB=(1/|B|)∫Bf(y)dy.
3. Proof of Theorems 1–4Proof of Theorem 1.
Without loss of generality, we may assume that α<0 and ω∈Apρ,θ. Pick any ball B=B(x0,r) and write
(40)f(x)=f0(x)+∑i=1∞fi(x),
where f0=χB(x0,2r)f,fi=χB(x0,2i+1r)∖B(x0,2ir)f for i≥1. Hence, we have
(41)(∫B(x0,r)|Tf(x)|pω(x)dx)1/p≤(∫B(x0,r)|Tf0(x)|pdx)1/pa+∑i=1∞(∫B(x0,r)|Tfi(x)|pω(x)dx)1/p.
By the Lωp boundedness of T (see [10]), we obtain
(42)∫B(x0,r)|Tf0(x)|pω(x)dx≤C(1+rρ(x0))-α×ω(B(x0,2r))λ∥f∥Lα,V,ωp,λ(ℝn)p.
By Lemma 8 and Corollary 10, as well as (5), there exist some positive constants q>1 and η such that
(43)∫B(x0,r)|Tfi(x)|pω(x)dx≤C∫B(x0,r)(∫B(x0,2i+1r)∖B(x0,2ir)|K(x,y)f(y)|dy)p×ω(x)dx≤Ckω(B(x0,2i+1r))λ-1∥f∥Lα,V,ωp,λ(ℝn)p×∫B(x0,r)(1+(2ir/p(x0)))-α+(l0+n+4)pθ(1+(2ir/ρ(x)))kpdx≤Ckω(B(x0,2i+1r))λ-1ω(B(x0,r))×(1+(2ir/p(x0)))-α+(l0+n+4)pθ(1+(2ir/ρ(x0)))kp/(l0+1)×∥f∥Lα,V,ωp,λ(ℝn)p≤Ck(ω(B(x0,2r))ω(B(x0,2i+1r)))1-λω(B(x0,2r))λ×(1+(2ir/p(x0)))-α+(l0+n+4)pθ(1+(2ir/ρ(x0)))kp/(l0+1)×∥f∥Lα,Vp,λ(ℝn)p≤Ck2-in(1-λ)/qω(B(x0,2r))λ×(1+(2ir/p(x0)))-α+(l0+n+4)pθ+η(1+(2ir/ρ(x0)))kp/(l0+1)×∥f∥Lα,Vp,λ(ℝn)p.
From (41), (42), and (43) with k=2([-α+(l0+n+4)pθ+η]+1)(l0+1), we obtain
(44)∥Tf∥Lα,V,ωp,λ(ℝn)≤C∥f∥Lα,V,ωp,λ(ℝn).
As for the case p=1, the proof can be given by replacing (42) with the corresponding weak estimate.
Proof of Theorem 2.
Without loss of generality, we may assume that α<0,b∈BMOθ(ρ). Pick any ball B=B(x0,r), as in the proof of Theorem 1 we write
(45)f(x)=f0(x)+∑i=1∞fi(x).
Set ω∈Apρ,∞ for p>1. By the Lωp boundedness of [b,T] (see [10]), we get
(46)∫B(x0,r)|[b,T]f0(x)|pω(x)dx≤C(1+rρ(x))-αω(B(x0,2r))λ∥f∥Lα,V,ωp,λ(ℝn)p.
Set
(47)br=1|B(x0,r)|∫B(x0,r)b(x)dx.
When i≥1, by Lemmas 7 and 9 and Corollaries 10 and 14, there exist some positive constants q>1 and η such that
(48)∫B(x0,r)|[b,T]fi(x)|pω(x)dx≤Ck(1+(2ir/ρ(x0)))kp/(l0+1)2-inp×{∫B(x0,r)|b(x)-br|pω(x)dx∫B(x0,2i+1r)|f(y)|dyaaa+ω(B(x0,r))∫B(x0,2i+1r)|b(y)-br||f(y)|dy}≤Ckipω(B(x0,2ir))1-λω(B(x0,2r))λ×(1+(2ir/ρ(x0)))-α+(l0+n+4)pθ(1+(2ir/ρ(x0)))kp/(l0+1)∥f∥Lα,V,ωp,λ(ℝn)p≤Cki(ω(B(x0,2r))ω(B(x0,2i+1r)))1-λω(B(x0,2r))λ×(1+(2ir/ρ(x0)))-α+(l0+n+4)pθ+η(1+(2ir/ρ(x0)))kp/(l0+1)∥f∥Lα,V,ωp,λ(ℝn)p≤Cki2-in(1-λ)/qω(B(x0,2r))λ×(1+(2ir/ρ(x0)))-α+(l0+n+4)pθ+η(1+(2ir/ρ(x0)))kp/(l0+1)∥f∥Lα,V,ωp,λ(ℝn)p.
If we take k=2([-α+(l0+n+4)pθ+η]+1)(l0+1) in (48), then we obtain
(49)∥[b,T]f∥Lα,V,ωp,λ(ℝn)≤C∥f∥Lα,V,ωp,λ(ℝn).
It remains to consider the case p=1. Set ω∈A1ρ,θ. From [10], we know that for any t>0(50)ω({y∈ℝn:|[b,T]f(y)|>t})≤C∫ℝn|f(x)|tln(2+|f(x)|t)ω(x)dx.
From this, we have
(51)ω(B(x0,2r))-λ(1+rρ(x0))α×ω({y∈B(x0,r):|[b,T]f0(y)|>t})≤C(1+rρ(x0))αω(B(x0,2r))×∫B(x0,2r)|f(x)|tln(2+|f(x)|t)ω(x)dx.
Set
(52)b2i+1r=1|B(x0,2i+1r)|∫B(x0,2i+1r)b(x)dx.
When i≥1, by Lemma 7, Corollary 10, and Proposition 12, note that ω∈A1ρ,θ, then there exist some positive constants q>1 and η such that
(53)ω(B(x0,2r))-λt-1(1+rρ(x0))α×∫B(x0,r)|[b,T]fi(y)ω(y)|dy≤Ck(1+(2ir/ρ(x0)))α+(l0+n+4)pθ+η(1+(r/ρ(x0)))k/(l0+1)×(2ir)-nω(B(x0,2r))-λt-1(1+rρ(x0))α×{∫B(x0,r)|b(y)-b2i+1r|ω(x)dx∫B(x0,2i+1r)|f(y)|dy+ω(B(x0,r))∫B(x0,2i+1r)|b(y)-b2i+1r||f(y)|dy}≤Cω(B(x0,2r))-λω(B)t-1(1+rρ(x0))-3k0×(∫Bi|f(y)|ω(y)dy∥b-b2i+1r∥expL,Bi∥f∥LlogL,Bi+iω(Bi)-1×∥b∥BMOθ(ρ)∫Bi|f(y)|ω(y)dy)≤Ciω(B(x0,2r))-λω(B)t-1(1+rρ(x0))-3k0×infγ>0{γ+γ|Bi|∫Bi|f(y)|γlog(2+|f(y)|γ)dy}+Ciω(B(x0,2r))-λω(B)ω(Bi)-1t-1(1+rρ(x0))-3k0×∫Bi|f(y)|ω(y)dy≤Ci(ω(2B)ω(Bi))1-λω(2Bi)-λ(1+rρ(x0))-k0×∫Bi|f(y)|tlog(2+|f(y)|t)ω(y)dy≤Ci2-in(1-λ)/qω(2Bi)-λ(1+rρ(x0))α×∫Bi|f(y)|tlog(2+|f(y)|t)ω(y)dy,
by taking k=(2[-α+(l0+n+4)pθ+η]+1)(l0+1) and k=4k0, where in the sixth inequality, we used the following facts (see [2]);
(54)∥f∥LlogL,B,ω=inf{γ>0:1|B|∫B|f(y)|γaaaa×log(2+|f(y)|γ)dy≤10},∥f∥expL,B=inf{γ>0:1|B|∫Bexp(|f(y)|γ)dy≤10};
the generalized Hölder inequality
(55)1|B|∫B|f(y)h(y)|dy≤C∥f∥LlogL,B∥h∥expL,B,∥f∥LlogL,B≤infλ>0{γ+γ|B|∫B|f(y)|γlog(2+|f(y)|γ)dy}≤2∥f∥LlogL,B.
Combing (51) and (53), we obtain that
(56)ω(B(x0,2r))-λ(1+rρ(x0))α×ω({y∈B(x,r):|[b,T]f(y)|>t})≤CsupB(x,r)⊂ℝn(1+rρ(x0))αω(B(x0,2r))-λ×∫B(x,r)|f(y)|tln(2+|f(y)|t)ω(y)dy
holds for all balls B, where C is independent of x,r,t, and f.
Thus, Theorem 2 is proved.
Finally, we give some sketch proof of Theorems 3 and 4.
Proof of Theorems 3 and 4.
Let us denote B=L2(ℝ+,tdt) the set of measurable functions a:ℝ+→C with norm |a|B=(∫ℝ+|a(y,t)|2tdt)1/2<∞, ℳ(ℝn) the set of measurable functions defined on ℝn valued in ℂ, and ℳ(ℝn,B) the set of Bochner-measurable functions h:ℝn→B. The space Lp(ℝn,B)(ω) is the set of h∈ℳ(ℝn,B) with finite norm
(57)∥h∥Lp(ℝn,B)(ω)=(∫ℝn|h(x)|Bpω(x)dx)1/p.
We simply name the space as Lp(ℝn,B) when ω=1.
Thus, we can redefine the g as follows:
(58)g(f)(x)={gtf(x)≔Qtf(x)}t∈ℝ+,Qtf(x):=e-tL(f)(x),
which has associated kernel
(59)K(x,y)={Qt(x,y)}t∈ℝ+.
It is easy to see that
(60)|K(x,y)|B≤Ck(1+(|x-y|/ρ(x)))k1|x-y|n
for any k∈N.
Thus, we can adapt the same argument of Theorems 1 and 2 to prove Theorems 3 and 4; we omit the details here.
4. Proof of Theorems 5 and 6
We first need the following lemma.
Lemma 15 (see [5]).
Let kt(x,y) be as in (17). For any N∈ℕ, there exists a CN such that
(61)∫0∞t-(n-β)/2-1kt(x,y)dt≤CN(1+(|x0-y|/ρ(x0)))N1|x0-y|n-β
for all x∈B(x0,r) and y∈ℝn∖B(x0,2r).
Proof of Theorem 5.
Without loss of generality, we may assume that α<0 and ωq∈A1+q/p′ρ,θ(p>1). Pick any ball B=B(x0,r), as the proof of Theorem 1, we write
(62)f(x)=f0(x)+∑i=1∞fi(x).
Hence, we have
(63)(∫B(x0,r)|ℐβf(x)ω(x)|qdx)1/q≤(∫B(x0,r)|ℐβf0(x)ω(x)|qdx)1/q+∑i=1∞(∫B(x0,r)|ℐβfi(x)ω(x)|qdx)1/q.
Let ν=q/p. By the Lωpp→Lωqq boundedness of ℐβ (see [10]), we get
(64)∫B(x0,r)|ℐβf0(x)ω(x)|qdx≤C(1+rρ(x0))αω(2B)νλ∥f∥Lα,V,ωpp,λ(ℝn)q.
From (61), by Corollary 10, then there exists a constant q0>1 such that
(65)∫B(x0,r)|ℐβfi(x)ω(x)|qdx≤C∫B(x0,r)(∫0∞∫B(x0,2i+1r)∖B(x0,2ir)|f(y)kt(x,y)×f(y)|dytβ/2-1dt∫0∞∫B(x0,2i+1r)∖B(x0,2ir)|kt(x,y))q×ω(x)qdx≤CN∫B(x0,r)(2ir)-n(1+(2ir/ρ(x0)))Nq(∫B(x02i+2r)|f(y)|pdy)q/p×ω(x)qdx≤CN(ω(B(x0,2r))ω(B(x0,2i+1r)))1-λνω(B(x0,2r))-λν×(1+(2ir/ρ(x0)))-α+(l0+n+4)qθ(1+(2ir/ρ(x0)))Nq∥f∥Lα,V,ωpp,λ(ℝn)q≤CN2in(λν-1)/q0ω(B(x0,2r))-λν∥f∥Lα,V,ωpp,λ(ℝn)q,
where N=2([-α+(l0+n+4)qθ]+1).
Note that νλ<1. So
(66)∥ℐβf∥Lα,V,ωqq,νλ(ℝn)≤C∥f∥Lα,V,ωpp,λ(ℝn).
As for the case p=1, the proof is similar.
Proof of Theorem 6.
Without loss of generality, we may assume that α<0 and b∈BMOθ(ρ). Pick any ball B=B(x0,r), as in the proof of Theorem 1, we write
(67)f(x)=f0(x)+∑i=1∞fi(x).
For p>1. Set ω∈Ap∞,θ. Let ν=q/p. By the Lωpp→Lωqq boundedness of [b,Iβ] (see [10]), we get
(68)∫B(x0,r)|[b,ℐβ]f0(x)ω(x)|qdx≤C(1+rρ(x0))-αω(B(x0,2r))νλ∥f∥Lα,V,ωpp,λ(ℝn)q.
Set
(69)br=1|B(x0,r)|∫B(x0,r)b(x)dx.
When i≥1, by (61) and Corollary 10, then there exists some positive constant q0>1 such that
(70)∫B(x0,r)|[b,ℐβ]fi(x)ω(x)|qdx≤CN(1+(2ir/ρ(x0)))N(2ir)-(n-β)×{(∫B(x0,r)|b(x)-br|qωq(x)dx)∫B(x0,2i+1r)|f(y)|dyhhhhh+ω(B(x0,r))∫B(x0,2i+1r)|b(y)-br||f(y)|dy}≤CNi∥b∥BMOρ(ω(B(x0,2r))ω(B(x0,2i+1r)))1-λνω(B(x0,2r))-λν×(1+(2ir/ρ(x0)))-α+(l0+n+4)qθ(1+(2ir/ρ(x0)))Nq∥f∥Lα,V,ωpp,λ(ℝn)q≤CN2in(λν-1)/q0ω(B(x0,2r))-λν∥f∥Lα,V,ωpp,λ(ℝn)q,
by taking N=2([-α+(l0+n+4)qθ]+1).
Then,
(71)∥[b,ℐβ]f∥Lα,V,ωqq,νλ(ℝn)≤C∥f∥Lα,V,ωpp,λ(ℝn).
It remains to consider the case p=1 and q=n/(n-β). Set ω∈A1ρ,θ. From [10], we know
(72)ω({y∈ℝn:|[b,ℐβ]f(y)|>t})≤CΦ(∫ℝn|f(x)|tln(2+|f(x)|t)Θ(ω(x))dx),
where Φ(t)=[tlog(2+tβ/n)]n/(n-β) and Θ(t)=t1-β/nlog(e+t-β/n).
From this, we have
(73)ω(B(x0,2r))-λ(1+rρ(x0))α×ω({y∈B(x0,r):|[b,ℐβ]f0(y)|>t})≤CsupB(x0,r)⊂ℝn(1+rρ(x0))αω(B(x0,2r))-λ×Φ(∫B(x0,2r)|f(x)|tln(2+|f(x)|t)Θ(ω(x))dx).
Set
(74)b2i+1r=1|B(x0,2i+1r)|∫B(x0,2i+1r)b(x)dx.
When i≥1, by (61) and Corollary 10 and Proposition 12, note that ω∈A1ρ,θ, then there exist some positive constants q0>1 and η such that
(75)ω(B(x0,2r))-λt-q(1+rρ(x0))α×∫B(x0,r)|[b,ℐβ]fi(x)|qω(x)dx≤C(1+(2ir/ρ(x0)))Nq(2ir)-nω(B(x0,2r))-λ×(1+rρ(x0))αt-q×{∫B(x0,r)|b(x)-b2i+1r|qω(x)dx×(∫B(x0,2i+1r)|f(y)|dy)q+ω(B(x0,r))(∫B(x0,2i+1r)|b(y)-b2i+1r|aaaaaaaaaaaaaaaaaaaa×|f(y)|dy∫B(x0,2i+1r))q∫B(x0,2i+1r)}≤C(1+2irρ(x0))-3N0(2ir)-nω(B(x0,2r))1-λt-q×(|Bi|∥(b-b2i+1r)Φθ′∥expL,Bi∥f∥LlogL,Bi+i∫Bi|f(y)|dy∥(b-b2i+1r)Φθ′∥expL,Bi)q≤Ciq(1+2irρ(x0))-3N0|Bi|-1ω(B(x0,2r))1-λ×(infγ>0{γ+γ|Bi|∫Bi|f(y)|γlog(2+|f(y)|γ)dy}|Bi|t-1infγ>0{γ+γ|Bi|∫Bi|f(y)|γ×log(2+|f(y)|γ)dy})q+Ciq(1+2irρ(x0))-3N0|Bi|-1ω(2B)1-λ(∫Bi|f(y)|tdy)q≤Ciq(ω(2B)ω(Bi))1-λω(Bi)-λ(1+rρ(x0))-2N0×(∫Bi|f(y)|tlog(2+|f(y)|t)ω(y)1/qdy)q+Ciq(ω(2B)ω(Bi))1-λω(Bi)-λ(1+rρ(x0))-2N0×(∫Bi|f(y)|tω(y)1/qdy)q≤Ciq2-in(1-λ)/q0ω(Bi)-λ(1+rρ(x0))α×(∫Bi|f(y)|tlog(2+|f(y)|t)aaaaaa×ω(y)1/qdy∫Bi|f(y)|tlog)
by taking N0=[-α+(l0+n+4)qθ+η]+1 and N=4N0.
Hence,
(76)ω(2B)-λ(1+rρ(x0))α×ω({y∈B(x0,r):|[b,ℐβ](∑i=1∞fi)(y)|>t})≤C(∑i=1∞(ω(2B)-λt-q(1+rρ(x0))αaaaaa×∫B(x0,r)|[b,ℐβ]fi(y)ω(y)|qdy)1/q)q≤C(∑i=1∞(iq2-in(1-λ)/q0ω(Bi)-λ(1+rρ(x0))αaaaaa×(∫Bi|f(y)|tlog(2+|f(y)|t)aaaaaaaa×ω(y)1/qdy∫Bi|f(y)|t)q)1/q∑i=1∞)q≤CsupB(x0,r)⊂ℝn(1+rρ(x0))αω(2B)-λ×(∫B(x0,r)|f(x)|tln(2+|f(y)|t)aaaaaaa×ω(y)1/qdy∫B(x0,r)|f(x)|t)q.
By (73) and (76), we obtain the desired result. This completes the proof.
5. The Calderón-Zygmund Inequality
For the open set Ω⊂ℝn, ω∈Apρ,∞(ℝn)(1≤p<fz) and V∈Bn, we say f∈Lα,V,ωp,λ(Ω) if
(77)∥f∥Lα,Vp,λ(Ω)p=supB(x0,r)⊂ℝn(1+rρ(x0))αr-λ×∫B(x0,r)∩Ω|f(x)|pdx<∞.
In this section, we consider the behavior of the solution of the following Schrödinger equation:
(78)(-Δ+V)u=f(x),a.e.x∈Ω,
where f∈Lα,V,ωp,λ,ω(Ω), 1<p<∞, 0<λ<1, α∈(-∞,∞), and ω∈A∞ρ,∞.
Theorem 16.
Let Ω be an open set in ℝn and α∈(-∞,∞). If f∈Lα,V,ωp,λ(Ω), then there exists a function u∈Lα,V,ωq,νλ(Ω), where 1<p<n/2, 1/p-1/q=2/n, ν=q/p, 0<λ<1/ν, and ωq∈A1+q/p′ρ,∞, such that
(79)(-Δ+V)u=f(x),a.e.x∈Ω.
Furthermore,
(80)∥D2u∥Lα,V,ωp,λ(Ω)≤C∥f∥Lα,V,ωp,λ(Ω),
where 1<p<∞, 0<λ<n, and ω∈Apρ,∞;
(81)∥Du∥Lα,V,ωqq,ν1λ(Ω)≤C∥f∥Lα,V,ωpp,λ(Ω),
where 1<p<n,1/p-1/q=1/n,ν1=q/p, 0<λ<n/ν1, and ωq∈A1+q/p′ρ,∞;
(82)∥u∥Lα,V,ωqq,νλ(Ω)≤C∥f∥Lα,V,ωpp,λ(Ω),
where 1<p<n/2, 1/p-1/q=2/n, ν=q/p, 0<λ<1/ν, and ωq∈A1+q/p′ρ,∞.
Proof.
From the proof of Theorem 5, we have
(83)∥u∥Lα,V,ωqq,νλ(Ω)≤C∥ℐ2f∥Lα,V,ωqq,νλ(Ω)≤C∥f∥Lα,V,ωpp,λ(Ω).
From the proof of Theorem 1, we obtain
(84)∥D2u∥Lα,V,ωp,λ(Ω)≤C∥D2ℒ-1f∥Lα,V,ωp,λ(Ω)≤C∥f∥Lα,V,ωp,λ(Ω).
Thus, (80) and (82) hold.
From page 543 in [1], we know
(85)|DxΓ(x,y)|≤Ck(1+(|x-y|/ρ(x)))k1|x-y|n-1,
where Γ(x,y) is the fundamental solution for ℒ=-Δ+V.
Using (85) and adapting the argument for Theorem 5, we then have
(86)∥Du∥Lα,V,ωqq,ν1λ(Ω)≤C∥Dℒ-1f∥Lα,V,ωqq,ν1λ(Ω)≤C∥f∥Lα,V,ωpp,λ(Ω).
Thus, (81) holds. Hence, Theorem 16 is proved.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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