# Multilinear Commutators of Calderón-Zygmund Operator on Generalized Weighted Morrey Spaces

Authors: Vagif S. Guliyev; Farida Ch. Alizadeh
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

The boundedness of multilinear commutators of Calderón-Zygmund operatorTb→ on generalized weighted Morrey spaces Mp,φ(w) with the weight function w belonging to Muckenhoupt's class Ap is studied. When 1<p<∞ and b→=(b1,…,bm), bi∈BMO, i=1,…,m, the sufficient conditions on the pair (φ1,φ2) which ensure the boundedness of the operator Tb→ from Mp,φ1(w) to Mp,φ2(w) are found. In all cases the conditions for the boundedness of Tb→ are given in terms of Zygmund-type integral inequalities on (φ1,φ2), which do not assume any assumption on monotonicity of φ1(x,r),φ2(x,r) in r.


---

## Body

## 1. Introduction

LetT be a Calderón-Zygmund singular integral operator and b∈BMO(ℝn). A well known result of Coifman et al. [1] states that if b∈BMO(ℝn) and T is a Calderón-Zygmund operator, then the commutator operator [b,T]f=T(bf)-bTf is bounded on Lp(ℝn) for 1<p<∞. The commutators of Calderón-Zygmund operator play an important role in studying the regularity of solutions of elliptic, parabolic and ultraparabolic partial differential equations of second order (see, [2–7]).The classical Morrey spacesMp,λ were originally introduced by Morrey in [8] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [2–4, 8, 9].Letb→=(b1,…,bm),bj, and 1≤j≤m be locally integrable functions when we consider multilinear commutators as defined by
(1)Tb→f(x)=∫Rn∏j=1m(bj(x)-bj(y))K(x,y)f(y)dy,
where K(x,y) is Calderón-Zygmund kernel. That is, for all distinct x,y∈ℝn, and all z with 2|x-z|<|x-y|, there exist positive constant C and γ such that(i)
|K(x,y)|≤C|x-y|-n(ii)
|K(x,y)-K(z,y)|≤C(|x-z|γ/|x-y|n+γ); and(iii)
|K(y,x)-K(y,z)|≤C(|x-z|γ/|x-y|n+γ)when m=1, it is the classical commutator which was introduced by Coifman et al. in [1]. It is well known that Calderón-Zygmund operators play an important role in harmonic analysis (see [10–12]).We define the generalized weighed Morrey spaces as follows.Definition 1.
Let1≤p<∞, φ be a positive measurable function on ℝn×(0,∞) and w be non-negative measurable function on ℝn. We denote by Mp,φ(w) the generalized weighted Morrey space, the space of all functions f∈Lp,wloc⁡(ℝn) with finite norm
(2)∥f∥Mp,φ(w)=sup⁡x∈ℝn,r>0φ(x,r)-1w(B(x,r))-1/p∥f∥Lp,w(B(x,r)),
where Lp,w(B(x,r)) denotes the weighted Lp-space of measurable functions f for which
(3)∥f∥Lp,w(B(x,r))≡∥fχB(x,r)∥Lp,w(ℝn)=(∫B(x,r)|f(y)|pw(y)dy)1/p.
Furthermore, byWMp,φ(w) we denote the weak generalized weighted Morrey space of all functions f∈WLp,wloc⁡(ℝn) for which
(4)∥f∥WMp,φ(w)=sup⁡x∈ℝn,r>0φ(x,r)-1w(B(x,r))-1/p∥f∥WLp,w(B(x,r))<∞,
where WLp,w(B(x,r)) denotes the weak Lp,w-space of measurable functions f for which
(5)∥f∥WLp,w(B(x,r))≡∥fχB(x,r)∥WLp,w(ℝn)=sup⁡t>0t(∫{y∈B(x,r):|f(y)|>t}w(y)dy)1/p.Remark 2.
(1) Ifw≡1, then Mp,φ(1)=Mp,φ is the generalized Morrey space.
(2) Ifφ(x,r)≡w(B(x,r))(κ-1)/p, then Mp,φ(w)=Lp,κ(w) is the weighted Morrey space.
(3) Ifφ(x,r)≡v(B(x,r))κ/pw(B(x,r))-1/p, then Mp,φ(w)=Lp,κ(v,w) is the two weighted Morrey space.
(4) Ifw≡1 and φ(x,r)=r(λ-n)/p with 0<λ<n, then Mp,φ(w)=Lp,λ(ℝn) is the classical Morrey space and WMp,φ(w)=WLp,λ(ℝn) is the weak Morrey space.
(5) Ifφ(x,r)≡w(B(x,r))-1/p, then Mp,φ(w)=Lp,w(ℝn) is the weighted Lebesgue space.The commutators are useful in many nondivergence elliptic equations with discontinuous coefficients, [2–5]. In the recent development of commutators, Pérez and Trujillo-González [13] generalized these multilinear commutators and proved the weighted Lebesgue estimates. The weighted Morrey spaces Lp,κ(w) was introduced by Komori and Shirai [14]. Moreover, they showed that some classical integral operators and corresponding commutators are bounded in weighted Morrey spaces. Feng in [15] obtained the boundedness of the multilinear commutators in weighted Morrey spaces Lp,κ(w) for 1<p<∞ and 0<κ<1, where the symbol b→ belongs to bounded mean oscillation (BMO)n. Furthermore, was given the weighted weak type estimate of these operators in weighted Morrey spaces of Lp,κ(w) for p=1 and 0<κ<1.Recently, the generalized weighted Morrey spacesMp,φ(w) introduced by Guliyev [16, 17]. Moreover, in [16, 17] he studied the boundedness of the sublinear operators and their higher order commutators generated by Calderón-Zygmund operators and Riesz potentials in these spaces (see, also [18–20]).The following statement was proved in [18].Theorem A.
Let1≤p<∞, w∈Ap and (φ1,φ2) satisfies the condition
(6)∫r∞essinf⁡t<s<∞φ1(x,s)w(B(x,s))1/pw(B(x,t))1/pdtt≤Cφ2(x,r),
where C does not depend on x and r. Then the operator T is bounded from Mp,φ1(w) to Mp,φ2(w) for p>1 and from M1,φ1(w) to WM1,φ2(w).Remark 3.
Note that, Theorem A was proved in the casew≡1 in [21] and in the case w∈Ap and φ1(x,r)=φ2(x,r)≡w(B(x,r))(κ-1)/p in [14].Definition 4.
BMO
(
ℝ
n
) is the Banach space modulo constants with the norm ∥·∥* defined by
(7)∥b∥*=sup⁡x∈ℝn,r>01|B(x,r)|∫B(x,r)|b(y)-bB(x,r)|dy<∞,
where b∈L1loc⁡(ℝn) and
(8)bB(x,r)=1|B(x,r)|∫B(x,r)b(y)dy.In this paper, we prove the boundedness of the multilinear commutators of Calderón-Zygmund operatorTb→ from one generalized weighted Morrey space Mp,φ1(w) to another Mp,φ2(w) for 1<p<∞ and b→=(b1,…,bm), bi∈BMO, i=1,…,m.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. Main Results

In the following, main results are given. First, we present some estimates which are the main tools to prove our theorems, for the boundedness of the multilinear commutator operatorsTb→ on the generalized weighted Morrey spaces.Theorem 5.
Let1<p<∞, w∈Ap, b→=(b1,…,bm), bi∈BMO, i=1,…,m, and Tb→ be a multilinear commutators defined as (6). Then
(9)∥Tb→f∥Lp,w(B(x0,r))≤C∥b→∥*w(B(x0,r))1/p×∫2r∞ln⁡m(e+tr)∥f∥Lp,w(B(x0,t))w(B(x0,t))-1/pdtt
holds for any ball B=B(x0,r) and for all f∈Lp,wloc⁡(ℝn), where C does not depend on f, x0∈ℝn and r>0.Theorem 6.
Letw∈A1, b→=(b1,…,bm), bi∈BMO, i=1,…,m, and Tb→ be a multilinear commutators defined as (6). Then
(10)∥Tb→f∥WL1,w(B(x0,r))≤C∥b→∥*w(B(x0,r))×∫2r∞ln⁡m(e+tr)∥f∥LΦ,w(B(x0,t))w(B(x0,t))-1dtt
holds for any ball B=B(x0,r) and for all f∈L1loc⁡(ℝn), where C does not depend on f, x0∈ℝn and r>0, where Φ(t)=tln⁡m(e+t) and ∥f∥LΦ,w=∥Φ(|f|)∥L1,w.Now we give a theorem about the boundedness of the multilinear commutator operatorTb→ on the generalized weighted Morrey spaces.Theorem 7.
Let1<p<∞, w∈Ap, and (φ1,φ2) satisfies the condition
(11)∫r∞ln⁡m(e+tr)essinf⁡t<s<∞φ1(x,s)w(B(x,s))1/pw(B(x,t))1/pdtt≤Cφ2(x,r),
where C does not depend on x and r. Let b→=(b1,…,bm), bi∈BMO, i=1,…,m. Then the operator Tb→ is bounded from Mp,φ1(w) to Mp,φ2(w). Moreover,
(12)∥Tb→f∥Mp,φ2(w)≲∥b→∥*∥f∥Mp,φ1(w).Theorem 8.
Letw∈A1, and (φ1,φ2) satisfies the condition
(13)∫r∞ln⁡m(e+tr)essinf⁡t<s<∞φ1(x,s)w(B(x,s))w(B(x,t))dtt≤Cφ2(x,r),
where C does not depend on x and r. Let b→=(b1,…,bm), bi∈BMO, i=1,…,m. Then the operator Tb→ is bounded from MΦ,φ1(w) to WM1,φ2(w). Moreover,
(14)∥Tb→f∥WM1,φ2(w)≲∥b→∥*∥f∥MΦ,φ1(w),
where Φ(t)=tln⁡m(e+t) and ∥f∥MΦ,φ(w)=∥Φ(|f|)∥M1,φ(w).Whenφ1(x,r)=φ2(x,r)≡w(B(x,r))(κ-1)/p, from Theorem 7 we also get the following new result.Corollary 9.
Let1≤p<∞, 0<κ<1, w∈Ap, b→=(b1,…,bm), bi∈BMO, i=1,…,m. Then the operator Tb→ is bounded on Lp,κ(w) for p>1 and from LΦ,κ(w) to WM1,κ(w) for p=1, where Φ(t)=tln⁡m(e+t) and ∥f∥LΦ,κ(w)=∥Φ(|f|)∥L1,κ(w).Proof.
Let1≤p<∞, w∈Ap, 0<κ<1 and b→=(b1,…,bm), bi∈BMO, i=1,…,m. Then the pair (w(B(x,r))(κ-1)/p,w(B(x,r))(κ-1)/p) satisfies the condition (11) for p>1 and the condition (13) for p=1. Indeed, by Lemma 10 there exists C>0 and δ>0 such that for all x∈ℝn and t>r:
(15)w(B(x,t))≥C(tr)nδw(B(x,r)).
Then
(16)∫r∞ln⁡m(e+tr)essinf⁡t<s<∞w(B(x,s))κ/pw(B(x,t))1/pdtt=∫r∞ln⁡m(e+tr)w(B(x,t))(κ-1)/pdtt≲∫r∞ln⁡m(e+tr)((tr)nδw(B(x,r)))(κ-1)/pdtt=w(B(x,r))(κ-1)/p∫r∞ln⁡m(e+tr)(tr)nδ((κ-1)/p)dtt=w(B(x,r))(κ-1)/p∫1∞ln⁡m(e+τ)τnδ((κ-1)/p)dττ≈w(B(x,r))(κ-1)/p.Note that from Corollary9 was proved in [15].


## 3. Some Lemmas

Letℝn be the n-dimensional Euclidean space of points x=(x1,…,xn) with norm |x|=(∑i=1n‍xi2)1/2. For x∈ℝn and r>0, denote B(x,r) the open ball centered at x of radius r. Let B∁(x,r) be the complement of the ball B(x,r), and |B(x,r)| be the Lebesgue measure of B(x,r).A weight function is a locally integrable function onℝn which takes values in (0,∞) almost everywhere. For a weight w and a measurable set E, we define w(E)=∫Ew(x)dx, the Lebesgue measure of E by |E|, and the characteristic function of E by χE. Given a weight w, we say that w satisfies the doubling condition if there is a constant D>0 such that w(2B)≤Dw(B) for any ball B. When w satisfies the doubling condition, we denote w∈Δ2, for short.Ifw is a weight function, then we denote the weighted Lebesgue space by Lp(w)≡Lp(ℝn,w) with the norm
(17)∥f∥Lp,w=(∫ℝn|f(x)|pw(x)dx)1/p<∞,11111111111111111when1≤p<∞
and ∥f∥L∞,w=esssup⁡x∈ℝn|f(x)|w(x) when p=∞.We recall that a weight functionw is in the Muckenhoupt’s class Ap, 1<p<∞, if
(18)[w]Ap≔sup⁡B[w]Ap(B)=sup⁡B(1|B|∫Bw(x)dx)×(1|B|∫Bw(x)1-p′dx)p-1<∞,
where the sup is taken with respect to all the balls B and 1/p+1/p′=1. Note that, for all balls B we have
(19)[w]Ap(B)1/p=|B|-1∥w∥L1(B)1/p∥w-1/p∥Lp′(B)≥1
by Hölder’s inequality. For p=1, the class A1 is defined by the condition Mw(x)≤Cw(x) with [w]A1=sup⁡x∈ℝn(Mw(x)/w(x)), and for p=∞ we define A∞=⋃1≤p<∞‍Ap.Lemma 10 (see [22]).
We have the following:(1)
Ifw∈Ap for some 1≤p<∞, then w∈Δ2. Moreover, for all λ>1 we have
(20)w(λB)≤λnp[w]Apw(B).(2)
Ifw∈A∞, then w∈Δ2. Moreover, for all λ>1 we have
(21)w(λB)≤2λn[w]A∞λnw(B).(3)
Ifw∈Ap for some 1≤p≤∞, then there exist C>0 and δ>0 such that for any ball B and a measurable set S⊂B,
(22)w(S)w(B)≤C(|S||B|)δ.The following results are proved by Pérez and Trujillo-González [13].Lemma 11.
Let1<p<∞ and w∈Ap and suppose that b→=(b1,…,bm), bi∈BMO, i=1,…,m, then there exists a constant C>0 such that
(23)∫ℝn|Tb→f(x)|pw(x)dx≤C∫ℝn|f(x)|pw(x)dx.Although the commutators withBMO function are not of weak type (1,1), they have the following inequality.Lemma 12.
Letw∈A∞ and suppose that b→=(b1,…,bm), bi∈BMO, i=1,…,m, then there exists a constant C>0 such that
(24)sup⁡t>01Φ(1/t)w(x∈ℝn:|Tb→f(x)|>t)≤Csup⁡t>01Φ(1/t)w(x∈ℝn:|MΦ(∥b→∥f)(x)|>t),
where Φ(t)=tln⁡m(e+t).Lemma 13.
Letw∈A1 and suppose that b→=(b1,…,bm), bi∈BMO, i=1,…,m, then there exists a constant C>0 such that
(25)w(x∈ℝn:|Tb→f(x)|>λ)≤C∫ℝnΦ(|f(x)|)w(x)dx,
where Φ(t)=tln⁡m(e+t).In this paper, we need the following statement on the boundedness of the Hardy type operator(26)(H1g)(t)≔1t∫0tln⁡m(e+tr)g(r)dμ(r),0<t<∞,
where μ be a non-negative Borel measure on (0,∞).Theorem 14.
The inequality(27)esssupt>0w(t)H1g(t)≤cesssupt>0v(t)g(t)
holds for all non-negative and non-increasing g on (0,∞) if and only if
(28)A1≔sup⁡t>0w(t)t∫0tln⁡m(e+tr)dμ(r)esssup0<s<rv(s)<∞,
and c≈A1.Note that, Theorem14 is proved analogously to Theorem  4.3 in [21].Lemma 15 (see [23, Theorem 5, page 236]).
Letw∈A∞. Then the norm of BMO(ℝn) is equivalent to the norm of BMO(w), where
(29)BMO(w)={b:∥b∥*,w=sup⁡x∈ℝn,r>01w(B(x,r))×∫B(x,r)|b(y)-bB(x,r),w|w(y)dy<∞},bB(x,r),w=1w(B(x,r))∫B(x,r)b(y)w(y)dy.Remark 16.
(1) The John-Nirenberg inequality: there are constantsC1, C2>0, such that for all b∈BMO(ℝn) and β>0(30)|{x∈B:|b(x)-bB|>β}|≤C1|B|e-C2β/∥b∥*,∀B⊂ℝn.(2)
For1<p<∞ the John-Nirenberg inequality implies that
(31)∥b∥*≈sup⁡B(1|B|∫B|b(y)-bB|pdy)1/pand for 1≤p<∞ and w∈A∞(32)∥b∥*≈sup⁡B(1w(B)∫B|b(y)-bB|pw(y)dy)1/p.Indeed, it follows from the John-Nirenberg inequality and using Lemma10  (3), we get
(33)w({x∈B:|b(x)-bB|>β})≤Cw(B)e-C2βδ/∥b∥*
for some δ>0. Hence, this inequality implies that
(34)∫B|b(y)-bB|pw(y)dy=p∫0∞βp-1w({x∈B:|b(x)-bB|>β})dβ≤Cw(B)∫0∞βp-1e-C2βδ/∥b∥*dβ=Cw(B)∥b∥*p.To prove the requested equivalence we also need to have the right inequality, that is easily obtained using Hölder inequality, then we get (32). Note that (31) follows from (32) in the case w≡1.The following lemma was proved in [24].Lemma 17.
Letb be a function in BMO(ℝn). Let also 1≤p<∞, x∈ℝn, and r1,r2>0. Then
(35)(1|B(x,r1)|∫B(x,r1)|b(y)-bB(x,r2)|pdy)1/p≤C(1+|ln⁡r1r2|)∥b∥*,
where C>0 is independent of f, x, r1 and r2.The following lemma was proved in [17].Lemma 18.
(
i
) Let w∈A∞ and b be a function in BMO(ℝn). Let also 1≤p<∞, x∈ℝn, and r1,r2>0. Then
(36)(1w(B(x,r1))∫B(x,r1)|b(y)-bB(x,r2),w|pw(y)dy)1/p≤C(1+|ln⁡r1r2|)∥b∥*,
where C>0 is independent of f, x, r1 and r2.
(
i
i
) Let w∈Ap and b be a function in BMO(ℝn). Let also 1<p<∞, x∈ℝn, and r1,r2>0. Then
(37)(1w1-p′(B(x,r1))∫B(x,r1)|b(y)-bB(x,r2),w|p′×w(y)1-p′dy(1w1-p′(B(x,r1))∫B(x,r1)|b(y)-bB(x,r2),w|p′)1/p′≤C(1+|ln⁡r1r2|)∥b∥*,
where C>0 is independent of f, x, r1 and r2.


## 4. Proof of the Theorems

Proof of Theorem5.
Letp∈(1,∞). For arbitrary x0∈ℝn and r>0, set B=B(x0,r). Write f=f1+f2 with f1=fχ2B and f2=fχ(2B)∁. Hence
(38)∥Tb→f∥Lp,w(B)≤∥Tb→f1∥Lp,w(B)+∥Tb→f2∥Lp,w(B).
From the boundedness of Tb→ in Lp(w) (see Lemma 11) it follows that:
(39)∥Tb→f1∥Lp,w(B)≤∥Tb→f1∥Lp,w≲∥b→∥*∥f1∥Lp,w=∥b→∥*∥f∥Lp,w(2B).
For the term∥Tb→f2∥Lp,w(B), without loss of generality, we can assume m=2. Thus, the operator Tb→f2 can be divided into four parts
(40)Tb→f2(x)=(b1(x)-(b1)B,w)(b2(x)-(b2)B,w)×∫ℝnK(x,y)f2(y)dy+∫ℝnK(x,y)(b1(y)-(b1)B,w)×(b2(y)-(b2)B,w)f2(y)dy-(b1(x)-(b1)B,w)×∫ℝnK(x,y)(b2(y)-(b2)B,w)f2(y)dy-(b2(x)-(b2)B,w)×∫ℝnK(x,y)(b1(y)-(b1)B,w)f2(y)dy=I1(x)+I2(x)+I3(x)+I4(x).
Forx∈B we have
(41)|Tb→f2(x)|≤|I1(x)|+|I2(x)|+|I3(x)|+|I4(x)|≲|b1(x)-(b1)B,w||b2(x)-(b2)B,w|×∫(2B)∁|f(y)||x0-y|ndy+∫(2B)∁|b1(y)-(b1)B,w|×|b2(y)-(b2)B,w||f(y)||x0-y|ndy+|b1(x)-(b1)B,w|×∫(2B)∁|b2(y)-(b2)B,w||f(y)||x0-y|ndy+|b2(x)-(b2)B,w|×∫(2B)∁|b1(y)-(b1)B,w||f(y)||x0-y|ndy.
Then(42)∥Tb→f2∥Lp,w(B)≲(∫B(∫(2B)∁∏j=12|bi(y)-(bi)B,w||x0-y|n×|f(y)|dy(∫(2B)∁∏j=12|bi(y)-(bi)B,w||x0-y|n)pw(x)dx)1/p+((∫(2B)∁|b2(y)-(b2)B,w||x0-y|n|f(y)|dy)p∫B|b1(x)-(b1)B,w|×(∫(2B)∁|b2(y)-(b2)B,w||x0-y|n|f(y)|dy)p×w(x)dx(∫(2B)∁|b2(y)-(b2)B,w||x0-y|n|f(y)|dy)p)1/p+((∫(2B)∁|b1(y)-(b1)B,w||x0-y|n|f(y)|dy)p∫B|b2(x)-(b2)B,w|×(∫(2B)∁|b1(y)-(b1)B,w||x0-y|n|f(y)|dy)p×w(x)dx(∫(2B)∁|b1(y)-(b1)B,w||x0-y|n|f(y)|dy)p)1/p+(∫B(∫(2B)∁∏j=12|bi(x)-(bi)B,w||x0-y|n×|f(y)|dy(∫(2B)∁∏j=12|bi(x)-(bi)B,w||x0-y|n)pw(x)dx)1/p=I1+I2+I3+I4.
Let us estimate I1(43)I1=w(B)1/p∫(2B)∁∏j=12|bi(y)-(bi)B,w||x0-y|n|f(y)|dy≈w(B)1/p∫(2B)∁∏j=12|bi(y)-(bi)B,w|×|f(y)|∫|x0-y|∞dttn+1dy≈w(B)1/p∫2r∞∫2r≤|x0-y|≤t∏j=12|bi(y)-(bi)B,w|×|f(y)|dydttn+1≲w(B)1/p∫2r∞∫B(x0,t)∏j=12|bi(y)-(bi)B,w|×|f(y)|dydttn+1.
Applying Hölder’s inequality and by Lemma 18, we get
(44)I1≲w(B)1/p×∫2r∞∏j=12(∫B(x0,t)|bi(y)-(bi)B,w|2p′w(y)1-2p′dy)1/2p′×∥f∥Lp,w(B(x0,t))dttn+1≲∏j=12∥bj∥*w(B)1/p×∫2r∞(1+ln⁡⁡tr)2∥w-1/p∥Lp′(B(x0,t))∥f∥Lp,w(B(x0,t))dttn+1≲∥b→∥*w(B)1/p∫2r∞ln⁡2(e+tr)∥f∥Lp,w(B(x0,t))×w(B(x0,t))-1/pdtt.
Let us estimateI2(45)I2=(∫B|b1(x)-(b1)B,w|pw(x)dx)1/p×∫(2B)∁|b2(y)-(b2)B,w||x0-y|n|f(y)|dy≲∥b1∥*w(B)1/p∫(2B)∁|b2(y)-(b2)B,w||f(y)|×∫|x0-y|∞dttn+1dy≈∥b1∥*w(B)1/p×∫2r∞∫2r≤|x0-y|≤t|b2(y)-(b2)B,w||f(y)|dydttn+1≲∥b1∥*w(B)1/p×∫2r∞∫B(x0,t)|b2(y)-(b2)B,w||f(y)|dydttn+1.
Applying Hölder’s inequality and by Lemma18, we get
(46)I2≲∥b1∥*w(B)1/p×∫2r∞(∫B(x0,t)|b2(y)-(b2)B,w|p′w(y)1-p′dy)1/p′×∥f∥Lp,w(B(x0,t))dttn+1≲∏j=12∥bj∥*w(B)1/p∫2r∞(1+ln⁡⁡tr)∥w-1/p∥Lp′(B(x0,t))×∥f∥Lp,w(B(x0,t))dttn+1≲∥b→∥*w(B)1/p×∫2r∞ln⁡2(e+tr)∥f∥Lp,w(B(x0,t))w(B(x0,t))-1/pdtt.
In the same way, we shall get the result ofI3(47)I3≲∥b→∥*w(B)1/p∫2r∞ln⁡2(e+tr)∥f∥Lp,w(B(x0,t))×w(B(x0,t))-1/pdtt.
In order to estimateI4 note that
(48)I4=(∫B∏j=12|bi(x)-(bi)B,w|pw(x)dx)1/p×∫(2B)∁|f(y)||x0-y|ndy≤∏j=12(∫B|bi(x)-(bi)B,w|2pw(x)dx)1/2p×∫(2B)∁‍|f(y)||x0-y|ndy.
By Lemma 18, we get
(49)I4≲∥b→∥*w(B)1/p∫(2B)∁‍|f(y)||x0-y|ndy.
Applying Hölder’s inequality, we get
(50)∫(2B)∁|f(y)||x0-y|ndy≲∫2r∞∥f∥Lp,w(B(x0,t))∥w-1/p∥Lp′(B(x0,t))dttn+1≤[w]Ap1/p∫2r∞∥f∥Lp,w(B(x0,t))w(B(x0,t))-1/pdtt.
Thus, by (50)
(51)I4≲∥b→∥*w(B)1/p∫2r∞∥f∥Lp,w(B(x0,t))w(B(x0,t))-1/pdtt.
Summing up I1 and I4, for all p∈[1,∞) we get
(52)∥Tb→f2∥Lp,w(B)≲∥b→∥*w(B)1/p×∫2r∞ln⁡2(e+tr)∥f∥Lp,w(B(x0,t))×w(B(x0,t))-1/pdtt.
On the other hand,(53)∥f∥Lp,w(2B)≈|B|∥f∥Lp,w(2B)∫2r∞dttn+1≲|B|∫2r∞∥f∥Lp,w(B(x0,t))dttn+1≤w(B)1/p∥w-1/p∥Lp′(B)∫2r∞∥f∥Lp,w(B(x0,t))dttn+1≤w(B)1/p∫2r∞∥f∥Lp,w(B(x0,t))∥w-1/p∥Lp′(B(x0,t))dttn+1≤[w]Ap1/pw(B)1/p×∫2r∞∥f∥Lp,w(B(x0,t))w(B(x0,t))-1/pdtt.
Finally,(54)∥Tb→f∥Lp,w(B)≲∥b→∥*∥f∥Lp,w(2B)+∥b→∥*w(B)1/p×∫2r∞ln⁡m(e+tr)∥f∥Lp,w(B(x0,t))×w(B(x0,t))-1/pdtt,
and the statement of Theorem 5 follows by (53).Proof of Theorem6.
Letp=1. To deal with this result, we split f as above by f=f1+f2, which yields
(55)∥Tb→f∥WL1,w(B)≤∥Tb→f1∥WL1,w(B)+∥Tb→f2∥WL1,w(B).
From the boundedness of Tb→ from LΦ(w) to WL1,w (see Lemma 13) it follows that:
(56)∥Tb→f1∥WL1,w(B)≤∥Tb→f1∥WL1,w≲∥b→∥*∥f1∥LΦ,w=∥b→∥*∥f∥LΦ,w(2B).
For the last term∥Tb→f2∥WL1,w(B), without loss of generality, we still assume m=2. By homogeneity it is enough to assume λ/2=∥b1∥*=∥b2∥*=1 and hence, we only need to prove that
(57)w({x∈B:|Tb→f2(x)|>1})≲w(B)∫2r∞∥f∥LΦ,w(B(x0,t))w(B(x0,t))-1dtt
for all B=B(x0,r). In fact, by Lemma 12, we get
(58)w(x∈B:|Tb→f2(x)|>1)≤sup⁡t>01Φ(1/t)w(x∈ℝn:|Tb→f2(x)|>t)≲sup⁡t>01Φ(1/t)w(x∈B:MΦ(f2)(x)>t)=sup⁡t>01Φ(1/t)w(x∈B:M(Φ(f2))(x)>t),
where Φ(t)=tln⁡m(e+t). We use the Fefferman-Stein maximal inequality
(59)∫{x∈ℝn:Mf(x)>t}ϕ(t)dx≤Ct∫ℝn|f(x)|Mϕ(x)dx,
for any functions f and ϕ≥0. This yields
(60)w({x∈B:M(Φ(f2))(x)>t})≲1t∫{x∈ℝn:Φ(f2)(x)>t}χB(x)w(x)dx≲1t∫ℝnΦ(f2)(x)M(wχB)(x).
Then(61)w(x∈B:|Tb→f2(x)|>1)≲sup⁡t>01Φ(1/t)w(x∈B:M(Φ(f2))(x)>t)≲sup⁡t>01tΦ(1/t)∫ℝnΦ(f2)(x)M(wχB)(x).Proof of Theorem7.
By Theorem5 and Theorem 14 we have for p>1(62)∥Tb→f∥Mp,φ2(w)≲∥b→∥*sup⁡x∈ℝn,r>0φ2(x,r)-1×∫r∞ln⁡m(e+tr)∥f∥Lp,w(B(x,t))×w(B(x,t))-1/pdtt=∥b→∥*sup⁡x∈ℝn,r>0φ2(x,r)-1×∫0r-1ln⁡m(e+1tr)∥f∥Lp,w(B(x,t-1))×w(B(x,t-1))-1/pdtt=∥b→∥*sup⁡x∈ℝn,r>0φ2(x,r-1)-1r1r×∫0rln⁡m(e+rt)∥f∥Lp,w(B(x,t-1))×w(B(x,t-1))-1/pdtt≲∥b→∥*sup⁡x∈ℝn,r>0φ1(x,r-1)-1×w(B(x,r-1))-1/p∥f∥Lp,w(B(x,r-1))=∥b→∥*sup⁡x∈ℝn,r>0φ1(x,r)-1w(B(x,r))-1/p×∥f∥Lp,w(B(x,r))=∥b→∥*∥f∥Mp,φ1(w).



---

*Source: 710542-2014-01-21.xml*