Parabolic sublinear operators with rough kernel generated by parabolic calderön-zygmund operators and parabolic local campanato space estimates for their commutators on the parabolic generalized local morrey spaces

Ferit Gurbuz
From the journal Open Mathematics

Abstract

In this paper, the author introduces parabolic generalized local Morrey spaces and gets the boundedness of a large class of parabolic rough operators on them. The author also establishes the parabolic local Campanato space estimates for their commutators on parabolic generalized local Morrey spaces. As its special cases, the corresponding results of parabolic sublinear operators with rough kernel and their commutators can be deduced, respectively. At last, parabolic Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.

MSC 2010: 42B20; 42B25; 42B35

1 Introduction

Let ℝn be the n–dimensional Euclidean space of points x= (x1, …, xn) with norm |x|=(i=1nxi2)12. let B = B(x0, rB) denote the ball with the center x0 and radius rB. For a given measurable set E, we also denote the Lebesgue measure of E by |E|. For any given Ω ⊆ ℝn and 0 < p < ∞, denote by Lp (Ω) the spaces of all functions f satisfying

||f||Lp(Ω)=(Ω|f(x)|pdx)1p<.

Let Sn–1 = {x ∈ ℝn : |x| = 1} denote the unit sphere on ℝn (n ≥2) equipped with the normalized Lebesgue measure (x′), where x′ denotes the unit vector in the direction of x.

To study the existence and regularity results for an elliptic differential operator, i.e.

D=i,j=1nai,j2xixj

with constant coefficients {ai, j}, among some other estimates, one needs to study the singular integral operator T with a convolution kernel K (see [1] or [2]) satisfying

1. K(tx1,. . .,txn) = tnK(x), for any t > 0;

2. KC(ℝn \ {0});

3. Sn1K(x)dσ(x)=0.

Similarly, for the heat operator

D=x1j=2n2xj2,

the corresponding singular integral operator T has a kernel K satisfying

1. (a′)

K(t2x1, . . ., txn) = t–n–1K(x), for any t > 0;

2. (b′)

KC(ℝn) \ {0});

3. (c′)

Sn1K(x)(2x12+x22++xn2)dσ(x)=0.

To study the regularity results for a more general parabolic differential operator with constant coefficients, in 1966, Fabes and Riviére [3] introduced the following parabolic singular integral operator

T¯Pf(x)=p.υ.nK(y)f(xy)dy

with K satisfying

1. K(tα1x1,,tαnxn)=tαK(x1,,xn),t>0,x0,α=i=1nαi;

2. KC(ℝn) \ {0});

3. Sn1K(x)J(x)dσ(x)=0, where αi ≥ 1 (i = 1, . . ., n) and J(x)=α1x12++αnxn2.

Let ρ ∈ (0, ∞)and 0 ≤ φn–1 ≤ 2π, 0 ≤ φiπ, i = 1, . . ., n – 2. For any x ∈ ℝn, set

x1=ρα1cosφ1...cosφn2cosφn1,x2=ρα2cosφ1...cosφn2cosφn1,xn1=ραn1cosφ1sinφ2,xn=ραnsinφ1.

Then dx = ρα–1J(x′)dρdσ(x′), where α=i=1nαi,xSn1, is the element of area of Sn–1 and ρα–1J is the Jacobian of the above transform. In [3] Fabes and Riviére have pointed out that J(x′) is a C function on Sn–1 and 1 ≤J (x′) ≤ M, where M is a constant independent of x′. Without loss of generality, in this paper we may assume αnαn–1 ≥. ·· ≥ α1 ≥ 1. Notice that the above condition (i) can be written as (i′) K(Atx) = |det(At)|–1K(x), where At=diag[tα1,,tαn]=(tα100tαn) is a diagonal matrix.

Note that for each fixed x= (x1, . . ., xn) ∈ℝn the function

F(x,ρ)=i=1nxi2ρ2αi

is a strictly decreasing function of ρ > 0. Hence, there exists an unique t such that F(x, t) = 1. It has been proved in [3] that if we set ρ(0) = 0 and ρ(x) = t tsuch that F(x, t) = 1, then ρ is a metric on ℝn, and (ℝn, ρ) is called the mixed homogeneity space related to {αi}i=1n.

Remark 1.1

Many works have been done for parabolic singular integral operators, including the weak type estimates andLp(strong .p, p)) boundedness. For example, one can see references [46] for details.

Let P be a real n ×n matrix, whose all the eigenvalues have positive real part. Let At = tP(t > 0), and set γ = trP. Then, there exists a quasi-distance ρ associated with P such that (see [7])

(1 – 1) ρ (Atx) = (x), t > 0 for every x ∈ ℝn,

(1 –2) ρ (0) = 0, ρ(x – y) = ρ(y – x) ≥ 0, and ρ(x – y) ≤ k(ρ(x –z) + ρ (y –z)),

(1 – 3) dx = ργ–1dσ where ρ = ρ(x), w=Aρ1x and (w) is a measure on the unit ellipsoid {w : ρ(w) = 1}.

Then,{ℝn, ρ, dx} becomes a space of homogeneous type in the sense of Coifman-Weiss (see [7]) and a homogeneous group in the sense of Folland-Stein (see [8]). Moreover, we always assume that there hold the following properties of the quasi-distance ρ:

(1 – 4) For every x,

c1|x|α1ρ(x)c2|x|α2ifρ(x)1;c3|x|α3ρ(x)c4|x|α4ifρ(x)1.

and

ρ(θx)ρ(x)for0<θ<1,

with some positive constants ˛αi and ci(i = 1, . . ., 4). Similar properties also hold for the quasimetric ρ* associated with the adjoint matrix P*

The following are some important examples of the above defined matrices P and distances ρ:

1. Let (Px, x) ≥ (x, x) (x ∈ ℝn). In this case, ρ(x) is defined by the unique solution of |At1x|=1, and k = 1. This is the case studied by Calderón and Torchinsky in [9].

2. Let P be a diagonal matrix with positive diagonal entries, and let t = ρ (x) x ∈ ℝn be the unique solution of |At1x|=1.

1. When all diagonal entries are greater than or equal to 1, Besov et al. in [10] and Fabes and Riviére in [3] have studied the weak (1.1) and Lp (strong (p, p)) estimates of the singular integral operators on this space.

2. If there are diagonal entries smaller than 1, then ρ satisfies the above (1 – 1) – (1 – 4) with k ≥ 1.

It is a simple matter to check that ρ (xy) defines a distance between any two points x, y ∈ ℝn. Thus ℝn, endowed with the metric ρ, defines a homogeneous metric space [3, 10]. Denote by E(x, r) the ellipsoid with center at x and radius r, more precisely, E(x, r) = {y ∈ ℝn : ρ(xy) < r}. For k > 0, we denote kE(x, r) = {y ∈ ℝn : ρ(xy) < kr}. Moreover, by the property of ρ and the polar coordinates transform above, we have

|E(x,r)|=ρ(xy)<rdy=υρrα1++αn=υρrγ,

where |E(x, r)| stands for the Lebesgue measure of E(x, r) and υρ is the volume of the unit ellipsoid on ℝn . By EC(x, r) = ℝn\E(x, r), we denote the complement of E(x, r). Moreover, in the standard parabolic case P0 = diag[1, . . .,1,2] we have

ρ(x)=|x|2+|x|4+xn22,x=(x,xn).

Note that we deal not exactly with the parabolic metric, but with a general anisotropic metric ρ of generalized homogeneity, the parabolic metric being its particular case, but we keep the term parabolic in the title and text of the paper, the above existing tradition, see for instance [9].

Suppose that Ω(x) is a real-valued and measurable function defined on ℝn. Suppose that Sn–1 is the unit sphere on ℝn (n ≥ 2) equipped with the normalized Lebesgue surface measure .

Let Ω ∈ Ls(Sn–1) with 1 < s ≤ ∞ be homogeneous of degree zero with respect to At(Ω(x) is At-homogeneous of degree zero). We define s=ss1 for any s > 1. Suppose that TΩPrepresents a parabolic linear or

a parabolic sublinear operator, which satisfies that for any fL1(ℝn) with compact support and xsuppf

|TΩPf(x)|c0n|Ω(xy)|ρ(xy)γ|f(y)|dy,(1)

where c0 is independent of f and x.

We point out that the condition (1) in the case Ω ≡ 1 and P = I was first introduced by Soria and Weiss in [11] . The condition (1) is satisfied by many interesting operators in harmonic analysis, such as the parabolic Calderón–Zygmund operators, parabolic Carleson’s maximal operator, parabolic Hardy–Littlewood maximal operator, parabolic C. Fefferman’s singular multipliers, parabolic R. Fefferman’s singular integrals, parabolic Ricci– tein’s oscillatory singular integrals, parabolic the Bochner–Riesz means and so on (see [11, 12] for details).

Let Ω ∈ Ls(Sn–1) with 1 < s ≤ ∞ be homogeneous of degree zero with respect to At (Ω(x) is At-homogeneous of degree zero), that is,

Ω(Atx)=Ω(x),

for any t > 0, x ∈ ℝn and satisfies the cancellation(vanishing) condition

Sn1Ω(x)J(x)dσ(x)=0,

where x=x|x| for any x ≠ 0.

Let fLloc(ℝn). The parabolic homogeneous singular integral operator T¯ΩP and the parabolic maximal operator MΩP by with rough kernels are defined by

T¯ΩPf(x)=p.υ.nΩ(xy)ρ(xy)γf(y)dy,(2)
MΩPf(x)=supt>0|E(x,t)|1E(x,t)|Ω(xy)||f(y)|dy,

satisfy condition (1).

It is obvious that when Ω1,T¯ΩPT¯P and MΩPMP are the parabolic singular operator and the parabolic maximal operator, respectively. If P = I, then MΩIMΩ is the Hardy-Littlewood maximal operator with rough kernel, and T¯ΩIT¯Ω is the homogeneous singular integral operator. It is well known that the parabolic maximal and singular operators play an important role in harmonic analysis (see [8, 9] and [13, 14]). In particular, the boundedness of T¯ΩP on Lebesgue spaces has been obtained.

Theorem 1.2

Suppose that Ω ∈Ls(Sn–1), 1 < s ≤ ∞ is At-homogeneous of degree zero having mean value zero on Sn–1. If s′≤ p or p < s, then the operatorT¯ΩPis bounded on Lp(ℝn). Also, the operatorT¯ΩPis bounded from L1(ℝn) to WL1(ℝn). Moreover, we have for p > 1

T¯ΩPfLPC||f||Lp,

and for p = 1

T¯ΩPfWL1C||f||L1.
Corollary 1.3

Under the assumptions of Theorem 1.2, the operatorMΩPis bounded on Lp(ℝn). Also, the operatorMΩPis bounded from L1(ℝn) to WL1(ℝn). Moreover, we have for p > 1

MΩPfLpC||f||Lp,

and for p = 1

MΩPfWL1C||f||L1.

Proof. It suffices to refer to the known fact that

MΩPf(x)CγT¯ΩPf(x),Cγ=|E(0,1)|.

Note that in the isotropic case P = ITheorem 1.2 has been proved in [15].

Let b be a locally integrable function on ℝn, then we define commutators generated by parabolic maximal and singular integral operators with rough kernels and b as follows, respectively.

MΩ,bPf(x)=supt>0|E(x,t)|1E(x,t)|b(x)b(y)||Ω(xy)||f(y)|dy,[b,T¯ΩP]f(x)b(x)T¯ΩPf(x)T¯ΩP(bf)(x)=p.υ.Rn[b(x)b(y)]Ω(xy)ρ(xy)γf(y)dy.(3)

If we take α1 = ∙∙∙ = αn =1 and P = I, then obviously ρ(x)=|x|=(i=1nxi2)12,γ=n,(n,ρ)=(n,||),

EI(x, r) = B(x, r), At = tI and J(x′) ≡1. In this case, T¯ΩP defined as in (2) is the classical singular integral operator with rough kernel of convolution type whose boundedness in various function spaces has been well-studied by many authors (see [16-20], and so on). And also, in this case, [b,T¯ΩP] defined as in (3) is the classical commutator of singular integral operator with rough kernel of convolution type whose boundedness in various function spaces has also been well-studied by many authors (see [16-20], and so on).

The classical Morrey spaces Lp,λ have been introduced by Morrey in [21] to study the local behavior of solutions of second order elliptic partial differential equations(PDEs). In recent years there has been an explosion of interest in the study of the boundedness of operators on Morrey-type spaces. It has been shown that many properties of solutions to PDEs are concerned with the boundedness of some operators on Morrey-type spaces. In fact, better inclusion between Morrey and Hölder spaces allows to obtain higher regularity of the solutions to different elliptic and parabolic boundary problems.

Morrey has stated that many properties of solutions to PDEs can be attributed to the boundedness of some operators on Morrey spaces. For the boundedness of the Hardy–Littlewood maximal operator, the fractional integral operator and the Calderón–Zygmund singular integral operator on these spaces, we refer the readers to [2224]. For the properties and applications of classical Morrey spaces, see [25-28] and references therein. The generalized Morrey spaces Mp,φ are obtained by replacing rλ with a function φ(r) in the definition of the Morrey space. During the last decades various classical operators, such as maximal, singular and potential operators have been widely investigated in classical and generalized Morrey spaces.

We define the parabolic Morrey spaces Lp,λ, P(ℝn) via the norm

||f||Lp,λ,P=supxn,r>0rλp||f||Lp(E(x,r))<,

where fLploc(n),0λγ and 1 ≤ p ≤ ∞

Note that Lp,0,P = Lp(ℝn) and Lp, γ, P = L(ℝn). If λ < 0 or λ > γ, then Lp = Θ, where Θ is the set of all functions equivalent to 0 on ℝn.

We also denote by WLp,λ,PWLp,λ, P(ℝn) the weak parabolic Morrey space of all functions fWLploc(n) for which

||f||WLp,λ,P||f||WLp,λ,P(n)=supxn,r>0rλp||f||WLp(E(x,r))<,

where WLp(E(x, r)) denotes the weak Lp-space of measurable functions f for which

||f||WLp(E(x,r))||fχE(x,r)||WLp(n)=supt>0t|{yE(x,r):|f(y)|>t}|1/p=sup0<t|E(x,r)|t1/p(fχE(x,r))*(t)<,

where g* denotes the non-increasing rearrangement of a function g.

Note that WLp(ℝn) = WLp,0,P(ℝn),

Lp,λ,P(n)WLp,λ,P(n)and||f||WLp,λ,P||f||Lp,λ,P.

If P = I, then Lp,λ, I(ℝn) ≡ Lp,λ(ℝn) is the classical Morrey space.

It is known that the parabolic maximal operator MPis also bounded on Lp,λ, P for all 1 < p < ∞ and 0 < λ < γ(see, e.g. [29]), whose isotropic counterpart has been proved by Chiarenza and Frasca [23].

In this paper, we prove the boundedness of the parabolic sublinear operators with rough kernel TΩPsatisfying condition (1) generated by parabolic Calderón-Zygmund operators with rough kernel from one parabolic generalized local Morrey space LMp,φ1,P{x0} to another one LMp,φ2,P{x0}, 1 < p < ∞, and from the space LM1,φ1,P{x0} to the weak space WLM1,φ2,P{x0}. In the case of bLCp2,λ,P{x0} (parabolic local Campanato space) and [b,TΩP] is a sublinear operator, we find the sufficient conditions on the pair (φ1, φ2) which ensures the boundedness of the commutator operators [b,TΩP] from LMp1,φ1,P{x0} to LMp,φ2,P{x0},1<p<,1p=1p1+1p2 and 0λ<1γ.

By AB we mean that ACB with some positive constant C independent of appropriate quantities. If AB and BA, we write AB and say that A and B are equivalent.

2 Parabolic generalized local Morrey spaces

Let us define the parabolic generalized Morrey spaces as follows.

Definition 2.1 (parabolic generalized Morrey space)

Let φ(x, r) be a positive measurable function onn × (0, ∞)and 1 ≤ p < ∞. We denote by Mp,φ, PMp,φ, P(ℝn) the parabolic generalized Morrey space, the space of all functionsfLploc(n)with finite quasinorm

||f||Mp,φ,P=supxRn,r>0φ(x,r)1|E(x,r)|1p||f||Lp(E(x,r))<.

Also byWMp,φ, PWMp,φ, P(ℝn) we denote the weak parabolic generalized Morrey space of all functionsfWLploc(n)for which

||f||WMp,φ,P=supxn,r>0φ(x,r)1|E(x,r)|1p||f||WLp(E(x,r))<.

According to this definition, we recover the parabolic Morrey space Lp,λ,P and the weak parabolic Morrey space WLp,λ,P under the choice φ(x,r)=rλγp:

Lp,λ,P=Mp,φ,P|φ(x,r)=rλγp,WLp,λ,P=WMp,φ,P|φ(x,r)=rλγp.

Inspired by the above Definition 2.1, [16] and the Ph.D. thesis of Gurbuz [17], we introduce the parabolic generalized local Morrey spaces LMp,φ,P{x0} by the following definition.

Definition 2.2 (parabolic generalized local Morrey space)

Let φ(x, r) be a positive measurable function onn ×(0, ) and 1 ≤ p < ∞ For any fixed x0 ∈ ℝn we denote byLMp,φ,P{x0}LMp,φ,P{x0}(n)the parabolic generalized local Morrey space, the space of all functionsfLploc(n)with finite quasinorm

||f||LMp,φ,P{x0}=supr>0φ(x0,r)1|E(x0,r)|1p||f||Lp(E(x0,r))<.

Also byWLMp,φ,P{x0}WLMp,φ,P{x0}(n)we denote the weak parabolic generalized local Morrey space of all functionsfWLploc(n)for which

||f||WLMp,φ,P{x0}=supr>0φ(x0,r)1|E(x0,r)|1p||f||WLp(E(x0,r))<.

According to this definition, we recover the local parabolic Morrey space LLp,λ,P{x0} and weak local parabolic Morrey space WLLp,λ,P{x0} under the choice φ(x0,r)=rλγp:

LLp,λ,P{x0}=LMp,φ,P{x0}|φ(x0,r)=rλγp,WLLp,λ,P{x0}=WLMp,φ,P{x0}|φ(x0,r)=rλyp.

Furthermore, we have the following embeddings:

Mp,φ,PLMp,φ,P{x0},||f||LMp,φ,P{xo}||f||Mp,φ,P,WMp,φ,PWLMp,φ,P{x0},||f||WLMp,φ,P{x0}||f||WMp,φ,P.

In [30] the following statement has been proved for parabolic singular operators with rough kernel T¯ΩP, containing the result in [3133].

Theorem 2.3

Suppose that ΩLs(Sn-1), 1 < s ≤ ∞ is At-homogeneous of degree zero and has mean value zero on Sn-1. Let1s<p<(s=ss1)and φ(x, r) satisfies conditions

c1φ(x,r)φ(x,t)cφ(x,r)(4)

whenever rt ≤ 2r where c (≥ 1) does not depend on t, r, x ∈ ℝnand

rφ(x,t)pdttCφ(x,r)p,(5)

where C does not depend on x and r. Then the operatorT¯ΩPis bounded on Mp,φ, P

The results of [3133] imply the following statement.

Theorem 2.4

Let 1 ≤ p < ∞ and φ(x, r) satisfies conditions (4) and (5). Then the operators MP andTP are bounded on Mp,φ, Pfor p > 1 and fromM1,φ, PtoWM1,φ,Pand for p = 1.

The following statement, containing the results obtained in [3133] has been proved in [34, 35] (see also [36-39] and [40, 41]).

Theorem 2.5

Let 1 ≤ p < ∞ and the pair (φ1, φ2) satisfies the condition

rφ1(x,t)dttCφ2(x,r),(6)

where C does not depend on x and r. Then the operatorTPis bounded fromMp,φ1,PtoMp,φ2,Pfor p > 1 and fromM1,φ1,PtoWM1,φ2,Pforp = 1.

Finally, inspired by the Definition 2.2, [16] and the Ph.D. thesis of Gurbuz [17] in this paper we consider the boundedness of parabolic sublinear operators with rough kernel on the parabolic generalized local Morrey spaces and give the parabolic local Campanato space estimates for their commutators.

3 Parabolic sublinear operators with rough kernel generated by parabolic Calderón-Zygmund operators on the spaces LMp,φ, P{x0}

In this section, we will prove the boundedness of the operator TΩP on the parabolic generalized local Morrey spaces LMp,φ,P{x0} by using the following statement on the boundedness of the weighted Hardy operator

Hωg(t):=tg(s)ω(s)ds,0<t<,

where ω is a fixed non-negative function and measurable on (0, ∞).

Theorem 3.1

([16, 17, 42])

Let v1, v2and ω be positive almost everywhere and measurable functions on (0, ∞). The inequality

esssupt>0υ2(t)Hωg(t)Cesssupt>0υ1(t)g(t)(7)

holds for some C > 0 for all non-negative and non-decreasing functions g on (0, ∞) if and only if

B:=supt>0υ2(t)tω(s)dsesssups<τ<υ1(τ)<.(8)

Moreover, the value C = B is the best constant for (7).

We first prove the following Theorem 3.2.

Theorem 3.2

Let x0 ∈ ℝn, 1 ≤ p < ∞ and ΩLs(Sn-1), 1 < s ≤ ∞, be At-homogeneous of degree zero. LetTΩPbe a parabolic sublinear operator satisfying condition (1), bounded on Lp(n) for p > 1, and bounded from L1(ℝn) to WL1(ℝn).

If p > 1and s′ ≤ p, then the inequality

TΩPfLp(E(x0,r))<˜rγp2krtγp1||f||Lp(E(x0,t))dt

holds for any ellipsoid E(x0, r) and for allfLploc(n).

If p > 1and p < s, then the inequality

TΩPfLp(E(x0,r))<˜rγpγs2krtγsγp1||f||Lp(E(x0,t))dt

holds for any ellipsoid E(x0, r) and for allfLploc(n).

Moreover, for s > 1 the inequality

TΩPfWLq(E(x0,r))<˜rγ2krtγ1||f||L1(E(x0,t))dt(9)

holds for any ellipsoid E(x0, r) and for allfL1loc(n).

Proof. Let 1 < p < ∞ and s′ ≤ p. Set E = E(x0, r) for the parabolic ball (ellipsoid) centered at x0 and of radius r and 2kE = E(x0, 2kr). We represent f as

f=f1+f2,f1(y)=f(y)χ2kE(y),f2(y)=f(y)χ(2kE)C(y),r>0

and have

TΩPfLp(E)TΩPf1Lp(E)+TΩPf2Lp(E).

Since f1Lp(n),TΩPf1Lp(n) and from the boundedness of TΩP on Lp(ℝn) (see Theorem 1.2) it follows that:

TΩPf1Lp(E)TΩPf1Lp(n)C||f1||Lp(n)=C||f||Lp(2kE),

where constant C > 0 is independent of f.

It is clear that xE, y ∈ (2kE)C implies 12kρ(x0y)ρ(xy)3k2ρ(x0y). We get

|TΩPf2(x)|2γc1(2kE)C|f(y)||Ω(xy)|ρ(x0y)γdy.

By the Fubini’s theorem, we have

(2kE)C|f(y)||Ω(xy)|ρ(x0y)γdy(2kE)C|f(y)||Ω(xy)|ρ(x0y)dttγ+1dy2kr2krρ(x0y)t|f(y)||Ω(xy)|dydttγ+12krE(x0,t)|f(y)||Ω(xy)|dydttγ+1.

Applying the Hölder’s inequality, we get

(2kE)C|f(y)||Ω(xy)|ρ(x0y)dy<˜2kr||f||Lp(E(x0,t))||Ω(x)||Ls(E(x0,t))|E(x0,y)|11p1sdttγ+1.(10)

For xE(x0, t), notice that Ω is At-homogenous of degree zero and Ω ∈ Ls(Sn-1), s > 1. Then, we obtain

(E(x9,t)|Ω(xy)|sdy)1s=(E(xx0,t)|Ω(z)|sdz)1s(E(0,t+|xx0|)|Ω(z)|sdz)1s(E(0,2t)|Ω(z)|sdz)1s=(Sn102t|Ω(z)|sdσ(z)rn1dr)1s=C||Ω||Ls(Sn1)|E(x0,2t)|1s.(11)

Thus, by (11), it follows that:

|TΩPf2(x)|<˜2kr||f||Lp(E(x0,t))dttγp+1.

Moreover, for all p ∈ [1, ∞) the inequality

TΩPf2Lp(E)<˜rγp2kr||f||Lp(E(x0,t))dttγp+1(12)

is valid. Thus, we obtain

TΩPfLp(E)<˜||f||Lp(2kE)+rγp2kr||f||Lp(E(x0,t))dttγp+1.

On the other hand, we have

||f||Lp(2kE)rγp||f||Lp(2kE)2krdttγp+1rγp2kr||f||Lp(E(x0,t))dttγp+1.(13)

By combining the above inequalities, we obtain

TΩPf2Lp(E)<˜rγp2kr||f||Lp(E(x0,t))dttγp+1.

Let 1 < p < s. Similarly to (11), when yB(x0, t), it is true that

(E(x0,r)|Ω(xy)|sdy)1sC||Ω||Ls(Sn1)|E(x0,32t)|1s.(14)

By the Fubini’s theorem, the Minkowski inequality and (14), we get

TΩPf2Lp(E)(E|2krE(x0,t)|f(y)||Ω(xy)|dydttγ+1|pdx)1p2krE(x0,t)|f(y)|||Ω(y)||Lp(E)dydttγ+1|E(x0,r)|1p1s2krE(x0,t)|f(y)|||Ω(y)||Ls(E)dydttγ+1<˜rγpγs2kr||f||L1(E(x0,t))|E(x0,32t)|1sdttγ+1<˜rγpγs2krtγsγp1||f||Lp(E(x0,t))dt.

Let p = 1 < s ≤ ∞. From the weak (1, 1) boundedness of TΩP and (13) it follows that:

TΩPf1WL1(E)TΩPf1WL1(n)<˜||f1||L1(n)=||f||L1(2kE)<˜rγ2kr||f||L1(E(x0,t))dttγ+1.(15)

Then from (12) and (15) we get the inequality (9), which completes the proof. □

In the following theorem (our main result), we get the boundedness of the operator TΩP satisfying condition (1) on the parabolic generalized local Morrey spaces LMp,φ,P{x0}.

Theorem 3.3

Let x0 ∈ ℝn, 1 ≤ p < ∞ and Ω ∈ Ls(Sn–1), 1 < s∞, be At-homogeneous of degree zero. LetTΩPbe a parabolic sublinear operator satisfying condition (1), bounded on Lp(ℝn) for p > 1, and bounded from L1(n) to WL1(n). Let also, for s′ ≤ p, p ≠1, the pair (φ1, φ2) satisfies the condition

ressinft<τ<φ1(x0,τ)τγptγp+1dtCφ2(x0,r),(16)

and for 1 < p < s the pair (φ1, φ2) satisfies the condition

ressinft<τ<φ1(x0,τ)τγptγpγs+1dtCφ2(x0,r)rγs,(17)

where C does not depend on r.

Then the operatorTΩPis bounded fromLMp,φ1,P{x0}toLMp,φ2,P{x0}for p > 1and fromLM1,φ1,P{x0}toWLM1,φ2,P{x0}for p = 1. Moreover, we have for p > 1

TΩPfLMp,φ2,P{x0}<˜||f||LMp,φ1,P{x0},(18)

and for p = 1

TΩPfWLM1,φ2,P{x0}<˜||f||LM1,φ1,P{x0}.(19)

Proof. Let 1 < p < ∞and s′ ≤ p. By Theorem 3.2 and Theorem 3.1 with v2(r) = φ2(x0, r)–1, v1 = φ1(x0,r)1rγp,w(r)=rγp1 and g(r)=||f||Lp(E(x0,r)), we have

TΩPfLMp,φ2,P{x0}<˜supr>0φ2(x0,r)1r||f||Lp(E(x0,t))dttγp+1<˜supr>0φ1(x0,r)1rγp||f||Lp(E(x0,r))=||f||LMp,φ1,P{x0},

where the condition (8) is equivalent to (16), then we obtain (18).

Let 1 < p < s. By Theorem 3.2 and Theorem 3.1 with v2(r) = φ2(x0, r)–1, υ1=φ1(x0,r)1rγp+γs,w(r)=rγp+γs1 and g(r)=||f||Lp(E(x0,r)), we have

TΩPfLMp,φ2,P{x0}<˜supr>0φ2(x0,r)1rγsr||f||Lp(E(x0,t))dttγpγs+1<˜supr>0φ1(x0,r)1rγp||f||Lp(E(x0,r))=||f||LMp,φ1,P{x0},

where the condition (8) is equivalent to (17). Thus, we obtain (18).

Also, for p = 1 we have

TΩPfWLM1,φ2,P{x0}<˜supr>0φ2(x0,r)1r||f||L1(E(x0,t))dttγ+1<˜supr>0φ1(x0,r)1rγ||f||L1(E(x0,r))=||f||LM1,φ1,P{x0}.

Hence, the proof is completed. □

In the case of s = ∞ from Theorem 3.3, we get

Corollary 3.4

Let x0 ∈ ℝn, 1 ≤ p < and the pair (φ1, φ2) satisfies condition (16). Then the operators MP andTPare bounded fromLMp,φ1,P{x0}toLMp,φ2,P{x0}for p > 1 and fromLM1,φ1,P{x0}toWLM1,φ2,P{x0}for p =1.

Corollary 3.5

Let x0 ∈ ℝn, 1 ≤ p < ∞and Ω ∈ Ls(Sn–1), 1 < s∞, be At-homogeneous of degree zero. For s′ ≤ p, p ≠ 1, the pair (φ1, φ2) satisfies condition (16) and for 1 < p < s the pair (φ1, φ2) satisfies condition (17). Then the operatorsMΩPandT¯ΩPare bounded fromLMp,φ1,P{x0}toLMp,φ2,P{x0}for p > 1 and fromLM1,φ1,P{x0}toWLM1,φ2,P{x0}for p =1.

Corollary 3.6

Let x0 ∈ ℝn, 1 ≤ p < ∞ and Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, is homogeneous of degree zero. LetTΩPbe a parabolic sublinear operator satisfying condition (1), bounded on Lp(ℝn) for p > 1, and bounded from L1(ℝn) to WL1(ℝn). Let also, for s′p, p ≠ 1, the pair (φ1, φ2) satisfies the condition

ressinft<τ<φ1(x0,τ)τnptnp+1dtCφ2(x0,r),

and for 1 < p < s the pair (φ1, φ2) satisfies the condition

ressinft<τ<φ1(x0,τ)τnptnpns+1dtCφ2(x0,r)rns,

where C does not depend on r.

Then the operatorTΩPis bounded fromLMp,φ1,P{x0}toLMp,φ2,P{x0}for p > 1 and fromLM1,φ1,P{x0}toWLM1,φ2,P{x0}for p = 1. Moreover, we have for p > 1

TΩPfLMp,φ2,P{x0}<˜||f||LMp,φ1,P{x0},

and for p = 1

TΩPfWLM1,φ2,P{x0}<˜||f||LM1,φ1,P{x0}.
Remark 3.7

Note that, in the case of P= I Corollary 3.6 has been proved in [16, 17]. Also, in the case of P= I and s = ∞ Corollary 3.6 has been proved in [16, 17].

Corollary 3.8

Let 1 ≤ p < ∞and Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, be At-homogeneous of degree zero. LetTΩPbe a parabolic sublinear operator satisfying condition (1), bounded on Lp(ℝn) for p > 1, and bounded from L1(ℝn) to WL1(ℝn). Let also, for s′ ≤ p, p ≠ 1, the pair (φ12) satisfies the condition

ressinft<τ<φ1(x,τ)τγptγp+1dtCφ2(x,r),(20)

and for 1 < p < s the pair (φ1, φ2) satisfies the condition

ressinft<τ<φ1(x,τ)τγptγpγs+1dtCφ2(x,r)rγs,(21)

where C does not depend on x and r.

Then the operatorTΩPis bounded fromMp,φ1,PtoMp,φ2,Pfor p > 1 and fromM1,φ1,PtoWM1,φ2,P for p = 1. Moreover, we have for p > 1

TΩPfMp,φ2,P<˜||f||Mp,φ1,P,

and for p = 1

TΩPfWM1,φ2,P<˜||f||M1,φ1,P.

In the case of s = ∞ from Corollary 3.8, we get

Corollary 3.9

Let 1 ≤ p < ∞ and the pair (φ1, φ2) satisfies condition (20). Then the operators MP andTPare bounded fromMp,φ1,PtoMp,φ2,Pfor p > 1and fromM1,φ1,PtoWM1,φ2,P for p = 1.

Corollary 3.10

Let 1 ≤ p < ∞ and Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, be At-homogeneous of degree zero. Let also, for s′ ≤ p, p ≠ 1, the pair (φ1, φ2) satisfies condition (20) and for 1 < p < q the pair (φ1, φ2) satisfies condition (21). Then the operatorsMΩPandT¯ΩPare bounded fromMp,φ1toMp,φ2for p > 1 and fromM1,φ1toWM1,φ2for p = 1.

Remark 3.11

Condition (20) in Corollary 3.8 is weaker than condition (6) in Theorem 2.5. Indeed, if condition (6) holds, then

ressinft<τ<φ1(x,τ)τγptγp+1dtrφ1(x,t)dtt,r(0,),

so condition (20) holds.

On the other hand, the functions

φ1(r)=1χ(1,)(r)rγpβ,φ2(r)=rγp(1+rβ),0<β<γp

satisfy condition (20) but do not satisfy condition (6) (see [41, 43]).

Corollary 3.12

Let 1 ≤ p < ∞ and Ω ∈ Ls(Sn1), 1 < s ≤ ∞, be homogeneous of degree zero. LetTΩPbe a parabolic sublinear operator satisfying condition (1), bounded on Lp(ℝn) for p > 1, and bounded from L1(ℝn) to WL1(ℝn). Let also, for s′ ≤ p, p ≠ 1, the pair (φ1, φ2) satisfies the condition

ressinft<τ<φ1(x,τ)τnptnp+1dtCφ2(x,r),

and for 1 < p < s the pair (φ1, φ2) satisfies the condition

ressinft<τ<φ1(x,τ)τnptnpns+1dtCφ2(x,r)rns,

where C does not depend on x and r.

Then the operatorTΩPis bounded fromMp,φ1,PtoMp,φ2,Pfor p > 1 and fromM1,φ1,PtoWM1,φ2,Pfor p = 1. Moreover, we have for p > 1

TΩPfMp,φ2,P<˜||f||Mp,φ1,P,

and for p = 1

TΩPfWM1,φ2,P<˜||f||M1,φ1,P.
Remark 3.13

Note that, in the case of P = I Corollary 3.12 has been proved in [1618]. Also, in the case of P = I and s =Corollary 3.12 has been proved in [1618] and [41, 43].

4 Commutators of parabolic linear operators with rough kernel generated by parabolic Calderón-Zygmund operators and parabolic local Campanato functions on the spaces LMp,φ, P{x0}

In this section, we will prove the boundedness of the operators [b,TΩP] with bLCp2,λ,P{x0} on the parabolic generalized local Morrey spaces LMp,φ,P{x0} by using the following weighted Hardy operator

Hωg(r):=r(1+lntr)g(t)ω(t)dt,r(0,),

where ω is a weight function.

Let T be a linear operator. For a locally integrable function b on ℝn, we define the commutator [b, T] by

[b,T]f(x)=b(x)Tf(x)T(bf)(x)

for any suitable function f. Let T be a Calderón–Zygmund operator. A well known result of Coifman et al. [44] states that when K(x)=Ω(x)|x|n and Ω is smooth, the commutator [b, T]f = bTfT(bf) is bounded on Lp(ℝn), 1 < p < ∞, if and only if bBMO(ℝn).

Since BMO(n)p>1LCp,P{x0}(n), if we only assume bLCp,P{x0}(n), or more generally bLCp,λ,P{x0}(n), then [b, T] may not be a bounded operator on Lp(ℝn), 1 < p < . However, it has some boundedness properties on other spaces. As a matter of fact, Grafakos et al. [45, 46] have considered the commutator with bLCp,I{x0}(n) on Herz spaces for the first time. Morever, in [16, 17] and [19, 46], they have considered the commutators with bLCp,λ,I{x0}(n). The commutator of Calderón–Zygmund operators plays an important role in studying the regularity of solutions of elliptic partial differential equations of second order (see, for example, [2527]). The boundedness of the commutator has been generalized to other contexts and important applications to some non-linear PDEs have been given by Coifman et al. [47].

We introduce the parabolic local Campanato space LCp,λ,P{x0} following the known ideas of defining local Campanato space (see [16, 17, 42] etc).

Definition 4.1

Let 1 ≤ p < ∞ and0λ<1γ. A functionfLploc(n)is said to belong to theLCp,λ,P{x0}(n)(parabolic local Campanato space), if

||f||LCp,λ,P{x0}=supr>0(1|E(x0,r)|1+λpE(x0,r)|f(y)fE(x0,r)|pdy)1p<,(22)

where

fE(x0,r)=1|E(x0,r)|E(x0,r)f(y)dy.

Define

LCp,λ,P{x0}(n)={fLploc(n):||f||LCp,λ,P{x0}<}.
Remark 4.2

If two functions which differ by a constant are regarded as a function in the spaceLCp,λ,P{x0}(n), thenLCp,λ,P{x0}(n)becomes a Banach space. The spaceLCp,λ,P{x0}(n)when λ = 0 is just theLCp,P{x0}(n). Apparently, (22) is equivalent to the following condition:

supr>0infc(1|E(x0,r)|1+λpE(x0,r)|f(y)c|pdy)1p<.

In [48], Lu and Yang have introduced the central BMO space CBMOp(n)=LCp,0,I{0}(n). Also the space CBMO{x0}(n)=LC1,0,I{x0}(n) can be has been considered in other denotes in [49]. The space LCp,P{x0}(n) regarded as a local version of BMO(ℝn), the space of parabolic bounded mean oscillation, at the origin. But, they have quite different properties. The classical John-Nirenberg inequality shows that functions in BMO(ℝn) are locally exponentially integrable. This implies that, for any 1 ≤ p < ∞, the functions in BMO(ℝn) (parabolic BMO) can be described by means of the condition:

supxn,r>0(1|E(x,r)|E(x,r)|f(y)fE(x,r)|pdy)1p<,

where B denotes an arbitrary ball in ℝn. However, the space LCp,P{x0}(n) depends on p. If p1 < p2, then LCp2,P{x0}(n)LCp1,P{x0}(n). Therefore, there is no analogy of the famous John-Nirenberg inequality of BMO(ℝn) for the space LCp,P{x0}(n). One can imagine that the behavior of LCp,P{x0}(n) may be quite different from that of BMO(ℝn).

Theorem 4.3

([16, 17, 42])

Let v1, v2and ω be weigths on (0, ∞) and v1(t) be bounded outside a neighbourhood of the origin. The inequality

esssupr>0υ2(r)Hωg(r)Cesssupr>0υ1(r)g(r)(23)

holds for some C > 0 for all non-negative and non-decreasing functions g on (0, ∞) if and only if

esssupr>0υ2(r)Hωg(r)Cesssupr>0υ1(r)g(r)(24)

Moreover, the value C = B is the best constant for (23).

Remark 4.4

In (23) and (24) it is assumed that1=0and 0 · ∞ = 0.

Lemma 4.5

Let b be function inLCp,λ.P{x0}(n),1p<,0λ<1γand r1, r2 > 0. Then

(1|E(x0,r1)|1+λpE(x0,r1)|b(y)bE(x0,r2)|pdy)1pC(1+lnr1r2)||b||LCp,λ,P{x0},(25)

where C > 0 is independent of b, r1and r2.

From this inequality (25), we have

|bE(x0,r1)bE(x0,r2)|C(1+lnr1r2)|E(x0,r1)|λ||b||LCp,λ,P{x0},(26)

and it is easy to see that

||bbE||Lp(E)C(1+lnr1r2)rγp+γλ||b||LCp,λ,P{x0}.(27)

In [30] the following statements have been proved for the parabolic commutators of parabolic singular integral operators with rough kernel T¯ΩP, containing the result in [3133].

Theorem 4.6

Suppose that Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, is At-homogeneous of degree zero and bBMO(ℝn). Let1s<p<(s=ss1) and φ(x, r) satisfies the conditions (4) and (5). If the commutator operator[b,T¯ΩP]is bounded on Lp(ℝn), then the operator[b,T¯ΩP]is bounded on Mp,φ, P.

Theorem 4.7

Let 1 < p < ∞ bBMO(ℝn) and φ(x, t) satisfies conditions (4) and (5). Then the operatorsMbPand [b, TP ] are bounded on Mp,φ, P.

As in the proof of Theorem 3.3, it suffices to prove the following Theorem 4.8.

Theorem 4.8

Let x0 ∈ ℝn, 1 < p < ∞ and Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, be At-homogeneous of degree zero. LetTΩPbe a parabolic linear operator satisfying condition (1), bounded on Lp(ℝn) for 1 < p < ∞ Let also, bLCp2,λ,P{x0}(n),0λ<1γ and 1p=1p1+1p2.

Then, for s′ ≤ p, the inequality

[b,TΩP]fLp(E(x0,r))<˜||b||LCp2,λ,P{x0}rγp2kr(1+lntr)tγλγp11||f||Lp1(E(x0,t))dt

holds for any ellipsoid E(x0, r) and for allfLp1loc(n).

Also, for p1 < s, the inequality

[b,TΩP]fLp(E(x0,r))<˜||b||LCp2,λ,P{x0}rγpγs2kr(1+lntr)tγλγp1+γs1||f||Lp1(E(x0,t))dt

holds for any ellipsoid E(x0, r) and for allfLp1loc(n).

Proof. Let 1<p<,bLCp2,λ,P{x0}(n) and 1p=1p1+1p2. Set E=E(x0, r) for the parabolic ball (ellipsoid) centered at x0and of radius rand 2kE = E(x0, 2kr). We represent f as

f=f1+f2,f1(y)=f(y)χ2kE(y),f2(y)=f(y)χ(2kE)C(y),r>0

and have

[b,TΩP]f(x)=(b(x)bE)TΩPf1(x)TΩP((b()bE)f1)(x)+(b(x)bE)TΩPf2(x)TΩP((b()bE)f2)(x)J1+J2+J3+J4.

Hence we get

[b,TΩP]fLp(E)||J1||Lp(E)+||J2||Lp(E)+||J3||Lp(E)+||J4||Lp(E).

By the Hölder’s inequality, the boundedness of TΩP on Lp1(n) (see Theorem 1.2) it follows that:

||J1||Lp(E)(b()bE)TΩPf1()Lp(n)<˜||(b()bE)||Lp2(n)TΩPf1()Lp1(n)<˜||b||LCp2,λ,P{x0}rγp2+γλ||f1||Lp1(n)=||b||LCp2,λ,P{x0}rγp2+γp1+γλ||f1||Lp1(2kE)2krt1γp1dt<˜||b||LCp2,λ,P{x0}rγp2kr(1+lntr)tγλγp11||f||Lp1(E(x0,t))dt.

Using the the boundedness of TΩP on Lp(ℝn) (see Theorem 1.2), by the Hölder’s inequality for J2 we have

||J2||Lp(E)TΩP(b()bE)f1Lp(n)<˜||(b()bE)f1||Lp(n)<˜||(b()bE)Lp2(n)||f1||Lp1(n)<˜||b||LCp2,λ,P{x0}rγp2+γp1+γλ||f||Lp1(2kE)2krt1γp1dt<˜||b||LCp2,λ,P{x0}rγp2kr(1+lntr)tγλγp11||f||Lp1(E(x0,t))dt.

For J3, it is known that xE, y. ∈ (2kE)C, which implies 12kρ(x0y)ρ(xy)3k2ρ(x0y).

When s′ ≤ p1, by the Fubini’s theorem, the Hölder’s inequality and (11) we have

|TΩPf2(x)|c0(2kE)C|Ω(xy)||f(y)|ρ(x0y)dy2kr2kr<ρ(x0y)<t|Ω(xy)||f(y)|dyt1γdt<˜2krE(x0,t)|Ω(xy)||f(y)|dyt1γdt<˜2kr||f||Lp1(E(x0,t))||Ω(x)||Ls(E(x0,t))|E(x0,t)|11p11st1γdt<˜2kr||f||Lp1(E(x0,t))t1γp1dt.

Hence, we get

J3Lp(E)(b()bB)TΩPf2()Lp(n)<˜b()bELp(n)2krt1γp1fLp1(E(x0,  t))dt<˜(b()bE)Lp2(n)rγp12krt1γp1fLp1(E(x0,t))dt<˜bLCp2,λ,P{x0}rγp+γλ2kr(1+lntr)t1γp1fLp1(E(x0,t))dt<˜bLCp2,λ,P{x0}rγp2kr(1+lntr)tγλγp11fLp1(E(x0,t))dt

When p1 < s, by the Fubini’s theorem, the Minkowski inequality, the Hölder’s inequality and from (27), (14) we get

J3Lp(E)(E|2krE(x0,t)|f(y)||b(x)bE||Ω(xy)|dydttγ+1|pdx)1p2krE(x0,t)|f(y)|(b()bE)Ω(·y)Lp(E)dydttγ+12krE(x0,t)|f(y)|(b()bE)Lp2Ω(·y)Lp1(E)dydttγ+1<˜bLCp2,λ,P{x0}rγp+γλ|E|1p11s2krE(x0,t)|f(y)|||Ω(·y)||Ls(E)dydttγ+1<˜bLCp2,λ,P{x0}rγpγs+γλ2kr||f||L1(E(x0,t))|E(x0,32t)|1sdttγ+1<˜bLCp2,λ,P{x0}rγpγs+γλ2kr(1+lntr)||f||Lp1(E(x0,t))dttγp1γs+1<˜bLCp2,λ,P{x0}rγpγs2kr(1+lntr)tγλγp1+γs1||f||Lp1(E(x0,t))dt.

On the other hand, for J4, when s′≤ p, for xE, by the Fubini’s theorem, applying the Hölder’s inequality and from (26), (27), (11) we have

|TΩP((b()bB)f2)(x)|<˜(2kE)C|b(y)bE||Ω(xy)||f(y)|ρ(xy)γdy<˜(2kE)C|b(y)bE||Ω(xy)||f(y)|ρ(x0y)γdy2kr2kr<ρ(x0y)<t|b(y)bE||Ω(xy)||f(y)dydttγ+1<˜2krE(x0,t)|b(y)bE(x0,t)||Ω(xy)||f(y)dydttγ+1+2kr|bE(x0,r)bE(x0,t)|E(x0,t)|Ω(xy)||f(y)dydttγ+1<˜2kr||(b()bE(x0,t))f||Lp(E(x0,t))||Ω(y)||Ls(E(x0,t))|E(x0,t)|11p1sdttγ+1+2kr|(bE(x0,r)bE(x0,t))|||f||Lp1(E(x0,t))||Ω(y)||Ls(E(x0,t))|E(x0,t)|11p11stγ1dt<˜2kr||(b()bE(x0,t))||Lp2(E(x0,t))||f||Lp1(E(x0,t))t1γp1dt+||b||LCp2,λ,P{x0}2kr(1+lntr)||f||Lp1(E(x0,t))t1γp1+γλdt<˜+||b||LCp2,λ,P{x0}2kr(1+lntr)||f||Lp1(E(x0,t))t1γp1+γλdt.

Then, we have

J4Lp(E)=TΩP(b()bE)f2Lp(E)<˜bLCp2,λ,  P{x0}rnp2kr(1+lntr)tyλγp11fLp1(E(x0,t))  dt.

When p1 < s, by the Minkowski inequality, applying the Hölder’s inequality and from (26), (27), (14) we have

||J4||Lp(E)(E|2krE(x0,t)|b(y)bE(x0,t)||f(y)||Ω(xy)|dydttγ+1|pdx)1p+(E|2kr|bE(x0,r)bE(x0,t)|E(x0,t)|f(y)||Ω(xy)|dydttγ+1|dx)1p<˜2krE(x0,t)|b(y)bE(x0,t)||f(y)|||Ω(y)||Lp(E(x0,t))dydttγ+1+2kr|bE(x0,r)bE(x0,t)|E(x0,t)|f(y)|||Ω(y)||Lp(E(x0,t))dydttγ+1<˜|E|1p1s2krE(x0,t)|b(y)bE(x0,t)||f(y)|||Ω(y)||Ls(E(x0,t))dydttγ+1+|E|1p1s2kr|bE(x0,r)bE(x0,t)|E(x0,t)|f(y)|||Ω(y)||Ls(E(x0,t))dydttγ+1<˜rγpγs2kr||b()bE(x0,t)||Lp2(E(x0,t))||f||Lp1(E(x0,t))|E(x0,t)|11p|E(x0,32t)|1sdttγ+1+rγpγs2kr|bE(x0,r)bE(x0,t)|||f||Lp1(E(x0,t))||E(x0,32t)|1sdttγp1+1<˜rγpγs||b||LCp2,λ,P{x0}2kr(1+lntr)tγλγp1+γs1||f||Lp1(E(x0,t))dt.

Now combined by all the above estimates, we end the proof of this Theorem 4.8

Now we can give the following theorem (our main result).

Theorem 4.9

Let x0 ∈ ℝn, 1 < p < ∞and Ω ∈ Ls(Sn–1), 1 < s ≤ ∞,, be At-homogeneous of degree zero. LetTΩPbe a parabolic linear operator satisfying condition (1), bounded on Lp(ℝn) for 1 < p < ∞. LetbLCp2,λ,P{x0}(n),0λ<1γ and 1p=1p1+1p2. Let also, for s′ ≤ p the pair (φ1, φ2) satisfies the condition

r(1+lntr)t<τ<essinfφ1(x0,τ)τγp1tγp1+1γλdtCφ2(x0,  r),(28)

and for p1 < s the pair (φ1, φ2) satisfies the condition

r(1+lntr)t<τ<essinfφ1(x0,τ)τγp1tγp1γs+1γλdtCφ2(x0,  r)rγs,(29)

where C does not depend on r.

Then the operator[b,TΩP]is bounded fromLMp1,φ1,P{x0}toLMp,φ2,P{x0}. Moreover,

[b,  TΩP]fLMp,  φ2,  P{x0}bLCp2,  λ,  P{x0}fLMp1,  φ1,  P{x0}.(30)

Proof. Let p > 1and s′ ≤ p. By Theorem 4.8 and Theorem 4.3 with v2(r) =φ2(x0, r)–1, v1 = φ1(x0,r)1rγp1,w(r)=rγλγp11 and g(r)=||f||Lp1(E(x0,r)), we have

[b,TΩP]fLMp,φ2,P{x0}<˜supr>0φ2(x0,r)1||b||LCp2,λ,P{x0}r(1+lntr)tγλγp11||f||Lp1(E(x0,t))dt<˜||b||LCp2,λ,P{x0}supr>0φ1(x0,r)1rγp1||f||Lp1(E(x0,r))=||b||LCp2,λ,P{x0}||f||LMp1,φ1,P{x0},

where the condition (24) is equivalent to (28), then we obtain (30).

Let p > 1and p1 < s. By Theorem 4.8 and Theorem 4.3 with v2(r) = φ2(x0, r)–1, v1 = φ1(x0,r)1rγp1+γs,w(r)=rγλγp1+γs1 and g(r)=||f||Lp1(E(x0,r)), we have

[b,TΩP]fLMp,φ2,P{x0}<˜supr>0φ2(x0,r)1rγs||b||LCp2,λ,P{x0}r(1+lntr)tγλγp1+γs1||f||Lp1(E(x0,t))dt<˜||b||LCp2,λ,P{x0}supr>0φ1(x0,r)1rγp1||f||Lp1(E(x0,r))=||b||LCp2,λ,P{x0}||f||LMp1,φ1,P{x0},

where the condition (24) is equivalent to (29). Thus, we obtain (30).

Hence, the proof is completed. □

In the case of s = ∞ from Theorem 4.9, we get

Corollary 4.10

Let x0 ∈ ℝn, 1 < p < ∞, bLCp2,λ,P{x0}(n),0λ<1γ,1p=1p1+1p2and the pair (φ1, φ2) satisfies condition (28). Then the operatorsMbP and [b, TP] are bounded from LMp1,φ1,P{x0} to LMp,φ2,P{x0}.

Corollary 4.11

Let x0 ∈ ℝn, 1 < p < and Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, be At-homogeneous of degree zero. LetbLCp2,λ,P{x0}(n),0λ<1γ, 1p=1p1+1p2. Let also, for s′≤p, the pair (φ1, φ2) satisfies condition (28) and for p < s, the pair (φ1, φ2) satisfies condition (29). Then the operatorsMΩ,bPand[b,T¯ΩP]are bounded fromLMp1,φ1,P{x0}toLMp,φ2,P{x0}.

Corollary 4.12

Let x0 ∈ ℝn, 1 < p < ∞and Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, be homogeneous of degree zero. LetTΩPbe a parabolic linear operator satisfying condition (1), bounded on Lp(ℝn) for 1 < p < ∞ LetbLCp2,λ,P{x0}(n),0λ<1n,1p=1p1+1p2.Let also, for s′ ≤ p the pair (φ1, φ2) satisfies the condition

r(1+lntr)t<τ<essinfφ1(x0,τ)τnp1tnp1+1nλ  dtCφ2(x0,  r),(31)

and for p1 < s the pair (φ1, φ2) satisfies the condition

r(1+lntr)t<τ<essinfφ1(x0,τ)τnp1tnp1ns+1nλ  dtCφ2(x0,  r)rns,

where C does not depend on r.

Then the operator[b,TΩP]is bounded fromLMp1,φ1,P{x0}toLMp,φ2,P{x0}. Moreover,

[b,  TΩP]fLMp,  φ2,  P{x0}bLCp2,  λ,  P{x0}fLMp1,  φ1,  P{x0}.
Remark 4.13

Note that, in the case of P = I Corollary 4.12 has been proved in [16, 17]. Also, in the case of P =I and s = ∞ Corollary 4.12 has been proved in [16, 17].

Corollary 4.14

Let Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, be At-homogeneous of degree zero. LetTΩPbe a parabolic linear operator satisfying condition (1), bounded on Lp(ℝn) for 1 < p < ∞. Let 1 < p < ∞ and bBMO(ℝn) (parabolic bounded mean oscillation space). Let also, for s′ ≤ p the pair (φ1, φ2) satisfies the condition

r(1+lntr)t<τ<essinfφ1(x0,τ)τγptγp+1  dtCφ2(x,  r),(32)

and for p < s the pair (φ1, φ2) satisfies the condition

r(1+lntr)t<τ<essinfφ1(x0,τ)τγptγpγs+1  dtCφ2(x,  r)rγs,(33)

where C does not depend on x and r.

Then the operator[b,TΩP]is bounded fromMp,φ1,PtoMp,φ2,P. Moreover,

[b,  TΩP]fMp,  φ2,  P  bBMOfMp,  φ1,  P.

In the case of s = ∞from Corollary 4.14, we get

Corollary 4.15

Let 1 < p < ∞, bBMO (ℝn) and the pair (φ1, φ2) satisfies condition (32). Then the operatorsMbPand [b, TP] are bounded fromMp,φ1,PtoMp,φ2,P.

Corollary 4.16

Let Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, be At-homogeneous of degree zero. Let 1 < p < ∞ and bBMO(ℝn). Let also, for s′≤ p, the pair (φ1, φ2) satisfies condition (32) and for p < s, the pair (φ1, φ2) satisfies condition (33). Then the operatorsMΩ,bPand[b,T¯ΩP]are bounded fromMp,φ1,PtoMp,φ2,P.

Corollary 4.17

Let Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, be homogeneous of degree zero. Let 1 < p < ∞ and bBMO(ℝn). Let also, for s′ ≤ p the pair (φ1, φ2) satisfies the condition

r(1+lntr)t<τ<essinfφ1(x0,τ)τnptnp+1  dtCφ2(x,  r),

and for p < s the pair (φ1, φ2) satisfies the condition

r(1+lntr)t<τ<essinfφ1(x0,τ)τnptnpns+1  dtCφ2(x,  r)rns,

where C does not depend on x and r.

Then the operator[b,TΩP]is bounded fromMp,φ1,PtoMp,φ2,P. Moreover,

[b,  TΩP]fMp,  φ2,  PbBMOfMP,  φ1,  P.
Remark 4.18

Note that, in the case of P =I Corollary 4.17 has been proved in [1618]. Also, in the case of P =I and s = ∞ Corollary 4.17 has been proved in [1618] and [41, 43].

Now, we give the applications of Theorem 3.3 and Theorem 4.9 for the parabolic Marcinkiewicz operator.

Suppose that Ω(x) is a real-valued and measurable function defined on ℝn satisfying the following conditions:

1. Ω(x) is homogeneous of degree zero with respect to At, that is,

Ω(Atx)=Ω(x),  for  any  t>  0,  x    n\{0};
2. Ω(x) has mean zero on Sn–1, that is,

Sn1Ω(x')J(x')dσ(x')=0

where x=x|x| for any x ≠ 0.

3. Ω ∈ L1(Sn–1).

Then the parabolic Marcinkiewicz integral of higher dimension μΩγ is defined by

μΩγ(f)(x)=(0|FΩ,  t(f)(x)|2dtt3)1/2,

where

FΩ,  t(f)(x)=ρ(xy)tΩ(xy)ρ(xy)γ1f(γ)dy.

On the other hand, for a suitable function b, the commutator of the parabolic Marcinkiewicz integral μΩγ is defined

by

[b,  μΩγ](f)(x)=(0|FΩ,  t,  b(f)(x)|2dtt3)1/2,

where

FΩ,t,b(f)(x)=ρ(xy)tΩ(xy)ρ(xy)γ1[b(x)b(y)]f(y)dy.

We consider the space H={h:||h||=(0|h(t)|2dtt3)1/2<}. Then, it is clear μΩγ(f)(x)=||FΩ,t(x)||.

By the Minkowski inequality and the conditions on Ω we get

μΩγ(f)(x)n|Ω(xy)|ρ(xy)γ1|f(y)|(|xy|dtt3)1/2dyCn|Ω(xy)|ρ(xy)γ|f(y)|dy.

Thus, μΩγ satisfies the condition (1). When Ω ∈ Ls(Sn–1), (s < 1), It is known that μΩ is bounded on Lp(ℝn) for p > 1, and bounded from L1(ℝn) to WL1(ℝn) for p = 1 (see [50]), then from Theorems 3.3, 4.9 we get

Corollary 4.19

Let x0 ∈ ℝn, 1 ≤ p ∞, Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, Let also, for s′ ≤ p, p ≠ 1, the pair (φ1, φ2) satisfies condition (16) and for 1 < p < s the pair (φ1, φ2) satisfies condition (17) and Ω satisfies conditions (a)–(c). Then the operator μΩγ is bounded from LMp,φ1{x0} to LMp,φ2{x0} for p > 1 and from LM1,φ1{x0} to WLM1,φ2{x0}.

Corollary 4.20

Let 1 ≤ p < ∞, Ω ∈ Ls(Sn1), 1 < s ≤ ∞, Let also, for s′ ≤ p, p ≠ 1, the pair (φ1, φ2) satisfies condition (20) and for 1 < p < s the pair (φ1, φ2) satisfies condition (21) and Ω satisfies conditions (a)–(c). Then the operatorμΩγis bounded fromMp,φ1toMp,φ2for p > 1 and fromM1,φ1toWM1,φ2for p = 1.

Corollary 4.21

Let x0 ∈ ℝn (Sn–1), 1 < s ≤ ∞. Let 1 < p < ∞, bLCp2,λ{x0}(n),1p=1p1+1p2,0λ<1n.Let also, for s′ ≤ p the pair (φ1, φ2) satisfies condition (28) and for p1 < s the pair (φ1, φ2) satisfies condition (29) and Ω satisfies conditions (a)–(c). Then, the operator[b,μΩγ]is bounded fromLMp1,φ1{x0} to LMp,φ2{x0}.

Corollary 4.22

Let Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, 1 < p < ∞ and bBMO(ℝn). Let also, for s′ ≤ p the pair (φ1, φ2) satisfies condition (32) and for p < s the pair (φ1, φ2) satisfies condition (33) and Ω satisfies conditions (a)–(c). Then, the operator[b,μΩγ]is bounded fromMp,φ1toMp,φ2.

Remark 4.23

Obviously, if we take α1 = ∙∙∙ = αn and P = I, thenρ(x)=|x|=(i=1nxi2)1/2,γ=n, (ℝn, ρ) = (ℝn, |∙|), EI(x, r) = B (x, r). In this case,μΩγis just the classical Marcinkiewicz integral operator μΩ, which was first defined by Stein in 1958. In [51], Stein has proved that if Ω satisfies the Lipshitz condition of degree of α(0 < α ≤ 1) on Sn–1and the conditions (a), (b) (obviously, in the case At = tI and J(x′) ≡ 1), then μwΩis both of the type (p, p) (1 < p ≤ 2) and the weak type (1.1). (See also [52] for the boundedness of the classical Marcinkiewicz integral μΩ.)

Acknowledgement

The author would like to thank the Referees and Editors for carefully reading the manuscript and making several useful suggestions.

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