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# Open Mathematics

### formerly Central European Journal of Mathematics

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Volume 14, Issue 1

# 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
Published Online: 2016-05-19 | DOI: https://doi.org/10.1515/math-2016-0028

## 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 $|x|\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(\sum _{i=1}^{n}{x}_{i}^{2}\right)}^{\frac{1}{2}}.$ 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.

$D=∑i,j=1nai,j∂2∂xi∂xj$

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. $\underset{{S}^{n-1}}{\int }K\left(x\prime \right)d\sigma \left(x\prime \right)=0.$

Similarly, for the heat operator

$D=∂∂x1−∑j=2n∂2∂xj2,$

the corresponding singular integral operator T has a kernel K satisfying

• (a′)

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

• (b′)

KC(ℝn) \ {0});

• (c′)

$\underset{{S}^{n-1}}{\int }K\left(x\prime \right)\left(2x{\prime }_{1}^{2}+x{\prime }_{2}^{2}+\cdot \cdot \cdot +x{\prime }_{n}^{2}\right)d\sigma \left(x\prime \right)=0.$

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

$T¯Pf(x)=p.υ.∫ℝnK(y)f(x−y) dy$

with K satisfying

1. $K\left({t}^{{\alpha }_{1}}{x}_{1},\cdot \cdot \cdot ,{t}^{{\alpha }_{n}}{x}_{n}\right)={t}^{-\alpha }K\left({x}_{1},\cdot \cdot \cdot ,{x}_{n}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ne 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha =\sum _{i=1}^{n}{\alpha }_{i};$

2. KC(ℝn) \ {0});

3. $\underset{{S}^{n-1}}{\int }K\left(x\prime \right)J\left(x\prime \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}d\sigma \left(x\prime \right)=0,$ where αi ≥ 1 (i = 1, . . ., n) and $J\left(x\prime \right)={\alpha }_{1}x{\prime }_{1}^{2}+\cdot \cdot \cdot +{\alpha }_{n}x{\prime }_{n}^{2}.$

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

$x1=ρα1cos φ1...cosφn−2cosφn−1,x2=ρα2cos φ1...cosφn−2cosφn−1, ⋮xn−1=ραn−1cosφ1sinφ2,xn=ραnsinφ1.$

Then dx = ρα–1 J(x′)dρdσ(x′), where $\alpha =\sum _{i=1}^{n}{\alpha }_{i},x\prime \in {S}^{n-1},$ is the element of area of Sn–1 and ρα–1 J 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)|–1 K(x), where ${A}_{t}=diag\left[{t}^{{\alpha }_{1}},\cdot \cdot \cdot ,{t}^{{\alpha }_{n}}\right]=\left(\begin{array}{lll}{t}^{{\alpha }_{1}}\hfill & \hfill & 0\hfill \\ \hfill & \ddots \hfill & \hfill \\ 0\hfill & \hfill & {t}^{{\alpha }_{n}}\hfill \end{array}\right)$ 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 ${\left\{{\alpha }_{i}\right\}}_{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{n}.$

Many works have been done for parabolic singular integral operators, including the weak type estimates and Lp (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), $w={A}_{{\rho }^{-1}}x$ 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,

$c1|x|α1≤ρ(x)≤c2|x|α2if ρ(x)≥ 1;c3|x|α3≤ρ(x)≤c4|x|α4ifρ(x)≤ 1.$

and

$ρ(θx)≤ρ(x) for 0<θ<1,$

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 $|{A}_{{t}^{-1}}x|\text{\hspace{0.17em}}\text{\hspace{0.17em}}=1$, 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 $|{A}_{{t}^{-1}}x|\text{\hspace{0.17em}}\text{\hspace{0.17em}}=1$.

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(x−y)

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 $s\prime =\frac{s}{s-1}$ for any s > 1. Suppose that ${T}_{\text{Ω}}^{P}$represents a parabolic linear or

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

$|TΩPf(x)|≤c0∫ℝn|Ω(x−y)|ρ(x−y)γ|f(y)|dy,$(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

$∫Sn−1Ω(x′)J(x′)dσ(x′)=0,$

where $x\prime =\frac{x}{|x|}$ for any x ≠ 0.

Let fLloc(ℝn). The parabolic homogeneous singular integral operator ${\overline{T}}_{\text{Ω}}^{P}$ and the parabolic maximal operator ${M}_{\text{Ω}}^{P}$ by with rough kernels are defined by

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

satisfy condition (1).

It is obvious that when $\text{Ω}\equiv 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{T}}_{\text{Ω}}^{P}\equiv \text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{T}}^{P}$ and ${M}_{\text{Ω}}^{P}\equiv {M}^{P}$ are the parabolic singular operator and the parabolic maximal operator, respectively. If P = I, then ${M}_{\text{Ω}}^{I}\equiv {M}_{\text{Ω}}$ is the Hardy-Littlewood maximal operator with rough kernel, and ${\overline{T}}_{\text{Ω}}^{I}\equiv {\overline{T}}_{\text{Ω}}$ 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 ${\overline{T}}_{\text{Ω}}^{P}$ on Lebesgue spaces has been obtained.

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 operator ${\overline{T}}_{\text{Ω}}^{P}$ is bounded on Lp(ℝn). Also, the operator ${\overline{T}}_{\text{Ω}}^{P}$ is bounded from L1(ℝn) to WL1(ℝn). Moreover, we have for p > 1 $‖T¯ΩPf‖LP≤C||f||Lp,$

and for p = 1 $‖T¯ΩPf‖WL1≤C||f||L1.$

Under the assumptions of Theorem 1.2, the operator ${M}_{\text{Ω}}^{P}$is bounded on Lp(ℝn). Also, the operator ${M}_{\text{Ω}}^{P}$is bounded from L1(ℝn) to WL1(ℝn). Moreover, we have for p > 1 $‖MΩPf‖Lp≤C||f||Lp,$

and for p = 1 $‖MΩPf‖WL1≤C||f||L1.$

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 = I Theorem 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.

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

If we take α1 = ∙∙∙ = αn =1 and P = I, then obviously $\rho \left(x\right)=\text{\hspace{0.17em}}\text{\hspace{0.17em}}|x|\text{\hspace{0.17em}}\text{\hspace{0.17em}}={\left(\sum _{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{n}{x}_{i}^{2}\right)}^{\frac{1}{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\gamma =n,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({\text{ℝ}}^{n},\rho \right)=\left({\text{ℝ}}^{n},\text{\hspace{0.17em}}\text{\hspace{0.17em}}|\cdot |\right),$

EI(x, r) = B(x, r), At = tI and J(x′) ≡1. In this case, ${\overline{T}}_{\text{Ω}}^{P}$ 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, $\left[b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{T}}_{\text{Ω}}^{P}\right]$ 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

$||f||Lp,λ,P=supx∈ℝn,r>0r−λp||f||Lp(E(x,r))<∞,$

where $f\in {L}_{p}^{loc}\left({\text{ℝ}}^{n}\right),0\le \lambda \le \gamma$ 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 f$W{L}_{p}^{loc}\left({\text{ℝ}}^{n}\right)$ for which

$||f||WLp,λ,P≡ ||f||WLp,λ,P(ℝn)=supx∈ℝn,r>0r−λp||f||WLp(E(x,r))<∞,$

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(ℝn)=supt>0t|{y∈E(x,r): |f(y)| >t}|1/p=sup0

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

Note that WLp(ℝn) = WLp,0,P(ℝn),

$Lp,λ,P(ℝn)⊂WLp,λ,P(ℝn) and ||f||WLp,λ,P≤ | |f||Lp,λ,P.$

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 ${T}_{\text{Ω}}^{P}$satisfying condition (1) generated by parabolic Calderón-Zygmund operators with rough kernel from one parabolic generalized local Morrey space $L{M}_{p,{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to another one $L{M}_{p,{\phi }_{2},P}^{\left\{{x}_{0}\right\}}$, 1 < p < ∞, and from the space $L{M}_{1,{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to the weak space $WL{M}_{1,{\phi }_{2},P}^{\left\{{x}_{0}\right\}}$. In the case of $b\in L{C}_{{p}_{2,}\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$ (parabolic local Campanato space) and $\left[b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{\text{Ω}}^{P}\right]$ is a sublinear operator, we find the sufficient conditions on the pair (φ1, φ2) which ensures the boundedness of the commutator operators $\left[b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{\text{Ω}}^{P}\right]$ from $L{M}_{{p}_{1},{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to $L{M}_{p,{\phi }_{2},P}^{\left\{{x}_{0}\right\}},1 and $0\le \lambda <\frac{1}{\gamma }$.

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.

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 functions $f\in {L}_{p}^{loc}\left({\text{ℝ}}^{n}\right)$ with finite quasinorm $||f||Mp,φ,P=supx∈Rn,r>0⁡φ(x,r)−1|E(x,r)|−1p||f||Lp(E(x,r))<∞.$

Also by WMp,φ, PWMp,φ, P(ℝn) we denote the weak parabolic generalized Morrey space of all functions $f\in W{L}_{p}^{loc}\left({\text{ℝ}}^{n}\right)$ for which $||f||WMp,φ,P=supx∈ℝn,r>0φ(x,r)−1|E(x,r)|−1p||f||WLp(E(x,r)) <∞.$

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 $\phi \left(x,\text{\hspace{0.17em}}\text{\hspace{0.17em}}r\right)={r}^{\frac{\lambda -\gamma }{p}}$: $Lp,λ,P=Mp,φ,P|φ(x,r)=rλ−γp, WLp,λ,P=WMp,φ,P|φ(x,r)=rλ−γp.$

Inspired by the above Definition 2.1, [16] and the Ph.D. thesis of Gurbuz [17], we introduce the parabolic generalized local Morrey spaces $L{M}_{p,\phi ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$ by the following definition.

Let φ(x, r) be a positive measurable function onn ×(0, ) and 1 ≤ p < ∞ For any fixed x0 ∈ ℝn we denote by $L{M}_{p,\phi ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\equiv L{M}_{p,\phi ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$ the parabolic generalized local Morrey space, the space of all functions $f\in {L}_{p}^{loc}\left({\text{ℝ}}^{n}\right)$ with finite quasinorm $||f||LMp,φ,P{x0}=supr>0φ(x0,r)−1|E(x0,r)|−1p||f||Lp(E(x0,r))<∞.$

Also by $WL{M}_{p,\phi ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\equiv WL{M}_{p,\phi ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$ we denote the weak parabolic generalized local Morrey space of all functions $f\in W{L}_{p}^{loc}\left({\text{ℝ}}^{n}\right)$ 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 $L{L}_{p,\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$ and weak local parabolic Morrey space $WL{L}_{p,\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$ under the choice $\phi \left({x}_{0},r\right)={r}^{\frac{\lambda -\gamma }{p}}:$ $LLp,λ,P{x0}=LMp,φ,P{x0}|φ(x0,r)=rλ−γp, WLLp,λ,P{x0}=WLMp,φ,P{x0}|φ(x0,r)=rλ−yp.$

Furthermore, we have the following embeddings: $Mp,φ,P⊂LMp,φ,P{x0},||f||LMp,φ,P{xo}≤ | |f||Mp,φ,P,WMp,φ,P⊂WLMp,φ,P{x0},||f||WLMp,φ,P{x0}≤ ||f||WMp,φ,P.$

In [30] the following statement has been proved for parabolic singular operators with rough kernel ${\overline{T}}_{\text{Ω}}^{P}$, containing the result in [3133].

Suppose that ΩLs(Sn-1), 1 < s ≤ ∞ is At-homogeneous of degree zero and has mean value zero on Sn-1. Let $1\le s\prime and φ(x, r) satisfies conditions $c−1φ(x,r)≤φ(x,t)≤c φ(x,r)$(4)

whenever rt ≤ 2r where c (≥ 1) does not depend on t, r, x ∈ ℝn and $∫r∞φ(x,t)pdtt≤C φ(x,r)p,$(5)

where C does not depend on x and r. Then the operator ${\overline{T}}_{\text{Ω}}^{P}$ is bounded on Mp,φ, P

The results of [3133] imply the following statement.

Let 1 ≤ p < ∞ and φ(x, r) satisfies conditions (4) and (5). Then the operators MP and TP are bounded on Mp,φ, P for p > 1 and from M1,φ, P to WM1,φ,P and 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]).

Let 1 ≤ p < ∞ and the pair (φ1, φ2) satisfies the condition $∫r∞φ1(x,t)dtt≤C φ2(x,r),$(6)

where C does not depend on x and r. Then the operator TP is bounded from ${M}_{p,{\phi }_{1},\text{\hspace{0.17em}}P}$ to ${M}_{p,{\phi }_{2},\text{\hspace{0.17em}}P}$ for p > 1 and from ${M}_{1,{\phi }_{1},\text{\hspace{0.17em}}P}$ to $W{M}_{1,{\phi }_{2},\text{\hspace{0.17em}}P}$ for p = 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 $L{M}_{p,\phi ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$

In this section, we will prove the boundedness of the operator ${T}_{\text{Ω}}^{P}$ on the parabolic generalized local Morrey spaces $L{M}_{p,\phi ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$ by using the following statement on the boundedness of the weighted Hardy operator $Hωg(t):=∫t∞g(s)ω(s)ds, 0

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

([16, 17, 42])

Let v1, v2 and ω be positive almost everywhere and measurable functions on (0, ∞). The inequality $ess supt >0υ2(t)Hωg(t)≤C ess supt >0υ1(t)g(t)$(7)

holds for some C > 0 for all non-negative and non-decreasing functions g on (0, ∞) if and only if $B:=supt >0υ2(t)∫t∞ω(s)dsess sup s<τ<∞υ1(τ)<∞.$(8)

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

We first prove the following Theorem 3.2.

Let x0 ∈ ℝn, 1 ≤ p < ∞ and ΩLs(Sn-1), 1 < s ≤ ∞, be At-homogeneous of degree zero. Let ${T}_{\text{Ω}}^{P}$be 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 $‖TΩPf‖Lp(E(x0,r))<˜rγp∫2kr∞t−γp−1||f||Lp(E(x0,t))dt$

holds for any ellipsoid E(x0, r) and for all $f\in {L}_{p}^{loc}\left({\text{ℝ}}^{n}\right)$.

If p > 1and p < s, then the inequality $‖TΩPf‖Lp(E(x0,r))<˜rγpγs∫2kr∞tγsγp1||f||Lp(E(x0,t))dt$

holds for any ellipsoid E(x0, r) and for all $f\in {L}_{p}^{loc}\left({\text{ℝ}}^{n}\right)$.

Moreover, for s > 1 the inequality $‖TΩPf‖WLq(E(x0,r))<˜rγ∫2kr∞t−γ−1||f||L1(E(x0,t))dt$(9)

holds for any ellipsoid E(x0, r) and for all $f\in {L}_{1}^{loc}\left({\text{ℝ}}^{n}\right)$.

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

$f=f1+f2, f1(y)=f(y)χ2k E(y), f2(y)=f(y)χ(2k E)C(y), r>0$

and have

$‖TΩPf‖Lp(E)≤ ‖TΩPf1‖Lp(E)+‖TΩPf2‖Lp(E).$

Since ${f}_{1}\in {L}_{p}\left({\text{ℝ}}^{n}\right),{T}_{\text{Ω}}^{P}{f}_{1}\in {L}_{p}\left({\text{ℝ}}^{n}\right)$ and from the boundedness of ${T}_{\text{Ω}}^{P}$ on Lp(ℝn) (see Theorem 1.2) it follows that:

$‖TΩPf1‖Lp(E)≤ ‖TΩPf1‖Lp(ℝn)≤C||f1||Lp(ℝn)=C||f||Lp(2kE),$

where constant C > 0 is independent of f.

It is clear that xE, y ∈ (2kE)C implies $\frac{1}{2k}\rho \left({x}_{0}-y\right)\le \rho \left(x-y\right)\le \frac{3k}{2}\rho \left({x}_{0}-y\right).$ We get

$|TΩPf2(x)|≤2γc1 ∫(2k E)C|f(y)| |Ω(x−y)|ρ(x0−y)γdy.$

By the Fubini’s theorem, we have

$∫(2kE)C|f(y)||Ω(x−y)|ρ(x0−y)γdy≈∫(2kE)C|f(y)||Ω(x−y)|∫ρ(x0−y)∞dttγ+1dy≈∫2kr∞∫2kr≤ρ(x0−y)≤t|f(y)||Ω(x−y)|dydttγ+1≲∫2kr∞∫E(x0,t)|f(y)||Ω(x−y)|dydttγ+1.$

Applying the Hölder’s inequality, we get

$∫(2k E)C|f(y)| |Ω(x−y)|ρ(x0−y)dy<˜∫2k r∞||f||Lp(E(x0,t))||Ω(x−⋅)||Ls(E(x0,t)) |E(x0,y)|1−1p−1sdttγ+1.$(10)

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

$(∫E(x9,t)|Ω(x−y)|sdy)1s=(∫E(x−x0,t)|Ω(z)|sdz)1s≤(∫E(0,t+|x−x0|)|Ω(z)|sdz)1s≤(∫E(0,2t)|Ω(z)|sdz)1s=(∫Sn−1∫02t|Ω(z′)|sdσ(z′)rn−1dr)1s=C||Ω||Ls(Sn−1)|E(x0,2t)|1s.$(11)

Thus, by (11), it follows that:

$|TΩPf2(x)|<˜ ∫2k r∞||f||Lp(E(x0,t))dttγp+1.$

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

$‖TΩPf2‖Lp(E)<˜ rγp∫2k r∞||f||Lp(E(x0,t))dttγp+1$(12)

is valid. Thus, we obtain

$‖TΩPf‖Lp(E)<˜ ||f||Lp(2k E)+rγp∫2k r∞||f||Lp(E(x0,t))dttγp+1.$

On the other hand, we have

$||f||Lp(2k E)≈rγp||f||Lp(2k E)∫2k r∞dttγp+1≤rγp ∫2k r∞||f||Lp(E(x0,t))dttγp+1.$(13)

By combining the above inequalities, we obtain

$‖TΩPf2‖Lp(E)<˜ rγp∫2k r∞||f||Lp(E(x0,t))dttγp+1.$

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

$(∫E(x0,r)|Ω(x−y)|sdy)1s≤C||Ω||Ls(Sn−1)|E(x0,32t)|1s.$(14)

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

$‖TΩPf2‖Lp(E)≤(∫E|∫2k r∞∫E(x0,t)|f(y)| |Ω(x−y)|dydttγ+1|pdx)1p≤∫2k r∞∫E(x0,t)|f(y)| ||Ω(⋅ −y)||Lp(E)dydttγ+1≤|E(x0,r)|1p−1s∫2k r∞∫E(x0,t)|f(y)| ||Ω(⋅ −y)||Ls(E)dydttγ+1<˜rγp−γs∫2k r∞||f||L1(E(x0,t))|E(x0,32t)|1sdttγ+1<˜rγp−γs∫2k r∞tγs−γp−1||f||Lp(E(x0,t))dt.$

Let p = 1 < s ≤ ∞. From the weak (1, 1) boundedness of ${T}_{\text{Ω}}^{P}$ and (13) it follows that:

$‖TΩPf1‖WL1(E)≤‖TΩPf1‖WL1(ℝn)<˜ ||f1||L1(ℝn)= ||f||L1(2k E) <˜rγ∫2k r∞||f||L1(E(x0,t))dttγ+1.$(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 ${T}_{\text{Ω}}^{P}$ satisfying condition (1) on the parabolic generalized local Morrey spaces $L{M}_{p,\phi ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$.

Let x0 ∈ ℝn, 1 ≤ p < ∞ and Ω ∈ Ls(Sn–1), 1 < s∞, be At-homogeneous of degree zero. Let ${T}_{\text{Ω}}^{P}$ be 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 $∫r∞essinft<τ<∞φ1(x0,τ)τγptγp+1 dt≤Cφ2(x0,r),$(16)

and for 1 < p < s the pair (φ1, φ2) satisfies the condition $∫r∞essinft<τ<∞φ1(x0,τ)τγptγp−γs+1 dt≤Cφ2(x0,r)rγs,$(17)

where C does not depend on r.

Then the operator ${T}_{\text{Ω}}^{P}$ is bounded from $L{M}_{p,{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to $L{M}_{p,{\phi }_{2},P}^{\left\{{x}_{0}\right\}}$ for p > 1and from $L{M}_{1,{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to $WL{M}_{1,{\phi }_{2},P}^{\left\{{x}_{0}\right\}}$ for p = 1. Moreover, we have for p > 1 $‖TΩPf‖LMp,φ2,P{x0}<˜ ||f||LMp,φ1,P{x0},$(18)

and for p = 1 $‖TΩPf‖WLM1,φ2,P{x0}<˜ ||f||LM1,φ1,P{x0}.$(19)

Proof. Let 1 < p < ∞and s′ ≤ p. By Theorem 3.2 and Theorem 3.1 with v2(r) = φ2(x0, r)–1, v1 = ${\phi }_{1}{\left({x}_{0},r\right)}^{-1}{r}^{-\frac{\gamma }{p}},w\left(r\right)={r}^{-\frac{\gamma }{p}-1}$ and $g\left(r\right)=\text{\hspace{0.17em}}\text{\hspace{0.17em}}||f|{|}_{{L}_{p}\left(E\left({x}_{0},r\right)\right)}$, we have

$‖TΩPf‖LMp,φ2,P{x0}<˜supr >0 φ2(x0,r)−1∫r∞||f||Lp(E(x0,t))dttγp+1<˜supr >0 φ1(x0,r)−1r−γp||f||Lp(E(x0,r))= ||f||LMp,φ1,P{x0},$

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, ${\upsilon }_{1}={\phi }_{1}{\left({x}_{0},r\right)}^{-1}{r}^{-\frac{\gamma }{p}+\frac{\gamma }{s}},$ $w\left(r\right)={r}^{-\frac{\gamma }{p}+\frac{\gamma }{s}-1}$ and $g\left(r\right)=\text{\hspace{0.17em}}\text{\hspace{0.17em}}||f|{|}_{{L}_{p}\left(E\left({x}_{0},r\right)\right)}$, we have

$‖TΩPf‖LMp,φ2,P{x0}<˜supr >0 φ2(x0,r)−1r−γs∫r∞||f||Lp(E(x0,t))dttγp−γs+1<˜supr >0 φ1(x0,r)−1r−γp||f||Lp(E(x0,r))= ||f||LMp,φ1,P{x0},$

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

Also, for p = 1 we have

$‖TΩPf‖WLM1,φ2,P{x0}<˜supr >0 φ2(x0,r)−1∫r∞||f||L1(E(x0,t))dttγ+1<˜supr >0 φ1(x0,r)−1r−γ||f||L1(E(x0,r)) = ||f||LM1,φ1,P{x0}.$

Hence, the proof is completed. □

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

Let x0 ∈ ℝn, 1 ≤ p < and the pair (φ1, φ2) satisfies condition (16). Then the operators MP and TP are bounded from $L{M}_{p,{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to $L{M}_{p,{\phi }_{2},P}^{\left\{{x}_{0}\right\}}$ for p > 1 and from $L{M}_{1,{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to $WL{M}_{1,{\phi }_{2},P}^{\left\{{x}_{0}\right\}}$ for p =1.

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 operators ${M}_{\text{Ω}}^{P}$ and ${\overline{T}}_{\text{Ω}}^{P}$ are bounded from $L{M}_{p,{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to $L{M}_{p,{\phi }_{2},P}^{\left\{{x}_{0}\right\}}$ for p > 1 and from $L{M}_{1,{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to $WL{M}_{1,{\phi }_{2},P}^{\left\{{x}_{0}\right\}}$ for p =1.

Let x0 ∈ ℝn, 1 ≤ p < ∞ and Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, is homogeneous of degree zero. Let ${T}_{\text{Ω}}^{P}$ be 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 $∫r∞essinft<τ<∞φ1(x0,τ)τnptnp+1 dt≤Cφ2(x0,r),$

and for 1 < p < s the pair (φ1, φ2) satisfies the condition $∫r∞essinft<τ<∞φ1(x0,τ)τnptnp−ns+1 dt≤Cφ2(x0,r)rns,$

where C does not depend on r.

Then the operator ${T}_{\text{Ω}}^{P}$ is bounded from $L{M}_{p,{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to $L{M}_{p,{\phi }_{2},P}^{\left\{{x}_{0}\right\}}$ for p > 1 and from $L{M}_{1,{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to $WL{M}_{1,{\phi }_{2},P}^{\left\{{x}_{0}\right\}}$ for p = 1. Moreover, we have for p > 1 $‖TΩPf‖LMp,φ2,P{x0}<˜ ||f||LMp,φ1,P{x0},$

and for p = 1 $‖TΩPf‖WLM1,φ2,P{x0}<˜ ||f||LM1,φ1,P{x0}.$

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].

Let 1 ≤ p < ∞and Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, be At-homogeneous of degree zero. Let ${T}_{\text{Ω}}^{P}$be 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 $∫r∞essinft<τ<∞φ1(x, τ)τγptγp+1 dt≤Cφ2(x,r),$(20)

and for 1 < p < s the pair (φ1, φ2) satisfies the condition $∫r∞essinft<τ<∞φ1(x, τ)τγptγp−γs+1 dt≤Cφ2(x,r)rγs,$(21)

where C does not depend on x and r.

Then the operator ${T}_{\text{Ω}}^{P}$ is bounded from ${M}_{p,{\phi }_{1},\text{\hspace{0.17em}}P}$ to ${M}_{p,{\phi }_{2},\text{\hspace{0.17em}}P}$ for p > 1 and from ${M}_{1,{\phi }_{1},\text{\hspace{0.17em}}P}$ to $W{M}_{1,{\phi }_{2},\text{\hspace{0.17em}}P}$ for p = 1. Moreover, we have for p > 1 $‖TΩPf‖Mp,φ2,P<˜ ||f||Mp,φ1,P,$

and for p = 1 $‖TΩPf‖WM1,φ2,P<˜ ||f||M1,φ1,P.$

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

Let 1 ≤ p < ∞ and the pair (φ1, φ2) satisfies condition (20). Then the operators MP and TP are bounded from ${M}_{p,{\phi }_{1},\text{\hspace{0.17em}}P}$ to ${M}_{p,{\phi }_{2},\text{\hspace{0.17em}}P}$ for p > 1and from ${M}_{1,{\phi }_{1},\text{\hspace{0.17em}}P}$ to $W{M}_{1,{\phi }_{2},\text{\hspace{0.17em}}P}$ for p = 1.

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 operators ${M}_{\text{Ω}}^{P}$ and ${\overline{T}}_{\text{Ω}}^{P}$ are bounded from ${M}_{p,{\phi }_{1}}$ to ${M}_{p,{\phi }_{2}}$ for p > 1 and from ${M}_{1,{\phi }_{1}}$ to $W{M}_{1,{\phi }_{2}}$ for p = 1.

Condition (20) in Corollary 3.8 is weaker than condition (6) in Theorem 2.5. Indeed, if condition (6) holds, then $∫r∞essinft<τ<∞φ1(x,τ)τγptγp+1 dt≤∫r∞φ1(x,t)dtt, r∈(0,∞),$

so condition (20) holds.

On the other hand, the functions $φ1(r)=1χ(1,∞)(r)rγp−β, φ2(r)=r−γp(1+rβ), 0<β<γp$

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

Let 1 ≤ p < ∞ and Ω ∈ Ls(Sn1), 1 < s ≤ ∞, be homogeneous of degree zero. Let ${T}_{\text{Ω}}^{P}$ be 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 $∫r∞essinft<τ<∞φ1(x, τ)τnptnp+1 dt≤Cφ2(x,r),$

and for 1 < p < s the pair (φ1, φ2) satisfies the condition $∫r∞essinft<τ<∞φ1(x, τ)τnptnp−ns+1 dt≤Cφ2(x,r)rns,$

where C does not depend on x and r.

Then the operator ${T}_{\text{Ω}}^{P}$ is bounded from ${M}_{p,{\phi }_{1},\text{\hspace{0.17em}}P}$ to ${M}_{p,{\phi }_{2},\text{\hspace{0.17em}}P}$ for p > 1 and from ${M}_{1,{\phi }_{1},\text{\hspace{0.17em}}P}$ to $W{M}_{1,{\phi }_{2},\text{\hspace{0.17em}}P}$ for p = 1. Moreover, we have for p > 1 $‖TΩPf‖Mp,φ2,P<˜ ||f||Mp,φ1,P,$

and for p = 1 $‖TΩPf‖WM1,φ2,P<˜ ||f||M1,φ1,P.$

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 $L{M}_{p,\phi ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$

In this section, we will prove the boundedness of the operators $\left[b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{\text{Ω}}^{P}\right]$ with $b\in L{C}_{{p}_{2,}\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$ on the parabolic generalized local Morrey spaces $L{M}_{p,\phi ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$ 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 $K\left(x\right)=\frac{\text{Ω}\left(x\prime \right)}{|x{|}^{n}}$ 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 $BMO\left({\text{ℝ}}^{n}\right)\subset \underset{p>1}{\cap }L{C}_{p,P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right),$ if we only assume $b\in L{C}_{p,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$, or more generally b$L{C}_{p,\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$, 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 $b\in L{C}_{p,\text{\hspace{0.17em}}I}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$ on Herz spaces for the first time. Morever, in [16, 17] and [19, 46], they have considered the commutators with $b\in L{C}_{p,\lambda ,\text{\hspace{0.17em}}I}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$. 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 $L{C}_{p,\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$ following the known ideas of defining local Campanato space (see [16, 17, 42] etc).

Let 1 ≤ p < ∞ and $0\le \lambda <\frac{1}{\gamma }$. A function $f\in {L}_{p}^{loc}\left({\text{ℝ}}^{n}\right)$ is said to belong to the $L{C}_{p,\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$ (parabolic local Campanato space), if $||f||LCp,λ,P{x0}=supr >0(1|E(x0,r)|1+λp ∫E(x0,r)|f(y)−fE(x0,r)|pdy)1p<∞,$(22)

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

Define $LCp,λ,P{x0}(ℝn)={f∈Lploc(ℝn) : ||f||LCp,λ,P{x0}<∞}.$

If two functions which differ by a constant are regarded as a function in the space $L{C}_{p,\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$, then $L{C}_{p,\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$ becomes a Banach space. The space $L{C}_{p,\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$ when λ = 0 is just the $L{C}_{p,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$. Apparently, (22) is equivalent to the following condition: $supr >0infc ∈ ℂ(1|E(x0,r)|1+λp ∫E(x0,r)|f(y)−c|p dy)1p<∞.$

In [48], Lu and Yang have introduced the central BMO space $CBM{O}_{p}\left({\text{ℝ}}^{n}\right)=L{C}_{p,0,\text{\hspace{0.17em}}I}^{\left\{0\right\}}\left({\text{ℝ}}^{n}\right).$ Also the space $CBM{O}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)=L{C}_{1,0,\text{\hspace{0.17em}}I}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$ can be has been considered in other denotes in [49]. The space $L{C}_{p,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$ 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: $supx∈ℝn,r >0(1|E(x,r)| ∫E(x, r)|f(y)−fE(x,r)|p dy)1p<∞,$

where B denotes an arbitrary ball in ℝn. However, the space $L{C}_{p,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$ depends on p. If p1 < p2, then $L{C}_{{p}_{2},\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)\underset{\ne }{\subset }L{C}_{{p}_{1},\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right).$ Therefore, there is no analogy of the famous John-Nirenberg inequality of BMO(ℝn) for the space $L{C}_{p,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$. One can imagine that the behavior of $L{C}_{p,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right)$ may be quite different from that of BMO(ℝn).

([16, 17, 42])

Let v1, v2 and ω be weigths on (0, ∞) and v1(t) be bounded outside a neighbourhood of the origin. The inequality $ess supr >0υ2(r)Hωg(r)≤C ess supr >0υ1(r)g(r)$(23)

holds for some C > 0 for all non-negative and non-decreasing functions g on (0, ∞) if and only if $ess supr >0υ2(r)Hωg(r)≤C ess supr >0υ1(r)g(r)$(24)

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

In (23) and (24) it is assumed that $\frac{1}{\infty }=0$ and 0 · ∞ = 0.

Let b be function in $L{C}_{p,\lambda .\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right),1\le p<\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le \lambda <\frac{1}{\gamma }$ and r1, r2 > 0. Then $(1|E(x0,r1)|1+λp ∫E(x0,r1)|b(y)−bE(x0,r2)|pdy)1p≤C(1+lnr1r2)||b||LCp,λ,P{x0},$(25)

where C > 0 is independent of b, r1 and 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 $||b−bE||Lp(E) ≤C(1+lnr1r2)rγp+γλ||b||LCp,λ,P{x0}.$(27)

In [30] the following statements have been proved for the parabolic commutators of parabolic singular integral operators with rough kernel ${\overline{T}}_{\text{Ω}}^{P}$, containing the result in [3133].

Suppose that Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, is At-homogeneous of degree zero and bBMO(ℝn). Let $1\le s\prime and φ(x, r) satisfies the conditions (4) and (5). If the commutator operator $\left[b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{T}}_{\text{Ω}}^{P}\right]$ is bounded on Lp(ℝn), then the operator $\left[b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{T}}_{\text{Ω}}^{P}\right]$ is bounded on Mp,φ, P.

Let 1 < p < ∞ bBMO(ℝn) and φ(x, t) satisfies conditions (4) and (5). Then the operators ${M}_{b}^{P}$ and [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.

Let x0 ∈ ℝn, 1 < p < ∞ and Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, be At-homogeneous of degree zero. Let ${T}_{\text{Ω}}^{P}$be a parabolic linear operator satisfying condition (1), bounded on Lp(ℝn) for 1 < p < ∞ Let also, $b\in L{C}_{{p}_{2},\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right),0\le \lambda <\frac{1}{\gamma }$ and $\frac{1}{p}=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}$.

Then, for s′ ≤ p, the inequality $‖[b,TΩP]f‖Lp(E(x0,r))<˜ ||b||LCp2,λ,P{x0}rγp ∫2k r∞(1+lntr)tγλ−γp1−1||f||Lp1(E(x0,t))dt$

holds for any ellipsoid E(x0, r) and for all $f\in {L}_{{p}_{1}}^{loc}\left({\text{ℝ}}^{n}\right)$.

Also, for p1 < s, the inequality $‖[b,TΩP]f‖Lp(E(x0,r))<˜ ||b||LCp2,λ,P{x0}rγp−γs ∫2k r∞(1+lntr)tγλ−γp1+γs−1||f||Lp1(E(x0,t))dt$

holds for any ellipsoid E(x0, r) and for all $f\in {L}_{{p}_{1}}^{loc}\left({\text{ℝ}}^{n}\right)$.

Proof. Let $1 and $\frac{1}{p}=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}$. Set E= E(x0, r) for the parabolic ball (ellipsoid) centered at x0and of radius rand 2kE = E(x0, 2kr). We represent f as $f=f1+f2, f1(y)=f(y)χ2k E(y), f2(y)=f(y)χ(2k E)C(y), r>0$

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 $‖[b,TΩP]f‖Lp(E)≤ ||J1||Lp(E)+||J2||Lp(E)+||J3||Lp(E)+||J4||Lp(E).$

By the Hölder’s inequality, the boundedness of ${T}_{\text{Ω}}^{P}$ on ${L}_{{p}_{1}}\left({\text{ℝ}}^{n}\right)$ (see Theorem 1.2) it follows that: $||J1||Lp(E)≤‖(b(⋅)−bE)TΩPf1(⋅)‖Lp(ℝn)<˜ ||(b(⋅)−bE)||Lp2(ℝn)‖TΩPf1(⋅)‖Lp1(ℝn)<˜ ||b||LCp2,λ,P{x0}rγp2+γλ||f1||Lp1(ℝn)= ||b||LCp2,λ,P{x0}rγp2+γp1+γλ||f1||Lp1(2k E)∫2k r∞t−1−γp1dt<˜ ||b||LCp2,λ,P{x0}rγp ∫2k r∞(1+lntr)tγλ−γp1−1||f||Lp1(E(x0,t))dt.$

Using the the boundedness of ${T}_{\text{Ω}}^{P}$ on Lp(ℝn) (see Theorem 1.2), by the Hölder’s inequality for J2 we have $||J2||Lp(E)≤‖TΩP(b(⋅)−bE)f1‖Lp(ℝn)<˜ ||(b(⋅)−bE)f1||Lp(ℝn)<˜ ||(b(⋅)−bE)Lp2(ℝn)||f1||Lp1(ℝn)<˜||b||LCp2,λ,P{x0}rγp2+γp1+γλ||f||Lp1(2k E)∫2k r∞t−1−γp1dt<˜||b||LCp2,λ,P{x0}rγp∫2k r∞(1+lntr)tγλ−γp1−1||f||Lp1(E(x0,t))dt.$

For J3, it is known that xE, y. ∈ (2kE)C, which implies $\frac{1}{2k}\rho \left({x}_{0}-y\right)\le \rho \left(x-y\right)\le \frac{3k}{2}\rho \left({x}_{0}-y\right).$

When s′ ≤ p1, by the Fubini’s theorem, the Hölder’s inequality and (11) we have $|TΩPf2(x)|≤c0∫(2k E)C|Ω(x−y)||f(y)|ρ(x0−y)dy≈∫2k r∞∫2k r<ρ(x0−y)

Hence, we get $‖J3‖Lp(E)≤‖(b(⋅)−bB)TΩPf2(⋅)‖Lp(ℝn)<˜‖b(⋅)−bE‖Lp(ℝn)∫2kr∞t−1−γp1‖f‖Lp1(E(x0, t))dt<˜‖(b(⋅)−bE)‖Lp2(ℝn)rγp1∫2kr∞t−1−γp1‖f‖Lp1(E(x0,t))dt<˜‖b‖LCp2,λ,P{x0}rγp+γλ∫2kr∞(1+lntr)t−1−γp1‖f‖Lp1(E(x0,t))dt<˜‖b‖LCp2,λ,P{x0}rγp∫2kr∞(1+lntr)tγλ−γp1−1‖f‖Lp1(E(x0,t))dt$

When p1 < s, by the Fubini’s theorem, the Minkowski inequality, the Hölder’s inequality and from (27), (14) we get $‖J3‖Lp(E)≤(∫E|∫2kr∞∫E(x0,t)|f(y)||b(x)−bE||Ω(x−y)|dydttγ+1|pdx)1p≤∫2kr∞∫E(x0,t)|f(y)|‖(b(⋅)−bE)Ω(·−y)‖Lp(E)dydttγ+1≤∫2kr∞∫E(x0,t)|f(y)|‖(b(⋅)−bE)Lp2‖Ω(·−y)Lp1(E)dydttγ+1<˜‖b‖LCp2,λ,P{x0}rγp+γλ|E|1p1−1s∫2kr∞∫E(x0,t)|f(y)|||Ω(·−y)||Ls(E)dydttγ+1<˜‖b‖LCp2,λ,P{x0}rγp−γs+γλ∫2kr∞||f||L1(E(x0,t))|E(x0,32t)|1sdttγ+1<˜‖b‖LCp2,λ,P{x0}rγp−γs+γλ∫2kr∞(1+lntr)||f||Lp1(E(x0,t))dttγp1−γs+1<˜‖b‖LCp2,λ,P{x0}rγp−γs∫2kr∞(1+lntr)tγλ−γp1+γs−1||f||Lp1(E(x0,t))dt.$

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)| <˜∫(2k E)C|b(y)−bE| |Ω(x−y)||f(y)|ρ(x−y)γdy<˜∫(2k E)C|b(y)−bE| |Ω(x−y)||f(y)|ρ(x0−y)γdy≈∫2k r∞∫2k r<ρ(x0−y)

Then, we have $‖J4‖Lp(E)=‖TΩP(b(⋅)−bE)f2‖Lp(E)<˜‖b‖LCp2,λ, P{x0}rnp∫2kr∞(1+lntr)tyλ−γp1−1‖f‖Lp1(E(x0,t)) dt.$

When p1 < s, by the Minkowski inequality, applying the Hölder’s inequality and from (26), (27), (14) we have $||J4||Lp(E)≤(∫E|∫2k r∞∫E(x0,t)|b(y)−bE(x0,t)| |f(y)| |Ω(x−y)|dydttγ+1|pdx)1p+(∫E|∫2k r∞|bE(x0,r)−bE(x0,t)|∫E(x0,t)|f(y)| | Ω(x−y)|dydttγ+1| dx)1p<˜∫2k r∞∫E(x0,t)|b(y)−bE(x0,t)| |f(y)| ||Ω(⋅−y)||Lp(E(x0,t)) dydttγ+1+∫2k r∞|bE(x0,r)−bE(x0,t)|∫E(x0,t)|f(y)| | |Ω(⋅−y)||Lp(E(x0,t))dydttγ+1<˜ |E|1p−1s∫2k r∞∫E(x0,t)|b(y)−bE(x0,t)| |f(y)| ||Ω(⋅−y)||Ls(E(x0,t)) dydttγ+1+ |E|1p−1s∫2k r∞|bE(x0,r)−bE(x0,t)|∫E(x0,t)|f(y)| | |Ω(⋅−y)||Ls(E(x0,t))dydttγ+1<˜ rγp−γs∫2k r∞||b(⋅)−bE(x0,t)||Lp2(E(x0,t)) ||f||Lp1(E(x0,t))|E(x0,t)|1−1p|E(x0,32t)|1sdttγ+1+ rγp−γs∫2k r∞|bE(x0,r)−bE(x0,t)| ||f||Lp1(E(x0,t))||E(x0,32t)|1sdttγp1+1<˜ rγp−γs||b||LCp2,λ, P{x0}∫2k r∞(1+lntr)tγλ−γp1+γs−1||f||Lp1(E(x0,t))dt.$

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).

Let x0 ∈ ℝn, 1 < p < ∞and Ω ∈ Ls(Sn–1), 1 < s ≤ ∞,, be At-homogeneous of degree zero. Let ${T}_{\text{Ω}}^{P}$ be a parabolic linear operator satisfying condition (1), bounded on Lp(ℝn) for 1 < p < ∞. Let $b\in L{C}_{{p}_{2},\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right),0\le \lambda <\frac{1}{\gamma }$ and $\frac{1}{p}=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}$. Let also, for s′ ≤ p the pair (φ1, φ2) satisfies the condition $∫r∞(1+lntr)t<τ<∞essinfφ1(x0,τ)τγp1tγp1+1−γλdt≤Cφ2(x0, r),$(28)

and for p1 < s the pair (φ1, φ2) satisfies the condition $∫r∞(1+lntr)t<τ<∞essinfφ1(x0,τ)τγp1tγp1−γs+1−γλdt≤Cφ2(x0, r)rγs,$(29)

where C does not depend on r.

Then the operator $\left[b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{\text{Ω}}^{P}\right]$ is bounded from $L{M}_{{p}_{1},{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to$L{M}_{p,{\phi }_{2},\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$. Moreover, $‖[b, TΩP]f‖LMp, φ2, P{x0}≲‖b‖LCp2, λ, P{x0}‖f‖LMp1, φ1, P{x0}.$(30)

Proof. Let p > 1and s′ ≤ p. By Theorem 4.8 and Theorem 4.3 with v2(r) =φ2(x0, r)–1, v1 = ${\phi }_{1}{\left({x}_{0},r\right)}^{-1}{r}^{-\frac{\gamma }{{p}_{1}}},w\left(r\right)={r}^{\gamma \lambda -\frac{\gamma }{{p}_{1}}-1}$ and $g\left(r\right)=\text{\hspace{0.17em}}||f|{|}_{{L}_{{p}_{1}}\left(E\left({x}_{0},r\right)\right)}$, we have $‖[b,TΩP]f‖LMp,φ2, P{x0}<˜supr>0φ2(x0,r)−1||b||LCp2,λ, P{x0}∫r∞(1+lntr)tγλ−γp1−1||f||Lp1(E(x0,t))dt<˜ ||b||LCp2,λ, P{x0}supr>0φ1(x0,r)−1r−γp1||f||Lp1(E(x0,r))= ||b||LCp2,λ, P{x0}||f||LMp1,φ1, P{x0},$

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 = ${\phi }_{1}{\left({x}_{0},r\right)}^{-1}{r}^{-\frac{\gamma }{{p}_{1}}+\frac{\gamma }{s}},w\left(r\right)={r}^{\gamma \lambda -\frac{\gamma }{{p}_{1}}+\frac{\gamma }{s}-1}$ and $g\left(r\right)=\text{\hspace{0.17em}}||f|{|}_{{L}_{{p}_{1}}\left(E\left({x}_{0},r\right)\right)}$, we have $‖[b,TΩP]f‖LMp,φ2, P{x0}<˜supr>0φ2(x0,r)−1r−γs||b||LCp2,λ, P{x0}∫r∞(1+lntr)tγλ−γp1+γs−1||f||Lp1(E(x0,t))dt<˜ ||b||LCp2,λ, P{x0}supr>0φ1(x0,r)−1r−γp1||f||Lp1(E(x0,r))= ||b||LCp2,λ, P{x0}||f||LMp1,φ1, P{x0},$

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

Let x0 ∈ ℝn, 1 < p < ∞, $b\in L{C}_{{p}_{2},\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right),0\le \lambda <\frac{1}{\gamma },\frac{1}{p}=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}$ and the pair (φ1, φ2) satisfies condition (28). Then the operators ${M}_{b}^{P}$ and [b, TP] are bounded from $L{M}_{{p}_{1},{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to $L{M}_{p,{\phi }_{2},P}^{\left\{{x}_{0}\right\}}$.

Let x0 ∈ ℝn, 1 < p < and Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, be At-homogeneous of degree zero. Let $b\in L{C}_{{p}_{2},\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right),0\le \lambda <\frac{1}{\gamma }$, $\frac{1}{p}=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}$. 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 operators ${M}_{\text{Ω},b}^{P}$ and $\left[b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{T}}_{\text{Ω}}^{P}\right]$ are bounded from $L{M}_{{p}_{1},{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to $L{M}_{p,{\phi }_{2},\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$.

Let x0 ∈ ℝn, 1 < p < ∞and Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, be homogeneous of degree zero. Let ${T}_{\text{Ω}}^{P}$be a parabolic linear operator satisfying condition (1), bounded on Lp(ℝn) for 1 < p < ∞ Let $b\in L{C}_{{p}_{2},\lambda ,\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right),0\le \lambda <\frac{1}{n},\frac{1}{p}=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}.$ Let also, for s′ ≤ p the pair (φ1, φ2) satisfies the condition $∫r∞(1+lntr)t<τ<∞essinfφ1(x0,τ)τnp1tnp1+1−nλ dt≤Cφ2(x0, r),$(31)

and for p1 < s the pair (φ1, φ2) satisfies the condition $∫r∞(1+lntr)t<τ<∞essinfφ1(x0,τ)τnp1tnp1−ns+1−nλ dt≤Cφ2(x0, r)rns,$

where C does not depend on r.

Then the operator $\left[b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{\text{Ω}}^{P}\right]$ is bounded from $L{M}_{{p}_{1},{\phi }_{1},P}^{\left\{{x}_{0}\right\}}$ to $L{M}_{p,{\phi }_{2},\text{\hspace{0.17em}}P}^{\left\{{x}_{0}\right\}}$. Moreover, $‖[b, TΩP]f‖LMp, φ2, P{x0}≲‖b‖LCp2, λ, P{x0}‖f‖LMp1, φ1, P{x0}.$

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].

Let Ω ∈ Ls(Sn–1), 1 < s ≤ ∞, be At-homogeneous of degree zero. Let ${T}_{\text{Ω}}^{P}$ be 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 $∫r∞(1+lntr)t<τ<∞essinfφ1(x0,τ)τγptγp+1 dt≤Cφ2(x, r),$(32)

and for p < s the pair (φ1, φ2) satisfies the condition $∫r∞(1+lntr)t<τ<∞essinfφ1(x0,τ)τγptγp−γs+1 dt≤Cφ2(x, r)rγs,$(33)

where C does not depend on x and r.

Then the operator $\left[b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{\text{Ω}}^{P}\right]$ is bounded from ${M}_{p,{\phi }_{1},\text{\hspace{0.17em}}P}$ to ${M}_{p,{\phi }_{2},\text{\hspace{0.17em}}P}$. Moreover, $‖[b, TΩP]f‖Mp, φ2, P≲ ‖b‖BMO‖f‖Mp, φ1, P.$

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

Let 1 < p < ∞, bBMO (ℝn) and the pair (φ1, φ2) satisfies condition (32). Then the operators ${M}_{b}^{P}$ and [b, TP] are bounded from ${M}_{p,{\phi }_{1},\text{\hspace{0.17em}}P}$ to${M}_{p,{\phi }_{2},\text{\hspace{0.17em}}P}$.

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 operators ${M}_{\text{Ω},b}^{P}$ and $\left[b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{T}}_{\text{Ω}}^{P}\right]$ are bounded from ${M}_{p,{\phi }_{1},\text{\hspace{0.17em}}P}$ to ${M}_{p,{\phi }_{2},\text{\hspace{0.17em}}P}$.

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 $∫r∞(1+lntr)t<τ<∞essinfφ1(x0,τ)τnptnp+1 dt≤Cφ2(x, r),$

and for p < s the pair (φ1, φ2) satisfies the condition $∫r∞(1+lntr)t<τ<∞essinfφ1(x0,τ)τnptnp−ns+1 dt≤Cφ2(x, r)rns,$

where C does not depend on x and r.

Then the operator $\left[b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{\text{Ω}}^{P}\right]$ is bounded from ${M}_{p,{\phi }_{1},\text{\hspace{0.17em}}P}$ to ${M}_{p,{\phi }_{2},\text{\hspace{0.17em}}P}$. Moreover, $‖[b, TΩP]f‖Mp, φ2, P≲‖b‖BMO‖f‖MP, φ1, P.$

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, $Ω(Atx)=Ω(x), for any t> 0, x ∈ ℝn\{0};$

2. Ω(x) has mean zero on Sn–1, that is, $∫Sn−1Ω(x')J(x')dσ(x')=0$

where $x\prime =\frac{x}{|x|}$ for any x ≠ 0.

3. Ω ∈ L1(Sn–1).

Then the parabolic Marcinkiewicz integral of higher dimension ${\mu }_{\text{Ω}}^{\gamma }$ is defined by $μΩγ(f)(x)=(∫0∞|FΩ, t(f)(x)|2dtt3)1/2,$

where $FΩ, t(f)(x)=∫ρ(x−y)≤tΩ(x−y)ρ(x−y)γ−1f(γ)dy.$

On the other hand, for a suitable function b, the commutator of the parabolic Marcinkiewicz integral ${\mu }_{\text{Ω}}^{\gamma }$ is defined

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

where $FΩ,t,b(f)(x)=∫ρ(x−y)≤tΩ(x−y)ρ(x−y)γ−1[b(x)−b(y)]f(y)dy.$

We consider the space $H=\left\{h:\text{\hspace{0.17em}}\text{\hspace{0.17em}}||h||\text{\hspace{0.17em}}\text{\hspace{0.17em}}={\left(\underset{0}{\overset{\infty }{\int }}|h\left(t\right){|}^{2}\frac{dt}{{t}^{3}}\right)}^{1/2}<\infty \right\}.$ Then, it is clear ${\mu }_{\text{Ω}}^{\gamma }\left(f\right)\left(x\right)=\text{\hspace{0.17em}}||{F}_{\text{Ω},t}\left(x\right)||.$

By the Minkowski inequality and the conditions on Ω we get $μΩγ(f)(x)≤∫ℝn|Ω(x−y)|ρ(x−y)γ−1|f(y)|(∫|x−y|∞dtt3)1/2dy≤C∫ℝn|Ω(x−y)|ρ(x−y)γ|f(y)|dy.$

Thus, ${\mu }_{\text{Ω}}^{\gamma }$ 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

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 ${\mu }_{\text{Ω}}^{\gamma }$ is bounded from $L{M}_{p,{\phi }_{1}}^{\left\{{x}_{0}\right\}}$ to $L{M}_{p,{\phi }_{2}}^{\left\{{x}_{0}\right\}}$ for p > 1 and from $L{M}_{1,{\phi }_{1}}^{\left\{{x}_{0}\right\}}$ to $WL{M}_{1,{\phi }_{2}}^{\left\{{x}_{0}\right\}}$.

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 operator ${\mu }_{\text{Ω}}^{\gamma }$ is bounded from ${M}_{p,{\phi }_{1}}$ to ${M}_{p,{\phi }_{2}}$ for p > 1 and from ${M}_{1,{\phi }_{1}}$ to $W{M}_{1,{\phi }_{2}}$ for p = 1.

Let x0 ∈ ℝn (Sn–1), 1 < s ≤ ∞. Let 1 < p < ∞, $b\in L{C}_{{p}_{2},\lambda }^{\left\{{x}_{0}\right\}}\left({\text{ℝ}}^{n}\right),\frac{1}{p}=\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}},$ $0\le \lambda <\frac{1}{n}.$ 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 $\left[b,{\mu }_{\text{Ω}}^{\gamma }\right]$ is bounded from $L{M}_{{p}_{1},{\phi }_{1}}^{\left\{{x}_{0}\right\}}$ to $L{M}_{p,{\phi }_{2}}^{\left\{{x}_{0}\right\}}$.

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 $\left[b,{\mu }_{\text{Ω}}^{\gamma }\right]$ is bounded from ${M}_{p,{\phi }_{1}}$ to ${M}_{p,{\phi }_{2}}$.

Obviously, if we take α1 = ∙∙∙ = αn and P = I, then $\rho \left(x\right)=\text{\hspace{0.17em}}|x|\text{\hspace{0.17em}}\text{\hspace{0.17em}}={\left(\sum _{i=1}^{n}{x}_{i}^{2}\right)}^{1/2},\gamma =n,$ (ℝn, ρ) = (ℝn, |∙|), EI(x, r) = B (x, r). In this case,${\mu }_{\text{Ω}}^{\gamma }$ is 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–1 and the conditions (a), (b) (obviously, in the case At = tI and J(x′) ≡ 1), then μwΩ is 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.

## References

• 1

Calderón A.P. and Zygmund A., On the existence of certain singular integrals, Acta Math., 1952, 88, 85-139. Google Scholar

• 2

Calderón A.P. and Zygmund A., Singular integral operators and differential equations, Amer. J. Math., 1957, 79, 901-921. Google Scholar

• 3

Fabes E. and Riviére N., Singular integrals with mixed homogeneity, Stud. Math., 1966, 27, 19-38. Google Scholar

• 4

Hofmann S., A characterization of commutators of parabolic singular integrals, Fourier analysis and partial differential equations (Miraflores de la Siera, 1992), Stud. Adv. Math. CRC, Boca Raton, FL 1995, 195-210. Google Scholar

• 5

Hofmann S., Parabolic singular integrals of Calderón-type, rough operators, and caloric layer potentials, Duke Math. J., 1997, 90(2), 209-259.Google Scholar

• 6

Tao T., The weak-type (1,1) of L log L homogeneous convolution operators, Indiana Univ. Math. J., 1999, 48, 1547-1584. Google Scholar

• 7

Coifman R.R., Weiss G., Analyse harmonique non-commutative sur certains espaces homogènes, (French) Étude de certaines intégrales singuliàres, Lecture Notes in Mathematics, Vol. 242. Springer-Verlag, Berlin-New York, 1971. Google Scholar

• 8

Folland G.B., Stein E.M., Hardy Spaces on homogeneous groups, Math. Notes, 28, Princeton Univ. Press, Princeton, 1982. Google Scholar

• 9

Calderón A.P. and Torchinsky A., Parabolic maximal functions associated with a distribution, Adv. Math., 1975, 16, 1-64. Google Scholar

• 10

Besov O.V., II’in V.P., Lizorkin P.I., The Lp-estimates of a certain class of non-isotropically singular integrals, (Russian) Dokl. Akad. NaukSSSR., 1966, 169, 1250-1253. Google Scholar

• 11

Soria F, Weiss G., A remark on singular integrals and power weights, Indiana Univ. Math. J., 1994, 43, 187-204. Google Scholar

• 12

Lu G., Lu S.Z., Yang D.C., Singular integrals and commutators on homogeneous groups, Anal. Math., 2002, 28, 103-134. Google Scholar

• 13

Palagachev D.K., Softova L, Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s, Potential Anal., 2004, 20, 237-263. Google Scholar

• 14

Stein E.M., Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton NJ, 1993. Google Scholar

• 15

Muckenhoupt B. and Wheeden R.L., Weighted norm inequalities for singular and fractional integral. Trans. Amer. Math. Soc, 1971, 161, 249-258. Google Scholar

• 16

Balakishiyev A.S., Guliyev V.S., Gurbuz F and Serbetci A., Sublinear operators with rough kernel generated by Calderon-Zygmund operators and their commutators on generalized local Morrey spaces, J. Inequal. Appl. 2015, 2015:61. doi:10.1186/s13660-015-0582-y.

• 17

Gurbuz F, Boundedness of some potential type sublinear operators and their commutators with rough kernels on generalized local Morrey spaces [Ph.D. thesis], Ankara University, Ankara, Turkey, 2015. Google Scholar

• 18

Gurbuz F, Some estimates on generalized Morrey spaces for certain sublinear operators with rough kernel generated by Calderón-Zygmund operators and their commutators, submitted. Google Scholar

• 19

Fu Z.W., Lin Y and Lu S.Z., λ-Central BMO estimates for commutators of singular integral operators with rough kernel, Acta Math. Sin. (Engl. Ser.), 2008, 24, 373-386. Google Scholar

• 20

Lu S.Z., Ding Y, Yan D.Y, Singular Integrals and Related Topics, World Scientific, Singapore, 2007. Google Scholar

• 21

Morrey C.B., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc, 1938, 43, 126-166. Google Scholar

• 22

Adams D.R., A note on Riesz potentials, Duke Math. J., 1975, 42, 765-778. Google Scholar

• 23

Chiarenza F, Frasca M., Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat., 1987, 7, 273-279. Google Scholar

• 24

Peetre J., On the theory of Mp,λ, J. Funct. Anal., 1969, 4, 71-87. Google Scholar

• 25

Chiarenza F, Frasca M., Longo P., Interior W2, P-estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat., 1991, 40, 149-168. Google Scholar

• 26

Chiarenza F, Frasca M., Longo P., W2, P-solvability of Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc, 1993, 336, 841-853.Google Scholar

• 27

Fazio G.Di, Ragusa M.A., Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal., 1993, 112, 241-256. Google Scholar

• 28

Fazio G.Di, Palagachev D.K. and Ragusa M.A., Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients, J. Funct. Anal., 1999, 166, 179-196. Google Scholar

• 29

Meskhi A., Maximal functions, potentials and singular integrals in grand Morrey spaces, Complex Var. Elliptic Equ., 2011, 56, 1003-1019. Google Scholar

• 30

Ding Y, Yang D.C., Zhou Z., Boundedness of sublinear operators and commutators on Lp,ω (ℝn), Yokohama Math. J., 1998, 46, 15-27. Google Scholar

• 31

Mizuhara T, Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis (S. Igari, Editor), ICM 90 Satellite Proceedings, Springer - Verlag, Tokyo (1991), 183-189. Google Scholar

• 32

Nakai E., Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces, Math. Nachr., 1994, 166, 95-103.Google Scholar

• 33

Nakai E., The Campanato, Morrey and Hölder spaces on spaces of homogeneous type, Studia Math., 2006, 176, 1, 1-19. Google Scholar

• 34

Guliyev V.S., Integral operators on function spaces on the homogeneous groups and on domains in ℝn. Doctor’s degree dissertation, Mat. Inst. Steklov, Moscow, 1994, 329 pp. (in Russian). Google Scholar

• 35

Guliyev V.S., Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl. 2009, Art. ID 503948, 20 pp. Google Scholar

• 36

Burenkov V.I., Guliyev H.V., Guliyev V.S., Necessary and sufficient conditions for boundedness of the fractional maximal operators in the local Morrey-type spaces, J. Comput. Appl. Math., 2007, 208(1), 280-301. Google Scholar

• 37

Burenkov V.I., Guliyev V.S., Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces, Potential Anal., 2009, 30(3), 211-249.Google Scholar

• 38

Burenkov V.I., Gogatishvili A., Guliyev V.S., Mustafayev R.Ch., Boundedness of the fractional maximal operator in local Morrey-type spaces, Complex Var. Elliptic Equ., 2010, 55(8-10), 739-758.Google Scholar

• 39

Burenkov V.I., Gogatishvili A., Guliyev V.S., Mustafayev R.Ch., Boundedness of the Riesz potential in local Morrey-type spaces, Potential Anal., 2011, 35(1), 67-87. Google Scholar

• 40

Guliyev V.S., Mustafayev R.Ch., Boundedness of the anisotropic maximal and anisotropic singular integral operators in generalized Morrey spaces, Acta Math. Sin. (Engl. Ser.), 2011, 27(12), 2361-2370. Google Scholar

• 41

Guliyev VS., Aliyev S.S., Karaman T., Shukurov P., Boundedness of sublinear operators and commutators on generalized Morrey Space, Int. Eq. Op. Theory, 2011, 71(3), 327-355. Google Scholar

• 42

Guliyev V.S., Generalized local Morrey spaces and fractional integral operators with rough kernel, J. Math. Sci. (N.Y.), 2013, 193(2), 211-227. Google Scholar

• 43

Karaman T, Boundedness of some classes of sublinear operators on generalized weighted Morrey spaces and some applications [Ph.D. thesis], Ankara University, Ankara, Turkey, 2012. Google Scholar

• 44

Coifman R.R., Rochberg R., Weiss G., Factorization theorems for Hardy spaces in several variables, Ann. Math., 1976, 103(3), 611-635. Google Scholar

• 45

Grafakos L, Li X.W., Yang D.C., Bilinear operators on Herz-type Hardy spaces. Trans. Amer. Math. Soc, 1998, 350, 1249-1275. Google Scholar

• 46

Tao X.X., Shi Y.L., Multilinear commutators of Calderón-Zygmund operator on λ-central Morrey spaces, Adv. Math., 2011, 40, 47-59. Google Scholar

• 47

Coifman R.R., Lions P., Meyer Y, Semmes S., Compensated compactness and Hardy spaces. J. Math. Pures Appl., 1993, 72, 247-286. Google Scholar

• 48

Lu S.Z. and Yang D.C., The central BMO spaces and Littlewood-Paley operators, Approx. Theory Appl. (N.S.), 1995, 11, 72-94. Google Scholar

• 49

Rzaev R.M., On approximation of local summary functions by singular integrals in terms of mean oscillation and some applications, Preprint No. 1 of Inst. Physics of NAS of Azerb., 1992, pp. 1-43 (Russian). Google Scholar

• 50

Xue Q.Y, Ding Y and Yabuta K., Parabolic Littlewood-Paley g-function with rough kernel, Acta Math. Sin. (Engl. Ser.), 2008, 24(12) 2049-2060. Google Scholar

• 51

Stein E.M., On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc, 1958, 88, 430-466. Google Scholar

• 52

Torchinsky A. and Wang S., A note on the Marcinkiewicz integral, Colloq. Math., 1990, 60/61, 235-243. Google Scholar

Accepted: 2016-04-25

Published Online: 2016-05-19

Published in Print: 2016-01-01

Citation Information: Open Mathematics, Volume 14, Issue 1, Pages 300–323, ISSN (Online) 2391-5455,

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