Marcinkiewicz Integrals Research Articles

On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals

UDC 517.9 Let Ω be homogeneous of degree zero, have mean value zero, and integrable on the unit sphere. For m ∈ ℕ , let and let the higher-order commutator of the Marcinkiewicz integral μ Ω , b m be defined by μ Ω , b m ( f ) ( x ) = ( ∫ 0 ∞ | ∫ | x - y | ≤ t Ω ( x - y ) | x - y | n - 1 [ b ( x ) - b ( y ) ] m f ( y ) ⅆ y | 2 ⅆ t t 3 ) 1 2 . We establish a sparse domination of μ Ω , b m for Ω ∈ L i p ( 𝕊 n - 1 ) . Moreover, we also give Bloom-type characterizations of the two-weighted boundedness of the higher-order commutators μ Ω , b m , μ Ω , α , b * , m , and μ Ω , S , b m , where the higher-order commutators μ Ω , α , b * , m and μ Ω , S , b m are defined, respectively, by μ Ω , α , b * , m ( f ) ( x ) = ( ∫ ∫ ℝ + n + 1 ( t t + | x - y | ) n α | ∫ | y - z | ≤ t Ω ( y - z ) | y - z | n - 1 [ b ( x ) - b ( z ) ] m f ( z ) ⅆ z | 2 ⅆ y ⅆ t t n + 3 ) 1 2 , α > 1 , and μ Ω , S , b m ( f ) ( x ) = ( ∫ ∫ | x - y | < t | ∫ | y - z | ≤ t Ω ( y - z ) | y - z | n - 1 [ b ( x ) - b ( z ) ] m f ( z ) ⅆ z | 2 ⅆ y ⅆ t t n + 3 ) 1 2 .

Fractional type Marcinkiewicz integral and its commutator on grand variable Herz-Morrey spaces

Fractional type Marcinkiewicz integral and its commutator on grand variable Herz-Morrey spaces

Weighted Estimates of the Fractional Type Marcinkiewicz Integral and Its Commutator on Morrey–Guliyev Spaces

Weighted Estimates of the Fractional Type Marcinkiewicz Integral and Its Commutator on Morrey–Guliyev Spaces

On rough generalized Marcinkiewicz integrals along surfaces of revolution on product spaces

<abstract><p>In this paper, we prove the $ L^p $ boundedness of generalized Marcinkiewicz operators along surfaces of revolution on product spaces under very weak conditions on the the singular kernels. Our results generalize and improve many previously known results.</p></abstract>

On the Functions of Marcinkiewicz Integrals along Surfaces of Revolution on Product Domains via Extrapolation

In this paper, we establish certain Lp bounds for several classes of rough Marcinkiewicz integrals over surfaces of revolution on product spaces. By using these bounds and using an extrapolation argument, we obtain the Lp boundedness of these Marcinkiewicz integrals under very weak conditions on the kernel functions. Our results represent natural extensions and improvements of several known results on Marcinkiewicz integrals.

Multiple parametric Marcinkiewicz integrals with mixed homogeneity along surfaces

In this paper, the multiple parametric Marcinkiewicz integral operators with mixed homogeneity along surfaces are studied. The Lp-mapping properties for such operators are obtained under the rather weakened size conditions on the integral kernels both on the unit sphere and in the radial direction. The main results essentially improve and extend certain previous results.

Estimates for Generalized Parabolic Marcinkiewicz Integrals with Rough Kernels on Product Domains

We prove Lp estimates of a class of generalized Marcinkiewicz integral operators with mixed homogeneity on product domains. By using these estimates along with an extrapolation argument, we obtain the boundedness of our operators under very weak conditions on the kernel functions. Our results in this paper improve and extend several known results on both generalized Marcinkiewicz integrals and parabolic Marcinkiewicz integrals on product domains.

On weighted boundedness and compactness of commutators of Marcinkiewicz integral associated with Schrödinger operators

On weighted boundedness and compactness of commutators of Marcinkiewicz integral associated with Schrödinger operators

The multilinear Littlewood-Paley square operators and their commutators on weighted Morrey spaces

In this paper, we prove the boundedness of the multilinear Littlewood-Paley square operators and their commutators on weighted Morrey spaces, then we give the boundedness and weak-type $$L\log L$$ estimates for the commutators of multilinear Littlewood-Paley g-functions and multilinear Marcinkiewicz integrals on weighted Morrey spaces in the form of corollaries.

Calderón–Zygmund operators, Bochner–Riesz means, and parametric Marcinkiewicz integrals on Hardy–Morrey spaces with variable exponents

We obtain the boundedness of Calderón–Zygmund operators on the entire Hardy–Morrey spaces with variable exponents. We obtain this result by refining the extrapolation theory. This refined extrapolation theory also gives the boundedness of Fourier multipliers, Bochner–Riesz means, and parametric Marcinkiewicz integrals on Hardy–Morrey spaces with variable exponents.

Mixed radial‐angular integrability for rough maximal singular integrals and Marcinkiewicz integrals with mixed homogeneity

AbstractThis paper is devoted to studying maximal singular integrals and Marcinkiewicz integrals with rough kernels in a mixed homogeneity setting. Assuming that the kernels satisfy certain rather weak size conditions, the boundedness of such operators on the mixed radial‐angular spaces are established, respectively. Meanwhile, the corresponding vector‐valued versions are also given.

A Class of Rough Generalized Marcinkiewicz Integrals on Product Domains

In this article, suitable estimates for a class of rough generalized Marcinkiewicz integrals on product spaces are established. By these estimates, together with employing Yano’s extrapolation technique, we obtain the boundedness of the aforementioned integral operators under weak conditions on singular kernels. A number of known previous results on Marcinkiewicz as well as generalized Marcinkiewicz operators over a symmetric space are essentially improved or extended.

Boundedness of Commutators for Multilinear Marcinkiewicz Integrals with Generalized Campanato Functions on Generalized Morrey Spaces

Boundedness of Commutators for Multilinear Marcinkiewicz Integrals with Generalized Campanato Functions on Generalized Morrey Spaces

On Certain Estimates for Parabolic Marcinkiewicz Integrals Related to Surfaces of Revolution on Product Spaces and Extrapolation

In this paper, appropriate Lp bounds for particular classes of parabolic Marcinkiewicz integrals along surfaces of revolution on product spaces are obtained. These bounds allow us to use Yano’s extrapolation argument to obtain the Lp boundedness of the aforesaid integral operators under weak conditions on the kernels. These conditions on the kernels are the best possible among their respective classes. In this work, several previously known results on Marcinkiewicz integrals are fundamentally improved and extended.

A Note on a Class of Generalized Parabolic Marcinkiewicz Integrals along Surfaces of Revolution

In this article, certain sharp Lp estimates for a specific class of generalized Marcinkiewicz operators with mixed homogeneity associated to surfaces of revolution are established. By virtue of Yano’s extrapolation argument, beside these estimates, the Lp boundedness of the aforementioned operators under weaker assumptions on the kernels is confirmed. The obtained results in this article are fundamental extensions and improvements of numerous previously known results on parabolic generalized Marcinkiewicz integrals.

Parametric Marcinkiewicz Integral and Its Higher-Order Commutators on Variable Exponents Morrey-Herz Spaces

In this article, we prove the boundedness of the parametric Marcinkiewicz integral and its higher-order commutators generated by BMO spaces on the variable Morrey-Herz space. All the results are new even when α · is a constant.

On the Functions of Generalized Marcinkiewicz Integral Operators along Subvarieties via Extrapolation

Under weak conditions assumed on the kernels, appropriate Lp estimates for certain class of generalized Marcinkiewicz integrals over subvarieties are established. By the virtue of the obtained estimates along with Yano’s extrapolation argument, the Lp boundedness of the aforementioned integral operators under weaker conditions on the kernels is proved. The results in this paper represent essential extensions and improvements of many known results on the generalized Marcinkiewicz over symmetric spaces.

Sublinear operators on Herz–Hardy spaces with variable exponents

AbstractIn this paper, we establish the mapping properties of sublinear operators on Herz–Hardy spaces with variable exponents. We obtain these mapping properties by extending the extrapolation theory to Herz–Hardy spaces with variable exponents. In particular, we study the mapping properties of the multiplier operators, the Littlewood–Paley functions and the parametric Marcinkiewicz integrals on the Herz–Hardy spaces with variable exponents.

Fractional Type Marcinkiewicz Integral Operator Associated with Θ-Type Generalized Fractional Kernel and Its Commutator on Non-homogeneous Spaces

Abstract Let (𝒳, d, μ) be a non-homogeneous metric measure space satisfying the upper doubling and geometrically doubling conditions in the sense of Hytönen. Under assumption that θ and dominating function λ satisfy certain conditions, the authors prove that fractional type Marcinkiewicz integral operator M ˜ \tilde M α,lρ,q associated with θ-type generalized fractional kernel is bounded from the generalized Morrey space ℒ r,ϕp/r,κ (μ) into space ℒ p,ϕ,κ (μ), and bounded from the Lebesgue space Lr (μ) into space Lp (μ). Furthermore, the boundedness of commutator M ˜ \tilde M α,l,ρq,b generated by b ∈ R B M O ˜ ( μ ) b \in \widetilde {RBMO}\left( \mu \right) and the M ˜ \tilde M α,l,ρq,b on space ℒp (μ) and on space ℒ p , ϕ , κ (μ) is also obtained.

Weighted Norm Inequalities for Marcinkiewicz Integrals with Non-Smooth Kernels on Spaces of Homogeneous Type

Weighted Norm Inequalities for Marcinkiewicz Integrals with Non-Smooth Kernels on Spaces of Homogeneous Type