Parametric Marcinkiewicz Integrals Research Articles

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.

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.

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.

Operators on Orlicz-slice spaces and Orlicz-slice Hardy spaces

We establish the mapping properties of the singular integral operators, the Fourier integral operators and the geometric maximal operators on the Orlicz-slice spaces by using extrapolation. By extending the extrapolation theory to the Hardy Orlicz-slice spaces, we also obtain the mapping properties of the maximal Bochner-Riesz means, the parametric Marcinkiewicz integrals and the multiplier operators on the Hardy Orlicz-slice spaces.

On the mixed radial-angular integrability of Marcinkiewicz integrals with rough kernels

This paper studies the mixed radial-angular integrability of parametric Marcinkiewicz integrals along “polynomial curves”. Under the assumption that the kernels satisfy certain rather weak size conditions on the unit sphere with radial roughness, the authors prove that such operators are bounded on the mixed radial-angular spaces. Meanwhile, corresponding vector-valued versions are also obtained.

Sublinear operators on mixed-norm Hardy spaces with variable exponents

In this paper, we define and study the mixed-norm Hardy spaces with variable exponents. We establish some general principles for the mapping properties of sublinear operators on the mixed-norm Hardy spaces with variable exponents. By using these principles, we obtain the mapping properties of the Calderón–Zygmund operators, the oscillatory singular integral operators, the multiplier operators, the Littlewood–Paley functions, the intrinsic square functions, the parametric Marcinkiewicz integrals and the maximal Bochner–Riesz means on the mixed-norm Hardy spaces with variable exponents.

Inequalities for generalized parametric Marcinkiewicz integrals along polynomial compound curves

Inequalities for generalized parametric Marcinkiewicz integrals along polynomial compound curves

Boundedness of Generalized Parametric Marcinkiewicz Integrals Associated to Surfaces

In this article, the boundedness of the generalized parametric Marcinkiewicz integral operators M Ω , ϕ , h , ρ ( r ) is considered. Under the condition that Ω is a function in L q ( S n − 1 ) with q ∈ ( 1 , 2 ] , appropriate estimates of the aforementioned operators from Triebel–Lizorkin spaces to L p spaces are obtained. By these estimates and an extrapolation argument, we establish the boundedness of such operators when the kernel function Ω belongs to the block space B q 0 , ν − 1 ( S n − 1 ) or in the space L ( log L ) ν ( S n − 1 ) . Our results represent improvements and extensions of some known results in generalized parametric Marcinkiewicz integrals.

Boundedness and continuity of Marcinkiewicz integrals associated to homogeneous mappings on Triebel-Lizorkin spaces

We establish the boundedness and continuity of parametric Marcinkiewicz integrals associated to homogeneous compound mappings on Triebel-Lizorkin spaces and Besov spaces. Here the integral kernels are provided with some rather weak size conditions on the unit sphere and in the radial direction. Some known results are naturally improved and extended to the rough case.

A Note on Marcinkiewicz Integrals along Submanifolds of Finite Type

We study the parametric Marcinkiewicz integrals along submanifolds of finite type with rough kernels. The kernels of our operators are allowed to be very rough both on the unit sphere and in the radial direction. Under the rather weakened size conditions on the integral kernels, the Lp bounds will be established for such operators. As applications, the corresponding results for parametric Marcinkiewicz integrals related to area integrals and Littlewood-Paley gλ⁎ functions are also given.

Rough maximal functions supported by subvarieties on Triebel–Lizorkin spaces

In this paper, we establish the boundedness of a class of maximal functions related to rough singular integrals supported by compound subvarieties on Triebel–Lizorkin spaces and Besov spaces. As applications, several corresponding estimates of maximal functions related to parametric Marcinkiewicz integrals are also presented.

Integral operators of Marcinkiewicz type on Triebel–Lizorkin spaces

A systematic treatment is given of several classes of parametric Marcinkiewicz integrals. The boundedness on Triebel–Lizorkin spaces will be presented for these operators with rough kernels in , which relates to the Grafakos–Stefanov class. Moreover, the boundedness on Besov spaces for above operators is also considered.

Estimates of some integral operators with bounded variable kernels on the hardy and weak hardy spaces over ℝ n

In this paper, we first introduce \({L^{{\sigma _1}}}{\left( {\log L} \right)^{{\sigma _2}}}\) conditions satisfied by the variable kernels Ω(x, z) for 0 ≤ σ1 ≤ 1 and σ2 ≥ 0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators TΩ, fractional integrals TΩ,α and parametric Marcinkiewicz integrals μΩρ with variable kernels on the Hardy spaces Hp(Rn) and weak Hardy spaces WHp(Rn). Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy–Lorentz spaces Hp,q(Rn) for all p < q < ∞.

On Marcinkiewicz integrals associated to compound mappings with rough kernels

The Lp bounds for the parametric Marcinkiewicz integrals associated to compound mappings, which contain many classical surfaces as model examples, are given, where the kernels of our operators are rather rough on the unit sphere as well as in the radial direction. These results substantially improve and extend some known results.

BOUNDEDNESS OF SEVERAL INTEGRAL OPERATORS WITH BOUNDED VARIABLE KERNELS ON HARDY AND WEAK HARDY SPACES

In this paper, by using the atomic decomposition theory of Hardy space H1(ℝn) and weak Hardy space WH1(ℝn), we give the boundedness properties of some operators with variable kernels such as singular integral operators, fractional integrals and parametric Marcinkiewicz integrals on these spaces, under certain logarithmic type Lipschitz conditions assumed on the variable kernel Ω(x, z).

On the [formula omitted] boundedness for the multiple Littlewood–Paley functions with rough kernels

In this paper, we give a characterization of the L2 bounds for the Littlewood–Paley functions on product domains. As application, a new size condition, which implies the L2 boundedness of the multiple parametric Marcinkiewicz integrals with rough kernels and is weaker than the previous ones, is given.

Estimates for parametric Marcinkiewicz integrals in BMO and Campanato spaces

In this paper, the authors consider the behaviors of a class of parametric Marcinkiewicz integrals μΩρ, μΩ,λ*,ρ and μΩ,Sρ on BMO(ℝn) and Campanato spaces with complex parameter ρ and the kernel Ω in Llog+L(Sn−1). Here μΩ,λ*,ρ and μΩ,Sρ are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley gλ*-function and the Lusin area function S, respectively. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO(ℝn) or to a certain Campanato space, then [μΩ,λ*,ρ(f)]2, [μΩ,Sρ(f)]2 and [μΩρ(f)]2 are either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness are also established.

On certain estimates for Marcinkiewicz integrals and extrapolation

We obtainL p estimates for parametric Marcinkiewicz integrals associated to polynomial mappings and with rough kernels on the unit sphere as well as on the radial direction. These estimates will allow us to use an extrapolation argument to obtain some new and improved results on Marcinkiewicz integrals. Also, such estimates provide us with a unifying approach in dealing with Marcinkiewicz integrals when the kernel function ω belongs to the class of block spaces (0,α) B q (S n-1) as well as when ω belongs to the classL(logL)α (S n-1). Our conditions on the kernels are known to be the best possible in their respective classes.

On parametric marcinkiewicz integrals related to block spaces

In this paper, the L 2 -boundedness of a class of parametric Marcinkiewicz integral μ with kernel function Ω in B (S n−1) for for some q>1, and the radial function h(x) ∈ l ∞ (L s )(R+) for 1<s≤∞ are given. The L p (R n ) (2≤p<∞) boundedness of $$\mu _\Omega ^* ,{}^\backprime h^\rho .\lambda $$ and $$\mu _\Omega ^\rho ,h,S$$ with Ω in B (S n−1) and h (|x|) ∈ l ∞ (L s ) (R+) in application are obtained. Here $$\mu _\Omega ^* ,{}^\backprime h^\rho .\lambda $$ and $$\mu _\Omega ^\rho ,h,S$$ are parametric Marcinkiewicz integrals corresponding to the Littlewood-Paley g function and the Lusin area function S, respectively.