Marcinkiewicz Integral Operators Research Articles

Weighted $ L^{p} $ norms of Marcinkiewicz functions on product domains along surfaces

<abstract><p>We prove a weighted $ L^{p} $ boundedness of Marcinkiewicz integral operators along surfaces on product domains. For various classes of surfaces, we prove the boundedness of the corresponding operators on the weighted Lebsgue space $ L^{p}(\mathbb{R}^{n}\times\mathbb{R}^{m}, \, \omega _{1}(x)dx, \, \omega_{2}(y)dy) $, provided that the weights $ \omega_{1} $ and $ \omega_{2} $ are certain radial weights and that the kernels are rough in the optimal space $ L(\log L)(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1}) $. In particular, we prove the boundedness of Marcinkiewicz integral operators along surfaces determined by mappings that are more general than polynomials and convex functions. Also, in this paper we prove the weighted $ L^{p} $ boundedness of the related square and maximal functions. Our weighted $ L^{p} $ inequalities extend as well as generalize previously known $ L^{p} $ boundedness results.</p></abstract>

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.

Lp-Mapping Properties of a Class of Spherical Integral Operators

In this paper, we study a class of spherical integral operators IΩf. We prove an inequality that relates this class of operators with some well-known Marcinkiewicz integral operators by using the classical Hardy inequality. We also attain the boundedness of the operator IΩf for some 1<p<2 whenever Ω belongs to a certain class of Lebesgue spaces. In addition, we introduce a new proof of the optimality condition on Ω in order to obtain the L2-boundedness of IΩ. Generally, the purpose of this work is to set up new proofs and extend several known results connected with a class of spherical integral operators.

Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds

Abstract We establish certain delicate L p {L}^{p} bounds for a class of generalized Marcinkiewicz integral operators along submanifolds with rough kernels. These bounds allow us to use Yano’s extrapolation method to prove the L p {L}^{p} boundedness of the aforementioned integral operators under very weak assumptions on the kernels. Our results in this article improve and generalize many previously known 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 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.

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.

Atomic and Wavelet Characterization of Musielak–Orlicz Hardy Spaces for Generalized Orlicz Functions

The goal of this paper is to obtain atomic and wavelet characterizations of Musielak–Orlicz Hardy spaces. Recently, in 2018, Fu and Yang obtained wavelet characterizations of Musielak–Orlicz Hardy spaces for growth functions of uniformly upper type $$p_-$$ and of uniformly lower type $$p_+$$ with $$0<p_- \le p_+\le 1$$ . What is different from the existing works is that we merely assume $$0<p_- \le p_+<\infty $$ . One of the important tools that make it possible is to refine the convexity of Orlicz functions by obtaining canonical equivalent functions. As applications of the atomic characterization, we investigate the boundedness property of singular integral operators. Especially, we obtain the boundedness property of Marcinkiewicz integral operators acting on some Musielak–Orlicz Hardy spaces which are quasi-Banach spaces.

Commutators of log-Dini-type parametric Marcinkiewicz operators on non-homogeneous metric measure spaces

Let (mathcal{X}, d, mu ) be a non-homogeneous metric measure space, which satisfies the geometrically doubling condition and the upper doubling condition. In this paper, the authors prove the boundedness in L^{p} (mu ) of mth-order commutators mathcal{M}^{rho }_{b,m} generated by the Log-Dini-type parametric Marcinkiewicz integral operators with RBMO functions on (mathcal{X}, d, mu ). In addition, the boundedness of the mth-order commutators mathcal{M}^{rho }_{b,m} on Morrey spaces M^{q}_{p}(mu ), 1< p leq q< infty , is also obtained for the parameter 0<rho <infty .

Weighted estimates for Marcinkiewicz integrals with applications to angular integrability

We establish the weighted estimates for the parametric Marcinkiewicz integral operators with rough kernels along “polynomial curves” on $$\mathbb {R}^n$$ . As applications, we obtain that the above operators are bounded on the mixed radial-angular spaces, on the vector-valued mixed radial-angular spaces and on the vector-valued function spaces. Particularly, the above bounds are independent of the coefficients of the polynomials in the definition of the operators.

A boundedness result for Marcinkiewicz integral operator

Abstract We extend a boundedness result for Marcinkiewicz integral operator. We find a new space of radial functions for which this class of singular integral operators remains L p {L}^{p} -bounded when its kernel satisfies only the sole integrability condition.

Generalized parabolic Marcinkiewicz integrals associated with polynomial compound curves with rough kernels

AbstractIn this article, we study the generalized parabolic parametric Marcinkiewicz integral operators{ {\mathcal M} }_{{\Omega },h,{\Phi },\lambda }^{(r)}related to polynomial compound curves. Under some weak conditions on the kernels, we establish appropriate estimates of these operators. By the virtue of the obtained estimates along with an extrapolation argument, we give the boundedness of the aforementioned operators from Triebel-Lizorkin spaces toLpspaces under weaker conditions onΩandh. Our results represent significant improvements and natural extensions of what was known previously.

Sharp off-diagonal weighted weak type estimates for sparse operators

We prove sharp weak type weighted estimates for a class of sparse operators that includes majorants of standard $\alpha$-fractional singular integrals, fractional integral operators, Marcinkiewicz integral operators, and square functions. These bounds are knows to be sharp in many cases, and our main new result is the optimal bound $$[w]_{A_{p,q}}^{\frac{1}{q}}[w^{q}]_{A_{\infty}}^{\frac{1}{2}-\frac{1}{p}}\lesssim[w]_{A_{p,q}}^{\frac{1}{2}-\frac{\alpha}{d}}$$ for proper conditions which satisfy that three index $p$, $q$ and $\alpha$ ensure weak type norm of fractional square functions on $L^{q}(w^{q})$ with $p>2$.

Generalized Parabolic Marcinkiewicz Integral Operators Related to Surfaces of Revolution

In this work, the generalized parametric Marcinkiewicz integral operators with mixed homogeneity related to surfaces of revolution are studied. Under some weak conditions on the kernels, the boundedness of such operators from Triebel–Lizorkin spaces to L p spaces are established. Our results, with the help of an extrapolation argument, improve and extend some previous known results.

Generalized Weighted Morrey Estimates for Marcinkiewicz Integrals with Rough Kernel Associated with Schrödinger Operator and Their Commutators

Let L = −Δ + V(x) be a Schrodinger operator, where Δ is the Laplacian on ℝn, while nonnegative potential V(x) belonging to the reverse Holder class. The aim of this paper is to give generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schrodinger operator and their commutators. Moreover, the boundedness of the commutator operators formed by BMO functions and Marcinkiewicz integrals with rough kernel associated with Schroodinger operators is discussed on the generalized weighted Morrey spaces. As its special cases, the corresponding results of Marcinkiewicz integrals with rough kernel associated with Schroodinger operator and their commutators have been deduced, respectively. Also, Marcinkiewicz integral operators, rough Hardy-Littlewood (H-L for short) maximal operators, Bochner-Riesz means and parametric Marcinkiewicz integral operators which satisfy the conditions of our main results can be considered as some examples.

Lp Estimates of certain rough parametric Marcinkiewicz functions

We prove Lp estimates of a class of parametric Marcinkiewicz integral operators when their kernels satisfy only the \(L^1(\mathbb{S}^{n-1})\) integrability condition. The obtained Lp estimates resolve a problem left open in previous work. Our argument is based on duality technique and direct estimation of operators. As a consequence of our result, we deduce the Lp boundedness of a class of fractional Marcinkiewicz integral operators.

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.

$L^{p}$ bounds for a class of Marcinkiewicz integral operators

In this paper, we introduce a class of mappings that is more general than the class of polynomials as well as the class of convex functions. We prove $L^{p}$ estimates of Marcinkiewicz Mintegral operators along surfaces generated by mappings belong to this class. Moreover, we establish the $L^{p}$ boundedness of the corresponding maximal functions. Our results extend as well as improve previously known results on Marcinkiewicz integral operators and maximal functions.

A note on certain integrals along polynomial compound curves

In this paper we consider a singular integral operator and a parametric Marcinkiewicz integral operator with rough kernel. These operators have singularity along sets of the form curves {x=P(varphi (|y|))y'}, where P is a real polynomial satisfying P(0)=0 and φ satisfies certain smooth conditions. Under the conditions that varOmega in H^{1} (mathbf{S}^{n-1}) and hin Delta _{gamma }(mathbf{R}_{+}) for some gamma >1, we prove that the above operators are bounded on the Lebesgue space L^{2}( mathbf{R}^{n}). Moreover, the L^{2}-bounds of the maximal functions related to the above integrals are also established. Particularly, the bounds are independent of the coefficients of the polynomial P. In addition, we also present certain Hardy type inequalities related to these operators.

Higher-Order Commutators of Parametric Marcinkiewicz Integrals on Herz Spaces with Variable Exponent

LetΩ∈Ls(Sn-1)fors⩾1be a homogeneous function of degree zero andbbe BMO functions. In this paper, we obtain some boundedness of the parametric Marcinkiewicz integral operatorμΩρand its higher-order commutator[bm,μΩρ]on Herz spaces with variable exponent.