A note concerning Marcinkiewicz integral with rough kernel

In this paper, applying some properties of rough kernel and using some technical lemmas, we prove various norm inequalities concerning Marcinkiewicz integral with rough kernel. Indeed, we extend some known results on singular integrals to Marcinkiewicz integrals.

Marcinkiewicz integral operators along twisted surfaces

<p style='text-indent:20px;'>Marcinkiewicz integral operators on product domains defined by translates determined by twisted surfaces are introduced. Maximal functions along twisted surfaces are also introduced. Conditions on the underlined surfaces implying that the corresponding Marcinkiewicz integral operators map <inline-formula><tex-math id="M1">\begin{document}$ L^{p}\rightarrow L^{p} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ 1&lt;p&lt;\infty $\end{document}</tex-math></inline-formula> are obtained. A general method involving lacunary families of multi-indices is developed.</p>

Parameter θ -Type Marcinkiewicz Integral on Nonhomogeneous Weighted Generalized Morrey Spaces

Let X , d , μ be a nonhomogeneous metric measure space satisfying the upper doubling and geometrically doubling conditions in the sense of Hytönen. In this setting, the author proves that parameter θ -type Marcinkiewicz integral M θ ρ is bounded on the weighted generalized Morrey space L p , ϕ , τ ω for p ∈ 1 , ∞ . Furthermore, the boudedness of M θ ρ on weak weighted generalized Morrey space W L p , ϕ , τ ω is also obtained.

L2L^{2} boundedness for commutators of fractional differential type Marcinkiewicz integral with rough variable kernel and BMO Sobolev spaces

In this paper, for {0<\gamma<1} and {b\in I_{\gamma}(\mathrm{BMO})}, the authors give the {L^{2}({\mathbb{R}}^{n})} boundedness of {\mu_{\gamma;b}}, the commutator of a fractional differential type Marcinkiewicz integral with rough variable kernel, which is an extension of some known results.

A boundedness result for Marcinkiewicz integral operator

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} -bounded when its kernel satisfies only the sole integrability condition.

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

In 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 to Lp spaces under weaker conditions on Ω and h. Our results represent significant improvements and natural extensions of what was known previously.