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

Abstract

In this article, we study the generalized parabolic parametric Marcinkiewicz integral operators ℳΩ,h,Φ,λ(r){ {\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 L p spaces under weaker conditions on Ω and h. Our results represent significant improvements and natural extensions of what was known previously.

Keywords

• parametric marcinkiewicz integrals
• rough kernels
• triebel-lizorkin spaces
• extrapolation
• primary 42b20
• secondary 40b25, 47g10