A rough Marcinkiewicz integral along smooth curves

Abstract

We consider the boundedness of a kind of nonlinear integral operators on L p spaces. Including the parametric Marcinkiewicz integrals with rough kernels along compound curves {Φ(ϕ(|y|))y ; y ∈ R n } with Φ satisfying certain growth conditions and ϕ being differentiable function with monotonicity and some properties on the positive real line, we investigate the L p bounds of these operators under the integral kernels given by the sphere functions Ω in H 1 (S n−1 ) or Ω in L(log + L) 1/2 (S n−1 ) and the radial function h ∈ ∆ γ (R + ). As applications, the corresponding results for parametric Marcinkiewicz integral operators related to area integrals and Littlewood-Paley g * λ -functions are presented.

Sequential Quadratic Programming Method for Nonlinear Least Squares Estimation and Its Application

“…Unwrapping experiments using real data from the Jining area in China show that our proposed algorithm achieves more precise results than the least squares unwrapping algorithms. Quantitative indexes include differences in RMSE between rewrapped results and the original wrapped phase, computation time, and̃values [26][27][28]. The sequential quadratic programming method achieves better results with respect to three indices.…”

A Note on Marcinkiewicz Integrals Associated to Surfaces of Revolution

“…For example, see [8] for the case Ω ∈ H 1 (S n−1 ) (the Hardy space on the unit sphere; see [6,26]), [1] for the case Ω ∈ L(log + L) 1/2 (S n−1 ), [3] for the case Ω ∈ B (0,−1/2) q (S n−1 ) (the Block space generated by q-block), [5] for the case Ω ∈ F β (S n−1 ) (the Grafakos-Stefanov function class; see [16]). For relevant results on parametric Marcinkiewicz integral operator M h,Ω,ρ and other integral operators with rough kernels, we refer the readers to [10,17,18,21,24], among others. Recently, the investigation of the boundedness of the Marcinkiewicz integral operator on the Triebel-Lizorkin spaces has also attracted the attention of many authors.…”