A problem on rough parametric Marcinkiewicz functions

Abstract

In this note the authors give the L 2 (R") boundedness of a class of parametric Marcinkiewicz integral H P aA with kernel function ft in Z. log + L(5""') and radial function h(\x |) 6 /°°(L«)(R + )forl < q < oo. As its corollary, the Z/(R")(2 <p < oo) boundedness of fi'nxx and/x£ h s with £2 in Z.

$L^p$ BOUNDS FOR MARCINKIEWICZ INTEGRALS ALONG SURFACES AND EXTRAPOLATION

“…However, the authors of [14] showed that M ρ Ω,1 is still bounded on L p (R n ) for 1 < p < ∞ when Re(ρ) > 0 and Ω ∈ Lip α (S n−1 ) with 0 < α ≤ 1. These results were improved in [11]. In fact, the authors proved…”

“…f L p (R m ) (11) holds for |1/p − 1/2| < 1/γ ′ . Consequently, by (9) and (12), we finish the prrof of Theorem 1.…”

Parabolic Marcinkiewicz Integrals Associated to Polynomials Compound Curves and Extrapolation

“…For example, see [5,23,24,27] for the case ̺ ≡ 1, [4,17] for the case ̺ > 0, [12,21] for the case ̺ ∈ C with Re̺ > 0. On the other hand, the investigation of the parametric Marcinkiewicz integral operators µ ̺ Ω,h,Φ with rough kernels on the unit sphere as well as in the radial direction have also received a large amount of attention of many authors (see [2,9,10,11,13] et al). In particular, Al-Qassem and Pan [2] obtained the following result.…”

On certain estimates for Marcinkiewicz integrals and extrapolation

“…Reasoning as above, (3.32), (3.37), (3.39) and Lemma 3.3 provide 8). By using this argument repeatedly we ultimately obtain that…”

“…An improvement of the result in [14] in the case p = 2 was obtained in [8] as described in the following theorem:…”