On the function of Marcinkiewicz

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).…”

Marcinkiewicz integrals on product spaces

“…Historically, this is the case that had received the most amount of attention. For a sampling of past studies, see [2], [11], [13], [14], [18], [19]. Related results can also be found in [8], [15], and [17].…”

“…the underlying space is not a product space), the L p boundedness of µ Ω under the condition Ω ∈ L(log L) 1/2 was obtained first for p = 2 in [18], and then for all p ∈ (1, ∞) in [1]. Historically, this is the case that had received the most amount of attention.…”

On certain estimates for Marcinkiewicz integrals and extrapolation

“…The conclusion of Theorem A for p = 2 was first obtained by T. Walsh in [20]. Also, T. Walsh proved that the exponent 1/2 in L(log L) 1/2 (S n−1 ) cannot be replaced by any smaller number.…”

“…Subsequently, the study of M Ω and some of its extensions has attracted the attention of many authors. Readers may consult [20,7,1,3,2,6], among a large number of references for their development and applications. Before stating some known results relevant to our current study, we need to recall and introduce some definitions.…”