{${\large\bf L}^{\large\bf p}$ bounds for the function of Marcinkiewicz

A Marcinkiewicz integral type characterization of the Sobolev space

“…If n = 1 and Ω(y ) = sign y, we obtain integral (1.1). Stein proved in all dimensions that if Ω is odd, then µ Ω is bounded in L p , 1 < p < ∞, and if Ω is Hölder continuous with exponent 0 < α ≤ 1, then µ Ω is bounded in L p , 1 < p ≤ 2, and is of weak type (1,1). In the odd case the result was obtained as a consequence of the one dimensional result due to Waterman.…”

“…An optimal condition under which (1.4) is satisfied was discovered in [1]: (1.4) holds true provided Ω satisfies (1.2) and Ω ∈ L(log L) 1/2 (S n−1 ). A generalization of Theorem 1.2 to the case of weighted L p w (R n ) spaces, where w ∈ A p is a Muckenhoupt weight, was obtained by Sato in [23], see Theorem 4.2 below.…”

“…Actually it follows from the proof that (1.16) holds true under a weaker assumption that f ∈ W 1,1 loc (R n ) is such that…”

Fractional type Marcinkiewicz integral operators associated to surfaces

“…In the case α = 0, ρ = 1 and q = 2, it is again known in [1] that the conclusion in Theorem 2(iii) holds even when Ω ∈ L log L 1/2 (S n−1 ).…”

The existence and boundedness of multilinear Marcinkiewicz integrals on Companato spaces

“…Hörmander [6] 研究了一类参数型的 Marcinkiewicz 积分. 此后 的研究表明, 去掉核函数 Ω 的光滑性, 仍然可以保证 Marcinkiewicz 积分 µ Ω 的 L p (1 < p < ∞) 有界 性 (有关 Marcinkiewicz 积分研究进展可参见文献 [7][8][9][10][11][12][13][14][15][16][17] 以及综述文献 [18]).…”