加权平方函数不等式

Pub Date : 2018-01-01 DOI:10.5565/PUBLMAT6211804

A. Osȩkowski

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引用次数: 7

摘要

对于可积函数f在[0,1)d上，设S(f)和M f分别表示相应的并进平方函数和f的并进极大函数。本文包含下列陈述的证明。(i)如果w是[0,1)d上的二进A1权值，则||S(f)||L1(w)≤√5[w] 1/2 A1 ||M f||L1(w)。指数1/2是最好的。(ii)对于任何p > 1，不存在仅依赖于p的常数cp， αp，使得对于所有并矢Ap权值w在[0,1)d上，||S(f)| L1(w)≤cp[w] αp Ap |M f| L1(w)。

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Weighted square function inequalities

For an integrable function f on [0, 1)d, let S(f) and M f denote the corresponding dyadic square function and the dyadic maximal function of f, respectively. The paper contains the proofs of the following statements. (i) If w is a dyadic A1 weight on [0, 1)d, then ||S(f)||L1(w) ≤√ 5[w] 1/2 A1 ||M f||L1(w). The exponent 1/2 is shown to be the best possible. (ii) For any p > 1, there are no constants cp, αp epending only on p such that for all dyadic Ap weights w on [0, 1)d, ||S(f)||L1(w) ≤ cp[w] αp Ap ||M f||L1(w).

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