Abstract

In this paper we study the properties of the Lebesgue constant of the conjugate transforms. For conjugate Fejér means we will find necessary and sufficient condition on \(t\) for which the estimation \(E\left|\widetilde{\sigma}_{n}^{\left(t\right)}f\right|\leq cE\left|f\right|\) holds. We also prove that for dyadic irrational \(t\), \(L\log L\) is the maximal Orlicz space for which the estimation \(E\left|\widetilde{\sigma}_{n}^{\left(t\right)}f\right|\leq c_{1}+c_{2}E\left(\left|f\right|\log^{+}\left|f\right|\right)\) is valid.

中文翻译：

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对Dyadic组进行共轭变换

摘要

在本文中，我们研究了共轭变换的Lebesgue常数的性质。对于共轭Fejér而言，我们将在\（t \）上找到必要的充分条件，据此估计\（E \ left | \ widetilde {\ sigma} _ {n} ^ {\ left（t \ right）} f \ right | \ leq cE \ left | f \ right | \）成立。我们还证明，对于二元无理\（t \），\（L \ log L \）是最大Orlicz空间，对其估计\（E \ left | \ widetilde {\ sigma} _ {n} ^ {\ left （t \ right）} f \ right | \ leq c_ {1} + c_ {2} E \ left（\ left | f \ right | \ log ^ {+} \ left | f \ right | \ right）\）已验证。