L^p bounds for the function of Marcinkiewicz
L^p bounds for the function of Marcinkiewicz
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DOI：
10.4310/mrl.2002.v9.n5.a11
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发表时间：
2002-09
影响因子：
1
通讯作者：
Ahmad Al-Salman;H. Al-Qassem
中科院分区：
文献类型：
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作者：
Ahmad Al-Salman;H. Al-Qassem
上述算子是 E.M. Stein 在[7]中引入的，作为 Marcinkiewicz 函数概念从一维到更高维度的扩展。通过使用一维 Marcinkiewicz 函数的 L 有界性，Stein 证明只要 Ω 为奇数，μΩ 在 L(R) 上有界，且 1 < p < ∞。对于一般核函数Ω，μΩ的L有界性已经在Ω上的各种条件下建立。例如，Stein 证明，如果 Ω ∈ Lip(Sn−1)，则 μΩ 有界于 L(R)，且 1 < p ≤ 2。 Benedek、Calderon 和 Panzone 在[2]中证明，在 Ω ε C1(Sn−1) 的条件下，μΩ 的 L 有界性成立，1 < p < ∞。 1972 年，T. Walsh 证明，即使 Ω 相当粗糙，μΩ 的 L 有界性仍然成立。定理 1（沃尔什 [11]）。假设 p ∈ (1,∞)、r = min{p, p′} 且 Ω ∈ L(log L)(log log L)2(1−2/r )(Sn−1)。那么μΩ 就以 L(R) 为界。当 p = 2 时，定理 1 中的条件就是 Ω ∈ L(log L)1/2(Sn−1)，Walsh 证明这是最优的，因为 L(log L) 中的指数 1/2 不能被任何更小的数字替代。另一方面，当 p = 2 时，Walsh 并不认为他的条件在任何意义上都是最优的。事实上，通过与 Calderon 和 Zygmund 在奇异积分上的结果进行比较，我们很自然地会产生一个问题：即使当 p = 2 时，条件 Ω ∈ L(log L)1/2(Sn−1) 是否也足以满足 μΩ 的 L 有界性。这个问题由 Y. Ding 在[4]中正式提出，通过我们的下一个解决方案解决。定理。
The above operator was introduced by E.M. Stein in [7] as an extension of the notion of Marcinkiewicz function from one dimension to higher dimensions. By using the L boundedness of the 1-dimensional Marcinkiewicz function, Stein showed that μΩ is bounded on L(R) for 1 < p < ∞ whenever Ω is odd. For a general kernel function Ω, the L boundedness of μΩ has been established under various conditions on Ω. For example, Stein proved that μΩ is bounded on L(R) for 1 < p ≤ 2 if Ω ∈ Lip(Sn−1). Benedek, Calderon and Panzone proved in [2] that the L boundedness of μΩ holds for 1 < p < ∞ under the condition that Ω ∈ C1(Sn−1). In 1972 T. Walsh showed that the L boundedness of μΩ can still hold even if Ω is quite rough. Theorem 1 (Walsh [11]). Suppose that p ∈ (1,∞), r = min{p, p′}, and Ω ∈ L(log L)(log log L)2(1−2/r )(Sn−1). Then μΩ is bounded on L(R). When p = 2, the condition in Theorem 1 is simply Ω ∈ L(log L)1/2(Sn−1), which was shown by Walsh to be optimal in the sense that the exponent 1/2 in L(log L) cannot be replaced by any smaller numbers. On the other hand, Walsh did not consider his condition to be in any sense optimal when p = 2. Indeed, by comparing with the result of Calderon and Zygmund on singular integrals, one is naturally led to the question whether the condition Ω ∈ L(log L)1/2(Sn−1) is also sufficient for the L boundedness of μΩ even when p = 2. This problem, which was formally proposed by Y. Ding in [4], is resolved by our next theorem.