论文标题
$ l^p $估算的渐近行为，用于一类具有同质符号符号的乘数
Asymptotic behavior of $L^p$ estimates for a class of multipliers with homogeneous unimodular symbols
论文作者
论文摘要
我们研究与符号相关的傅立叶乘数运算符$匹配\ mapsto \ exp（ξ/|ξ|））$，其中$λ$是一个真实数字，$ ϕ $是一个实现的$ c^\ infty $在标准单位$ \ \ natbb {s} s}^n-1-1} $ sep n $ cy上的$ c^\ infty $函数。对于$ 1 <p <\ infty $，我们在$ l^p（\ mathbb {r}^n）$上研究这些运算符规范的渐近行为为$ |λ| \ to \ infty $。我们表明，这些规范始终为$ o（（（p^\ ast-1）|λ|^{n | 1/p-1/2 |}）$，其中$ p^\ ast $是$ p $及其偶联指数之间的较大数字。更重要的是，我们表明，这种界限在所有均值的欧几里得空间$ \ mathbb {r}^n $中都是锐利的。特别是，这对马兹亚提出的问题给出了负面答案。 属于研究类的具体操作员是形成二维Riesz组的乘数，由符号$ r \ exp（iφ）\ mapsto \ exp（iλ\cosφ）$给出。我们表明他们的$ l^p $规范与$（p^\ ast-1）|λ|^{2 | 1/p-1/2 |} $对于大$ |λ| $，在Dragičević，Petermichl和volberg的工作中肯定地解决了一个问题。
We study Fourier multiplier operators associated with symbols $ξ\mapsto \exp(iλϕ(ξ/|ξ|))$, where $λ$ is a real number and $ϕ$ is a real-valued $C^\infty$ function on the standard unit sphere $\mathbb{S}^{n-1}\subset\mathbb{R}^n$. For $1<p<\infty$ we investigate asymptotic behavior of norms of these operators on $L^p(\mathbb{R}^n)$ as $|λ|\to\infty$. We show that these norms are always $O((p^\ast-1) |λ|^{n|1/p-1/2|})$, where $p^\ast$ is the larger number between $p$ and its conjugate exponent. More substantially, we show that this bound is sharp in all even-dimensional Euclidean spaces $\mathbb{R}^n$. In particular, this gives a negative answer to a question posed by Maz'ya. Concrete operators that fall into the studied class are the multipliers forming the two-dimensional Riesz group, given by the symbols $r\exp(iφ) \mapsto \exp(iλ\cosφ)$. We show that their $L^p$ norms are comparable to $(p^\ast-1) |λ|^{2|1/p-1/2|}$ for large $|λ|$, solving affirmatively a problem suggested in the work of Dragičević, Petermichl, and Volberg.
