Asymptotic behavior of $L^p$ estimates for a class of multipliers with homogeneous unimodular symbols
Asymptotic behavior of $L^p$ estimates for a class of multipliers with homogeneous unimodular symbols
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一类具有齐次单模符号的乘法器的 $L^p$ 估计的渐近行为
DOI：
10.1090/tran/8883
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Vjekoslav Kovavc
中科院分区：
文献类型：
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作者：
Aleksandar Bulj;Vjekoslav Kovavc
我们研究与符号$\xi\mapsto \exp（i\lambda\phi（\xi/|11|））$，其中$\lambda$是一个真实的数，$\phi$是标准单位球面$\mathbb{S}^{n-1}\subset\mathbb{R}^n$上的一个实值$C^\infty$函数。对于$1<p<\infty$，我们研究了这些算子在$L^p（\mathbb{R}^n）$上的范数的渐近行为，|\lambda|\to\infty$。我们证明了这些范数总是O（（p^\ast-1））|\lambda| ^{n| 1/p-1/2|}）$，其中$p^\ast$是$p$和它的共轭指数之间的较大的数字。更重要的是，我们证明了这个界在所有偶数维欧氏空间$\mathbb{R}^n$中是尖锐的。特别是，这对Maz'ya提出的一个问题给出了否定的答案。属于所研究类的具体算子是形成二维Riesz群的乘子，由符号$r\exp（i\varphi）\mapsto \exp（i\lambda\cos\varphi）$给出。我们证明了它们的$L^p$范数与$（p^\ast-1）相当|\lambda| ^{2| 1/p-1/2|}$为大$|\lambda| $，解决肯定的一个问题，建议在工作中的Dragi\v{c}evi\'{c}，Petermichl，和Volberg。
We study Fourier multiplier operators associated with symbols $\xi\mapsto \exp(i\lambda\phi(\xi/|\xi|))$, where $\lambda$ is a real number and $\phi$ 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 $|\lambda|\to\infty$. We show that these norms are always $O((p^\ast-1) |\lambda|^{n|1/p-1/2|})$, 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 $r\exp(i\varphi) \mapsto \exp(i\lambda\cos\varphi)$. We show that their $L^p$ norms are comparable to $(p^\ast-1) |\lambda|^{2|1/p-1/2|}$ for large $|\lambda|$, solving affirmatively a problem suggested in the work of Dragi\v{c}evi\'{c}, Petermichl, and Volberg.