We consider $L^p$-$L^q$ estimates for the spherical harmonic projection operators and obtain sharp bounds on a certain range of $p$, $q$. As an application, we provide a proof of off-diagonal Carleman estimates for the Laplacian, which extends the earlier results due to Jerison and Kenig \cite{JK}, and Stein \cite{St-append}.
We study $L^p$-$L^q$ bounds on the spectral projection operator $Π_λ$ associated to the Hermite operator $H=|x|^2-Δ$ in $\mathbb R^d$. We are mainly concerned with a localized operator $χ_EΠ_λχ_E$ for a subset $E\subset\mathbb R^d$ and undertake the task of characterizing the sharp $L^p$--$L^q$ bounds. We obtain sharp bounds in extended ranges of $p,q$. First, we provide a complete characterization of the sharp $L^p$--$L^q$ bounds when $E$ is away from $\sqrtλ\mathbb S^{d-1}$. Secondly, we obtain the sharp bounds as the set $E$ gets close to $\sqrtλ\mathbb S^{d-1}$. Thirdly, we extend the rang…
Sharp power-weight estimates for the fractional spherical maximal operator
National Taiwan Normal University（台湾师范大学）；National Center for Theoretical Sciences, NTU（台湾理论科学中心）；National Taiwan University（台湾大学）；University of Pittsburgh（匹兹堡大学）
We prove $L^p\to L^q$ estimates for the local maximal operator associated with dilates of the Kóranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp $L^p\to L^q$ estimates for spherical means over the Korányi sphere, which can be used to improve the sparse domination bounds for the associated lacunary maximal operator.
我们考虑由$ A_{r} f (x) = \int_{\mathbb S ^{n-1}} f (x-r y) \; d σ(y)$给出的半径为$ 0< r< \infty $的球面上的函数$ f$在$ \mathbb R ^{n}$上的平均值，其中$ σ$是$ \mathbb S ^{n-1}$上的归一化旋转不变测度。我们证明了两个极大函数，第一个缺球极大函数，第二个完整的极大函数的稀疏边界的尖锐范围。$$ M_{lac} f = \sup_{j\in \mathbb Z } A_{2^j} f , \qquad M_{full} f = \sup_{ r>0 } A_{r} f . $$稀疏边界是这些极大函数的已知$L^p$边界的非常精确的变体。它们来自于已知的$ L ^{p}$，这些极大函数的本地化版本的改进估计，在我们的稀疏界的指数是尖锐的。我们得到新的加权不等式的权重在某些Muckenhoupt和反向Hölder类的交集。
On the finite field spherical restriction conjecture in four dimensions: the sharp endpoint and applications
Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与 interdisciplinary 科学研究院）；Institute of Mathematical Sciences, ShanghaiTech University（上海科技大学数学科学研究所）
In this article we study the Schrödinger equation associated with Harmonic oscillator in the form of Strichartz type inequality. We give simple proofs for Strichartz type inequalities using purely the $L^2 \to L^p$ operator norm estimates of the spectral projections associated harmonic oscillator proved in \cite{KT}. Our Strichartz type estimates are sharp in sense of regularity of initial data.
