arXiv subjects
Publications and source records attributed to Dunyan Yan.
We study long-time Strichartz estimates for the Schr\"{o}dinger equation on waveguide manifolds, and use them to establish upper bounds on the growth of Sobolev norms for the nonlinear Schr\"{o}dinger equation on three-dimensional waveguides.
We study the quantitative stability associated with the adjoint Fourier restriction inequality, focusing on the paraboloid and two-dimensional sphere cases. We show that these Strichartz-stability inequalities admit minimizers attaining their sharp constants, provided that these sharp constants are strictly smaller than the corresponding spectral-gap constants. Furthermore, for the two-dimensional sphere case, we obtain the existence of minimizers.
This article investigates the Fourier extension operator associated with the fractional surface $(\xi,|\xi|^{\alpha})$ for $\alpha\geq 2$. We show that the relevant $L^p\to L^q$ Fourier extension inequality possesses extremals for all exponents $p\in[1,2]$. Moreover, for all $p\in(1,2]$, the corresponding $L^p$-extremal sequences are precompact up to symmetries.
In this paper, we will prove the sharp bounds of various operators in mixed radial angular spaces on Heisenberg groups. It mainly includes the boundedness of linear transformation eigenvalue operator in mixed radial angular space; Sharp Bounds of Hilbert Operator and Hardy-Littlewood-P$\acute{o}$lya Operator in mixed radial-angular space
This paper is devoted to studying the boundedness of multilinear operartors and their commutators on generalized weighted Morrey spaces, which includes multilinear fractional maximal operator and multilinear fractional integral operator. Moreover, we show that two different characterizations for the boundedness of multilinear fractional maximal operators and their commutators on generalized weighted Morrey spaces under different conditions. As some inportant applications, we give the boundedness of multilinear fractional integral operator on generalized weighted Besov-Morrey spaces and also obtain two embedding theorems as well as apriori estimates for the sub-Laplacian $\mathcal L$.
We study the linear profile decomposition for the Airy type equation, where the associated Strichartz inequality corresponds to the Fourier extension inequality on the odd curve $\xi^{\ell}$. We also investigate an inhomogeneous case, modeled by the odd curve $\xi^3+\xi^5$ case. We note that, as observed by Frank and Sabin [Math. Ann., 2018], there is a two-profile phenomenon in the profile decomposition associated with odd curves.
In this paper, we study m-linear n-demensional Hardy-Littlewood-P\'{o}lya operator and m-linear n-demensional Hilbert operator on Heisenberg group BMO space. We obtain that the above two $m$-linear n-demensional operators is bounded in the BMO space of the Heisenberg group.
The main questions raised in this paper are to find the sufficient conditions that make multi-sublinear operators $T$ and their commutators ${T_{\prod \vec b }}$, ${T_{\sum {\vec b} }}$ to be bounded on three kinds of generalized weighted Morrey spaces. We give the main theorems of this paper to solve the above related questions. As corollaries of the main theorems, we give sufficient and necessary conditions for a class of multi-sublinear operators which are bounded on three kinds of generalized weighted Morrey spaces. As some inportant applications, we apply the main results to the multilinear vector-valued Calder\'on-Zygmund operators, multilinear Littlewood-Paley square operators, multilinear pseudo-differential operators and multilinear paraproducts.
In this paper, we obtained the sharp bounds for $m$-linear $n$-dimensional Hardy-Littlewood-P\'{o}lya operator and Hilbert operator in two power weighted Morrey space on Heisenberg group.
In this paper, we study sharp bound on higher-dimensional Lebesgue product space for Hardy operator on Heisenberg group, the constants of sharp bounds are obtained. In addition, we also give the boundedness for weighted Hardy operator and weighted Cesaro operator
In this paper, we will study $n$-dimensional Hardy operator and its dual in mixed radial-angular spaces on Heisenberg groups and obtain their sharp bounds by using the rotation method. Furthermore, the sharp bounds of $n$-dimensional weighted Hardy operator and weighted Ces\`{a}ro operator are also obtained.
In this paper, we will use the conclusions and methods in \cite{1} to obtain the sharp bounds for a class of integral operators with the nonnegative kernels in weighted-type spaces on Heisenberg group. As promotions, the sharp bounds of Hardy operator , Hardy Littlewood-P\'{o}lya operator and Hilbert operator are also obtained.
For $\alpha\geq 2$, we investigate a class of Fourier extension operators on fractional surfaces $(\xi,|\xi|^\alpha)$. For the corresponding $\alpha$-Strichartz inequalities, by applying the missing mass method and bilinear restriction theory, we characterize the precompactness of extremal sequences. Our result is valid in any dimension. In particular for dimension two, our result implies the existence of extremals for $\alpha \in [2,\alpha_0)$ with some $\alpha_0>5$.
Let $\alpha\in{\Bbb R}$, $0<p<\infty$ and $X$ be a ball quasi-Banach function space on ${\Bbb R}^n$. In this article, we introduce the Herz-type space $\dot{K}^{\alpha,p}_X({\Bbb R}^n)$ associated with $X$. We identify the dual space of $\dot{K}^{\alpha,p}_X({\Bbb R}^n)$, by which the boundedness of Hardy-Littlewood maximal operator on $\dot{K}^{\alpha,p}_X({\Bbb R}^n)$ is proved. By using the extrapolation theorem on ball quasi-Banach function spaces, we establish the extrapolation theorem on Herz-type spaces associated with ball quasi-Banach function spaces. Applying our extrapolation theorem, the boundedness of singular integral operators with rough kernels and their commutators, parametric Marcinkiewicz integrals, and oscillatory singular integral operators on $\dot{K}^{\alpha,p}_X({\Bbb R}^n)$ is obtained. As examples, we give some concrete function spaces which are members of Herz-type spaces associated with ball quasi-Banach function spaces.
In this paper, by using the rotation method, we calculate that the sharp bound for $n$-dimensional Hardy operator $\mathcal{H}$ on mixed radial-angular spaces. Furthermore, we also obtain the sharp bound for $n$-dimensional fractional Hardy operator $\mathcal{H}_\beta$ from $L^p_{|x|}L_{\theta}^{\bar{p}}({\Bbb R}^n)$ to $L^q_{|x|}L_{\theta}^{\bar{q}}({\Bbb R}^n)$, where $0<\beta<n$, $1<p,q,\bar{p},\bar{q}<\infty$ and $1/p-1/q=\beta/n$. By using duality, the corresponding results for the dual operators $\mathcal{H}^*$ and $\mathcal{H}^*_\beta$ are also established. In addition, the sharp weak-type estimate for $\mathcal{H}$ is also considered.
For decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero, we obtain the almost everywhere convergence results for sequences of Schr\"{o}dinger means $e^{it_{n}\Delta}f$, where $f \in H^{s}(\mathbb{R}^{N}), N\geq 2$. The convergence results are sharp up to the endpoints, and the method can also be applied to get the convergence results for the fractional Schr\"{o}dinger means and nonelliptic Schr\"{o}dinger means.
A necessary and sufficient condition on the precompactness of extremal sequences for one dimensional $\alpha$-Strichartz inequalities, equivalently $\alpha$-Fourier extension estimates, is established based on the profile decomposition arguments. One of our main tools is an operator-convergence dislocation property consequence which comes from the van der Corput Lemma. Our result is valid in asymmetric cases as well. In addition, we obtain the existence of extremals for non-endpoint $\alpha$-Strichartz inequalities.
This paper focuses on the operator norm of the truncated Hardy-Littlewood maximal operator $M^b_a$ and the strong truncated Hardy-Littlewood maximal operator $\tilde{M}^{\boldsymbol{b}}_{\boldsymbol{a}}$, respectively. We first present the $L^1$-norm of $M^b_a$, and then the $L^1$-norm of $\tilde{M}^{\boldsymbol{b}}_{\boldsymbol{a}}$ is given. Our study may have some enlightening significance for the research on sharp constant for the classical Hardy-Littlewood maximal inequality.