Related papers: On extremizers for Strichartz estimates for higher…
In this work we obtain sharp embedding inequalities for a family of conformally invariant integral extension operators. This family includes among others the classical Poisson extension operator and the extension operator with Riesz kernel.…
A well known conjecture states that constant functions are extremizers of the $L^2 \to L^6$ Tomas-Stein extension inequality for the circle. We prove that functions supported in a $\sqrt{6}/80$-neighbourhood of a pair of antipodal points on…
Sharp restriction theory and the finite field extension problem have both received a great deal of attention in the last two decades, but so far they have not intersected. In this paper, we initiate the study of sharp restriction theory on…
In this paper, we establish the linear profile decomposition for the one dimensional fourth order Schr\"odinger equation $$ iu_t-\mu\Delta u+\Delta^2u=0, t\in\mathbb{R}, x\in\mathbb{R}, u(0,x)=f(x)\in L^2, $$ where $\mu\ge 0$. As an…
Consider the adjoint restriction inequality associated with the hypersurface $\{ (\tau, \xi) \in \mathbb{R}^{d+1} : \tau = |\xi|^2 \} \cup \{(\tau, \xi) \in \mathbb{R}^{d+1} : \tau - \tau_0 = |\xi - \xi_0|^2\}$ for any $(\tau_0, \xi_0) \neq…
Our main purpose in this paper is to establish the existence and nonexistence of extremal functions for sharp inequality of Adimurthi-Druet type for fractional dimensions on the entire space. Precisely, we extend the sharp Trudinger-Moser…
In this note, we study maximizers for Fourier extension inequalities on the sphere. We prove that constant functions are local maximizers for the $L^p(\mathbb{S}^{d-1})$ to $L^p(\mathbb{R}^d)$ Fourier extension estimates in the same range…
We investigate the sharp endpoint extension inequality for the moment curve in finite fields. We determine the optimal constant and characterize the maximizers in two complementary regimes: (i) low dimensions $d\leq 20$; (ii) large field…
In this article, we prove that all global, nonendpoint Fourier restriction inequalities for the paraboloid in $\mathbb R^{1+d}$ have extremizers and that $L^p$-normalized extremizing sequences are precompact modulo symmetries. This result…
In this paper, we establish the existence of extremals for two kinds of Stein-Weiss inequalities on the Heisenberg group. More precisely, we prove the existence of extremals for the Stein-Weiss inequalities with full weights in Theorem 1.1…
Fixed-point equations with Lipschitz operators have been studied for more than a century, and are central to problems in mathematical optimization, game theory, economics, and dynamical systems, among others. When the Lipschitz constant of…
We prove the existence of maximizers for Strichartz inequalities for the wave equation in dimensions $d\geq 3$. Our approach follows the scheme given by Shao, which obtains the existence of maximizers in the context of the Schr\"odinger…
In this note we investigate complete non-selfadjointness for all maximally dissipative extensions of a Schr\"odinger operator on a half-line with dissipative bounded potential and dissipative boundary condition. We show that all maximally…
An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically…
We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schr\"{o}dinger operators. Such refinements begin to characterize the near-optimizers of the Strichartz estimate, and play a pivotal part in the…
We derive a lower bound on the location of global extrema of eigenfunctions for a large class of non-local Schr\"odinger operators in convex domains under Dirichlet exterior conditions, featuring the symbol of the kinetic term, the strength…
This paper is devoted to study the sharp Moser-Trudinger type inequalities in whole space $\mathbb R^N$, $N \geq 2$ in more general case. We first compute explicitly the \emph{normalized vanishing limit} and the \emph{normalized…
In this paper, we will use a suitable tranform to investigate the sharp constants and optimizers for the following Caffarelli-Kohn-Nirenberg inequalities for a wide range of parameters $(r,p,q,s,\mu,\sigma)$ and $0\leq a\leq1$:…
We consider the Schr\"odinger operator on a combinatorial graph consisting of a finite graph and a finite number of discrete half-lines, all jointed together, and compute an asymptotic expansion of its resolvent around the threshold $0$.…
We consider the adjoint restriction inequality associated to the hypersurface $\{(\tau, \xi) : \tau = \pm|\xi|^2, \;\xi \in \mathbb{R}^d\}$ at the Stein-Tomas exponent. Extremizers exist in all dimensions and extremizing sequences are…