On Birman's sequence of Hardy–Rellich-type inequalities

In 1961, Birman proved a sequence of inequalities $\{I_n\}$, for $n\in \mathbb{N}$, valid for functions in $C_0^n((0,\infty ))?L^2((0,\infty ))$. In particular, $I_1$ is the classical (integral) Hardy inequality and $I_2$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_n([0,\infty ))$ of functions defined on $[0,\infty )$. Moreover, $f\in H_n([0,\infty ))$ implies $f^{\prime }\in H_{n?1}([0,\infty ))$; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite $b>0$, these inequalities hold on the standard Sobolev space $H_0^n((0,b))$. Furthermore, in all cases, the Birman constants ${[(2n?1)!!]}^2/2^{2n}$ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in $L^2((0,\infty ))$ (resp., $L^2((0,b))$). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.     

Sharp threshold of blow-up and scattering for the fractional Hartree equation

We consider the fractional Hartree equation in the $L^2$-supercritical case, and find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $M{[u_0]}^{\frac{s?s_c}{s_c}}E[u_0]<M{[Q]}^{\frac{s?s_c}{s_c}}E[Q]$ and $M{[u_0]}^{\frac{s?s_c}{s_c}}{6u_{0}6}_{{\dot{H}}^s}^2<M{[Q]}^{\frac{s?s_c}{s_c}}{6Q6}_{{\dot{H}}^s}^2$, then the solution $u(t)$ is globally well-posed and scatters; if $M{[u_0]}^{\frac{s?s_c}{s_c}}E[u_0]<M{[Q]}^{\frac{s?s_c}{s_c}}E[Q]$ and $M{[u_0]}^{\frac{s?s_c}{s_c}}{6u_{0}6}_{{\dot{H}}^s}^2>M{[Q]}^{\frac{s?s_c}{s_c}}{6Q6}_{{\dot{H}}^s}^2$, the solution $u(t)$ blows up in finite time. This condition is sharp in the sense that the solitary wave solution $e^{it}Q(x)$ is global but not scattering, which satisfies the equality in the above conditions. Here, Q is the ground-state solution for the fractional Hartree equation.     

Pointwise asymptotic behavior of modulated periodic reaction–diffusion waves

By working with the periodic resolvent kernel and the Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction–diffusion equations. With our linearized estimates together with a nonlinear iteration scheme developed by Johnson–Zumbrun, we obtain $L^p$-behavior ($p?1$) of a nonlinear solution to a perturbation equation of a reaction–diffusion equation with respect to initial data in $L^1\cap H^2$ recovering and slightly sharpening results obtained by Schneider using weighted energy and renormalization techniques. We obtain also pointwise nonlinear estimates with respect to two different initial perturbations $|u_0|?E_0{e}^{?{|x|}^2/M}$, ${|u_0|}_{H^2}?E_0$ and $|u_0|?E_0{(1+|x|)}^{?r}$, $r>2$, ${|u_0|}_{H^2}?E_0$ respectively, $E_0>0$ sufficiently small and $M>1$ sufficiently large, showing that behavior is that of a heat kernel. These pointwise bounds have not been obtained elsewhere, and do not appear to be accessible by previous techniques.     

A sharp Balian–Low uncertainty principle for shift-invariant spaces

A sharp version of the Balian–Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators ${\{f_k\}}_{k=1}^K?L^2({\mathbb{R}}^d)$ are translated along a lattice to form a frame or Riesz basis for a shift-invariant space V, and if V has extra invariance by a suitable finer lattice, then one of the generators $f_k$ must satisfy ${\int }_{{\mathbb{R}}^d}|x||f_k(x){|}^{2}dx=\infty$, namely, $\stackrel{?}{f_k}?H^{1/2}({\mathbb{R}}^d)$. Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space $H^{d/2+?}({\mathbb{R}}^d)$; our results provide an absolutely sharp improvement with $H^{1/2}({\mathbb{R}}^d)$. Our results are sharp in the sense that $H^{1/2}({\mathbb{R}}^d)$ cannot be replaced by $H^s({\mathbb{R}}^d)$ for any $s<1/2$.     

Translational absolute continuity and Fourier frames on a sum of singular measures

A finite Borel measure μ in ${\mathbb{R}}^d$ is called a frame-spectral measure if it admits an exponential frame (or Fourier frame) for $L^2(\mu )$. It has been conjectured that a frame-spectral measure must be translationally absolutely continuous, which is a criterion describing the local uniformity of a measure on its support. In this paper, we show that if any measures ν and λ without atoms whose supports form a packing pair, then $\nu ?\lambda +{\delta }_t?\nu$ is translationally singular and it does not admit any Fourier frame. In particular, we show that the sum of one-fourth and one-sixteenth Cantor measure ${\mu }_4+{\mu }_{16}$ does not admit any Fourier frame. We also interpolate the mixed-type frame-spectral measures studied by Lev and the measure we studied. In doing so, we demonstrate a discontinuity behavior: For any anticlockwise rotation mapping $R_{\theta }$ with $\theta \ne \pm \pi /2$, the two-dimensional measure ${\rho }_{\theta }(?):=({\mu }_4\times {\delta }_0)(?)+({\delta }_0\times {\mu }_{16})(R_{\theta }^{?1}?)$, supported on the union of x-axis and $y=(\cot ?\theta )x$, always admit a Fourier frame. Furthermore, we can find ${\{e^{2\pi i〈\lambda ,x〉}\}}_{\lambda \in {\mathrm{\Lambda }}_{\theta }}$ such that it forms a Fourier frame for ${\rho }_{\theta }$ with frame bounds independent of θ. Nonetheless, ${\rho }_{\pm \pi /2}$ does not admit any Fourier frame.