Let $\mathcal H=-\Delta+|x|^2$ be the Hermite operator on $\mathbb R^2$, and let $\Pi_\lambda$ denote the spectral projection corresponding to $\lambda=2N+2$. We prove the sharp log-free endpoint estimate $||\Pi_\lambda||_{L^2(\mathbb R^2)\to L^{10/3}(\mathbb R^2)}\lesssim\lambda^{-1/10}$. The proof uses a spectral decomposition in polar coordinates and combines Koch-Tataru localized spectral projection bounds with a Liouville-Green representation, van der Corput estimates for exponential sums, and a weighted $TT^*$ argument across radial scales.
Let $1<p<n$ and let $v(x)=(1+|x|^{p/(p-1)})^{-(n-p)/p}$ be the standard radial extremal for the sharp Sobolev inequality. We determine all eigenvalues and eigenspaces of the linearized $p$-Laplacian at $v$, defined by its closed quadratic form in $L^2(\mathbb{R}^n,v^{p^*-2} dx)$. After decomposition into spherical harm...
We study the periodic fractional stochastic Burgers equation $(\partial_t+\Lambda^\gamma)u=\partial_x(u^2)+|\partial_x|^{1-\alpha}\xi$, where $1<\gamma\leq 2$ and $\xi$ is space-time white noise. Under the condition $\alpha>\max{(7-4\gamma)/2,(15-8\gamma)/6}$, we establish pathwise $L^1$, energy, and Besov estimates fo...
We prove sharp $L^{4}$ Strichartz estimates for two-dimensional periodic Schr\"{o}dinger flows with integral diagonal symbols $ak_{1}^2+bk_2^2$ where $a,b \in \mathbb Z \setminus \{0\}$. The estimates exhibit a sharp arithmetic dichotomy according to whether $-ab$ is a square: for frequencies bounded by $N$, the optima...
We study quantitative unique continuation for bi-Laplace equations \[ \Delta^2u+W(x)\Delta u+V(x)\cdot\nabla u=0 \] with $C^2$ lower-order coefficients. Using the weighted-frequency function, we establish an $L^\infty$ three-ball inequality whose exponential factor has coefficient dependence $\|W\|_{C^2}^{1/2}+\|V\|_{C...
We prove an $L^4$ Strichartz estimate for the dispersion $n_1^2+\alpha n_2^2$ on $\mathbb T^2$, for every $\alpha>0$. The loss is controlled by the Shannon entropy of the normalized squared Fourier coefficients and the cardinality of their support $S$. In particular, we obtain a norm loss of $(\log\#S)^{1/2}$ on fixed...
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
Matthew J. Colbrook, Siavash Sadeghi· 0 citations
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