In this paper, we establish the boundedness of the bilinear Bochner-Riesz means $\mathcal{B}^{\alpha}_R$ on the complex sphere $\mathbb{S}$. More precisely, we prove that $\mathcal{B}^{\alpha}_R$ is bounded from $L^{p_1}(\mathbb{S}) \times L^{p_2}(\mathbb{S}) \to L^p(\mathbb{S})$ where $1/p_1+1/p_2=1/p$ and $1\leq p_1, p_2 \leq \infty$, for an admissible range of exponents, with the required smoothness parameter $\alpha$ described in terms of the topological dimension of $\mathbb{S}$. To facilitate our proof, we establish several analytic estimates, including restriction-type estimates, weighted Plancherel estimates with large power of weights and bilinear weighted Plancherel estimates, which are derived from the ground up in our setting and may be considered of independent interest.
We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform \[ \mathcal{C}_2f(x) := \sup_{\lambda} \left|\operatorname{p.v.} \int_{\mathbb{R}}f(x-y)e^{2 \pi i \lambda y^2} \frac{\mathrm{d} y }{y} \right| \] and its lacunary counterpart obtained by restricting $\lambda$ to $2^{\mathbb{Z}}...
A. Fragkos, Ben Krause, Michael T. Lacey· 0 citations
Let $(\mathcal N,\nu)$ be a finite tracial von Neumann algebra and set $\mathcal{K}_r(P,Q)=\nu((P^{1/2}QP^{1/2})^r)$. We prove that $$ [\mathcal{K}_r(P_i,P_j)]_{i,j=1}^3\ge0, \qquad 1/2\le r\le1. $$ The range $1/2<r<1$ is new even for matrices, and order three is sharp for $1/2\le r<1$. This single mechanism resolves t...
Let $X=\mathbb{P}^3$ and let \[ \alpha_n=(0,1,-\tfrac12,\tfrac16-n)\in H^{\mathrm{even}}(X,\mathbb{Q}) \] with respect to the basis $1,H,H^2,H^3$, where $H=c_1(\mathcal{O}_X(1))$. We prove that every Gieseker semistable sheaf on $X$ with Chern character $\alpha_n$ is stable and uniquely of the form $\iota_{P*}\mathcal{...
R. Anderson· Journal of Geometry and Phys...· 0 citations
Let $K \subset \mathbb{R}^{m \times n}$ be a compact $C^1$-submanifold with boundary, $p \in (1,\infty)$ and $Q := (0,1)^n$. We prove that $K$ satisfies a rigidity estimate of the form $\|Du - (Du)_{Q}\|_{L^p} \leq C \|\mathrm{dist}_K(Du)\|_{L^p}$, $u \in W^{1,p}(Q,\mathbb{R}^m)$, if and only if $K$ satisfies sequentia...
A classical theorem of Maillet asserts that every nonconstant rational function over $\mathbb{Q}$ maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether a transcendental entire function can have the same property. We prove a strong smooth counterpart: writing $\mathscr{L}$ for the set of Liouville n...
A classical theory of Amrouche, Girault and Giroire (1994) resolves the Laplace equation on $\mathbb{R}^n$ through an isomorphism between weighted Sobolev spaces. Inspired by their framework, we develop the corresponding $L^p$ theory for the fractional Laplacian: we introduce weighted fractional Sobolev spaces $\Lambda...
