Related papers: Sharp pointwise and uniform estimates for $\bar\pa…
We investigate Hermitian metrics on the anti-canonical bundle of a rational surface obtained by blowing up the projective plane at nine points. For that purpose, we pose a modified variant of an argument made by Ueda on the complex analytic…
In this note we show that the weighted $L^{2}$-Sobolev estimates obtained by P. Charpentier, Y. Dupain & M. Mounkaila for the weighted Bergman projection of the Hilbert space $L^{2}\left(\Omega,d\mu_{0}\right)$ where $\Omega$ is a smoothly…
In this paper we prove the uniqueness of the critical point for stable solutions of the Robin problem \[ \begin{cases} -\Delta u=f(u)&\text{in }\Omega\\ u>0&\text{in }\Omega\\ \partial_\nu u+\beta u=0&\text{on }\partial\Omega, \end{cases}…
The following problem originated from a question due to Paul Turan. Suppose $\Omega$ is a convex body in Euclidean space $\RR^d$ or in $\TT^d$, which is symmetric about the origin. Over all positive definite functions supported in $\Omega$,…
We prove essentially sharp incidence estimates for a collection of $\delta$-tubes and $\delta$-balls in the plane, where the $\delta$-tubes satisfy an $\alpha$-dimensional spacing condition and the $\delta$-balls satisfy a…
We introduce a new integral representation formula in the d-bar Neumann Theory on weakly pseudoconvex domains which satisfies certain estimates analogous to the basic L^2 estimate. It is expected that more complete estimates can be obtained…
We consider the 3D Boltzmann equation for the Maxwellian particle and soft potential with an angular cutoff. We prove sharp global well-posedness with initial data small in the scaling-critical space. The solution also remains in $L^{1}$ if…
We introduce a positive scalar function $\rho(a, \Omega)$ for a domain $\Omega$ of a complex manifold $X$ with a global holomorphic frame of the cotangent bundle by closed Abelian differentials, which heuristically measure the distance from…
We prove sharp pointwise blow-up estimates for finite-energy sign-changing solutions of critical equations of Schr\"odinger-Yamabe type on a closed Riemannian manifold $(M,g)$ of dimension $n \ge 3$. This is a generalisation of the…
We prove the boundedness of Bergman type projections in two different analytic function spaces in bounded strongly pseudoconvex domains with the smooth boundary. Our results were previously well-known in the case of the unit disk.
We analyse the maximum achievable rate of sustained computation for a given convex region of three dimensional space subject to geometric constraints on power delivery and heat dissipation. We find a universal upper bound across both…
Quantitative formulations of Fefferman's counterexample for the ball multiplier are naturally linked to square function and vector-valued estimates for directional singular integrals. The latter are usually referred to as Meyer-type lemmas…
We generalize to intersection of strictly $c$ -convex domains in Stein manifold, $ L^{r}-L^{s}$ and Lipschitz estimates for the solutions of the $ \bar \partial $ equation done by Ma and Vassiliadou for domains in $ {\mathbb{C}}^{n}.$ For…
A class of pseudoconvex domains in $\mathbb{C}^{n}$ generalizing the Hartogs triangle is considered. The $L^p$ boundedness of the Bergman projection associated to these domains is established, for a restricted range of $p$ depending on the…
In this paper, we prove the existence of solutions of the Poincar\'e-Lelong equation $\sqrt{-1}\partial\bar{\partial}u=f$ on a strictly convex bounded domain $\Omega\subset\mathbb{C}^n$ $(n\geq1)$, where $f$ is a $d$-closed $(1,1)$ form and…
We observe that the continuity assumption on $f$ for the uniform estimates of the canonical solution to $\bar\partial u = f$ on products of $C^2$ bounded planar domains in \cite{DPZ} can be reduced to the boundedness assumption. This…
In this paper we investigate the regularity properties of weighted Bergman projections for smoothly bounded pseudo-convex domains of finite type in $\mathbb{C}^{n}$. The main result is obtained for weights equal to a non negative rational…
This paper studies the heat equation $u_t=\Delta u$ in a bounded domain $\Omega\subset\mathbb{R}^{n}(n\geq 2)$ with positive initial data and a local nonlinear Neumann boundary condition: the normal derivative $\partial u/\partial n=u^{q}$…
In this paper, we establish uniform a priori estimates for positive solutions to the (higher) critical order superlinear Lane-Emden system in bounded domains with Navier boundary conditions in arbitrary dimensions $n\geq3$. First, we prove…
Estimating the norm of the solution of the linear difference equation $u(\theta)-u(\theta+\omega)=v(\theta)$ plays a fundamental role in KAM theory. Optimal (in certain sense) estimates for the solution of this equation were provided by…