Related papers: $L^2$-estimates for the $d$-operator acting on sup…

Let $H_1$ and $H_2$ be two Hilbert spaces, $K$ and $L$ be bounded operatrors on $H_1$ and $H_2$ respectively. In this paper we study the relationship between $K$-frames for $H_1$ and $L$-frames for $H_2$ and $K\oplus L$-frames for…

We investigate the analogues of certain classical estimates of Littlewood for the Riemann zeta-function in the context of quadratic Dirichlet $L$-functions over function fields. In some situations, we are actually able to establish finer…

In this new version, we correct some typos. For the readers' convenience, we also added some footnotes and more details for certain lemmas and theorems.

We prove $L^{2}$ estimates and solvability for a variety of simply characteristic constant coefficient partial differential equations $P(D)u=f$. These estimates \[||u||_{L^2(D_{r})}\le C\sqrt{d_{r}d_{s}} ||f||_{_{L^2(D_{s})}}\] depend on…

We introduce a systematic method to solve a type of Cartan's realization problem. Our method builds upon a new theory of Lie algebroids and Lie groupoids with structure group and connection. This approach allows to find local as well as…

We present a refined, improved $L^2$-theory for the $\bar{\partial}$-operator for $(0,q)$ and $(n,q)$-forms on Hermitian complex spaces of pure dimension $n$ with isolated singularities. The general philosophy is to use a resolution of…

We offer an example of the second order Kawaguchi metric function the extremal flow of which generalizes the flat space-time model of the semi-classical spinning particle to the framework of the pseudo-Riemannian space-time. The general…

The eigenfunctions of the Laplacian are a central object from the realms of analytic number theory to geometric analysis. We prove that H\"ormander $L^2$-$L^{\infty}$ estimates are equivalent to restriction estimates to small geodesic…

We introduce a notion of a para-K\"{a}hler strict Lie 2-algebra, which can be viewed as a categorification of a para-K\"{a}hler Lie algebra. In order to study para-K\"{a}hler strict Lie 2-algebra in terms of strict pre-Lie 2-algebras, we…

We prove a quantum ergodic restriction (QER) theorem for real hypersurfaces $\Sigma \subset X,$ where $X$ is the Grauert tube associated with a real-analytic, compact Riemannian manifold. As an application, we obtain $h$ independent upper…

We consider supersymmetric N=2 solutions with non-vanishing NS three-form. Building on worldsheet results, we reduce the problem to a single generalized Monge-Ampere equation on the generalized Kaehler potential K recently interpreted…

In this article, we focus on $L^{2}(\mathbb{R}^d)\times\cdots\times L^{2}(\mathbb{R}^d)\rightarrow L^{2/m}(\mathbb{R}^d)$ estimates for multilinear maximal averages over non-degenerate hypersurfaces. Our findings is new for $m$-linear…

Some key features of the symmetries of the Schr\"odinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the algebra/superalgebra duality involving first and…

The coadjoint orbits of compact Lie groups each carry a canonical (positive definite) K\"ahler structure, famously used to realize the group's irreducible representations in holomorphic sections of appropriate line bundles (Borel-Weil…

In this paper, first, we introduce a notion of modified Rota-Baxter Lie algebras of weight $\mathrm{\lambda}$ with derivations (or simply modified Rota-Baxter LieDer pairs) and their representations. Moreover, we investigate cohomologies of…

Hypergeometric functions and their generalizations play an important r\^{o}les in diverse applications. Many authors have been established generalizations of hypergeometric functions by a number ways. In this paper, we aim at establishing…

This paper is devoted mainly to the global existence problem for the two-dimensional parabolic-parabolic Keller-Segel in the full space. We derive a critical mass threshold below which global existence is ensured. Using carefully energy…

We use duality in the manner of Serre to generalize a theorem of Hedenmalm on solution of the $\bar \partial $ equation with inverse of the weight in H\"ormander $\displaystyle L^{2}$ estimates.\

We study the problem of estimating the $L^2$ norm of Laplace eigenfunctions on a compact Riemannian manifold $M$ when restricted to a hypersurface $H$. We prove mass estimates for the restrictions of eigenfunctions $\phi_h$, $(h^2 \Delta -…

Let $X$ be a Fano manifold of dimension at least $2$ and $D$ be a smooth divisor in a multiple of the anticanonical class, $\frac1\alpha(-K_X)$ with $\alpha>1$. It is well-known that K\"ahler-Einstein metrics on $X$ with conic singularities…