Related papers: $L^2$ estimates for the $\bar \partial$ operator
We discuss compactness of the d-bar-Neumann operator in the setting of weighted L^2-spaces on b C^n. In addition we describe an approach to obtain the compactness estimates for the d-bar-Neumann operator. For this purpose we have to define…
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 prove L2 x L2 to weak L1 estimates for some novel bilinear maximal operators of Kakeya and lacunary type thus extending to this setting, the works of Cordoba and of Nagel, Stein and Wainger.
The $\bar{\partial}$-Neumann problem is the fundamental boundary value problem in several complex variables. It features an elliptic operator coupled with non-coercive boundary conditions. The problem is globally regular on many, but not…
In this paper, we extend the uniform $L^2$-estimate of $\bar{\partial}$-equations for flat nontrivial line bundles, proved for compact K\"ahler manifolds in the previous work, to compact complex manifolds. In the proof, by tracing the…
This paper is the second part of our series of works to establish $L^2$ estimates and existence theorems for the $\overline{\partial}$ operators in infinite dimensions. In this part, we consider the most difficult case, i.e., the underlying…
We apply methods from complex analysis, in particular the d-bar-Neumann operator, to investigate spectral properties of Pauli operators.
As an application of a new characterization of compactness of the $\bar\partial $-Neumann operator we derive a sufficient condition for compactness of the $\bar\partial $- Neumann operator on $(0,q)$-forms in weighted $L^2$-spaces on…
In this paper, we introduce twisted Rota-Baxter operators on Lie algebras as an operator analogue of twisted r-matrices. We construct a suitable $L_\infty$-algebra whose Maurer-Cartan elements are given by twisted Rota-Baxter operators.…
In this work we introduce and study fractional measure theoretic elliptic operators on the torus and a new stochastic process named W-Brownian motion. We establish some regularity and spectral results related to the operators cited above,…
An integral solution operator for $\bar\partial$ is constructed on product domains that include the punctured bidisc. This operator is shown to satisfy $L^p$ estimates for all $1\leq p <\infty$, though with non-standard -- relative to…
We give a survey on L^2-invariants such as L^2-Betti numbers and L^2-torsion taking an algebraic point of view. We discuss their basic definitions, properties and applications to problems arising in topology, geometry, group theory and…
We study a complex valued version of the Sobolev inequalities and its relationship to compactness of the d-bar-Neumann operator. For this purpose we use an abstract characterization of compactness derived from a general description of…
Let $P(\Delta)$ be a polynomial of the Laplace operator $\Delta=\sum_{j=1}^n\frac{\partial^2}{\partial x^2_j}$ on $\mathbb{R}^n$. We prove the existence of weak solutions of the equation $P(\Delta)u=f$ and the existence of a bounded right…
We prove L^p estimates for a class of two-dimensional multilinear forms that naturally generalize (dyadic variants of) both classical paraproducts and the twisted paraproduct introduced in [5] and studied in [1] and [6]. The method we use…
We prove a bilinear $L^2(\R^d) \times L^2(\R^d) \to L^2(\R^{d+1})$ estimate for a pair of oscillatory integral operators with different asymptotic parameters and phase functions satisfying a transversality condition. This is then used to…
New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalisations as well as 4-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all…
We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schr\"{o}dinger operators. Such refinements begin to characterize the near-optimizers of the Strichartz estimate, and play a pivotal part in the…
This paper surveys recent work on Lie algebras of differential operators and their application to the construction of quasi-exactly solvable Schroedinger operators.
We prove pointwise variational Lp bounds for a bilinear Fourier integral operator in a large but not necessarily sharp range of exponents. This result is a joint strengthening of the corresponding bounds for the classical Carleson operator,…