Related papers: H\"ormander's solution of the $\bar\partial$ -equa…

The note concerns the $\bar\partial$ problem on product domains in $\mathbb C^2$. We show that there exists a bounded solution operator from $C^{k, \alpha}$ into itself, $k\in \mathbb Z^+\cup \{0\}, 0<\alpha< 1$. The regularity result is…

We study the $\bar{\partial}_b$-Neumann problem for domains $\Omega$ contained in a strictly pseudoconvex manifold M^{2n+1} whose boundaries are noncharacteristic and have defining functions depending solely on the real and imaginary parts…

This article contains a characterization of when certain weighted Sobolev spaces on $\Bbb R^n$ embed compactly into $L^2(\mathbb R^n, \varphi)$. The characterization is in terms of derivatives of the weight function $\varphi$ and involves…

We analyze the properties that manifest Hamiltonian nature of the Schr\"odinger equation and show that it can be considered as originating from singular Lagrangian action (with two second class constraints presented in the Hamiltonian…

We consider Gabor frames $\{e^{2\pi i bm \cdot} g(\cdot-ak)\}_{m,k \in \mathbb{Z}}$ with translation parameter $a=L/2$, modulation parameter $b \in (0,2/L)$ and a window function $g \in C^n(\mathbb{R})$ supported on $[x_0,x_0+L]$ and…

We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c l c} -\Delta u & = &|u|^{q-2}u & \textrm{in }\Omega, \\ u_n & = & 0 & \textrm{on }\partial\Omega,\end{array}\right. \end {equation*} where $\Omega \subset…

Let $\omega$ be a continuous weight on $\mathbb R^+$ and let $L^1(\omega)$ be the corresponding convolution algebra. By results of Gr{\o}nb{\ae}k and Bade & Dales the continuous derivations from $L^1(\omega)$ to its dual space…

In this paper we state the weighted Hardy inequality \begin{equation*} c\int_{{\mathbb R}^N}\sum_{i=1}^n \frac{\varphi^2 }{|x-a_i|^2}\, \mu(x)dx\le \int_{{\mathbb R}^N} |\nabla\varphi|^2 \, \mu(x)dx +k \int_{\mathbb{R}^N}\varphi^2 \,…

The Hardy--Littlewood inequality for complex homogeneous polynomials asserts that given positive integers $m\geq2$ and $n\geq1$, if $P$ is a complex homogeneous polynomial of degree $m$ on $\ell_{p}^{n}$ with $2m\leq p\leq\infty$ given by…

We consider on a closed Riemannian spin manifold $(M^n,g,\sigma)$ the spinorial Yamabe type equation $D_g\varphi=\lambda|\varphi|^{\frac{2}{n-1}}\varphi$, where $\varphi$ is a spinor field and $\lambda$ is a positive constant. For a…

We study in this work the existence of minimizing solutions to the critical-power type equation $\triangle_{\textbf{g}}u+h.u=f .u^{\frac{n+2}{n-2}} $ on a compact riemannian manifold in the limit case normally not solved by variational…

Let $\mathfrak g$ be either the Lie superalgebra $\mathfrak{gl}(V)\oplus\mathfrak{gl}(V)$ where $V:=\mathbb C^{m|n}$ or the Lie superalgebra $\mathfrak{gl}(V)$ where $V:=\mathbb C^{m|2n}$. Furthermore, let $W$ be the $\mathfrak g$-module…

The original proof of Dacorogna-Moser theorem on the prescribed Jacobian PDE, $\text{det}\,\nabla\varphi=f$, can be modified in order to obtain control of support of the solutions from that of the initial data, while keeping optimal…

We consider the problem of finding and describing minimisers of the Rayleigh quotient \[ \Lambda_\infty \, :=\, \inf_{u\in \mathcal{W}^{2,\infty}(\Omega)\setminus\{0\} }\frac{\|\Delta u\|_{L^\infty(\Omega)}}{\|u\|_{L^\infty(\Omega)}}, \]…

We study products of eigenfunctions of the Laplacian $-\Delta \phi_{\lambda} = \lambda \phi_{\lambda}$ on compact manifolds. If $\phi_{\mu}, \phi_{\lambda}$ are two eigenfunctions and $\mu \leq \lambda$, then one would perhaps expect their…

For a bounded open set Omega in a complete oriented Riemannian n-manifold and a positive finite Borel measure mu with support contained in Omega, we define an associated Krein-Feller operators (or Laplacian) Delta_mu by assuming the…

We deduce various norm equivalences, and convolution estimates for the modulation space $M^{\sharp ,q}_{(\omega )}$ consisting of all $f\in M^{\infty ,q}_{(\omega )}$ such that $|V_\phi f \cdot \omega |$ satisfies a mild vanishing condition…

The endpoint of this series of papers is to construct the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $\omega$ is a suitable closed 2-form. In the first paper, we…

We are concerned with an harmonic analysis in Hilbert spaces $L^2(\mu)$, where $\mu$ is a probability measure on $\br^n$. The unifying question is the presence of families of orthogonal (complex) exponentials $e_\lambda(x) = \exp(2\pi i…

We study the cohomology $H^*_{\lambda \omega}(G/\Gamma, {\mathbb C})$ of the deRham complex $\Lambda^*(G/\Gamma)\otimes{\mathbb C}$ of a compact solvmanifold $G/\Gamma$ with a deformed differential $d_{\lambda \omega}=d + \lambda\omega$,…