Narrow Results

Let (M,g) be a smooth compact Riemannian n-manifold, n ≥ 4, and h be a Holdër continuous function on M. We prove multiplicity of changing sign solutions for equations like Δ_g u + hu = |u|^{2* - 2} u, where Δ_g is the Laplace–Beltrami operator and 2* = 2n/(n - 2) is critical from the Sobolev viewpoint.

Let ϕ(z) = (ϕ₁(z),…,ϕ_n(z)) be a holomorphic self-map of B and ψ(z) a holomorphic function on B, where B is the unit ball of ℂⁿ. Let 0 < p, s < +∞, -n - 1 < q < +∞, q+s > -1 and α ≥ 0, this paper characterizes boundedness and compactness of weighted composition operator W_ψ,ϕ induced by ϕ and ψ between the space F(p, q, s) and α-Bloch space .

In this paper, the authors study the compactness for higher order commutators of oscillatory singular integral operators with rough kernels satisfying L^q-Dini conditions.

These notes are concerned with the $L^2$-Sobolev theory of the complex Green operator on pseudoconvex, oriented, bounded and closed Cauchy–Riemann (CR)-submanifolds of ${ℂ}^n$ of hypersurface type. This class of submanifolds generalizes that of boundaries of pseudoconvex domains. We first discuss briefly the CR-geometry of general CR-submanifolds and then specialize to this class. Next, we review the basic $L^2$-theory of the tangential CR operator and the associated complex Green operator(s) on these submanifolds. After these preparations, we discuss recent results on compactness and regularity in Sobolev spaces of the complex Green operator(s).

We discuss compactness of the $\overline{\partial }$-Neumann operator in the setting of weighted $L^2$-spaces on ${ℂ}^n.$ In addition we describe an approach to obtain the compactness estimates for the $\overline{\partial }$-Neumann operator. For this purpose we have to define appropriate weighted Sobolev spaces and prove an appropriate Rellich–Kondrachov lemma.

In this paper, we obtain compactness theorems for Yang–Mills–Higgs fields on vector bundle $E$ over compact Riemannian manifold $M$, $\mathrm{\text{dim}}M>4$, with general Higgs-like potential $W:ℝ\to [0,\infty )$.