We establish the comparison principle in the class ${}_p\left(f\right)$. The result obtained is applied to the Dirichlet problem in ${}_p\left(f\right)$.

We prove some existence results for the complex Monge-Ampère equation $\left(d{d}^{c}u\right)ⁿ=gd\lambda$ in ℂⁿ in a certain class of homogeneous functions in ℂⁿ, i.e. we show that for some nonnegative complex homogeneous functions g there exists a plurisubharmonic complex homogeneous solution u of the complex Monge-Ampère equation.

Let f:ℝ² → ℝ be a polynomial mapping with a finite number of critical points. We express the degree at infinity of the gradient ∇f in terms of the real branches at infinity of the level curves {f(x,y) = λ} for some λ ∈ ℝ. The formula obtained is a counterpart at infinity of the local formula due to Arnold.

For any holomorphic function F in the unit polydisc Uⁿ of ℂⁿ, we consider its restriction to the diagonal, i.e., the function in the unit disc U of ℂ defined by F(z) = F(z,...,z), and prove that the diagonal mapping maps the mixed norm space $H^{p,q,\alpha }\left(Uⁿ\right)$ of the polydisc onto the mixed norm space $H^{p,q,\left|\alpha \right|+(p/q+1)(n-1)}\left(U\right)$ of the unit disc for any 0 < p < ∞ and 0 < q ≤ ∞.

Let $A\subset {ℝ}^n$ be a set-germ at $0\in {ℝ}^n$ such that $0\in \overline{A}$. We say that $r\in S^{n-1}$ is a direction of $A$ at $0\in {ℝ}^n$ if there is a sequence of points $\left\{x_i\right\}\subset A\setminus \left\{0\right\}$ tending to $0\in {ℝ}^n$ such that $\frac{x_i}{\parallel x_i\parallel }\to r$ as $i\to \infty$. Let $D\left(A\right)$ denote the set of all directions of $A$ at $0\in {ℝ}^n$.Let $A,B\subset {ℝ}^n$ be subanalytic set-germs at $0\in {ℝ}^n$ such that $0\in \overline{A}\cap \overline{B}$. We study the problem of whether the dimension of the common direction set, $dim\left(D\right(A)\cap D(B\left)\right)$ is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of $A$ and $B$ are also subanalytic. In particular if two subanalytic...