We prove the Penney-Fujiwara Plancerel Formula associated to a monomial representation of a nilpotent Lie group. We give also a short proof of a theorem due to Corwin and Greenleaf about the algebra of differential operators on certain nilpotent homogeneous space.

New regularity bounds for subschemes of codimension two are established. They rest on the Serre correspondence to reflexive sheaves, a study of cohomological annihilators and the strong restriction property. Examples are described where the bounds are sharp.

With a symbol in $OpS_{1,1}^{m}$ or its kernel-distribution, one has a great difficulty to study the continuity or other properties, here one use the wavelets bases which come from the Beylkin-Coifman-Rokhlin (B-C-R) algorithm to study such operators. Each operator in $OpS_{1,1}^{m}$ corresponds to its wavelet coefficients; withth is idea, one characterizes $OpS_{1,1}^m$ with a discrete space, and characterizes $OpS_{1,1}^{0}$ with a kernel-distribution space. As an application, theorem 1 of chapter 9 in tome II of [8] is a corollary of two theorems of this paper.

In this paper we classify all regular surfaces of general type with $p_{g}=3$ and non-birational bicanonical map which do not have a pencil of curves of genus 2.

We characterize generalized functions including distributions and ultradistributions which are rotation invariant and homogeneous as follows:

If $u$ is a generalized function in $\mathbf{R}^{n}$ with $n \geq 2$ which is rotation invariantand homogeneous of real degree $k$ then it can be written as \[ \begin{array}{ccc} u&=&\left\{ \begin{array}{cl}a|x|^{k}+b\Delta ^{\frac{-n-k}{2}}\delta , & \text{if } -n-k \text{ is an even nonnegative integer,}\\ a|x|^{k}, & \text{otherwise.}\end{array}\right. \end{array} \] In addition, we find a structure theorem of rotation invariant ultradistributions with support at the origin.

In this note, we investigate elliptic surfaces in char. $p>0$ with multiple fibers of a supersingular elliptic curve. In particular, we show that wild fibers on elliptic surfaces over $\mathbf{P}^{1}$ with $c_{2}=0$ and the general fiber being a supersingular elliptic curve can be reduced to tame fibers by taking a purely inseparable covering of degree $p$ successively and as an application of it, we show that if such elliptic surface has only one multiple fiber, then its multiplicity is equal to $p$.