We investigate whether a Stein manifold$M$which allows proper holomorphic embedding into ℂ^{$n$}can be embedded in such a way that the image contains a given discrete set of points and in addition follow arbitrarily close to prescribed tangent directions in a neighbourhood of the discrete set. The infinitesimal version was proven by Forstnerič to be always possible. We give a general positive answer if the dimension of$M$is smaller than$n$/2 and construct counterexamples for all other dimensional relations. The obstruction we use in these examples is a certain hyperbolicity condition.

In this paper we investigate Riesz transforms$R$_{μ}^{($k$)}of order$k$≥1 related to the Bessel operator Δ_{μ}$f$($x$)=-$f$”($x$)-((2μ+1)/$x$)$f$’($x$) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We obtain that for every$k$≥1,$R$_{μ}^{($k$)}is a principal value operator of strong type ($p$,$p$),$p$∈(1,∞), and weak type (1,1) with respect to the measure$d$λ($x$)=$x$^{2μ+1}$dx$in (0,∞). We also characterize the class of weights ω on (0,∞) for which$R$_{μ}^{($k$)}maps$L$^{$p$}(ω) into itself and$L$^{1}(ω) into$L$^{1,∞}(ω) boundedly. This class of weights is wider than the Muckenhoupt class $\mathcal{A}_{p}^\mu$ of weights for the doubling measure$d$λ. These weighted results extend the ones obtained by Andersen and Kerman.

We give an elementary proof of the existence of an asymptotic expansion in powers of$k$of the Bergman kernel associated to$L$^{$k$}, where$L$is a positive line bundle over a compact complex manifold. We also give an algorithm for computing the coefficients in the expansion.

We discuss smoothing effects of homogeneous dispersive equations with constant coefficients. In the case where the characteristic root is positively homogeneous, time-global smoothing estimates are known. It is also known that a dispersiveness condition is necessary for smoothing effects. We show time-global smoothing estimates where the characteristic root is not necessarily homogeneous. Our results give a sufficient condition so that lower order terms can be absorbed by the principal part, and also indicate that smoothing effects may be caused by lower order terms in the case where the dispersiveness condition fails to hold.

The main purpose of this paper is to study the asymptotic property of one class of number-theoretic functions and give a sharp hybrid mean-value formula by using the generalized Bernoulli numbers, Gauss sums and the mean-value theorems of Dirichlet$L$-functions.