In this paper we give a complete description of the power weighted inequalities, of strong, weak and restricted weak type for the pair of Riesz transforms associated with the Laguerre function system $\{\mathcal{L}_k^{\alpha}\}$, for any given α>-1. We achieve these results by a careful estimate of the kernels: near the diagonal we show that they are local Calderón–Zygmund operators while in the complement they are majorized by Hardy type operators and the maximal heat-diffusion operator. We also show that in all the cases our results are sharp.

We study the stability of Gorenstein preenvelopes and precovers in the cases of H-extensions and smash products with H, where H is a Hopf algebra. We use these to define Gorenstein dimensions and give new examples of the so-called Gorenstein categories.

In this paper a Wiener-type characterization is presented of those boundary points of a Denjoy domain where the Green function is Lipschitz continuous. This property is linked with the splitting of a Euclidean boundary point into two minimal Martin boundary points.

In this paper, we give a new characterization of Carleson measures for the generalized Bergman spaces. We show first that this problem is equivalent to a T(1)-type problem. Using an idea of Verdera (see [V]), we introduce a sort of curvature in the unit ball adapted to our kernel. We establish a good λ inequality which then yields the solution of this T(1)-type problem.

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.

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.

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.

In this paper we investigate Riesz transforms R_μ^(k) of order k≥1 related to the Bessel operator Δ_μf(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 dλ(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 L¹(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 dλ. These weighted results extend the ones obtained by Andersen and Kerman.

We investigate whether a Stein manifold M which allows proper holomorphic embedding into ℂⁿ 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.

We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows (X,μ, T_t) and f∈L^p(X,μ), there is a set X_f⊂X of probability one, so that for all x∈X_f, $\lim_{s\downarrow0}\int_{s<|t|<1/s}e^{i\theta t} f(\textup{T}_tx)\,\frac{dt}t\quad\text{exists for all}\ \theta.$ The proof is by way of establishing an appropriate oscillation inequality which is itself an extension of Carleson’s theorem.