The canonical solution operator to restricted to Bergman spaces

We first show that the canonical solution operator to restricted to -forms with holomorphic coefficients can be expressed by an integral operator using the Bergman kernel. This result is used to prove that in the case of the unit disc in the canonical solution operator to restricted to -forms with holomorphic coefficients is a Hilbert-Schmidt operator. In the sequel we give a direct proof of the last statement using orthonormal bases and show that in the case of the polydisc and the unit ball in 1,$"> the corresponding operator fails to be a Hilbert-Schmidt operator. We also indicate a connection with the theory of Hankel operators.

     

Exotic smooth structures on

We construct exotic and as a corollary of recent results of I. Dolgachev and C. Werner concerning a numerical Godeaux surface. We also construct another exotic using the surgery techniques of R. Fintushel and R. J. Stern. We show that these 4-manifolds are irreducible by computing their Seiberg-Witten invariants.

     

A remark on the existence of suitable vector fields related to the dynamics of scalar semi-linear parabolic equations

In 1992, P. Polácik showed that one could linearly imbed any vector field into a scalar semi-linear parabolic equation on with Neumann boundary condition provided that there exists a smooth vector field on such that

In this short paper, we give a classification of all the domains on which one may find such a type of vector field.

     

Spectrum of the -Neumann Laplacian on polydiscs

The spectrum of the -Neumann Laplacian on a polydisc in is explicitly computed. The calculation exhibits that the spectrum consists of eigenvalues, some of which, in particular the smallest ones, are of infinite multiplicity.

     

Linear functionals and the duality principle for harmonic functions

Let ${\mathcal H}$ be the class of complex‐valued harmonic functions in the unit disk |z| < 1 and ${\mathcal H}_1$ the set of all functions $f\in {\mathcal H}$ such that f(0) = 0, f_z(0) = 1 and $f_{\overline{z}}(0)=0$. For $V \subset {\mathcal H}_1$, its dual V* is where * denotes the Hadamard product for harmonic functions. The set V is a dual class if V = W* for some $W \subset {\mathcal H}_1.$ In the present paper, the duality principle is extended to ${\mathcal H}_1$ by means of the Hadamard product. Counterparts of the dual classes are introduced and their structural properties studied.     

Hadamard Products of Convex Harmonic Mappings

Functions f in the class $ K_H $ are convex, univalent, harmonic, and sense preserving in the unit disk. Such functions can be expressed as $ f = h + \overline {g} $ where h and g are analytic functions. If $ f \in K_H $ has $ h(0) = 0, g(0) = 0, h'(0) = 1$ , and $ g'(0) = 0 $ , then $ f \in K_H^0 $ . For $ f \in K_H^0 $ and } analytic in the unit disk, an integral representation for $ f\tilde {*}\varphi = h*\varphi + \overline {g*\varphi } $ is found. With } a strip mapping, $ f\tilde {*}\varphi $ is shown to be in $ K_H^0 $ . In a 1958 paper, Pólya and Schoenberg conjectured that if f and g are conformal mappings of the unit disk onto convex domains, then the Hadamard product f 2 g of f and g has the same property. It is known that the analogue of that result for harmonic mappings is false. In this paper, some examples are given in which the property of convexity is preserved for Hadamard products of certain convex harmonic mappings. In addition, an integral formula is used to determine the geometry of the Hadamard product from the geometry of the factors. This is true in particular for the convolution of strip mappings with certain functions $ f_n \in K_H^0 $ which take the unit disk to regular n -gons.     

Stein compacts in Levi-flat hypersurfaces

We explore connections between geometric properties of the Levi foliation of a Levi-flat hypersurface and holomorphic convexity of compact sets in , or bounded in part by . Applications include extendability of Cauchy-Riemann functions, solvability of the -equation, approximation of Cauchy-Riemann and holomorphic functions, and global regularity of the -Neumann operator.

     

On the -invariant of rational surface singularities

We show that for rational surface singularities with odd determinant the -invariant defined by W. Neumann is an obstruction for the link of the singularity to bound a rational homology 4-ball. We identify the -invariant with the corresponding correction term in Heegaard Floer theory.

     

Exotic smooth structures on , Part II

We construct exotic and using the surgery techniques of R. Fintushel and R.J. Stern. We show that these 4-manifolds are irreducible by computing their Seiberg-Witten invariants.

     

On symplectic fillings of lens spaces

Let be the contact structure naturally induced on the lens space by the standard contact structure on the three-sphere . We obtain a complete classification of the symplectic fillings of up to orientation-preserving diffeomorphisms. In view of our results, we formulate a conjecture on the diffeomorphism types of the smoothings of complex two-dimensional cyclic quotient singularities.

     

On the Bochner theorem on orthogonal operators

We prove the following theorem. Any isometric operator , that acts from the Hilbert space with nonnegative weight to the Hilbert space with nonnegative weight , allows for the integral representation

where the kernels and satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.

     

Analyticity for singular sums of squares of degenerate vector fields

Recently J. J. Kohn (2005) proved hypoellipticity for

(the negative of) a singular sum of squares of complex vector fields on the complex Heisenberg group, an operator which exhibits a loss of derivatives. Subsequently, M. Derridj and D. S. Tartakoff proved analytic hypoellipticity for this operator using rather different methods going back to earlier methods of Tartakoff. Those methods also provide an alternate proof of the hypoellipticity given by Kohn.

In this paper, we consider the equation

for which the underlying manifold is only of finite type, and prove analytic hypoellipticity using methods of Derridj and Tartakoff. This operator is also subelliptic with large loss of derivatives, but the exact loss plays no role for analytic hypoellipticity. Nonetheless, these methods give a proof of hypoellipticity with precise loss as well, which is to appear in a forthcoming paper by A. Bove, M. Derridj, J. J. Kohn and the author.

     

The Distortion Theorems for Harmonic Mappings with Negative Coefficient Analytic Parts

Some sharp estimates for coefficients, distortion and the growth order are obtained for harmonic mappings $f \in TL^{\alpha}_H,$ which are locally univalent harmonic mappings in the unit disk $\mathbb{D}:=\{z:|z|<1\}.$ Moreover, denoting the subclass $TS^{\alpha}_H$ of the normalized univalent harmonic mappings, we also estimate the growth of $|f|,$ $f \in TS^α_H,$ and their covering theorems.     

On the use of the topological degree theory in broken orbits analysis

Dynamical systems in are studied. Let be a bounded open set. We will be interested in those periodic orbits such that at least one of its points lies inside and at least one of its points lies outside ; the orbits with this property are called -broken. Information about the structure of the set of -broken orbits is suggested; results are formulated in terms of topological degree theory.

     

On the characterization of the kernel of the geodesic X-ray transform

Let be a compact manifold with boundary. We consider covariant symmetric tensor fields of order two that belong to a Sobolev space . We prove, under the assumption that the metric is simple, that solenoidal tensor fields that belong to the kernel of the geodesic X-ray transform are smooth up to the boundary. As a corollary we obtain that they form a finite-dimensional set in .

     

A $\overline{\partial}$ -Theoretical Proof of Hartogs’ Extension Theorem on Stein Spaces with Isolated Singularities

Let X be a connected normal Stein space of pure dimension d≥2 with finitely many isolated singularities. By solving a weighted -equation with compact support on a desingularization of X, we derive Hartogs’ Extension Theorem on X by the -idea due to Ehrenpreis.      

Heegner points and Mordell-Weil groups of elliptic curves over large fields

Let be an elliptic curve defined over of conductor and let be the absolute Galois group of an algebraic closure of . For an automorphism , we let be the fixed subfield of under . We prove that for every , the Mordell-Weil group of over the maximal Galois extension of contained in has infinite rank, so the rank of is infinite. Our approach uses the modularity of and a collection of algebraic points on - the so-called Heegner points - arising from the theory of complex multiplication. In particular, we show that for some integer and for a prime prime to , the rank of over all the ring class fields of a conductor of the form is unbounded, as goes to infinity.

     

Distinguishing embedded curves in rational complex surfaces

We construct many pairs of smoothly embedded complex curves with the same genus and self-intersection number in the rational complex surfaces with the property that no self-diffeomorphism of sends one to the other. In particular, as a special case we answer a question originally posed by R. Gompf (1995) concerning genus two curves of self-intersection number 0 in .

     

Some generalizations of Chirka's extension theorem

In this paper, we generalize Chirka's theorem on the extension of functions holomorphic in a neighbourhood of - where is the open unit disc in and is the graph of a continuous valued function on - to higher dimensions, for certain classes of graphs 1$">. In particular, we show that Chirka's extension theorem generalizes to configurations in 1$">, involving graphs of (non-holomorphic) polynomial maps with small coefficients.

     

The limiting case of the Marcinkiewicz integral: growth for convex sets

The Marcinkiewicz integral

plays a well-known and prominent role in harmonic analysis. In this paper, we estimate the growth of it in the limiting case . Throughout, we assume that is convex; it is interesting that this condition cannot be dropped.