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Our main result introduces a new way to characterize two-dimensional finite ball quotients by algebraicity of their Bergman kernels. This characterization is particular to dimension two and fails in higher dimensions, as is illustrated by a counterexample in dimension three constructed in this paper. As a corollary of our main theorem, we prove, e.g., that a smoothly bounded strictly pseudoconvex domain G in C 2 has rational Bergman kernel if and only if there is a rational biholomorphism from G to B 2 . [ABSTRACT FROM AUTHOR]

In this article, we use a particular case of convolution as an operator to discuss a number of problems concerning multiplier results between function spaces such as Hardy and B p -spaces. As a consequence, we extend certain well-known results on fractional derivatives and fractional integrals. Also, we find condition on the parameters b , c such that 풫 b , c in B p . [ABSTRACT FROM AUTHOR]

In this work, we study some algebraic and topological properties of the ring of global analytic functions on the normalization of a reduced complex analytic space. If is a Stein space, we characterize in terms of the (topological) completion of the integral closure of the ring of global holomorphic functions on X (inside its total ring of fractions) with respect to the usual Fréchet topology of. This shows that not only the Stein space but also its normalization is completely determined by the ring of global analytic functions on X. This result was already proved in 1988 by Hayes–Pourcin when is an irreducible Stein space, whereas in this paper we afford the general case. We also analyze the real underlying structures and of a reduced complex analytic space and its normalization. We prove that the complexification of provides the normalization of the complexification of if and only if is a coherent real analytic space. Roughly speaking, coherence of the real underlying structure is equivalent to the equality of the following two combined operations: (1) normalization + real underlying structure + complexification, and (2) real underlying structure + complexification + normalization. [ABSTRACT FROM AUTHOR]

In this work we present the concept of C-semianalytic subset of a real analytic manifold and more generally of a real analytic space. C-semianalytic sets can be understood as the natural generalization to the semianalytic setting of global analytic sets introduced by Cartan ( C-analytic sets for short). More precisely S is a C-semianalytic subset of a real analytic space $$(X,{\mathcal {O}}_X)$$ if each point of X has a neighborhood U such that $$S\cap U$$ is a finite boolean combinations of global analytic equalities and strict inequalities on X. By means of paracompactness C-semianalytic sets are the locally finite unions of finite boolean combinations of global analytic equalities and strict inequalities on X. The family of C-semianalytic sets is closed under the same operations as the family of semianalytic sets: locally finite unions and intersections, complement, closure, interior, connected components, inverse images under analytic maps, sets of points of dimension k, etc. although they are defined involving only global analytic functions. In addition, we characterize subanalytic sets as the images under proper analytic maps of C-semianalytic sets. We prove also that the image of a C-semianalytic set S under a proper holomorphic map between Stein spaces is again a C-semianalytic set. The previous result allows us to understand better the structure of the set N( X) of points of non-coherence of a C-analytic subset X of a real analytic manifold M. We provide a global geometric-topological description of N( X) inspired by the corresponding local one for analytic sets due to Tancredi and Tognoli (Riv Mat Univ Parma (4) 6:401-405, 1980), which requires complex analytic normalization. As a consequence it holds that N( X) is a C-semianalytic set of dimension $${\le }\dim (X)-2$$ . [ABSTRACT FROM AUTHOR]

In this work we present the concept of amenable C- semianalytic subset of a real analytic manifold M and study the main properties of this type of sets. Amenable C-semianalytic sets can be understood as globally defined semianalytic sets with a neat behavior with respect to Zariski closure. This fact allows us to develop a natural definition of irreducibility and the corresponding theory of irreducible components for amenable C-semianalytic sets. These concepts generalize the parallel ones for: complex algebraic and analytic sets, C-analytic sets, Nash sets and semialgebraic sets. [ABSTRACT FROM AUTHOR]