Related papers: Which powers of holomorphic functions are integrab…

We prove that the pointwise product of two holomorphic functions of the upper half-plane, one in the Hardy space $\mathcal H^1$, the other one in its dual, belongs to a Hardy type space. Conversely, every holomorphic function in this space…

We prove that there is a continuous non-negative function $g$ on the unit sphere in $\cd$, $d \geq 2$, whose logarithm is integrable with respect to Lebesgue measure, and which vanishes at only one point, but such that no non-zero bounded…

There is a known generalization of the classical Schwarz lemma to holomorphic functions from the polydisk to the disk. In this paper, we characterize those functions which satisfy equality everywhere in this generalized inequality: they are…

We relate analytically defined deformations of modular curves and modular forms from the literature to motivic periods via cohomological descriptions of deformation theory. Leveraging cohomological vanishing results, we prove the existence…

In this paper we obtain the complete description of all indecomposable characters (central positive-definite functions) of inductive limits of the symmetric groups under block diagonal embedding. As a corollary we obtain the full…

Let us have a family of ideals of holomorphic functions vanishing at N distinct points of a complex manifold, all tending to a single point. As is known, convergence of the ideals does not guarantee the convergence of the pluricomplex Green…

Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld's upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note we show that the…

Let U be the closed unit disc in C and let p be a point on the unit circle. Let f be a continuous function on U which extends holomorphically from each circle contained in U and centered at the origin, and from each circle contained in U…

We study noncommutative versions of holomorphic and harmonic functions on the unit disk.

Given an uncountable algebraically closed field $K$, we proved that if partially defined function $f\colon K \times \dots \times K \dashrightarrow K$ defined on a Zariski open subset of the $n$-fold Cartesian product $K \times \dots \times…

In this paper studied isometries of $F$-spaces of integrable functions with logarithm. In particular, using passports of Boolean algebra, a necessary and sufficient condition of isometry $F$-spaces of integrable functions of logarithm with…

We establish an edge of the wedge theorem for the sheaf of holomorphic functions with exponential growth at infinity and construct the sheaf of Laplace hyperfunctions in several variables. We also study the fundamental properties of the…

In this report, we show that all n-variable Boolean function can be represented as polynomial threshold functions (PTF) with at most $0.75 \times 2^n$ non-zero integer coefficients and give an upper bound on the absolute value of these…

We generalize the formula for the log canonical threshold(LCT) of plane curves over the complex numbers to arbitrary characteristics. Our proof relies purely on valuation theory, instead of on the theory of $D$-modules.

We prove that the graph of a continuous function $f$, defined on a domain of ${\mathbb C}^n$, is pluripolar if and only if $f$ is holomorphic.

Let ${\nu}_q(n)$ be the p-adic valuation of $n$. We show that the power series with coefficients ${\nu}_q(n)$, respectively ${\nu}_p(n)(\mathrm{ mod\;} k)$, are non-holonomic and not algebraic in characteristic 0. We find infinitely many…

By using Cauchy integral formula in the theory of complex functions, the authors establish some integral representations for the principal branches of several complex functions involving the logarithmic function, find some properties, such…

From some observations on the linear differential operators occurring in the Lattice Green function of the d-dimensional face centred and simple cubic lattices, and on the linear differential operators occurring in the n-particle…

We characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. We then generalize this result to a characterization of normal functions…

In this note, we show that the values of integrals of the log-tangent function with respect to any square-integrable function on $\left[0 , \frac{\pi}{2} \right]$ may be determined by a finite or infinite sum involving the Riemann…