Compact Operators on the Bergman Spaces with Variable Exponents on the Unit Disc of C

We study the compactness of some classes of bounded operators on the Bergman space with variable exponent. We show that via extrapolation, some results on boundedness of the Toeplitz operators with general L1 symbols and compactness of bounded operators on the Bergman spaces with constant exponents can readily be extended to the variable exponent setting. In particular, if S is a finite sum of finite products of Toeplitz operators with symbols from class BT, then S is compact if and only if the Berezin transform of S vanishes on the boundary of the unit disc

Mixed norm spaces of analytic functions as spaces of generalized fractional derivatives of functions in hardy type spaces

The aim of the paper is twofold. First, we present a new general approach to the definition of a class of mixed norm spaces of analytic functions A(q;X)(D), 1 <= q < infinity on the unit disc D. We study a problem of boundedness of Bergman projection in this general setting. Second, we apply this general approach for the new concrete cases when X is either Orlicz space or generalized Morrey space, or generalized complementary Morrey space. In general, such introduced spaces are the spaces of functions which are in a sense the generalized Hadamard type derivatives of analytic functions having l(q) summable Taylor coefficients.Russian Fund of Basic Research [15-01-02732]; SFEDU grant [07/2017-31]info:eu-repo/semantics/publishedVersio

Zeros of Random Holomorphic Sections of Semipositive Line Bundles on Punctured Riemann Surfaces

With early works dating back to the 1930’s until today, here is a growing interest in the theory of asymptotic distributions of expected zeros of random polynomials when their degree grows indefinitely. A natural geometric generalization of random polynomials are random sections of a holomorphic line bundle over a complex manifold. In 1999, Shiffman and Zelditch proved that on a compact K¨ahler manifold, the zeros of sections of high tensor powers of a holomorphic line bundle asymptotically equidistribute with respect to the normalized curvature of the line bundle. Their result has numerous applications in mathematical physics and was generalized in many different directions. In this thesis we generalize their result to a semipositively curved holomorphic line bundle over a punctured Riemann surface. To achieve this, we discuss many tools that have proven themselves to represent an appropriate framework to study statistical properties of ensembles of zeros on complex manifolds. We start by proving the existence of spectral gap for the Kodaira Laplacian that is associated to the line bundle. We use this result, together with the technique of analytic localization by Ma and Marinescu, to prove a pointwise global on-diagonal asymptotic expansion of the associated Bergman kernel in our setting. Moreover, we show locally uniform estimates on the Bergman kernel and its derivatives. We use these estimates to prove the locally uniform convergence of the induced Fubini-Study metrics and their potentials to the global curvature and its potential, respectively. We conclude by showing that the expected zeros of holomorphic sections equidistribute with respect to the normalized curvature of the line bundle. Moreover, we apply the theory of meromorphic transforms by Dinh and Sibony estimate the speed of convergence in our equidistribution result