The main goal of this paper is to present an alternative, real variable proof of the T(1)-Theorem for the Cauchy Integral. We then prove that the estimate from below of analytic capacity in terms of total Menger curvature is a direct consequence of the T(1)-Theorem. An example shows that the L 1-BMO estimate for the Cauchy Integral does not follow from L 2 boundedness when the underlying measur...

We study solutions of the Riemann boundary value problem in class Cauchy type integrals with densities from grand Lebesgue spaces domains nonsmooth boundaries. The case piecewise conjugacy coefficients is studied.

Two new inequalities for Riemann--Stieltjes integral are introduced functions of bounded $p$-variation and H\"{o}lder continuous integrators.

We study the local solvability problem for a class of semilinear equations whose linear part is the Kohn Laplacian, acting on top degree forms. We also study the validity of the Poincaré lemma, in top degree, for semilinear perturbations of the tangential Cauchy-Riemann complex.

Many naturally occurring phenomena require theoretical treatment utilizing complex analysis by methods such as the Cauchy-Riemann relations using hyper-geometrical spaces which treat inherently nonlinear, non-dispersive, collective nonlocal resonant states of a quantum system, so as to be consistent with the nonlinearity inherent in General Relativity. Typical quantum approaches form linear app...

We consider the motion of a thin filament of viscous fluid in a HeleShaw cell. The appropriate thin film analysis and use of Lagrangian variables leads to the Cauchy-Riemann system in a surprisingly direct way. We illustrate the inherent ill-posedness of these equations in various contexts.

in the Chow ring with rational coefficients CH(S)Q = ⊕nCH (S)Q. Here ch is the Chern character and Td(TX), Td(TS) stand for the Todd power series evaluated at the Chern classes of the tangent bundle of X, respectively S. Since both sides of (1.1) take values in CH(S)Q := CH (S)⊗Q, only information modulo torsion about the Chern classes of f∗[F ] can be obtained from this identity. The goal of o...

Classically there are two points of view in the study of global existence problems in the theory of functions of a complex variable. One is to piece together local solutions (such as power series), always staying within the category of holomorphic functions. This method seems to have been initiated by Weierstrass; in the theory of several complex variables it has been implemented by the study o...

We discuss the origins of the tangential Cauchy Riemann equation beginning with W. Wirtinger in 1926, and trace the largely unknown early developments until the emergence of the ∂b−Neumann complex in the 1960s. Vienna is a most appropriate venue for a program centered on the ∂− Neumann Problem. Not only did the calculus of the differential operators ∂/∂zj and ∂/∂zj originate in the work of Wilh...