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We develop a method for proving sup-norm and Holder estimates for $overline{partial}$ on wide class of finite type pseudoconvex domains in $mathbb{C}^n$. A fundamental obstruction to proving sup-norm estimates is the possibility of singular complex curves with exceptionally high order of contact with the boundary. Our method handles this problem, and in $mathbb{C}^3$, we prove sup-norm and Holder estimates for all bounded, pseudoconvex domains with real-analytic boundary.
We study Sobolev estimates for solutions of the inhomogenous Cauchy-Riemann equations on annuli in $cx^n$, by constructing exact sequences relating the Dolbeault cohomology of the annulus with respect to Sobolev spaces of forms with those of the enve
We consider parabolic systems with nonlinear dynamic boundary conditions, for which we give a rigorous derivation. Then, we give them several physical interpretations which includes an interpretation for the porous-medium equation, and for certain re
Consider a $2$-nondegenerate constant Levi rank $1$ rigid $mathcal{C}^omega$ hypersurface $M^5 subset mathbb{C}^3$ in coordinates $(z, zeta, w = u + iv)$: [ u = Fbig(z,zeta,bar{z},bar{zeta}big). ] The Gaussier-Merker model $u=frac{zbar{z}+ frac{1}{2}
The goal of this article is to prove the Sum of Squares Conjecture for real polynomials $r(z,bar{z})$ on $mathbb{C}^3$ with diagonal coefficient matrix. This conjecture describes the possible values for the rank of $r(z,bar{z}) |z|^2$ under the hypot
In the Gaussian white noise model, we study the estimation of an unknown multidimensional function $f$ in the uniform norm by using kernel methods. The performances of procedures are measured by using the maxiset point of view: we determine the set o