<p>For many questions of complex analysis of several variables classical potential theory does not provide suitable tools and is replaced by pluripotential theory. The latter got many important applications within complex analysis and related fields. Pluripolar sets play a special role in pluripotential theory. These are the exceptional sets this theory. Complete pluripolar sets are especially important. In the thesis we study complete pluripolar sets and pluripolar hulls. We show that in some sense there are many complete pluripolar sets. We show that on each closed subset of the complex plane there is continuous function whose graph is complete pluripolar. On the other hand we study the propagation of pluripolar sets, equivalently we study pluripolar hulls. We relate the pluripolar hull of a graph to fine analytic continuation of the function. Fine analytic continuation of an analytic function over the unit disk is related to the fine topology introduced by Cartan and to the previously known notion of finely analytic functions. We show that fine analytic continuation implies non-triviality of the pluripolar hull. Concerning the inverse direction, we show that the projection of the pluripolar hull is finely open. The difficulty to judge from non-triviality of the pluripolar hull about fine analytic continuation lies in possible multi-sheetedness. If however the pluripolar hull contains the graph of a smooth extension of the function over a fine neighborhood of a boundary point we indeed obtain fine analytic continuation.</p>

For many questions of complex analysis of several variables classical potential theory does not provide suitable tools and is replaced by pluripotential theory. The latter got many important applications within complex analysis and related fields. Pluripolar sets play a special role in pluripotential theory. These are the exceptional sets this theory. Complete pluripolar sets are especially important. In the thesis we study complete pluripolar sets and pluripolar hulls. We show that in some sense there are many complete pluripolar sets. We show that on each closed subset of the complex plane there is continuous function whose graph is complete pluripolar. On the other hand we study the propagation of pluripolar sets, equivalently we study pluripolar hulls. We relate the pluripolar hull of a graph to fine analytic continuation of the function. Fine analytic continuation of an analytic function over the unit disk is related to the fine topology introduced by Cartan and to the previously known notion of finely analytic functions. We show that fine analytic continuation implies non-triviality of the pluripolar hull. Concerning the inverse direction, we show that the projection of the pluripolar hull is finely open. The difficulty to judge from non-triviality of the pluripolar hull about fine analytic continuation lies in possible multi-sheetedness. If however the pluripolar hull contains the graph of a smooth extension of the function over a fine neighborhood of a boundary point we indeed obtain fine analytic continuation.

This thesis consists of a comprehensive summary and six scientific papers dealing with the boundary behaviour of functions related to parabolic partial differential equations and several complex variables. Paper I concerns solutions to non-linear parabolic equations of linear growth. The main results include a backward Harnack inequality, and the Hölder continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary of an NTA cylinder. It is also shown that the Riesz measure associated with such solutions has the doubling property. Paper II is concerned with solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a weight in the Muckenhoupt class 1+2/n. Two main results are that non-negative solutions which vanish continuously on the lateral boundary of an NTA cylinder satisfy a backward Harnack inequality and that the quotient of two such functions is Hölder continuous up to the boundary. Another result is that the parabolic measure associated to such equations has the doubling property. In Paper III, it is shown that a bounded pseudoconvex domain whose boundary is α-Hölder for each 0&lt;α&lt;1, is hyperconvex. Global estimates of the exhaustion function are given. In Paper IV, it is shown that on the closure of a domain whose boundary locally is the graph of a continuous function, all plurisubharmonic functions with continuous boundary values can be uniformly approximated by smooth plurisubharmonic functions defined in neighbourhoods of the closure of the domain. Paper V studies  Poletsky’s notion of plurisubharmonicity on compact sets. It is shown that a function is plurisubharmonic on a given compact set if, and only if, it can be pointwise approximated by a decreasing sequence of smooth plurisubharmonic functions defined in neighbourhoods of the set. Paper VI introduces the notion of a P-hyperconvex domain. It is shown that in such a domain, both the Dirichlet problem with respect to functions plurisubharmonic on the closure of the domain, and the problem of approximation by smooth plurisubharmoinc functions in neighbourhoods of the closure of the domain have satisfactory answers in terms of plurisubharmonicity on the boundary.