ABSTRACT. We study projections onto non-degenerate one-dimensional families of lines and planes in R 3 . Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most 1/2-dimensional sets B ⊂ R 3 is typically preserved under one-dimensional families of projections onto lines. We improve the result by an ε, proving that if dim H B = s > 1/2, then the packing dimension of the projections is almost surely at least σ(s) > 1/2. For projections onto planes, we obtain a similar bound, with the threshold 1/2 replaced by 1. In the special case of self-similar sets K ⊂ R 3 without rotations, we obtain a full Marstrand type projection theorem for oneparameter families of projections onto lines. The dim H K ≤ 1 case of the result follows from recent work of M. Hochman, but the dim H K > 1 part is new: with this assumption, we prove that the projections have positive length almost surely. CONTENTS

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Abstract. We prove analogs of classical almost sure dimension theorems for Euclidean projection mappings in the first Heisenberg group, equipped with a sub-Riemannian metric.

Abstract. We propose a method by modulus of curve families to identify extremal quasiconformal mappings in the Heisenberg group. This approach allows to study minimizers not only for the maximal distortion but also for a mean distortion functional, where the candidate for the extremal map is not required to have constant distortion. As a counterpart of a classical Euclidean extremal problem, we consider the class of quasiconformal mappings between two spherical annuli in the Heisenberg group. Using logarithmic-type coordinates we can define an analog of the classical Euclidean radial stretch map and discuss its extremal properties both with respect to the maximal and the mean distortion. We prove that our stretch map is a minimizer for a mean distortion functional and it minimizes the maximal distortion within the smaller subclass of sphere-preserving mappings.

We study singular integral operators induced by 3-dimensional Calderón-Zygmund kernels in the Heisenberg group. We show that if such an operator is L 2 bounded on vertical planes, with uniform constants, then it is also L 2 bounded on all intrinsic graphs of compactly supported C 1,α functions over vertical planes.In particular, the result applies to the operator R induced by the kernel, the horizontal gradient of the fundamental solution of the sub-Laplacian. The L 2 boundedness of R is connected with the question of removability for Lipschitz harmonic functions. As a corollary of our result, we infer that the intrinsic graphs mentioned above are non-removable. Apart from subsets of vertical planes, these are the first known examples of non-removable sets with positive and locally finite 3-dimensional measure.