On the complexification of the Weierstrass non-differentiable function.
On the complexity of some -ideals of -P-porous sets
Let be a porosity-like relation on a separable locally compact metric space . We show that the -ideal of compact --porous subsets of (under some general conditions on and ) forms a -complete set in the hyperspace of all compact subsets of , in particular it is coanalytic and non-Borel. Our general results are applicable to most interesting types of porosity. It is shown in the cases of the -ideals of -porous sets, --porous sets, -strongly porous sets, -symmetrically porous sets...
On the complexity of sums of Dirichlet measures
Let be the set of all Dirichlet measures on the unit circle. We prove that is a non Borel analytic set for the weak* topology and that is not norm-closed. More precisely, we prove that there is no weak* Borel set which separates from (or even , the set of all measures singular with respect to every measure in . This extends results of Kaufman, Kechris and Lyons about and and gives many examples of non Borel analytic sets.
On the conformal gauge of a compact metric space
In this article we study the Ahlfors regular conformal gauge of a compact metric space , and its conformal dimension . Using a sequence of finite coverings of , we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute using the critical exponent associated to the combinatorial modulus.
On the connection between Hausdorff measures and generalized capacity
On the continuity of certain non-additive set functions
On the continuity of the Faber mapping
On the covering by small random intervals
On the density maxima of a function
On the derivation and covering properties of a differentiation basis
On the descriptive sot theory of the lexicographic square
On the descriptive theory of sets
On the difference property of Borel measurable functions
If an atomlessly measurable cardinal exists, then the class of Lebesgue measurable functions, the class of Borel functions, and the Baire classes of all orders have the difference property. This gives a consistent positive answer to Laczkovich's Problem 2 [Acta Math. Acad. Sci. Hungar. 35 (1980)]. We also give a complete positive answer to Laczkovich's Problem 3 concerning Borel functions with Baire-α differences.
On the difference property of families of measurable functions
We show that, generally, families of measurable functions do not have the difference property under some assumption. We also show that there are natural classes of functions which do not have the difference property in ZFC. This extends the result of Erdős concerning the family of Lebesgue measurable functions.
On the differentiation of integrals of functions from Lφ(L)
On the differentiation of integrals of functions from Orlicz classes
On the differentiation theorem in metric groups
On the dimension of an irrigable measure
On the dimension of limit sets of geometrically finite Möbius groups.
On the dimension of -harmonic measure in space
Let , and let , be given. In this paper we study the dimension of -harmonic measures that arise from non-negative solutions to the -Laplace equation, vanishing on a portion of , in the setting of -Reifenberg flat domains. We prove, for , that there exists small such that if is a -Reifenberg flat domain with , then -harmonic measure is concentrated on a set of -finite -measure. We prove, for , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of -harmonic measure...