Abstract Let $$g(z)=\int _0^zp(t)\exp (q(t))\,dt+c$$ g ( z ) = ∫ 0 p t exp q ...

In the following paper, one studies, given a bounded, connected open set $\Omega$ $\subseteq$ R n , $\kappa$ > 0, positive Radon measure $\mu$ 0 in and (signed) on satisfying $\mu$($\Omega$) = |$\mu$| $\kappa$$\mu$ possibility of solving equation div u by vector field |u| $\kappa$w (where w is an integrable weight only related to geometry 0), together with mild boundary condition. This extends ...

Let λ(X) denote Lebesgue measure. If X ⊆ [0, 1] and r ∈ (0, 1) then the r-Hausdorff capacity of X is denoted by H(X) and is defined to be the infimum of all ∑ ∞ i=0 λ(Ii) r where {Ii}i∈ω is a cover of X by intervals. The r Hausdorff capacity has the same null sets as the r-Hausdorff measure which is familiar from the theory of fractal dimension. It is shown that, given r < 1, it is possible to ...

There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine’s theorem and Jarńık’s theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that ...

We consider several natural situations where the union or intersection of an uncountable family of measurable (in various senses) sets with a “good” additional structure is again measurable or may fail to be measurable. We primarily deal with Lebesgue measurable sets and sets with the Baire property. In particular, uncountable unions of sets homeomorphic to a closed Euclidean simplex are consid...

Let Γ be a closed set in Rn with the Lebesgue measure |Γ| = 0. The first aim of the paper is to give a Fourier analytical characterization of the Hausdorff dimension of Γ. Let 0 < d < n. If there exist a Borel measure μ with supp μ ⊂ Γ and constants c1 > 0 and c2 > 0 such that c1r ≤ μ(B(x, r)) ≤ c2r for all 0 < r < 1 and all x ∈ Γ, where B(x, r) is a ball with centre x and radius r, then Γ is c...

We introduce local grand Lebesgue spaces, over a quasi-metric measure space \( ( X,d, \mu ) \), where the is “aggrandized” not everywhere but only at given closed set F of zero. show that such spaces coincide for different choices aggrandizers if their Matuszewska–Orlicz indices are positive. Within framework we study maximal operator, singular operators with standard kernel, and potential type...

Abstract In this paper we show that generic continuous Lebesgue measure-preserving circle maps have the s-limit shadowing property. addition, obtain is a property also for maps. particular, implies classical shadowing, periodic and limit are in these two settings as well.