arXiv subjects
Publications and source records attributed to Peter Hästö.
In this paper we explore the potential of the double phase functional in an image processing context. To this end, we study minimizers of the double phase energy for functions with bounded variation and show that this energy can be obtained by $Γ$-convergence or relaxation of regularized functionals. A central tool is a capped fractional maximal function of the derivative of $BV$ functions.
We prove that a $Φ$-function can be extended from a domain $Ω$ to all of $\mathbb R^n$ while preserving crucial properties for harmonic analysis on the generalized Orlicz space $L^Φ$.
We prove higher integrability of the spatial gradient of weak solutions to parabolic systems with $ϕ$-growth, where $φ=φ(t)$ is a general Orlicz function. The parabolic systems need be neither degenerate nor singular. Our result is a generalized version of the one of J. Kinnunen and J. Lewis [Duke Math. J. 102 (2000), no. 2, 253--271] for the parabolic $p$-Laplace systems.
We consider proper subdomains $G$ of $\mathbb{R}^n$ and their images $G'=f(G)$ under quasiconformal mappings $f$ of $\mathbb{R}^n$. We compare the distance ratio metrics of $G$ and $G'$; as an application we show that $φ$-uniform domains are preserved under quasiconformal mappings of $\mathbb{R}^n$. A sufficient condition for $φ$-uniformity is obtained in terms of the quasi-symmetry condition. We give a geometric condition for uniformity: If $G\subset\mathbb{R}^n$ is $ϕ$-uniform and satisfies the twisted cone condition, then it is uniform. We also construct a planar $ϕ$-uniform domain whose complement is not $ψ$-uniform for any $ψ$.
The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz-Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.
We prove optimal integrability results for solutions of the $p(\cdot)$-Laplace equation in the scale of (weak) Lebesgue spaces. To obtain this, we show that variable exponent Riesz and Wolff potentials maps $L^1$ to variable exponent weak Lebesgue spaces.
We obtain explicit and simple conditions which in many cases allow one decide, whether or not a Denjoy domain endowed with the Poincare or quasihyperbolic metric is Gromov hyperbolic. The criteria are based on the Euclidean size of the complement. As a corollary, the main theorem allows to deduce the non-hyperbolicity of any periodic Denjoy domain.
In this article we introduce Triebel--Lizorkin spaces with variable smoothness and integrability. Our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years. Vector-valued maximal inequalities do not work in the generality which we pursue, and an alternate approach is thus developed. Applying it, we give molecular and atomic decomposition results and show that our space is well-defined, i.e., independent of the choice of basis functions. As in the classical case, a unified scale of spaces permits clearer results in cases where smoothness and integrability interact, such as Sobolev embedding and trace theorems. As an application of our decomposition, we prove optimal trace theorems in the variable indices case.
Packing density is a permutation occurrence statistic which describes the maximal number of permutations of a given type that can occur in another permutation. In this article we focus on containment of sets of permutations. Although this question has been tangentially considered previously, this is the first systematic study of it. We find the packing density for various special sets of permutations and study permutation and pattern co-occurrence.
In this article we generalize packing density problems from permutations to patterns with repeated letters and generalized patterns. We are able to find the packing density for some classes of patterns and several other short patterns.