Holder Singularity Research Articles

Multifractal Topography: A Tool to Measure Tectonic Complexity in the Zagros Mountain Range

In this research, it is tried to measure tectonic complexity from the multifractal attributes of the Earth’s surface topography in the Zagros Mountain range. For this aim the multifractal characteristics of topography have been achieved using the strange attractor formalism. The approach is based on wavelet–fractal calculation of one-dimensional topography cross-sections. Wavelet coefficients have been employed for the multifractal formalism in order to obtain the multifractal Holder singularity and generalized fractal dimension spectrums. The linked shape parameters of the spectrums are considered to determine the multifractal attributes of topography data sets. Calculations indicate the topography is scale invariant and likely to be chaotic; however, every region has its own multifractal characteristics. The obtained results suggest a relatively heterogeneous topography for the middle parts of the simple-folded belt and a quite homogeneous one for the central portions of the high-Zagros thrust belt. Correspondingly, it is concluded the tectonic processes involved in the central parts of the simple-folded belt are relatively intricate and heterogeneous. While in the central portions of the high-Zagros belt, those processes are fairly homogeneous and act in a simple way. Our findings are in correspondence with the tectonic background of the Zagros fold and thrust belt reported in the literature. This study offers a quantitative measure and provides comparative results to evaluate the topography and superficial tectonic features of the crust in the Zagros Mountain range, using a wavelet based strange attractor formalism.

On the traces of Sobolev functions on the boundary of an anisotropic cusp

Given a “zero” cusp \(G_{\vec \lambda } \) with an anisotropic Holder singularity at the vertex, we consider various embedding theorems for the Sobolev spaces \(W_p^1 (G_{\vec \lambda } )\) and the questions of embeddings of the traces of Sobolev functions into Lebesgue classes on the boundary of the cusp.

Singularity Spectrum of Generic α-Hölder Regular Functions After Time Subordination

A question of Yves Meyer motivated the research concerning “time” subordinations of real functions. Denote by \({\mathcal {B}}_{1}^{\alpha}\) the metric space of functions with Lipschitz constant 1 defined on [0,1], equipped with the complete metric defined via the supremum norm. Given a function \(g\in {\mathcal {B}}_{1}^{\alpha}\) one obtains a time subordination of g simply by considering the composite function Z=g○f, where f∈ℳ:={f:f(0)=0, f(1)=1 and f is a continuous nondecreasing function on [0,1]}. The metric space \(\mathcal {E}^{\alpha}=\mathcal {M}\times {\mathcal {B}}_{1}^{\alpha}\) equipped with the product supremum metric is a complete metric space. In this paper for all α∈[0,1) multifractal properties of g○f are investigated for a generic (typical) element (f,g)∈ℰ α . In particular we determine the generic Holder singularity spectrum of g○f.

Weighted Sobolev spaces and embedding theorems

In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A p -condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. The proposed method is based on the concept of 'generalized' quasiconformal homeomorphisms (homeomorphisms with bounded mean distortion). The choice of the homeomorphism type depends on the choice of the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition operators for weighted Sobolev spaces. With the help of these homeomorphism classes the embedding problem for non-smooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic Holder singularities demonstrate the sharpness of our machinery comparatively with known results.

Singularity sets of Lévy processes

We completely describe the size and large intersection properties of the Holder singularity sets of Levy processes. We also study the set of times at which a given function cannot be a modulus of continuity of a Levy process. The Holder singularity sets of the sample paths of certain random wavelet series are investigated as well.

Traces of functions of generalized Sobolev classes

Considering the Sobolev type function classes on a metric space equipped with a Borel measure we address the question of compactness of embeddings of the space of traces into Lebesgue spaces on the sets of less “dimension.” Also, we obtain compactness conditions for embeddings of the traces of the classical Sobolev spaces W 1 on the “zero” cusp with a Holder singularity at the vertex.

On the traces of Sobolev functions on the boundary of a cusp with a Hölder singularity

For the Sobolev classes W p 1 on a “zero” cusp with a Holder singularity at the vertex, we consider the question of compactness of the embedding of the traces of Sobolev functions into the Lebesgue classes on the boundary of the cusp.

Homeomorphisms that induce monomorphisms of Sobolev spaces

LetG, G′ be domains in ℝn. We obtain a geometrical description of the class of all homeomorphisms ϕ:G→ G′ that induce bounded operators ϕ* from the seminormed Sobolev spaceL p 1 (G′) toL p 1 (G) by the rule ϕ* u =u o ϕ. Forp-Poincare domains the same classes of homeomorphisms induce bounded operators for classical Sobolev spacesW p 1 . These classes of homeomorphisms are natural generalizations of the class of quasiconformal homeomorphisms that correspond to the casep=n. We demonstrate some applications of our results for embedding theorems in domains with Holder singularities.