The Lebesgue-Borel Measure and Hausdorff Measures

2017 ◽  
pp. 45-69
Keyword(s):  
Borel Measure ◽  
Hausdorff Measures

scholarly journals Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Changbao Pang ◽  
Antti Perälä ◽  
Maofa Wang
Keyword(s):  
Borel Measure ◽  
Bergman Spaces ◽  
Embedding Theorem ◽  
Weighted Bergman Spaces ◽  
Embedding Theorems ◽  
Positive Borel Measure ◽  
Doubling Property ◽  
Area Operator ◽  
Special Cases ◽  
Upper Half Plane

AbstractWe establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure $$d\omega (y)dx$$ d ω ( y ) d x with the doubling property $$\omega (0,2t)\le C\omega (0,t)$$ ω ( 0 , 2 t ) ≤ C ω ( 0 , t ) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

scholarly journals On density with respect to equivalent measures

2010 ◽  
Vol 43 (1) ◽  
Author(s):  
Artur Bartoszewicz ◽  
Małgorzata Filipczak ◽  
Tadeusz Poreda
Keyword(s):  
Metric Space ◽  
Borel Measure ◽  
Borel Subset ◽  
Density Point

AbstractIn the paper there is disscussed a notion of a density point of a Borel subset of a metric space with respect to a Borel measure

The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(\Omega,f)$

2021 ◽  
Vol 127 (2) ◽  
pp. 287-316
Author(s):  
Ayoub El Gasmi
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Borel Measure ◽  
Positive Borel Measure ◽  
Hessian Equation ◽  
Hessian Operator ◽  
Hyperconvex Domain ◽  
Complex Hessian Operator

Let $\Omega\subset \mathbb{C}^{n}$ be a bounded $m$-hyperconvex domain, where $m$ is an integer such that $1\leq m\leq n$. Let $\mu$ be a positive Borel measure on $\Omega$. We show that if the complex Hessian equation $H_m (u) = \mu$ admits a (weak) subsolution in $\Omega$, then it admits a (weak) solution with a prescribed least maximal $m$-subharmonic majorant in $\Omega$.

scholarly journals The Hausdorff dimension and exact Hausdorff measure of random recursive sets with overlapping

2002 ◽  
Vol 31 (1) ◽  
pp. 11-21 ◽  
Author(s):  
Hongwen Guo ◽  
Dihe Hu
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Intersection Property ◽  
Random Sets ◽  
Hausdorff Measures ◽  
Open Set Condition ◽  
Finite Intersection ◽  
Open Set ◽  
Finite Intersection Property

We weaken the open set condition and define a finite intersection property in the construction of the random recursive sets. We prove that this larger class of random sets are fractals in the sense of Taylor, and give conditions when these sets have positive and finite Hausdorff measures, which in certain extent generalize some of the known results, about random recursive fractals.

Embeddings of Müntz Spaces in

2019 ◽  
Vol 62 (1) ◽  
pp. 1-9
Author(s):  
Ihab Al Alam ◽  
Pascal Lefèvre
Keyword(s):  
Borel Measure ◽  
Essential Norm ◽  
Weak Compactness ◽  
Positive Borel Measure ◽  
Absolutely Continuous ◽  
Borel Measures ◽  
Embedding Operator ◽  
Müntz Spaces

AbstractIn this paper, we discuss the properties of the embedding operator $i_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6EC}}:M_{\unicode[STIX]{x1D6EC}}^{\infty }{\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707})$, where $\unicode[STIX]{x1D707}$ is a positive Borel measure on $[0,1]$ and $M_{\unicode[STIX]{x1D6EC}}^{\infty }$ is a Müntz space. In particular, we compute the essential norm of this embedding. As a consequence, we recover some results of the first author. We also study the compactness (resp. weak compactness) and compute the essential norm (resp. generalized essential norm) of the embedding $i_{\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2}}:L^{\infty }(\unicode[STIX]{x1D707}_{1}){\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707}_{2})$, where $\unicode[STIX]{x1D707}_{1}$, $\unicode[STIX]{x1D707}_{2}$ are two positive Borel measures on [0, 1] with $\unicode[STIX]{x1D707}_{2}$ absolutely continuous with respect to $\unicode[STIX]{x1D707}_{1}$.

scholarly journals Flows of measures generated by vector fields

2018 ◽  
Vol 148 (4) ◽  
pp. 773-818
Author(s):  
Emanuele Paolini ◽  
Eugene Stepanov
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Borel Measure ◽  
Weak Sense ◽  
Positive Borel Measure ◽  
Smooth Structure ◽  
Smooth Vector ◽  
Integral Curves ◽  
The Given ◽  
Measurable Vector

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

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