Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1605.08790

A simple characterization of homogeneous Young measures and weak $L^1$ convergence of their densities

Piotr Puchała
View original source ↗Read source document ↗Export metadata ↓

Abstract

We formulate a simple characterization of homogeneous Young measures associated with measurable functions. It is based on the notion of the quasi-Young measure introduced in the previous article published in this Journal. First, homogeneous Young measures associated with the measurable functions are recognized as the constant mappings defined on the domain of the underlying function with values in the space of probability measures on the range of these functions. Then the characterization of homogeneous Young measures via image measures is formulated. Finally, we investigate the connections between weak convergence of the homogeneous Young measures understood as elements of the Banach space of scalar valued measures and the weak* L1 sequential convergence of their densities. A scalar case of the smooth functions and their Young measures being Lebesgue-Stieltjes measures is also analyzed.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Piotr Puchała. 2016-12-25. A simple characterization of homogeneous Young measures and weak $L^1$ convergence of their densities. https://doi.org/10.1080/02331934.2016.1269261

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Conditional expectation operators on $C(X)$

At the COSAEF conference in 2021, several participants asked the question whether a conditional expectation operator in the sense of Kuo, Labaushagne and Watson could be constructed in vector lattices other than $\mathcal{L}_p$ spaces and in particular in $C(X)$. This work answers positively to this question and participates in an old discussion on integrals in $C(X)$ space.

math.FA↗
arXiv2026

Fixed Point Rigidity of the Operator $Γ_pΠ_p^\ast$ and the LYZ Conjecture

We characterize the fixed points of the operator $Γ_pΠ_p^\ast$ for $n\geq 3$ and $1 0$ if and only if $K$ is an origin-centered ellipsoid, thereby settling the Lutwak--Yang--Zhang fixed-point conjecture in this range. Our proof is based on a variational analysis along linear reflection shadow systems. To address the nonlinear structure of the $L_p$ setting, we introduce the $L_p$-Projection Rolodex, which provides a dimensional reduction of the volume of the polar $L_p$-projection body to weighted lower-dimensional sectional functionals. A suitable change of variables, together with Ball's harmonic Prékopa--Leindler inequality, yields the convexity needed along the shadow system. Under the fixed-point condition, a first-variation identity then forces $\operatorname{vol}_n(Π_p^\ast K_t)$ to remain constant throughout the deformation. The rigidity statement follows from the equality characterization under Steiner symmetrization.

math.FA↗
arXiv2026

Logarithmic oscillatory multipliers and log-subdyadic square functions

We develop square-function estimates for Fourier multipliers whose local oscillation scale is \[ ρ(R)=\frac{R}{(\log R)^{γ-1}}, \qquad γ>1. \] This scale lies strictly between the dyadic scale and every fixed power-subdyadic scale at high frequency. For high-frequency symbols satisfying a localized Sobolev condition on balls of radius comparable to $ρ(R)$, we prove a pointwise square-function estimate and a weighted $L^2$ multiplier inequality. After adjoining a smooth compactly supported low-frequency part, we derive unweighted $L^p$ bounds. The weighted estimate is governed by a logarithmic geometric maximal operator which is strongly bounded above the critical $L^r$ threshold, satisfies weak type at the critical equality, and fails even weak type below it. As a model application, consider \[ L(ξ)=\frac12\log(e^2+|ξ|^2), \qquad m_{γ,β}(ξ)=L(ξ)^{-β}e^{iL(ξ)^γ}. \] For $p=2$, the associated multiplier is bounded on $L^2$ for every $β\geq0$. For $1 d(γ-1)\left|\frac12-\frac1p\right|. \] At the critical equality we obtain the corresponding Lorentz endpoint estimates.

math.FA↗