Mean Oscillation Research Articles

Lipschitz continuity for elliptic free boundary problems with Dini mean oscillation coefficients

We establish local interior Lipschitz continuity of the solutions of a class of free boundary elliptic problems assuming the coefficients of the equation of Dini mean oscillation in at least one direction. The novelty in this regularity result lies in the fact that it allows discontinuous coefficients in all but one variable.

Tailoring Optoelectronic Properties of ZnO Nanoparticles via Incorporation of Multiwalled Carbon Nanotube Contents on of ZnO/MWCNT Nanocomposites

The present study demonstrates the tailoring of optoelectronic properties of ZnO nanoparticles (NPs) by adding multi-walled carbon nanotubes (MWCNTs) at different weight ratios. ZnO/MWCNT nanocomposites were successfully prepared using the solution blending method, and the optical spectra of all samples were determined using a UV–vis spectrometer. An increase in the optical absorption coefficient (α) and extinction coefficient (k) was observed with an increase in MWCNT content, resulting in the significant ability to attenuate incident light in ZnO/MWCNT nanocomposites compared to pure ZnO NPs. The optical band gap of pure ZnO NPs decreased from 3.26 eV to 2.58 eV in the ZnO/50 wt% MWCNT nanocomposite. The refractive index values ranged from 1.66 to 1.61 as the wavelength varied between 400 nm and 700 nm. Furthermore, the incorporation of MWCNTs in the ZnO/MWCNT nanocomposite had an impact on various dispersion parameters related to the refractive index. With increasing MWCNT content, the dielectric constant (ε 0) decreased from 2.40 to 1.96, while the mean oscillator wavelength (λ 0) increased from 157.7 nm to 164.2 nm and the oscillator strength (S0) decreased from 5.93 × 10−5 nm−2 to 3.70 × 10−6 nm−2. The localized density of state (N/m*) increased from 7.45 × 1057 to 20.8 × 1057 (m3 Kg) −1, and the long wavelength refractive index (ε ∞) rose from 3.683 to 4.745. Moreover, the plasma frequency (ω p) of the electron exhibited an increase from 2.15 × 1031 to 6.01 × 1031 Hz. These findings highlight the potential of tailoring the optoelectronic properties of ZnO NPs through the incorporation of MWCNTs, paving the way for the development of novel materials with improved optoelectronic characteristics for a wide range of applications.

The John–Nirenberg inequality for Orlicz–Lorentz spaces in a probabilistic setting

The John–Nirenberg inequality for Orlicz–Lorentz spaces in a probabilistic setting

Vanishing mean oscillation and continuity of rearrangements

We study the decreasing rearrangement of functions in VMO, and show that for rearrangeable functions, the mapping f↦f⁎ preserves vanishing mean oscillation. Moreover, as a map on BMO, while bounded, it is not continuous, but continuity holds at points in VMO (under certain conditions). This also applies to the symmetric decreasing rearrangement. Many examples are included to illustrate the results.

A sharp [formula omitted] estimate for the total variation process

For a given p≥2, let X be an Lp bounded martingale and let Y be a martingale of bounded mean oscillation. The paper contains the proof of the estimate ‖∫0∞|d〈X,Y〉t|‖p≤p‖X‖p‖Y‖bmo. The inequality is sharp for each p and the range p≥2 cannot be expanded without additional assumptions on X and Y. The proof rests on the existence of a certain special function, enjoying appropriate size and concavity requirements.

Stability and boundedness of regular solutions for a Sisko flow in an infinite annular porous space

The present article is intended to study the boundedness of solutions for an unsteady non-Newtonian flow, whose strain–stress relationship is provided by the Sisko fluid model. Such kind of flow can appear in different scenarios, but we make it particular to the field of magnetohydrodynamics, as it is a remarkable area of research in its own right. The ideas exposed in this work can be extended, in a similar manner, to a wide range of applications. To make our fluid further general, we use the Darcy’s law to characterize the porous space in which the fluid is flowing. We develop the boundedness criteria provided that the velocity w and the function g=−∂w/∂r satisfy w,g,∂g/∂r,∂2g/∂r2∈L2(0,T;BMO), where BMO means bounded mean oscillation space. For this purpose, we will consider energy estimates in Sobolev spaces and develop the boundedness criteria for the resulting unsteady parabolic nonlinear equation.

On isolated singularities of mappings with finite length distortion

Here we give a survey of various integral criteria in terms of inner dilatations $K_{I}$ for the removability of isolated singularities of mappings with finite length distortion in $\mathbb{R}^{n}$, $n\geq 2$, that are a natural extension of the well-known Martio-Vaisal a mappings with bounded length distortion. In particular, the survey includes many effective integral criteria of the types of BMO, bounded mean oscillation by John-Nirenberg, FMO, finite mean oscillation, Calderon-Zygmund, Lehto, and Orlicz.

Time fractional parabolic equations with partially SMO coefficients

We present the unique solvability in Sobolev spaces of time fractional parabolic equations in divergence and non-divergence forms. The leading coefficients are merely measurable in (t,x1) for aij, 1≤i,j≤d, (i,j)≠(1,1). The coefficient a11 is merely measurable locally either in t or x1. As functions of the remaining variables, the coefficients have small mean oscillations. We consider mixed norm Sobolev spaces with Muckenhoupt weights. Our results generalize previous work on parabolic equations with time fractional derivatives to a much larger class of coefficients and solution spaces.

The commutator of the Bergman projection on strongly pseudoconvex domains with minimal smoothness

Consider a bounded, strongly pseudoconvex domain D⊂Cn with minimal smoothness (namely, the class C2) and let b be a locally integrable function on D. We characterize boundedness (resp., compactness) in Lp(D),p>1, of the commutator [b,P] of the Bergman projection P in terms of an appropriate bounded (resp. vanishing) mean oscillation requirement on b. We also establish the equivalence of such notion of BMO (resp., VMO) with other BMO and VMO spaces given in the literature. Our proofs use a dyadic analog of the Berezin transform and holomorphic integral representations going back (for smooth domains) to N. Kerzman & E. M. Stein, and E. Ligocka.

Bergman projection and BMO in hyperbolic metric: improvement of classical result

The Bergman projection Pα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_\\alpha $$\\end{document}, induced by a standard radial weight, is bounded and onto from L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document} to the Bloch space B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}$$\\end{document}. However, Pα:L∞→B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_\\alpha : L^\\infty \\rightarrow \\mathcal {B}$$\\end{document} is not a projection. This fact can be emended via the boundedness of the operator Pα:BMO2(Δ)→B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_\\alpha :\ extrm{BMO}_2(\\Delta )\\rightarrow \\mathcal {B}$$\\end{document}, where BMO2(Δ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{BMO}_2(\\Delta )$$\\end{document} is the space of functions of bounded mean oscillation in the Bergman metric. We consider the Bergman projection Pω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_\\omega $$\\end{document} and the space BMOω,p(Δ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{BMO}_{\\omega ,p}(\\Delta )$$\\end{document} of functions of bounded mean oscillation induced by 1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<\\infty $$\\end{document} and a radial weight ω∈M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega \\in \\mathcal {M}$$\\end{document}. Here M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {M}$$\\end{document} is a wide class of radial weights defined by means of moments of the weight, and it contains the standard and the exponential-type weights. We describe the weights such that Pω:BMOω,p(Δ)→B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_\\omega :\ extrm{BMO}_{\\omega ,p}(\\Delta )\\rightarrow \\mathcal {B}$$\\end{document} is bounded. They coincide with the weights for which Pω:L∞→B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_\\omega : L^\\infty \\rightarrow \\mathcal {B}$$\\end{document} is bounded and onto. This result seems to be new even for the standard radial weights when p≠2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\ e 2$$\\end{document}.

Fractional Gagliardo–Nirenberg interpolation inequality and bounded mean oscillation

Fractional Gagliardo–Nirenberg interpolation inequality and bounded mean oscillation

Nonstationary Venttsel problems with discontinuous data

The paper deals with Venttsel boundary problems for second-order linear and quasilinear parabolic operators with discontinuous principal coefficients. These are supposed to be functions of vanishing mean oscillation with respect to the space variables, while only measurability is required in the time-variable. We derive a priori estimates in composite Sobolev spaces for the strong solutions, and develop maximal regularity and strong solvability theory for such problems.

Locally uniform domains and extension of bmo functions

We prove that for a domain \(\Omega\subset\mathbb{R}^n\), being \((\epsilon,\delta)\) in the sense of Jones is equivalent to being an extension domain for bmo, the nonhonomogeneous version of the space of functions of bounded mean oscillation on \(\Omega\). Such domains, which can be identified as local versions of uniform domains (defined by requiring the presence of length cigars between nearby points), allow a definition of bmo\((\Omega)\) in terms of "small" and "large" cubes contained in \(\Omega\), where the scale is closely tied to the geometry of the domain.

Fractional maximal functions and mean oscillation on bounded doubling metric measure spaces

Let (X,d,μ) be a doubling metric measure space. We consider the behavior of the fractional maximal function Mα for 0≤α<Q, where Q is the doubling dimension, acting on functions of bounded mean oscillation (BMO) and vanishing mean oscillation (VMO). For α>0, we additionally assume that the space is bounded. We show that Mα is bounded from BMO to BLO, a subclass of BMO, and maps VMO to itself when μ has the annular decay property. We also show by means of examples that the action of Mα is not continuous on these function spaces.

Calderón‐Zygmund estimates for higher order elliptic equations from composite material

We study Calderón‐Zygmund estimates for the weak solution to divergence‐form higher order elliptic equations. We assume that the domain is composed of disjoint subdomains of ‐Reifenberg flat condition and the coefficients are merely measurable in one variable and have small bounded mean oscillation (BMO) in the other variables. Based on our new understanding for the relation between these assumptions and composite material, we establish estimates for ‐th order elliptic equations.

Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents

Abstract In this article, we obtain some sufficient conditions for the boundedness of commutators of p p -adic Hardy-Littlewood operators with symbols in central bounded mean oscillation space and Lipschitz space on the p p -adic function spaces with variable exponents such as the p p -adic local central Morrey, p p -adic Morrey-Herz, and p p -adic local block spaces with variable exponents.

On functions of bounded β-dimensional mean oscillation

Abstract In this paper, we define a notion of β-dimensional mean oscillation of functions u : Q 0 ⊂ ℝ d → ℝ {u:Q_{0}\subset\mathbb{R}^{d}\to\mathbb{R}} which are integrable on β-dimensional subsets of the cube Q 0 {Q_{0}} : ∥ u ∥ BMO β ⁢ ( Q 0 ) := sup Q ⊂ Q 0 ⁡ inf c ∈ ℝ ⁡ 1 l ⁢ ( Q ) β ⁢ ∫ Q | u - c | ⁢ 𝑑 ℋ ∞ β , \displaystyle\|u\|_{\mathrm{BMO}^{\beta}(Q_{0})}\vcentcolon=\sup_{Q\subset Q_{% 0}}\inf_{c\in\mathbb{R}}\frac{1}{l(Q)^{\beta}}\int_{Q}|u-c|\,d\mathcal{H}^{% \beta}_{\infty}, where the supremum is taken over all finite subcubes Q parallel to Q 0 {Q_{0}} , l ⁢ ( Q ) {l(Q)} is the length of the side of the cube Q, and ℋ ∞ β {\mathcal{H}^{\beta}_{\infty}} is the Hausdorff content. In the case β = d {\beta=d} we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every β ∈ ( 0 , d ] {\beta\in(0,d]} one has a dimensionally appropriate analogue of the John–Nirenberg inequality for functions with bounded β-dimensional mean oscillation: There exist constants c , C &gt; 0 {c,C&gt;0} such that ℋ ∞ β ⁢ ( { x ∈ Q : | u ⁢ ( x ) - c Q | &gt; t } ) ≤ C ⁢ l ⁢ ( Q ) β ⁢ exp ⁡ ( - c ⁢ t ∥ u ∥ BMO β ⁢ ( Q 0 ) ) \displaystyle\mathcal{H}^{\beta}_{\infty}(\{x\in Q:|u(x)-c_{Q}|&gt;t\})\leq Cl(Q)% ^{\beta}\exp\biggl{(}-\frac{ct}{\|u\|_{\mathrm{BMO}^{\beta}(Q_{0})}}\biggr{)} for every t &gt; 0 {t&gt;0} , u ∈ BMO β ⁢ ( Q 0 ) {u\in\mathrm{BMO}^{\beta}(Q_{0})} , Q ⊂ Q 0 {Q\subset Q_{0}} , and suitable c Q ∈ ℝ {c_{Q}\in\mathbb{R}} . Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.

Hydrodynamic normalization conditions in the theory of degenerate Beltrami equations

We study the existence of normalized homeomorphic solutions for the degenerate Beltrami equation fz = μ(z )f in the whole complex plane C , assuming that its measurable coefficient μ(z ), | μ(z ) |&lt;1 a. e., has compact support and the degeneration of the equation is controlled by the tangential dilatation quotient KT μ (z , z0) . We show that if KT μ (z , z0) has bounded or finite mean oscillation dominants, or satisfies the Lehto type integral divergence condition, then the Beltrami equation admits a regular homeomorphic W1,1loc solution f with the hydrodynamic normalization at infinity. We also give integral criteria of Calderon-Zygmund or Orlicz types for the existence of the normalized solutions in terms of KT μ (z , z0) and the maximal dilatation Kμ (z ) .

The Helmholtz Decomposition of a BMO Type Vector Field in a Slightly Perturbed Half Space

We introduce a space of L^2 vector fields with bounded mean oscillation whose “normal” component to the boundary is well-controlled. We establish its Helmholtz decomposition in the case when the domain is a perturbed C^3 half space in {textbf{R}}^n(n ge 3) with small perturbation.

Gaussian analytic functions of bounded mean oscillation

We consider random analytic functions given by a Taylor series with independent, centered complex Gaussian coefficients. We give a new sufficient condition for such a function to have bounded mean oscillations. Under a mild regularity assumption this condition is optimal. Using a theorem of Holland and Walsh, we give as a corollary a new bound for the norm of a random Gaussian Hankel matrix. Finally, we construct some exceptional Gaussian analytic functions which in particular disprove the conjecture that a random analytic function with bounded mean oscillations always has vanishing mean oscillations.