scholarly journals Oscillation and variation for the Riesz transform associated with Bessel operators

2018 ◽  
Vol 149 (1) ◽  
pp. 169-190 ◽  
Author(s):  
Huoxiong Wu ◽  
Dongyong Yang ◽  
Jing Zhang
Keyword(s):  
Riesz Transform ◽  
Bessel Operator ◽  
Bessel Operators ◽  
Variation Operator ◽  
Image Position ◽  
Oscillation Operator

Let λ > 0 and letbe the Bessel operator on ℝ+ := (0,∞). We show that the oscillation operator 𝒪(RΔλ,∗) and variation operator 𝒱ρ(RΔλ,∗) of the Riesz transform RΔλ associated with Δλ are both bounded on Lp(ℝ+, dmλ) for p ∈ (1,∞), from L1(ℝ+, dmλ) to L1,∞(ℝ+, dmλ), and from L∞(ℝ+, dmλ) to BMO(ℝ+, dmλ), where ρ ∈ (2,∞) and dmλ(x) := x2λ dx. As an application, we give the corresponding Lp-estimates for β-jump operators and the number of up-crossings.

scholarly journals REAL-VARIABLE CHARACTERIZATIONS OF HARDY SPACES ASSOCIATED WITH BESSEL OPERATORS

2011 ◽  
Vol 09 (03) ◽  
pp. 345-368 ◽  
Author(s):  
DACHUN YANG ◽  
DONGYONG YANG
Keyword(s):  
Maximal Function ◽  
Hardy Spaces ◽  
Riesz Transform ◽  
Real Variable ◽  
Bessel Operator ◽  
Bessel Operators ◽  
Lusin Area Function ◽  
Radial Maximal Function ◽  
Nontangential Maximal Function

Let λ > 0, p ∈ ((2λ + 1)/(2λ + 2), 1], and [Formula: see text] be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces Hp((0,∞),dmλ) associated with △λ in terms of the radial maximal function, the nontangential maximal function, the grand maximal function, the Littlewood–Paley g-function and the Lusin-area function, where dmλ(x) ≡ x2λ dx. As an application, the authors further obtain the Riesz transform characterization of these Hardy spaces.

scholarly journals Compactness of Riesz transform commutator associated with Bessel operators

2018 ◽  
Vol 135 (2) ◽  
pp. 639-673 ◽  
Author(s):  
Xuan Thinh Duong ◽  
Ji Li ◽  
Suzhen Mao ◽  
Huoxiong Wu ◽  
Dongyong Yang
Keyword(s):  
Riesz Transform ◽  
Bessel Operators

scholarly journals On the Domains of Bessel Operators

2021 ◽  
Author(s):  
Jan Dereziński ◽  
Vladimir Georgescu
Keyword(s):  
Sobolev Space ◽  
Schrödinger Operator ◽  
Second Order ◽  
The Other ◽  
Bessel Operator ◽  
Bilinear Forms ◽  
Minimal Realization ◽  
Schrodinger Operator ◽  
Bessel Operators ◽  
Other Hand

AbstractWe consider the Schrödinger operator on the halfline with the potential $$(m^2-\frac{1}{4})\frac{1}{x^2}$$ ( m 2 - 1 4 ) 1 x 2 , often called the Bessel operator. We assume that m is complex. We study the domains of various closed homogeneous realizations of the Bessel operator. In particular, we prove that the domain of its minimal realization for $$|\mathrm{Re}(m)|<1$$ | Re ( m ) | < 1 and of its unique closed realization for $$\mathrm{Re}(m)>1$$ Re ( m ) > 1 coincide with the minimal second-order Sobolev space. On the other hand, if $$\mathrm{Re}(m)=1$$ Re ( m ) = 1 the minimal second-order Sobolev space is a subspace of infinite codimension of the domain of the unique closed Bessel operator. The properties of Bessel operators are compared with the properties of the corresponding bilinear forms.

Riesz transform and g-function associated with Bessel operators and their appropriate Banach spaces

2007 ◽  
Vol 157 (1) ◽  
pp. 259-282 ◽  
Author(s):  
Jorge J. Betancor ◽  
Juan Carlos Fariña ◽  
Teresa Martínez ◽  
José Luis Torrea
Keyword(s):  
Banach Spaces ◽  
Riesz Transform ◽  
Bessel Operators

Boundedness and compactness characterizations of Riesz transform commutators on Morrey spaces in the Bessel setting

2018 ◽  
Vol 17 (01) ◽  
pp. 145-178 ◽  
Author(s):  
Suzhen Mao ◽  
Huoxiong Wu ◽  
Dongyong Yang
Keyword(s):  
Hardy Space ◽  
Riesz Transform ◽  
Morrey Spaces ◽  
Bessel Operator ◽  
Factorization Theorem ◽  
Commutator Formula ◽  
Weak Factorization

Let [Formula: see text] and [Formula: see text] be the Bessel operator on [Formula: see text]. In this paper, the authors show that [Formula: see text] (or [Formula: see text], respectively) if and only if the Riesz transform commutator [Formula: see text] is bounded (or compact, respectively) on Morrey spaces [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text]. A weak factorization theorem for functions belonging to the Hardy space [Formula: see text] in the sense of Coifman–Rochberg–Weiss in Bessel setting, via [Formula: see text] and its adjoint, is also obtained.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
Image Position

A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

Optimizing conditions for x-ray microchemical analysis in analytical electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 548-549 ◽  
Author(s):  
N. J. Zaluzec
Keyword(s):  
Solid Angle ◽  
Analytical Electron Microscopy ◽  
Incident Beam ◽  
Thin Foil ◽  
Experimental Conditions ◽  
X Ray ◽  
Microchemical Analysis ◽  
Analytical Electron ◽  
Image Position ◽  
Incident Beam Energy

The ultimate sensitivity of microchemical analysis using x-ray emission rests in selecting those experimental conditions which will maximize the measured peak-to-background (P/B) ratio. This paper presents the results of calculations aimed at determining the influence of incident beam energy, detector/specimen geometry and specimen composition on the P/B ratio for ideally thin samples (i.e., the effects of scattering and absorption are considered negligible). As such it is assumed that the complications resulting from system peaks, bremsstrahlung fluorescence, electron tails and specimen contamination have been eliminated and that one needs only to consider the physics of the generation/emission process.The number of characteristic x-ray photons (Ip) emitted from a thin foil of thickness dt into the solid angle dΩ is given by the well-known equation

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