An Onofri-type Inequality on the Sphere with Two Conical Singularities

2012 ◽  
Vol 55 (3) ◽  
pp. 663-672
Author(s):  
Chunqin Zhou
Keyword(s):  
Gaussian Curvature ◽  
Type Inequality ◽  
Conical Singularities ◽  
Image Position

AbstractIn this paper, we give a new proof of the Onofri-type inequalityon the sphere S with Gaussian curvature 1 and with conical singularities divisor for β ∈ (–1, 0); here p1 and p2 are antipodal.

A representation of distributions supported on smooth hypersurfaces of Rn

1995 ◽  
Vol 117 (1) ◽  
pp. 153-160
Author(s):  
Kanghui Guo
Keyword(s):  
Gaussian Curvature ◽  
Dual Space ◽  
Smooth Hypersurface ◽  
Zero Gaussian Curvature ◽  
Schwartz Class ◽  
Image Position

Let S(Rn) be the space of Schwartz class functions. The dual space of S′(Rn), S(Rn), is called the temperate distributions. In this article, we call them distributions. For 1 ≤ p ≤ ∞, let FLp(Rn) = {f:∈ Lp(Rn)}, then we know that FLp(Rn) ⊂ S′(Rn), for 1 ≤ p ≤ ∞. Let U be open and bounded in Rn−1 and let M = {(x, ψ(x));x ∈ U} be a smooth hypersurface of Rn with non-zero Gaussian curvature. It is easy to see that any bounded measure σ on Rn−1 supported in U yields a distribution T in Rn, supported in M, given by the formula

scholarly journals Time-Harmonic Solutions for Maxwell’s Equations in Anisotropic Media and Bochner–Riesz Estimates with Negative Index for Non-elliptic Surfaces

2021 ◽  
Author(s):  
Rainer Mandel ◽  
Robert Schippa
Keyword(s):  
Gaussian Curvature ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Smooth Surfaces ◽  
Negative Index ◽  
Elliptic Surfaces ◽  
Conical Singularities ◽  
Fourier Restriction ◽  
Time Harmonic ◽  
Homogeneous Media

AbstractWe solve time-harmonic Maxwell’s equations in anisotropic, spatially homogeneous media in intersections of $$L^p$$ L p -spaces. The material laws are time-independent. The analysis requires Fourier restriction–extension estimates for perturbations of Fresnel’s wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds but without flat points, and surfaces with conical singularities. Our estimates are based on new Bochner–Riesz estimates with negative index for non-elliptic surfaces.

A weighted weak-type bound for Haar multipliers

2018 ◽  
Vol 148 (3) ◽  
pp. 643-658
Author(s):  
Adam Osȩkowski
Keyword(s):  
Maximal Operator ◽  
Function Method ◽  
Dual Problem ◽  
Weak Type ◽  
Analogous Result ◽  
Type Inequality ◽  
Bellman Function ◽  
Weak Type Inequality ◽  
Image Position

We study a weighted maximal weak-type inequality for Haar multipliers that can be regarded as a dual problem of Muckenhoupt and Wheeden. More precisely, if Tε is the Haar multiplier associated with the sequence ε with values in [−1, 1], and is the r-maximal operator, then for any weight w and any function f on [0, 1) we havefor some constant Cr depending only on r. We also show that the analogous result does not hold if we replace by the dyadic maximal operator Md. The proof rests on the Bellman function method; using this technique we establish related weighted Lp estimates for p close to 1, and then deduce the main result by extrapolation arguments.

Weighted norm inequalities for positive operators restricted on the cone of λ-quasiconcave functions

2019 ◽  
Vol 150 (1) ◽  
pp. 17-39 ◽  
Author(s):  
Amiran Gogatishvili ◽  
Júlio S. Neves
Keyword(s):  
Sufficient Conditions ◽  
Type Inequality ◽  
Weighted Norm ◽  
Hardy Type Inequality ◽  
Weighted Function ◽  
Hardy Type ◽  
Quasiconcave Functions ◽  
Measurable Functions ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet ρ be a monotone quasinorm defined on ${\rm {\frak M}}^ + $, the set of all non-negative measurable functions on [0, ∞). Let T be a monotone quasilinear operator on ${\rm {\frak M}}^ + $. We show that the following inequality restricted on the cone of λ-quasiconcave functions $$\rho (Tf) \les C_1\left( {\int_0^\infty {f^p} v} \right)^{1/p},$$where $1\les p\les \infty $ and v is a weighted function, is equivalent to slightly different inequalities considered for all non-negative measurable functions. The case 0 < p < 1 is also studied for quasinorms and operators with additional properties. These results in turn enable us to establish necessary and sufficient conditions on the weights (u, v, w) for which the three weighted Hardy-type inequality $$\left( {\int_0^\infty {{\left( {\int_0^x f u} \right)}^q} w(x){\rm d}x} \right)^{1/q} \les C_1\left( {\int_0^\infty {f^p} v} \right)^{1/p},$$holds for all λ-quasiconcave functions and all 0 < p, q ⩽ ∞.

The Fourier Transforms of Smooth Measures on Hypersurfaces of Rn + 1

1986 ◽  
Vol 38 (2) ◽  
pp. 328-359 ◽  
Author(s):  
Bernard Marshall
Keyword(s):  
Fourier Transform ◽  
Asymptotic Expansion ◽  
Unit Sphere ◽  
Gaussian Curvature ◽  
Fourier Transforms ◽  
Surface Measure ◽  
Positive Gaussian Curvature ◽  
Smooth Measures ◽  
The Fourier Transform ◽  
Image Position

The Fourier transform of the surface measure on the unit sphere in Rn + 1, as is well-known, equals the Bessel functionIts behaviour at infinity is described by an asymptotic expansionThe purpose of this paper is to obtain such an expression for surfaces Σ other than the unit sphere. If the surface Σ is a sufficiently smooth compact n-surface in Rn + 1 with strictly positive Gaussian curvature everywhere then with only minor changes in the main term, such an asymptotic expansion exists. This result was proved by E. Hlawka in [3]. A similar result concerned with the minimal smoothness of Σ was later obtained by C. Herz [2].

On Korn's first inequality with non-constant coefficients

2002 ◽  
Vol 132 (1) ◽  
pp. 221-243 ◽  
Author(s):  
Patrizio Neff
Keyword(s):  
Lipschitz Domain ◽  
Large Deformations ◽  
Type Inequality ◽  
Two Dimensional ◽  
Smooth Part ◽  
Bounded Lipschitz Domain ◽  
Elasto Plasticity ◽  
Constant Coefficients ◽  
Image Position ◽  
Dimensional Lebesgue Measure

In this paper we prove a Korn-type inequality with non-constant coefficients which arises from applications in elasto-plasticity at large deformations. More precisely, let Ω ⊂ R3 be a bounded Lipschitz domain and let Γ ⊂ ∂Ω be a smooth part of the boundary with non-vanishing two-dimensional Lebesgue measure. Define and let be given with det Fp(x) ≥ μ+ > 0. Moreover, suppose that Rot . Then Clearly, this result generalizes the classical Korn's first inequality which is just our result with Fp = 11. With slight modifications, we are also able to treat forms of the type

scholarly journals A reverse Hölder type inequality for the logarithmic mean and generalizations

2000 ◽  
Vol 41 (3) ◽  
pp. 401-409 ◽  
Author(s):  
John Maloney ◽  
Jack Heidel ◽  
Josip Pečarić
Keyword(s):  
Type Inequality ◽  
Logarithmic Mean ◽  
Image Position ◽  
Reverse Hölder

AbstractAn inequality involving the logarithmic mean is established. Specifically, we show thatwhere . Then several generalizations are given.

scholarly journals Optimal Hardy inequalities in cones

2016 ◽  
Vol 147 (1) ◽  
pp. 89-124 ◽  
Author(s):  
Baptiste Devyver ◽  
Yehuda Pinchover ◽  
Georgios Psaradakis
Keyword(s):  
Distance Function ◽  
Type Inequality ◽  
Hardy Type Inequality ◽  
Hardy Inequalities ◽  
Smoothness Assumption ◽  
Hardy Type ◽  
Definite Sense ◽  
Image Position

Let Ω be an open connected cone in ℝn with vertex at the origin. Assume that the Operatoris subcritical in Ω, where δΩ is the distance function to the boundary of Ω and μ ⩽ 1/4. We show that under some smoothness assumption on Ω the improved Hardy-type inequalityholds true, and the Hardy-weight λ(μ)|x|–2 is optimal in a certain definite sense. The constant λ(μ) > 0 is given explicitly.

Multivariate Landau–Kolmogorov-type inequality

1989 ◽  
Vol 105 (2) ◽  
pp. 335-350 ◽  
Author(s):  
Z. Ditzian
Keyword(s):  
Banach Spaces ◽  
Type Inequality ◽  
Kolmogorov Type ◽  
Image Position

AbstractAssuming that the nth iterate of the Laplacian Δnf belongs to L∞(ℝ), we show for 0 < k < 2n thatwhere ∂/∂ξi is the derivative in the ei direction. The result is also extended to other Banach spaces of functions on ℝd.

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