Unit Sphere Sn Research Articles

On the injectivity of the shifted Funk–Radon transform and related harmonic analysis

AbstractNecessary and sufficient conditions are obtained for injectivity of the shifted Funk–Radon transform associated with k-dimensional totally geodesic submanifolds of the unit sphere Sn in ℝn+1. This result generalizes the well known statement for the spherical means on Sn and is formulated in terms of zeros of Jacobi polynomials. The relevant harmonic analysis is developed, including a new concept of induced Stiefel (or Grassmannian) harmonics, the Funk–Hecke type theorems, addition formula, and multipliers. Some perspectives and conjectures are discussed.

Elliptic inequalities with nonlinear convolution and Hardy terms in cone-like domains

We study the inequality −Δu−μ|x|2u≥(|x|−α⁎up)uq in an unbounded cone CΩρ⊂RN (N≥2) generated by a subdomain Ω of the unit sphere SN−1⊂RN, p,q,ρ>0, μ∈R and 0≤α<N. In the above, |x|−α⁎up denotes the standard convolution operator in the cone CΩρ. We discuss the existence and nonexistence of positive solutions in terms of N,p,q,α,μ and Ω. Extensions to systems of inequalities are also investigated.

First stability eigenvalue of singular hypersurfaces with constant mean curvature in the unit sphere

In this paper, we study the first eigenvalue of the stability operator on an integral n n -varifold with constant mean curvature in the unit sphere S n + 1 \mathbb {S}^{n+1} . We find the optimal upper bound and prove a rigidity result characterizing the case when it is attained. This gives a new characterization for certain singular Clifford tori.

A sharp regularity estimate for the Schrödinger propagator on the sphere

Let ΔSn denote the Laplace-Beltrami operator on the n-dimensional unit sphere Sn, n≥2. In this paper we show that‖eitΔSnf‖L4([0,2π)×Sn)≤C‖f‖Wα,4(Sn) holds if α>(n−2)/4. The range of α is sharp in the sense that the estimate fails for α<(n−2)/4. As a consequence, we obtain space-time Lp-estimates for eitΔSn for 2≤p≤∞. We also prove that the maximal operator f→sup0≤t<2π⁡|eitΔSnf| is bounded from Wα,2(Sn) to L6n/(3n−2)(Sn) for α>1/3 whenever f are zonal functions on Sn.

Jump and variational inequalities for hypersingular integrals with rough kernels

In this paper, we study the jump function and variation of hypersingular integral operators with rough kernelsTΩ,α,εf(x)=∫|y|>εΩ(y)|y|n+αf(x−y)dy, where α≥0, Ω is an integrable function on the unit sphere Sn−1 satisfying certain cancellation conditions. More precisely, we first show that for 1<p<∞, the jump function and variation of the family of truncated hypersingular integrals {TΩ,α,ε}ε>0 extends to a bounded operator from the Sobolev space Lαp to the Lebesgue space Lp with Ω belonging to the Hardy space Hq(Sn−1) where q=n−1n−1+α, which gives a positive answer to an open problem proposed by Ding-Hong-Liu [15].

On Minimal Hypersurfaces of a Unit Sphere

Minimal compact hypersurface in the unit sphere Sn+1 having squared length of shape operator A2&lt;n are totally geodesic and with A2=n are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in Sn+1. One finds a naturally induced vector field w called the associated vector field and a smooth function ρ called support function on the hypersurface M of Sn+1. It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in S5 to be totally geodesic is that the support function ρ is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in Sn+1, (n&gt;2), provided the scalar curvature τ is a constant on integral curves of w. Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in Sn+1 is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field Aw.

On the spherical slice transform

We study the spherical slice transform [Formula: see text] which assigns to a function f on the unit sphere Sn in [Formula: see text] the integrals of f over cross-sections of Sn by k-dimensional affine planes passing through the north pole (0,…, 0, 1). These transforms are known when k = n. We consider all 2 ≤ k ≤ n and obtain an explicit formula connecting [Formula: see text] with the classical (k − 1)-plane Radon–John transform [Formula: see text] on [Formula: see text]. Using this connection, known facts for Rk−1, like inversion formulas, support theorems, representation on zonal functions, and some others, are reformulated for [Formula: see text].

The Minkowski norm and Hessian isometry induced by an isoparametric foliation on the unit sphere

Let Mt be an isoparametric foliation on the unit sphere (Sn−1(1), gst) with d principal curvatures. Using the spherical coordinates induced by Mt, we construct a Minkowski norm with the representation \(F = r\sqrt {2f(t)} \), which generalizes the notions of (α, β)-norm and (α1, α2)-norm. Using the technique of the spherical local frame, we give an exact and explicit answer to the question when \(F = r\sqrt {2f(t)} \) really defines a Minkowski norm. Using the similar technique, we study the Hessian isometry Φ between two Minkowski norms induced by Mt, which preserves the orientation and fixes the spherical ξ-coordinates. There are two ways to describe this Φ, either by a system of ODEs, or by its restriction to any normal plane for Mt, which is then reduced to a Hessian isometry between Minkowski norms on ℝ2 satisfying certain symmetry and (d)-properties. When d > 2, we prove that this Φ can be obtained by gluing positive scalar multiplications and compositions of the Legendre transformation and positive scalar multiplications, so it must satisfy the (d)-property for any orthogonal decomposition ℝn = V′ + V″, i.e., for any nonzero x = x′ + x″ and \(\Phi (x) = \bar x = \bar x\prime + \bar x\prime \prime \) with \(x\prime ,\bar x\prime \in {\bf{V}}\prime \) and x″, \(x\prime \prime ,\bar x\prime \prime \in {\bf{V}}\prime \prime \), we have \(g_x^{{F_1}}(x\prime \prime ,x) = g_{\bar x}^{{F_2}}(\bar x\prime \prime ,\bar x)\). As byproducts, we prove the following results. On the indicatrix (SF, g), where F is a Minkowski norm induced by Mt and g is the Hessian metric, the foliation Nt = SF ∩ ℝ>0M0 is isoparametric. Laugwitz Conjecture is valid for a Minkowski norm F induced by Mt, i.e., if its Hessian metric g is flat on ℝn{0} with n > 2, then F is Euclidean.

Continuous valuations on the space of Lipschitz functions on the sphere

We study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere Sn−1. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded with respect to the Lipschitz norm. This fact, in combination with measure theoretical arguments, will yield an integral representation for continuous and rotation invariant valuations on the space of Lipschitz functions over the 1-dimensional sphere.

Invariant hypersurfaces with linear prescribed mean curvature

Our aim is to study invariant hypersurfaces immersed in the Euclidean space Rn+1, whose mean curvature is given as a linear function in the unit sphere Sn depending on its Gauss map. These hypersurfaces are closely related with the theory of manifolds with density, since their weighted mean curvature in the sense of Gromov is constant. In this paper we obtain explicit parametrizations of constant curvature hypersurfaces, and also give a classification of rotationally invariant hypersurfaces.

The Orlicz version of the Lp Minkowski problem for −n < p < 0

Given a function f on the unit sphere Sn−1, the Lp Minkowski problem asks for a convex body K whose Lp surface area measure has density f with respect to the standard (n−1)-Hausdorff measure on Sn−1. In this paper we deal with the generalization of this problem which arises in the Orlicz-Brunn-Minkowski theory when an Orlicz function φ substitutes the Lp norm and p is in the range (−n,0). This problem is equivalent to solve the Monge-Ampere equationφ(h)det⁡(∇2h+hI)=f on Sn−1, where h is the support function of the convex body K.

The Vertical Slice Transform on the Unit Sphere

Abstract The vertical slice transform in spherical integral geometry takes a function on the unit sphere Sn to integrals of that function over spherical slices parallel to the last coordinate axis. This transform was investigated for n = 2 in connection with inverse problems of spherical tomography. The present article gives a survey of some methods which were originally developed for the Radon transform, hypersingular integrals, and the spherical mean Radon-like transforms, and can be adapted to obtain new inversion formulas and singular value decompositions for the vertical slice transform in the general case n ≥ 2 for a large class of functions.

Beckner inequalities for Moebius measures on spheres

In this paper, we consider the Moebius measures μxn indexed by dimension n and |x| &lt; 1 on the unit sphere Sn−1 in ℝn (n ≥ 3), and provide a precise two-sided estimate on the order of the Beckner inequality constant with exponent p ∈ [1, 2) in the three parameters. As special cases for p = 1 and p tending to 2, our results cover those in Barthe et al. [Forum Math. (submitted for publication)] for n ≥ 3 and explore an interesting phenomenon.

A note on the spectral gap for general harmonic measures on spheres

In this paper, we consider general harmonic measures μxn,β on the unit sphere Sn−1 in Rn, where x∈Rn with 0≤|x|<1,β∈R and n≥3. Following the idea in Barthe et al. (2014) , we obtain the lower bound for the spectral gap of μxn,β. For the harmonic measure (β=0), we improve the lower bound of the spectral gap in Barthe et al. (2014) and we also improve those in Milman (2015) for general β∈R.

Large deviation principle for some beta ensembles

Let L L be a positive line bundle over a projective complex manifold X X , L p L^p its tensor power of order p p , H 0 ( X , L p ) H^0(X,L^p) the space of holomorphic sections of L p L^p , and N p N_p the complex dimension of H 0 ( X , L p ) H^0(X,L^p) . The determinant of a basis of H 0 ( X , L p ) H^0(X,L^p) , together with some given probability measure on a weighted compact set in X X , induces naturally a β \beta -ensemble, i.e., a random N p N_p -point process on the compact set. Physically, depending on X X and the value of β \beta , this general setting corresponds to a gas of free or interacting fermions on X X and may admit an interpretation in terms of some random matrix models. The empirical measures, associated with such β \beta -ensembles, converge almost surely to an equilibrium measure when p p goes to infinity. We establish a large deviation theorem (LDT) with an effective speed of convergence for these empirical measures. Our study covers a large class of β \beta -ensembles on a compact subset of the unit sphere S n ⊂ R n + 1 \mathbb {S}^n\subset \mathbb {R}^{n+1} or of the Euclidean space R n \mathbb {R}^n .

Smoothness of a Holomorphic Function in a Ball and of its Modulus on the Sphere

Let a function f be holomorphic in the unit ball 𝔹 n , continuous in the closed ball $$ {\overline{\mathbb{B}}}^n $$ , and let f(z) ≠ 0, z ∈ 𝔹 n . Assume that |f| belongs to the α-Holder class on the unit sphere S n , 0 < α ≤ 1. The present paper is devoted to the proof of the statement that f belongs to the α/2-Holder class on $$ {\overline{\mathbb{B}}}^n $$ .

Extrapolation Problem for Continuous Time Periodically Correlated Isotropic Random Fields

The problem of optimal linear estimation of functionals depending on the unknown values of a random fieldζ(t,x), which is mean-square continuous periodically correlated with respect to time argumenttє R and isotropic on the unit sphere Sn with respect to spatial argumentxєSn. Estimates are based on observations of the fieldζ(t,x) +Θ(t,x) at points (t,x) :t&lt; 0;xєSn, whereΘ(t,x) is an uncorrelated withζ(t,x) random field, which is mean-square continuous periodically correlated with respect to time argumenttє R and isotropic on the sphereSnwith respect to spatial argumentxєSn. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case of spectral certainty where the spectral densities of the fields are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given.

On M-Projective Curvature Tensor of Lorentzian &lt;i&gt;α&lt;/i&gt;-Sasakian Manifolds

In this paper, we study the nature of Lorentzianα-Sasakian manifolds admitting M-projective curvature tensor. We show that M-projectively flat and irrotational M-projective curvature tensor of Lorentzian α-Sasakian manifolds are locally isometric to unit sphere Sn(c) , wherec = α2. Next we study Lorentzianα-Sasakian manifold with conservative M-projective curvature tensor. Finally, we find certain geometrical results if the Lorentzianα-Sasakian manifold satisfying the relation M(X,Y)⋅R=0.

When does a discrete-time random walk in $\mathbb{R}^{n}$ absorb the origin into its convex hull?

We connect this question to a problem of estimating the probability that the image of certain random matrices does not intersect with a subset of the unit sphere Sn−1Sn−1. In this way, the case of a discretized Brownian motion is related to Gordon’s escape theorem dealing with standard Gaussian matrices. We show that for the random walk BMn(i),i∈NBMn(i),i∈N, the convex hull of the first CnCn steps (for a sufficiently large universal constant CC) contains the origin with probability close to one. Moreover, the approach allows us to prove that with high probability the π/2π/2-covering time of certain random walks on Sn−1Sn−1 is of order nn. For certain spherical simplices on Sn−1Sn−1, we prove an extension of Gordon’s theorem dealing with a broad class of random matrices; as an application, we show that CnCn steps are sufficient for the standard walk on ZnZn to absorb the origin into its convex hull with a high probability. Finally, we prove that the aforementioned bound is sharp in the following sense: for some universal constant c>1c>1, the convex hull of the nn-dimensional Brownian motion conv{BMn(t):t∈[1,cn]}conv⁡{BMn(t):t∈[1,cn]} does not contain the origin with probability close to one.

Holomorphic Campanato spaces on the unit ball

As outlined below, this paper is devoted to a Carleson-type-measure-based study of the holomorphic Campanato 2-space on the open unit ball Bn of Cn, comprising all Hardy 2-functions whose oscillations in non-isotropic metric balls on the compact unit sphere Sn are proportional to some power of the radius other than the dimension n≥1.