Twisted spherical means in annular regions in CnC ^nCn and support theorems (original) (raw)

Spherical means in annular regions in the n -dimensional real hyperbolic spaces

Proceedings of The Indian Academy of Sciences-mathematical Sciences

Let Z r,R be the class of all continuous functions f on the annulus Ann(r, R) in the real hyperbolic space mathbbBn\mathbb B^nmathbbBn with spherical means M s f(x) = 0, whenever s > 0 and xinmathbbBnx\in\mathbb B^nxinmathbbBn are such that the sphere S s (x) ⊂ Ann(r, R) and Br(o)subseteqBs(x).B_r(o)\subseteq B_s(x).Br(o)subseteqBs(x). In this article, we give a characterization for functions in Z r,R . In the case R = ∞, this result gives a new proof of Helgason’s support theorem for spherical means in the real hyperbolic spaces.

Spherical means and the restriction phenomenon

The Journal of Fourier Analysis and Applications, 2001

LetP beasmoothcompactconvexplanarcurvewitharclengthdmandletdcr = ap dm where ap is a cutoff function. For | ~ SO(2) set cr| = (r(OE) for any measurable planar set E. Then, for suitable functions f in ~2, the inequality represents an average over rotations, of the Stein-Tomas restriction phenomenon. We obtain best possible indices for the above inequality when F is any convex curve and under various geometric assumptions.

Regularity of spherical means and localization of spherical harmonic expansions

Journal of the Australian Mathematical Society, 1986

Let G/K be a compact symmetric space, and let G = KAK be a Cartan decomposition of G. For f in L1(G) we define the spherical means f(g, t) = ∫k∫k ∫(gktk′) dk dk′, g ∈ G, t ∈ A. We prove that if f is in Lp(G), 1 ≤ p ≤ 2, then for almost every g ∈ G the functions t → f(g, t) belong to certain Soblev spaces on A. From these regularity results for the spherical means we deduce, if G/K is a compact rank one symmetric space, a theorem on the almost everywhere localization of spherical harmonic expansions of functions in L2 (G/K).

The support theorem for the single radius spherical mean transform

2009

Let f ∈ L p (R n ) and R > 0. The transform is considered that integrates the function f over (almost) all spheres of radius R in R n . This operator is known to be non-injective (as one can see by taking Fourier transform). However, the counterexamples that can be easily constructed using Bessel functions of the 1st kind, only belong to L p if p > 2n/(n − 1). It has been shown previously by S. Thangavelu that for p not exceeding the critical number 2n/(n − 1), the transform is indeed injective.

Injectivity of the Spherical Mean Operator and Related Problems

2000

The problem of injectivity for a Radon transform over level sets of poly- nomials in Rn is studied. The main results concern the spherical mean operator dened on compactly supported continuous functions. Related problems and more general transforms are discussed.

On a subclass of analytic functions involving harmonic means

Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2015

In the present paper, we consider a generalised subclass of analytic functions involving arithmetic, geometric and harmonic means. For this function class we obtain an inclusion result, Fekete-Szegö inequality and coefficient bounds for bi-univalent functions.

On certain harmonic measures on the unit disk

Colloquium Mathematicum, 1997

We will denote by clos E the closure of the set E ⊂ C and by ω(z,E, D) the harmonic measure at z of the set clos E ∩ clos D relative to the component of D \ clos E that contains z. Beurling in his dissertation (see [1], pp. 58–62) proved the following theorem: Theorem 1 (Beurling’s shove theorem). Let K be the union of a finite number of intervals on the radius (0, 1) of the unit disk D. Let l be the total logarithmic measure of K. Then