Regularity of spherical means (original) (raw)
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.
On multivalent functions of bounded radius rotations
Applied Mathematics Letters, 2011
In this work, we consider the classes of p-valent analytic functions with bounded radius and bounded boundary rotations. The results proved are sharp and improve some of the known results to a generalized form. We also solve completely an open problem posed by Nunokawa et al. (2007) [5] as a special case of our main result.
2011
We prove that if the Hausdorff dimension of a compact subset of R d is greater than d+1 2 , then the set of angles determined by triples of points from this set has positive Lebesgue measure. Sobolev bounds for bi-linear analogs of generalized Radon transforms and the method of stationary phase play a key role. These results complement those of V. Harangi, T. Keleti, G. Kiss, P. Maga, P. Mattila and B. Stenner in ([8]). We also obtain new upper bounds for the number of times an angle can occur among N points in R d , d ≥ 4, motivated by the results of Apfelbaum and Sharir ([1]) and Pach and Sharir ([13]). We then use this result to establish sharpness results in the continuous setting. Another sharpness result relies on the distribution of lattice points on large spheres in higher dimensions.
On Certain Operators and Related Topics in Geometric Function Theory
2009
Certificate I hereby declare that this project neither as a whole nor as a part there of has been copied out from any source. It is further declared that I have developed this thesis on the basis of my personal efforts made under the sincere guidance of my supervisor. No portion of the work presented in this thesis has been submitted in support of any other degree or qualification of this or any other University or Institute of learning, if found I shall stand responsible.
Polynomial Approximation in the Mean with Respect to Harmonic Measure on Crescents
Transactions of the American Mathematical Society, 1987
For 1 < s < oo and "nice" crescents G, this paper gives a necessary condition (Theorem 2.6) and a sufficient condition (Theorem 2.5) for density of the polynomials in the generalized Hardy space Ha(G). These conditions are easily tested and almost equivalent. The problem of polynomial approximation in the mean with respect to area measure on crescents has been extensively researched by such authors as J. Brennan [1] and S. Mergelyan [6]. This author explores the same problem except that area measure is replaced by harmonic measure. Hence, the question here may be rephrased: given 1 < s < oo, for which crescents G are the polynomials dense in the generalized Hardy space HS(G) (the collection of all analytic functions / on G for which |/|s has a harmonic majorant on G. For z0 in G, ||/||So = Uf(zo)1^8, where uj is the least harmonic majorant of |/|s on G)? 1. Preliminaries. Suppose G is a bounded Dirichlet region in the complex plane (that is, G is a bounded, open, connected set on which the Dirichlet problem is solvable). If z EG, then define P*:GR(rJG)^R by pz(f)=f(z), where / is the solution to the Dirichlet problem on G with boundary values /. By the maximum principle, pz is a bounded, positive linear functional on Gr,(<9G). Since dG is compact, the Riesz representation theorem gives a unique Borel measure ui = u>(-,G,z) (indeed a probability measure by the maximum principle and the fact that Pz(l)-I) with support(o;) Ç <9G such that pz(f) = / /duj for all / in Cr(öG). This measure is called harmonic measure for G at z. To familiarize oneself with some of the properties of harmonic measure one may consult [4]. Throughout this paper, "m" denotes Lebesgue measure on R. 1.1 DEFINITION. A crescent is a region (in the complex plane C) bounded by two Jordan curves which intersect in a single point such that one of the Jordan curves is internal to the other. If G is a crescent, then denote the outer boundary of G by dooG = dG~(Gb eing the polynomially convex hull of the closure of G) and the inner boundary