The principle of general localization on unit sphere

Abstract

In this paper we study the general localization principle for Fourier-Laplace series on unit sphere S N ⊂ R N+1 . Weak type (1, 1) property of maximal functions is used to establish the estimates of the maximal operators of Riesz means at critical index N−1 2 . The properties Jacobi polynomials are used in estimating the maximal operators of spectral expansions in L 2 (S N ). For extending positive results on critical line α = (N − 1)( 1 p − 1 2 ), 1 p 2, we apply interpolation theorem for the family of the linear operators of weak types. The generalized localization principle is established by the analysis of spectral expansions in L 2 . We have proved the sufficient conditions for the almost everywhere convergence of Fourier-Laplace series by Riesz means on the critical line.

The generalized localization for multiple Fourier integrals

“…The same result for the spectral expansions of the Laplace-Beltrami operator on the unit sphere was obtained in [2]. It should be noted, if s > (N − 1)(1/p − 1/2),…”

“…Il'in [7] was the first to introduce the concept of generalized principle of localization for an arbitrary eigenfunction expansions. Following Il'in we say that the generalized localization principle for E λ holds in L p (R N ), if for any function f ∈ L p (R N ) the equality lim λ→∞ E λ f (x) = 0 (2) holds almost-everywhere on R N \ supp( f ).…”

“…Observe, unlike the classical Riemann localization principle, here it suffices the equality (2) to be hold only almosteverywhere (not everywhere) on R N \ supp( f ).…”

Asymptotic formula for the Riesz means of the spectral functions of Laplace-Beltrami operator on unit sphere

“…With properties found in [4], [5] and [7], we are able to overcome previous restrictions and obtain the asymptotic formula for the Riesz means kernel of the spectral expansions of order α > 0, α ∈ . Some recent works on the Riesz means of spectral functions can be found in the following articles [10][11][12][13][14][15][16][17][18][19] and more.…”

On the sufficient conditions of the localization of the Fourier-Laplace series of distributions from liouville classes

“…In the case of Spectral expansions of the integrable functions on unit sphere the same problems are considered in the papers [1], [2], [3], [6], [8], [12] and [15] . For more reference related to the problems of the convergence of the Fourier-Laplace series on unit sphere we refer the readers to [20].…”