Maximal estimates for a generalized spherical mean Radon transform acting on radial functions

Abstract

We study a generalized spherical means operator, viz. generalized spherical mean Radon transform, acting on radial functions. As the main results, we find conditions for the associated maximal operator and its local variant to be bounded on power weighted Lebesgue spaces. This translates, in particular, into almost everywhere convergence to radial initial data results for solutions to certain Cauchy problems for classical Euler-Poisson-Darboux and wave equations. Moreover, our results shed some new light to the interesting and important question of optimality of the yet known L p boundedness results for the maximal operator in the general non-radial case. It appears that these could still be notably improved, as indicated by our conjecture of the ultimate sharp result. t f |. Obtaining such results requires a different, in fact more tricky, approach from that used in [2]. It is worth emphasizing that both our works, [2] and this one, were to large extent motivated by connections of the generalized spherical means with solutions to a number of classical initial-value PDE problems being of physical and practical importance; see e.g. [2, Section 7] and references given there.Let n ≥ 2 and consider the generalized spherical means transformationwhere F is the Fourier transform in R n and the radial multiplier is given via m β (s) = 2 β+n/2−1 Γ(β + n/2) J β+n/2−1 (s) s β+n/2−1 , s > 0, 2010 Mathematics Subject Classification. Primary: 44A12; Secondary: 42B37, 35L15, 35B07, 35L05, 35Q05.

Endpoint estimates and optimality for the generalized spherical maximal operator on radial functions

“…Both M α,β t and M α,β * have been extensively investigated recently by Ciaurri, Nowak and Roncal [5,6]. Boundedness of M α,β * on power weighted L p spaces for 1 < p < ∞ was studied in [6].…”

“…Both M α,β t and M α,β * have been extensively investigated recently by Ciaurri, Nowak and Roncal [5,6]. Boundedness of M α,β * on power weighted L p spaces for 1 < p < ∞ was studied in [6]. Sufficient conditions for the boundedness were obtained in [6, Theorem 1.5], while necessary conditions can be found in [6,Proposition 1.6].…”

“…This is possible thanks to a comprehensive analysis of the kernel K α,β t (x, z) done in [5]. We note that, in general, the proofs of Theorems A and B refer heavily to the notation and reasonings from [6], so in this sense the present paper is not self-contained.…”

“…With the above corollaries, we can state a sharp improvement of [6, Corollary 1.8] and an analogous result for weak and restricted weak type inequalities. First, however, we must confess that the statement of [6,Corollary 1.8] missed the global condition β > 1/p + (1 − n)/2 coming from [6, (1.3)]. In consequence, [6, Figure 1] and [6, Conjecture 1.9] should also be revised, which is done below, see Figure 1 and Conjecture 1.7.…”

“…In what follows we use some notation from [6] without much comment. For the readers' convenience, Figure 2 below visualizes the relevant regions in R 3 + determining the splitting of the kernel…”