Two-Weight Mixed Norm Estimates for a Generalized Spherical Mean Radon Transform Acting on Radial Functions

Abstract

Abstract. We investigate a generalized spherical means operator, viz. generalized spherical mean Radon transform, acting on radial functions. We establish an integral representation of this operator and find precise estimates of the corresponding kernel. As the main result, we prove two-weight mixed norm estimates for the integral operator, with general power weights involved. This leads to weighted Strichartz type estimates for solutions to certain Cauchy problems for classical Euler-Poisson-Darboux and wave equations with radial initial data.

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

“…This paper is a natural continuation of our recent research from [2]. We study a generalized spherical means operator acting on radial functions.…”

“…We study a generalized spherical means operator acting on radial functions. In [2] we viewed this operator as a family of integral transforms {M α,β t : t > 0} acting on profile functions on R + and found fairly precise estimates of the associated integral kernels K α,β t (x, z). This enabled us to prove two-weight L p − L q (L r t ) estimates for f → M α,β t f , with 1 ≤ p, q ≤ ∞ and 1 ≤ r < ∞.…”

“…In the present work we focus on the more subtle limiting case r = ∞ and restrict to p = q, which in the above language of mixed norm estimates corresponds to weighted L p -boundedness of the maximal operator f → sup t>0 |M α,β 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.…”

“…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.…”

“…Kernel estimates. We first rephrase [2, Theorem 3.3] in a way more convenient for our present purposes; see also [2,Section 2.3]. In particular, we neglect parts concerning sharpness of the bounds for certain α and β.…”