Bochner-riesz Means with Respect to a 2 by 2 Cylinder

The generalized Bochner-Riesz operator SR,λ may be defined as Sf(x) = F−1 [( 1− ρ R )λ + f̂ ] (x) where ρ is an appropriate distance function and F−1 is the inverse Fourier transform. The sharp bound ‖Sf‖L4(R2×R2) ≤ C‖f‖L4(R2×R2) is shown for the distance function ρ(ξ′, ξ”) = max{|ξ′|, |ξ′′|}. This is a rough distance function corresponding to the R4 cylinder analog {(x1, x2, x3, x4) ∈ R4, x1+ x2 ≤ 1, x3 + x4 ≤ 1}. Introduction Bochner-Riesz means were conceived as a method for addressing the convergence of the inverse Fourier transform. We know the inverse Fourier transform (denoted F−1) will allow us to recover f from the Fourier transform of f (denoted f̂) if f is in Schwartz space (smooth, rapidly decaying at ∞, denoted S) or if f ∈ L. However, for more general f ∈ L, f̂ will typically not be integrable and the inverse Fourier transform will not represent a convergent Lebesgue integral. Bochner-Riesz means allow one to check the convergence of the integral as a limit. We define distance functions to be functions ρ which are continuous on R and satisfy ρ(tx) = tρ(x), t > 0, ρ(x) > 0 if x 6= 0. We define the generalized Bochner-Riesz operator S as follows: (0.1) Sf(x) = F−1[ ( 1− ρ R )λ + f̂ ](x). Here (g(ξ))+ = max{g(ξ), 0} is the positive part. Note that as R →∞, Sf → f for f ∈ S. Also note that Sf → f for f ∈ L by Plancherel’s theorem. On general principals, the question of convergence on other L spaces is equivalent to the question of boundedness of the operators S. By scaling, we may also assume that R = 1. From now on we will focus on the boundedness of S = S on L. Standard Bochner-Riesz means, where ρ(ξ) = |ξ|, have been studied extensively. This case will be referred to as spherical means, as the multiplier is supported on a spherical ball. In 1971, Fefferman [2] showed that for spherical means to be Date: July 21, 2008. 2000 Mathematics Subject Classification. Primary 42B15; Secondary 42B25.