Bochner-Riesz means with respect to a rough distance function

The generalized Bochner-Riesz operator S-R,S-lambda may be defined as S(R,lambda)f(x)=F-1[(1-rho/R)(+)(lambda)(f) over cap](x) where. is an appropriate distance function and F-1 is the inverse Fourier transform. The behavior of S-R,S-lambda on (LP)-P-p (R-d x R) is described for rho(xi', xi(d+1)) = max{vertical bar xi'vertical bar, vertical bar xi(d+1)vertical bar}, a rough distance function. We conjecture that this operator is bounded on R-d x R when lambda > max{d(1/2 - 1/p)- 1/2, 0} and p < 6/d-3,and unbounded when p >= 2 + 6/d-3. This conjecture is verified for large ranges of p.