SHARP ESTIMATES FOR THE BOCHNER-RIESZ OPERATOR OF NEGATIVE ORDER IN R-2

The Bochner-Riesz operator T-alpha on R-n of order a is defined by (T( alpha)f)boolean AND(xi) = (1-\xi\(2))(+)(alpha)/Gamma(alpha+1)(f) over cap(xi) where boolean AND denotes the Fourier transform and r(+)(alph a) = r(alpha) if r > 0, and r(+)(alpha) = 0 if r less than or equal to 0. We determine all pairs (p, q) such that T-alpha on R-2 of negative order is bounded from L-p(R-2) to L-q(R-2). TO be more precise, we pr ove that for 0 < delta < 3/2 the estimate \\T(-delta)f\\(Lq(R2)) less than or equal to C\\f\\(Lp(R2)) holds if and only if (1/p, 1/q) is an element of Delta(-delta), where Delta(-delta) = {(1/p, 1/q) is an elem ent of [0, 1] x [0, 1]: 1/p-1/q greater than or equal to 2 delta/3, 1/ p > 1/4 + delta/2, 1/q < 3/4-delta/2}. We also obtain some weak-type r esults for T-alpha.