Jodeit's extensions for bilinear multipliers (Bull. London Math Soc. 40 (2008) 937-944)

Abstract

We are writing this note to correct the above-mentioned paper and provide errata for some of the calculations in the paper. Unfortunately, these shortcomings were overlooked by the authors. Errata(1) Page 941, line 12 should be replaced by Observe that in the above proof, essentially we have proved the boundedness of the following extension, given by the symbolwhere1 2 ]ξ η}. First, the set P n should be written asFurther, we do not know if this assertion is true. Indeed, if this were so, and Theorem 3.5 held as stated, these would then give us another proof of the boundedness of the bilinear Hilbert transform for p 3 1. To be precise, consider φ n = δ 0 (n) where δ 0 is Dirac mass at 0. Thenwhich is the case for p 3 1 (it is the Fourier transform of the constant function 1). Hence, by dilation one can get that χ {(ξ,η):ξ>η} belongs to M p3 p1,p2 (R). Next, Theorems 3.3 and 3.5 can be restated with a suitable, but crucial modification in the constants. The constants now depend on the L 1 norms of the kernels. In order to prove this, we use Lacey's and Thiele's result [2]. We restate these theorems and give the proof in detail.