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Boundedness of smooth bilinear square functions and applications to some bilinear pseudo-differential operators. (English) Zbl 1242.47040

The authors prove the following L p estimates for smooth bilinear square functions. Let Ω:=(ω) ωΩ be a well-distributed collection of intervals of the same length and equidistant. Then, for exponents p 1 ,p 2 ,p 3 [2,] satisfying 0<1/p 3 =(1/p 1 )+(1/p 2 ), there exists a constant C, independent of the collection Ω, such that for all f,g𝒮(), one has

ωΩ |T χω (f,g)| 2 1/2 L p-3 () Cf L p 1 () g L p 2 () ·

Here T σ (f,g)(x):= 2 e ix(ξ+η f ^(ξ)g ^(η)σ(x,ξ,η)dξdη, for symbols in the exotic “class” B 0,0 0 . The boundedness of some bilinear pseudo-differential operators associated with symbols belonging to the subclass BS 0,0 0 is deduced.

MSC:
47G30Pseudodifferential operators
42B15Multipliers, several variables
42C10Fourier series in special orthogonal functions
35S99Pseudodifferential operators
47A07Forms on topological linear spaces