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Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators. (English) Zbl 1110.47039

The present paper deals with bilinear pseudodifferential operators, defined a priori from S( n )×S( n ) into S ' ( n ), of the form

T σ (f,g)(x)= n n σ(x,ξ,η)f ^(ξ)g ^(η)e ix(ξ+η) dξdη,

where their symbols σ satisfy estimates of the form (the class BS ρ,δ m )

| x α ξ β η γ σ(x,ξ,η)|C αβγ (1+|ξ|+|η|) m+δ|α|-ρ(|β|+|γ|) ,

or (the class BS ρ,δ;θ m )

| x α ξ β η γ σ(x,ξ,η)|C αβγ;θ (1+|η-ξtanθ|) m+δ|α|-ρ(|β|+|γ|)

for all (x,ξ,η) 3n , all multi-indices α,β and γ, and some constants C αβγ or, respectively, C αβγ;θ . It is assumed that θ(-π 2,π 2), with the convention that θ=π/2 corresponds to the decay in terms of 1+|ξ| only. S( n ) denotes the Schwartz space of functions and S ' ( n ) is the space of tempered distributions. By f ^ is denoted the Fourier transform of the function fS( n )·

The authors study mainly the boundedness properties of such operators T σ . Among many other results, the authors prove that every operator T σ with a symbol in the class BS 1,1 m , m0, has a bounded extension from L m+s p ×L m+s q into L s r , provided that 1/p+1/q=1/r, 1<p,q,r<, and s>0· Moreover,

T σ (f,g) L s r C(p,q,r,s,n,m,σ)f L m+s p g L q +f L p g L m+s q ·

A symbolic calculus for the transposes of bilinear pseudodifferential operators and for the composition of linear pseudodifferential operators is also presented.


MSC:
47G30Pseudodifferential operators
42B15Multipliers, several variables
42C10Fourier series in special orthogonal functions
35S99Pseudodifferential operators
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