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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.

47G30Pseudodifferential operators
42B15Multipliers, several variables
42C10Fourier series in special orthogonal functions
35S99Pseudodifferential operators
[1]Bényi, Á. Bilinear singular integrals and pseudodifferential operators, PhD Thesis, University of Kansas, Lawrence, (2002).
[2]Bényi, Ä. Bilinear pseudodifferential operators with forbidden symbols on Lipschitz and Besov spaces,J. Math. Anal. Appl. 284, 97–103, (2003). · Zbl 1037.35113 · doi:10.1016/S0022-247X(03)00245-2
[3]Bényi, Á., Gröchenig, K., Heil, C., and Okoudjou, K. Modulation spaces and a class of bounded multilinear pseudodifferential operators,J. Operator Theory to appear.
[4]Bényi, Á. and Torres, R. H. Symbolic calculus and the transposes of bilinear pseudodifferential operators,Comm. Partial Differential Equations 28, 1161–1181, (2003). · Zbl 1103.35370 · doi:10.1081/PDE-120021190
[5]Bényi, Á. and Torres, R. H. Almost orthogonality and a class of bounded bilinear pseudodifferential operators,Math. Res. Lett. 11, 1–11, (2004). · doi:10.4310/MRL.2004.v11.n1.a1
[6]Bourdaud, G. Lp-estimates for certain nonregular pseudo-differential operators,Comm. Partial Differential Equations 7, 1023–1033, (1982). · Zbl 0499.35097 · doi:10.1080/03605308208820244
[7]Brown, R. M. and Torres, R. H. Uniqueness in the inverse conductivity problem for conductivities with 3/2 derivatives inL P,p &gt; 2n,J. Fourier Anal. Appl. 9(6), 563–574, (2003). · Zbl 1051.35105 · doi:10.1007/s00041-003-0902-3
[8]Chae, D. On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces,Comm. Pure Appl. Math. 55, 654–678, (2002). · Zbl 1025.35016 · doi:10.1002/cpa.10029
[9]Christ, F. M. and Weinstein, M. I. Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation,J. Func. Anal. 100, 87–109, (1991). · Zbl 0743.35067 · doi:10.1016/0022-1236(91)90103-C
[10]Coifman, R. R. and Grafakos, L. Hardy space estimates for multilinear operators, I,Rev. Mat. Iberoamericana 8, 45–67, (1992). · Zbl 0785.47025 · doi:10.4171/RMI/116
[11]Coifman, R. R., Lions, P. L., Meyer, Y., and Semmes, S. Compensated compactness and Hardy spaces,J. Math. Pures Appl. (9) 72, 247–286, (1993).
[12]Coifman, R. R. and Meyer, Y. On commutators of singular integrals and bilinear singular integrals,Trans. Amer. Math. Soc. 212, 315–331, (1975). · doi:10.1090/S0002-9947-1975-0380244-8
[13]Coifman, R. R. and Meyer, Y. Commutateurs d’intégrales singulièrs et opérateurs multilinéaires,Ann. Inst. Fourier (Grenoble) 28, 177–202, (1978). · Zbl 0368.47031 · doi:10.5802/aif.708
[14]Coifman, R.R. and Meyer, Y. Au-delà des opérateurs pseudo-diffeŕentiels,Astérisque 57, Société Math. de France, (1978).
[15]Gilbert, J. and Nahmod, A. Hardy spaces and a Walsh model for bilinear cone operators,Trans. Amer. Math. Soc. 351, 3267–3300, (1999). · Zbl 0919.42009 · doi:10.1090/S0002-9947-99-02490-3
[16]Gilbert, J. and Nahmod, A. Boundedness of bilinear operators with nonsmooth symbols,Math. Res. Lett. 7, 767–778, (2000). · doi:10.4310/MRL.2000.v7.n6.a9
[17]Gilbert, J. and Nahmod, A. Bilinear operators with nonsmooth symbols. IJ. Fourier Anal. Appl. 7(5), 435–467, (2001). · Zbl 0994.42014 · doi:10.1007/BF02511220
[18]Gilbert, J. and Nahmod, A. Lp-boundedness of time-frequency paraproducts, II,J. Fourier Anal. Appl. 8(2), 109–172, (2002). · Zbl 1028.42013 · doi:10.1007/s00041-002-0006-5
[19]Grafakos, L. Hardy space estimates for multilinear operators, II,Rev. Mat. Iberoamericana 8, 69–92, (1992). · Zbl 0785.47026 · doi:10.4171/RMI/117
[20]Grafakos, L. and Torres, R. H. Discrete decompositions for bilinear operators and almost diagonal conditions,Trans. Amer. Math. Soc. 354, 1153–1176, (2002). · Zbl 0988.42013 · doi:10.1090/S0002-9947-01-02912-9
[21]Grafakos, L. and Torres, R. H. Multilinear Calderón-Zygmund theory,Adv. Math. 165, 124–164, (2002). · Zbl 1032.42020 · doi:10.1006/aima.2001.2028
[22]Gröchenig, K.Foundations of Time-Frequency Analysis, Birkhäuser, Boston, (2001).
[23]Hörmander, L. Pseudo-differential operators,Comm. Pure Appl. Math. 18, 501–517, (1965). · Zbl 0125.33401 · doi:10.1002/cpa.3160180307
[24]Kato, T. and Ponce, G. Commutator estimates and the Euler and Navier-Stokes equations,Comm. Pure Appl. Math. 41, 891–907, (1988). · Zbl 0671.35066 · doi:10.1002/cpa.3160410704
[25]Kenig, C., Ponce, G., and Vega, L. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,Comm. Pure Appl. Math. 46, 527–620, (1993). · Zbl 0808.35128 · doi:10.1002/cpa.3160460405
[26]Kenig, C. and Stein, E. Multilinear estimates and fractional integration,Math. Res. Lett. 6, 1–15, (1999). · doi:10.4310/MRL.1999.v6.n1.a1
[27]Kohn, J. J. and Nirenberg, L. An algebra of pseudodifferential operators,Comm. Pure Appl. Math. 18, 269–305, (1965). · Zbl 0171.35101 · doi:10.1002/cpa.3160180121
[28]Lacey, M. and Thiele, C. Lp-bounds for the bilinear Hilbert transform, 2 &lt;p &lt; Ann. of Math. (2) 146, 693–724, (1997). · Zbl 0914.46034 · doi:10.2307/2952458
[29]Lacey, M. and Thiele, C. Calderón’s conjecture,Ann. of Math. (2) 149, 475–496, (1999). · Zbl 0934.42012 · doi:10.2307/120971
[30]Meyer, Y. Remarques sur un théorème de J. M. Bony, Prépub. Dept. Math. Univ Paris-Sud, 91405 Orsay, France, (1980).
[31]Muscalu, C., Pipher, J., Tao, T., and Thiele, C. Bi-parameter paraproducts,Acta Math. 193, 269–296, (2004). · Zbl 1087.42016 · doi:10.1007/BF02392566
[32]Muscalu, C., Tao, T., and Thiele, C. Multilinear operators given by singular multipliers,J. Amer. Math. Soc. 15, 469–496, (2002). · Zbl 0994.42015 · doi:10.1090/S0894-0347-01-00379-4
[33]Okoudjou, K. Embeddings of some classical Banach spaces into modulation spaces,Proc. Amer. Math. Soc. 132, 1639–1647, (2004). · Zbl 1044.46030 · doi:10.1090/S0002-9939-04-07401-5
[34]Runst, T. and Sickel, W.Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin, (1996).
[35]Stein, E. M.Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, (1993).
[36]Wainger, S. Special trigonometric series in k-dimensions,Mem. Amer. Math. Soc. 59, (1965).