zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 $S (\Bbb R^{n}) \times S (\Bbb R^{n})$ into $S' (\Bbb R^{n}),$ of the form $$T_{\sigma} (f,g) (x) = \int_{\Bbb R^{n}} \int_{\Bbb R^{n}} \sigma (x,\xi,\eta) \widehat{f} (\xi) \widehat{g} (\eta) e^{i x(\xi + \eta)} d \xi\, d \eta,$$ where their symbols $\sigma$ satisfy estimates of the form (the class $B S_{\rho,\delta}^{m})$ $$\vert \partial_{x}^{\alpha} \partial_{\xi}^{\beta} \partial_{\eta}^{\gamma} \sigma (x,\xi,\eta) \vert \leq C_{\alpha \beta \gamma} (1 + \vert \xi \vert + \vert \eta \vert )^{m + \delta \vert \alpha \vert - \rho (\vert \beta \vert + \vert \gamma \vert)},$$ or (the class $B S_{\rho,\delta;\theta}^{m}$) $$\vert \partial_{x}^{\alpha} \partial_{\xi}^{\beta} \partial_{\eta}^{\gamma} \sigma (x,\xi,\eta) \vert \leq C_{\alpha \beta \gamma;\theta} (1 + \vert \eta - \xi \tan \theta \vert )^{m + \delta \vert \alpha \vert - \rho (\vert \beta \vert + \vert \gamma \vert)}$$ for all $(x,\xi,\eta) \in \Bbb R^{3n}$, all multi-indices $\alpha,\beta$ and $\gamma,$ and some constants $C_{\alpha \beta \gamma}$ or, respectively, $C_{\alpha \beta \gamma;\theta}$. It is assumed that $\theta \in (-\frac{\pi}{2},\frac{\pi}{2})$, with the convention that $\theta = \pi/2$ corresponds to the decay in terms of $1 + \vert \xi \vert$ only. $S (\Bbb R^{n})$ denotes the Schwartz space of functions and $S' (\Bbb R^{n})$ is the space of tempered distributions. By $\widehat{f}$ is denoted the Fourier transform of the function $f\in S (\Bbb R^{n}).$ The authors study mainly the boundedness properties of such operators $T_{\sigma}$. Among many other results, the authors prove that every operator $T_{\sigma}$ with a symbol in the class $B S_{1,1}^{m}$, $m \geq 0,$ has a bounded extension from $L_{m + s}^{p} \times L_{m + s}^{q}$ into $L_{s}^{r},$ provided that $1/p + 1/q = 1/r$, $1 < p,q,r < \infty,$ and $s > 0.$ Moreover, $$\Vert T_{\sigma} (f,g) \Vert_{L_{s}^{r}} \leq C (p,q,r,s,n,m,\sigma) \left(\Vert f \Vert_{L_{m + s}^{p}} \Vert g \Vert_{L^{q}} + \Vert f \Vert_{L^{p}} \Vert g \Vert_{L_{m + s}^{q}}\right).$$ 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
Full Text: DOI
[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. · Zbl 1106.47041
[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). · Zbl 1067.47062 · 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). · Zbl 0864.42009
[12] Coifman, R. R. and Meyer, Y. On commutators of singular integrals and bilinear singular integrals,Trans. Amer. Math. Soc. 212, 315--331, (1975). · Zbl 0324.44005 · 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). · Zbl 0987.42017 · 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). · Zbl 0966.42020
[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). · Zbl 0952.42005 · 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). · Zbl 0873.35001
[35] Stein, E. M.Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, (1993). · Zbl 0821.42001
[36] Wainger, S. Special trigonometric series in k-dimensions,Mem. Amer. Math. Soc. 59, (1965). · Zbl 0136.36601