Bényi, Árpád; Torres, Rodolfo H. Symbolic calculus and the transposes of bilinear pseudodifferential operators. (English) Zbl 1103.35370 Commun. Partial Differ. Equations 28, No. 5-6, 1161-1181 (2003). Summary: A symbolic calculus for the transposes of a class of bilinear pseudodifferential operators is developed. The calculus is used to obtain boundedness results on products of Lebesgue spaces. A larger class of pseudodifferential operators that does not admit a calculus is also considered. Such a class is the bilinear analog of the so-called exotic class of linear pseudodifferential operators and fail to produce bounded operators on products of Lebesgue spaces. Nevertheless, the operators are shown to be bounded on products of Sobolev spaces with positive smoothness, generalizing the Leibniz rule estimates for products of functions. Cited in 2 ReviewsCited in 52 Documents MSC: 35S05 Pseudodifferential operators as generalizations of partial differential operators 47G30 Pseudodifferential operators PDFBibTeX XMLCite \textit{Á. Bényi} and \textit{R. H. Torres}, Commun. Partial Differ. Equations 28, No. 5--6, 1161--1181 (2003; Zbl 1103.35370) Full Text: DOI References: [1] DOI: 10.1080/03605308208820244 · Zbl 0499.35097 [2] DOI: 10.1080/03605308808820568 · Zbl 0659.35115 [3] DOI: 10.1016/0022-1236(91)90103-C · Zbl 0743.35067 [4] DOI: 10.1090/S0002-9947-1975-0380244-8 [5] Coifman R. R., Ann. Inst. Fourier, Grenoble 28 pp 177– (1978) · Zbl 0368.47031 [6] Coifman R. R., Astérisque 57 pp 1– (1978) [7] Folland, G. B. 1983.Lectures on Partial Differential Operators160Bombay: Tata Institute of Fundamental Research, Springer-Verlag. [8] DOI: 10.1006/aima.2001.2028 · Zbl 1032.42020 [9] DOI: 10.1002/cpa.3160180307 · Zbl 0125.33401 [10] Journé J.-L., Lecture Notes in Mathematics 994 pp 128 pp.– (1983) [11] DOI: 10.1002/cpa.3160410704 · Zbl 0671.35066 [12] Kenig C., Math. Res. Lett. 6 pp 1– (1999) · Zbl 0952.42005 [13] DOI: 10.1002/cpa.3160180121 · Zbl 0171.35101 [14] Meyer, Y. 1980.Remarques sur un théorème de J. M. Bony24 pp91405 Orsay, France: Prépub. Dept. Math. Univ. Paris-Sud. [15] Petersen, B. E. 1983.Introduction to the Fourier Transform and Pseudo-Differential Operatorspp. 256Boston, Massachusetts: Pitman (Advanced Publishing Program). [16] Stein, E. M. 1993.Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integralspp. 695Princeton, New Jersey: Princeton University Press. [17] Taylor, E. M. 1981.Pseudodifferential Operatorspp. 452Princeton, New Jersey: Princeton University Press. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.