 Author
 Michael Ruzhansky (UGent) and Mitsuru Sugimoto
 Organization
 Abstract
 This paper describes an approach to global smoothing problems for nondispersive equations based on ideas of comparison principle and canonical transformation established in authors' previous paper (Ruzhansky and Sugimoto, Proc Lond Math Soc, 105:393423, 2012), where dispersive equations were treated. For operators a(Dx) of order m satisfying the dispersiveness condition del a(xi) not equal 0 for xi not equal 0, the global smoothing estimate parallel to < x >(s)vertical bar Dx vertical bar((m1)/2)e(ila(Dx))phi(x)parallel to(L2(Rt x Rxn)) <= C parallel to phi parallel to(L2(Rxn)) (s > 1/2) is wellknown, while it is also known to fail for nondispersive operators. For the case when the dispersiveness breaks, we suggest the estimate in the form parallel to < x >(s)vertical bar del a(Dx)vertical bar(1/2)e(ita(Dx))phi(x)parallel to(L2(Rt x Rxn)) <= c parallel to phi parallel to(L2(Rxn)) (s > 1/2) which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator a(Dx) . We show that this estimate and its variants do continue to hold for a variety of nondispersive operators , where del a(xi) may become zero on some set. Moreover, other types of such estimates, and the case of timedependent equations are also discussed.
 Keywords
 NONLINEAR SCHRODINGEREQUATIONS, TIMEDEPENDENT COEFFICIENTS, DISPERSIVE EQUATIONS, CONSTANTCOEFFICIENTS, REGULARITY, DECAY, SYSTEMS, CONVERGENCE, PRINCIPLE, PROPERTY
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Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU8585370
 Chicago
 Ruzhansky, Michael, and Mitsuru Sugimoto. 2015. “Smoothing Estimates for Nondispersive Equations.” Mathematische Annalen 365 (12): 241–269.
 APA
 Ruzhansky, M., & Sugimoto, M. (2015). Smoothing estimates for nondispersive equations. MATHEMATISCHE ANNALEN, 365(12), 241–269.
 Vancouver
 1.Ruzhansky M, Sugimoto M. Smoothing estimates for nondispersive equations. MATHEMATISCHE ANNALEN. 2015;365(12):241–69.
 MLA
 Ruzhansky, Michael, and Mitsuru Sugimoto. “Smoothing Estimates for Nondispersive Equations.” MATHEMATISCHE ANNALEN 365.12 (2015): 241–269. Print.
@article{8585370, abstract = {This paper describes an approach to global smoothing problems for nondispersive equations based on ideas of comparison principle and canonical transformation established in authors' previous paper (Ruzhansky and Sugimoto, Proc Lond Math Soc, 105:393423, 2012), where dispersive equations were treated. For operators a(Dx) of order m satisfying the dispersiveness condition del a(xi) not equal 0 for xi not equal 0, the global smoothing estimate parallel to < x >(s)vertical bar Dx vertical bar((m1)/2)e(ila(Dx))phi(x)parallel to(L2(Rt x Rxn)) <= C parallel to phi parallel to(L2(Rxn)) (s > 1/2) is wellknown, while it is also known to fail for nondispersive operators. For the case when the dispersiveness breaks, we suggest the estimate in the form parallel to < x >(s)vertical bar del a(Dx)vertical bar(1/2)e(ita(Dx))phi(x)parallel to(L2(Rt x Rxn)) <= c parallel to phi parallel to(L2(Rxn)) (s > 1/2) which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator a(Dx) . We show that this estimate and its variants do continue to hold for a variety of nondispersive operators , where del a(xi) may become zero on some set. Moreover, other types of such estimates, and the case of timedependent equations are also discussed.}, author = {Ruzhansky, Michael and Sugimoto, Mitsuru}, issn = {00255831}, journal = {MATHEMATISCHE ANNALEN}, keywords = {NONLINEAR SCHRODINGEREQUATIONS,TIMEDEPENDENT COEFFICIENTS,DISPERSIVE EQUATIONS,CONSTANTCOEFFICIENTS,REGULARITY,DECAY,SYSTEMS,CONVERGENCE,PRINCIPLE,PROPERTY}, language = {eng}, number = {12}, pages = {241269}, title = {Smoothing estimates for nondispersive equations}, url = {http://dx.doi.org/10.1007/s0020801512811}, volume = {365}, year = {2015}, }
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