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Smoothing estimates for non-dispersive equations

(2015) MATHEMATISCHE ANNALEN. 365(1-2). p.241-269
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Abstract
This paper describes an approach to global smoothing problems for non-dispersive equations based on ideas of comparison principle and canonical transformation established in authors' previous paper (Ruzhansky and Sugimoto, Proc Lond Math Soc, 105:393-423, 2012), where dispersive equations were treated. For operators a(D-x) 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 D-x vertical bar((m-1)/2)e(ila(Dx))phi(x)parallel to(L2(Rt x Rxn)) <= C parallel to phi parallel to(L2(Rxn)) (s > 1/2) is well-known, while it is also known to fail for non-dispersive operators. For the case when the dispersiveness breaks, we suggest the estimate in the form parallel to < x >(-s)vertical bar del a(D-x)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(D-x) . We show that this estimate and its variants do continue to hold for a variety of non-dispersive operators , where del a(xi) may become zero on some set. Moreover, other types of such estimates, and the case of time-dependent equations are also discussed.
Keywords
NONLINEAR SCHRODINGER-EQUATIONS, TIME-DEPENDENT COEFFICIENTS, DISPERSIVE EQUATIONS, CONSTANT-COEFFICIENTS, REGULARITY, DECAY, SYSTEMS, CONVERGENCE, PRINCIPLE, PROPERTY

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Citation

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Chicago
Ruzhansky, Michael, and Mitsuru Sugimoto. 2015. “Smoothing Estimates for Non-dispersive Equations.” Mathematische Annalen 365 (1-2): 241–269.
APA
Ruzhansky, M., & Sugimoto, M. (2015). Smoothing estimates for non-dispersive equations. MATHEMATISCHE ANNALEN, 365(1-2), 241–269.
Vancouver
1.
Ruzhansky M, Sugimoto M. Smoothing estimates for non-dispersive equations. MATHEMATISCHE ANNALEN. 2015;365(1-2):241–69.
MLA
Ruzhansky, Michael, and Mitsuru Sugimoto. “Smoothing Estimates for Non-dispersive Equations.” MATHEMATISCHE ANNALEN 365.1-2 (2015): 241–269. Print.
@article{8585370,
  abstract     = {This paper describes an approach to global smoothing problems for non-dispersive equations based on ideas of comparison principle and canonical transformation established in authors' previous paper (Ruzhansky and Sugimoto, Proc Lond Math Soc, 105:393-423, 2012), where dispersive equations were treated. For operators a(D-x) 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 D-x vertical bar((m-1)/2)e(ila(Dx))phi(x)parallel to(L2(Rt x Rxn)) <= C parallel to phi parallel to(L2(Rxn)) (s > 1/2) 
is well-known, while it is also known to fail for non-dispersive operators. For the case when the dispersiveness breaks, we suggest the estimate in the form 
parallel to < x >(-s)vertical bar del a(D-x)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(D-x) . We show that this estimate and its variants do continue to hold for a variety of non-dispersive operators , where del a(xi) may become zero on some set. Moreover, other types of such estimates, and the case of time-dependent equations are also discussed.},
  author       = {Ruzhansky, Michael and Sugimoto, Mitsuru},
  issn         = {0025-5831},
  journal      = {MATHEMATISCHE ANNALEN},
  keywords     = {NONLINEAR SCHRODINGER-EQUATIONS,TIME-DEPENDENT COEFFICIENTS,DISPERSIVE EQUATIONS,CONSTANT-COEFFICIENTS,REGULARITY,DECAY,SYSTEMS,CONVERGENCE,PRINCIPLE,PROPERTY},
  language     = {eng},
  number       = {1-2},
  pages        = {241--269},
  title        = {Smoothing estimates for non-dispersive equations},
  url          = {http://dx.doi.org/10.1007/s00208-015-1281-1},
  volume       = {365},
  year         = {2015},
}

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