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This study examines single-particle electron motions in both a plane electromagnetic wave and a Gaussian focus in vacuum. Exact, explicit analytic expressions for relativistic electron trajectories in a plane wave are obtained, using the proper time as a parameter, in the general case of arbitrary initial positions and velocities. It is shown that previous analyses can be completed using the proper-time parameter. The conditions under which localized oscillatory motions ('figure-of-eight' orbits) occur are derived from the new solutions. The general solutions are also connected with the figure-of-eight orbits by a Lorentz transformation. The analytic solutions for arbitrary initial conditions and an arbitrary initial field phase can be used to determine the ranges of electron ejection angle and emerging electron energy in a vacuum laser accelerator, in which electrons are ejected externally, and provide a basis for explaining the spectrum of nonlinear Thomson scattering radiation. Numerical solutions are used for electron motions in the focus of a Gaussian laser beam, and the mean motion allows one to test a new expression for the relativistic ponderomotive force. It is suggested that plane wave solutions can provide a basis for approximating the orbital motion of particles in Gaussian beams.

The local maximal inequality for the Schrödinger operators of order α > 1 is shown to be bounded from H s (R 2 ) to L 2 for any s > 3 8 . This improves the previous result of Sjölin on the regularity of solutions to fractional order Schrödinger equations. Our method is inspired by Bourgain's argument in case of α = 2. The extension from α = 2 to general α > 1 confronts three essential obstacles: the lack of Lee's reduction lemma, the absence of the algebraic structure of the symbol and the inapplicable Galilean transformation in the deduction of the main theorem. We get around these difficulties by establishing a new reduction lemma at our disposal and analyzing all the possibilities in using the separateness of the segments to obtain the analogous bilinear L 2 −estimates.To compensate the absence of Galilean invariance, we resort to Taylor's expansion for the phase function. The Bourgain-Guth inequality in [BG] is also rebuilt to dominate the solution of fractional order Schrödinger equations.2000 Mathematics Subject Classification. 42B25; 35Q41.

In this paper, a class of refinable functions is given by smoothening pseudo-splines in order to get divergence free and curl free wavelets. The regularity and stability of them are discussed. Based on that, the corresponding Riesz wavelets are constructed.

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This work concerns linear and nonlinear wave equations in three space dimensions in a radial setting. We first prove new Strichartz estimates for the linear equation, in weighted Sobolev spaces which were introduced in a preceding article of Tristan Roy and the first author. These spaces are not Hilbert space. However they are local, thus adapted to finite speed of propagation, and related to a conservation law of the linear wave equation. We also construct the adapted profile decomposition. Our main motivation is to study the radial semilinear wave equation in three space dimensions with a power-like nonlinearity which is greater than cubic, and not quintic. We prove that the equation is locally well-posed in the scale-invariant weighted Sobolev space mentioned above, and that the norm of any non-scattering solution goes to infinity at the maximal time of existence. This gives a refinement on known results on energy-subcritical and energy-supercritical wave equation, with a unified proof.

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