We consider two types of multilinear pseudodifferential operators. First, we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in weighted Lebesgue spaces. These results generalise earlier work of the present authors concerning linear pseudo-pseudodifferential operators. Secondly, we investigate the boundedness of bilinear pseudodifferential operators with symbols in the Hörmander S m ρ,δ classes. These results are new in the case ρ < 1, that is, outwith the scope of multilinear Calderón-Zygmund theory.

Abstract. In this paper we prove weighted norm inequalities with weights in the Ap classes, for pseudodifferential operators with symbols in the class S n(ρ−1) ρ,δ that fall outside the scope of Calderón-Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy-Littlewood type maximal functions. Our weighted norm inequalities also yield L p boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in OPS m ρ,δ .

We prove weighted norm inequalities for pseudodifferential operators with amplitudes which are only measurable in the spatial variables. The result is sharp, even for smooth amplitudes. Nevertheless, in the case when the amplitude contains the oscillatory factor ξ → e i|ξ | 1−ρ , the result can be substantially improved. We extend the L p -boundedness of pseudo-pseudodifferential operators to certain weights. End-point results are obtained when the amplitude is either smooth or satisfies a homogeneity condition in the frequency variable. Our weighted norm inequalities also yield the boundedness of commutators of these pseudodifferential operators with functions of bounded mean oscillation.

The present article establishes connections between the structure of the deterministic Navier-Stokes equations and the structure of (similarity) equations that govern self-similar solutions as expected values of certain naturally associated stochastic cascades. A principle result is that explosion criteria for the stochastic cascades involved in the probabilistic representations of solutions to the respective equations coincide. While the uniqueness problem itself remains unresolved, these connections provide interesting problems and possible methods for investigating symmetry breaking and the uniqueness problem for Navier-Stokes equations. In particular, new branching Markov chains, including a dilogarithmic branching random walk on the multiplicative group (0, ∞), naturally arise as a result of this investigation.

In a recent paper Calogero and Alcantara [1] derived a Lorentz-invariant Fokker-Planck equation, which corresponds to the evolution of a particle distribution associated with relativistic Brownian Motion. We study the "one and one-half" dimensional version of this problem with nonlinear electromagnetic interactions -the relativistic Vlasov-Maxwell-Fokker-Planck system -and obtain the first results concerning well-posedness of solutions. Specifically, we prove the global-in-time existence and uniqueness of classical solutions to the Cauchy problem and a gain in regularity of the distribution function in its momentum argument.

In this paper we extend the results of the research started in [6] and [7], in which Karlin-McGregor diagonalization of certain reversible Markov chains over countably infinite general state spaces by orthogonal polynomials was used to estimate the rate of convergence to a stationary distribution.We use a method of Koornwinder [5] to generate a large and interesting family of random walks which exhibits a lack of spectral gap, and a polynomial rate of convergence to the stationary distribution. For the Chebyshev type subfamily of Markov chains, we use asymptotic techniques to obtain an upper bound of order O log t √ t and a lower bound of order O 1 √ t on the distance to the stationary distribution regardless of the initial state. Due to the lack of a spectral gap, these results lie outside the scope of geometric ergodicity theory [8].

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In this paper we prove local well-posedness for Quasi-linear Scrhödinger equations with initial data in unweighted Sobolev Spaces. For small initial data with minimal smoothness this has addressed by J. Marzuola, J. Metcalfe and D. Tataru [15], [16]. This work does not attempt to address the minimal regularity for initial data, but instead builds on the previous results of C. Kenig, G. Ponce, and L. Vega [13], [12] to remove the smallness condition in unweighted spaces. This is accomplished by developing a non-centered version of Doi's Lemma, which allows one to prove Kato type smoothing estimates. These estimates make it possible to achieve the necessary a priori linear results.