In this paper we prove several results for the scattering phase (spectral shift function) related with perturbations of the electromagnetic field for the Dirac operator in the Euclidean space.

Many accurate results are now available for perturbations of the Schrödinger operator, in the high energy regime or in the semi-classical regime. Here we extend these results to the Dirac operator. There are several technical problems to overcome because the Dirac operator is a system, its symbol is a 4×4 matrix, and its continuous spectrum has positive and negative values. We show that we can separate positive and negative energies to prove high energy asymptotic expansion and we construct a semi-classical Foldy-Wouthuysen transformation in the semi-classical case. We also prove an asymptotic expansion for the scattering phase when the speed of light tends to infinity (non-relativistic limit).

In this paper we study the (L^p(u^p), L^q(v^q)) boundedness of the higher order commutators T ${}_{Ω,α,b}^{m}$ and M ${}_{Ω,α,b}^{m}$ formed by the fractional integral operator T_Ω,α, the fractional maximal operator M_Ω,α, and a function b(x) in BMO(v), respectively.

Our results imporve and extend the corresponding results obtained by Segovia and Torrea in 1993 [9].

We study the solvability problem for the multidimensional Riccati equation $−Δu=|∇u|^q+ω$, where $q>1$ and $ω$ is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation $−Δu−ωu=0$ with nonnegative potential $ω$. We establish explicit criteria for the existence of global solutions on $\mathbf{R}^n$ in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type $−Lu=f(x, u, ∇u)+ω$ where $f(x, u, ∇u)\asymp a(x)|∇u|^{q_1} +b(x)|u|^{q_2}$, and $L$ is a uniformly elliptic operator.

We consider the Newton equation $$\ddot{x} = F(x), \quad F(x) = -∇ \upsilon (x), \quad x \in \mathbf{R}^d,\\ \text{where }\upsilon \in C^2(\mathbf{R}^d, \mathbf{R}), |∂^j_x v(x) |\le c_{|j|}(1+|x|)^{-(\alpha+|j|)}$$ for |j|≤2 and some α>1.

We give estimates and asymptotics for scattering solutions and scattering data for the equation (*) for the case of small angle scattering. We show that scattering data at high energies uniquely determine the X-ray transforms PF and Pv. Applying results on inversion of the X-ray transform P we obtain that for d≥2 scattering data at high energies uniquely determine F and v. For the case of potentials with compact support we give a connection between boundary value data and scattering data and for d≥2 we obtain, using known results, a uniqueness theorem in the inverse scattering problem at fixed energy.

Let ϕ be an analytic function defined on the unit disk D, with ϕ(D)⊂D, ϕ(0)=0, and ϕ′(0)=λ≠0. Then by a classical result of G. Kœnigs, the sequence of normalized iterates Φ_n/λⁿ converges uniformly on compact subsets of D to a function σ analytic in D which satisfies σ°φ=λσ. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy space H_p, the sequence Φ_n/λⁿ converges to σ in the norm of H_p. We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ϕ is univalent.

When ϕ is inner, P. Bourdon and J. Shapiro have shown that σ does not belong to the Nevanlinna class, in particular it does not belong to any H_p. It is natural to ask, how bad can the growth of σ be in this case? As a partial answer we show that σ always belongs to some Bergman space L ${}_{a}^{p}$ .

LetI be a union of finitely many closed intervals in [−1, 0). Let I^↞ be a single interval of the form [−1, −a] chosen to have the same logarithmic length as I. Let D be the unit disc. Then, Beurling [8] has shown that the harmonic measure of the circle ∂D at the origin in the slit disc D/I is increased if I is replaced by I^↞. We prove a number of cognate results and extensions. For instance, we show that Beurling's result remains true if the intervals in I are not just one-dimensional, but if they in fact constitute polar rectangles centred on the negative real axis and having some fixed constant angular width. In doing this, we obtain a new proof of Beurling's result. We also discuss a conjecture of Matheson and Pruss [25] and some other open problems.