We obtain sharp integral potential bounds for gradients of solutions to a wide class of quasilinear elliptic equations with measure data. Our estimates are global over bounded domains that satisfy a mild exterior capacitary density condition. They are obtained in Lorentz spaces whose degrees of integrability lie below or near the natural exponent of the operator involved. As a consequence, nonlinear Calderón–Zygmund type estimates below the natural exponent are also obtained for $\mathcal{A}$-superharmonic functions in the whole space ℝⁿ. This answers a question raised in our earlier work (On Calderón–Zygmund theory for p- and $\mathcal{A}$-superharmonic functions, to appear in Calc. Var. Partial Differential Equations, DOI 10.1007/s00526-011-0478-8) and thus greatly improves the result there.

We provide characterizations of $p\mspace {1mu}$-nilpotency for fusion systems and $p\mspace {1mu}$-local finite groups that are inspired by known result for finite groups. In particular, we generalize criteria by Atiyah, Brunetti, Frobenius, Quillen, Stammbach and Tate.

Suppose μ is a positive measure on $\mathbb{R}^{2}$ given by μ=ν×λ, where ν and λ are Radon measures on $\mathcal{S}^{1}$ and $\mathbb{R}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .1ex\hbox {$\scriptscriptstyle +$}} {\scriptscriptstyle +}}}$, respectively, which do not vanish on any open interval. We prove that if for either ν or λ there exists a set of positive measure A in its domain for which the upper and lower s-densities, 0< s≤1, are positive and finite for every x∈A then the uncentered Hardy–Littlewood maximal operator M_μ is weak-type (1,1) if and only if ν is doubling and λ is doubling away from the origin. This generalizes results of Vargas concerning rotation-invariant measures on $\mathbb{R}^{n}$ when n=2.

We prove the existence of a resonance-free region in scattering by a strictly convex obstacle $\mathcal{O}$ with the Robin boundary condition $\partial_{\nu}u+\gamma u|_{\partial\mathcal{O}}=0$. More precisely, we show that the scattering resonances lie below a cubic curve ℑζ=−S|ζ|^1/3+C. The constant S is the same as in the case of the Neumann boundary condition γ=0. This generalizes earlier results on cubic pole-free regions obtained for the Dirichlet boundary condition.

Let $(R, \frak{m}, k_{R})$ be a regular local k-algebra satisfying the weak Jacobian criterion, and such that k_R/k is an algebraic field extension. Let $\mathcal{D}_{R}$ be the ring of k-linear differential operators of R. We give an explicit decomposition of the $\mathcal{D}_{R}$-module $\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}_{R}^{n+1}$ as a direct sum of simple modules, all isomorphic to $\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}$, where certain “Pochhammer” differential operators are used to describe generators of the simple components.

For smooth compact oriented Riemannian manifolds M of dimension 4k+2, k≥0, with or without boundary, and a vector bundle F on M with an inner product and a flat connection, we construct a modification of the Hodge star operator on the middle-dimensional (parabolic) cohomology of M twisted by F. This operator induces a canonical complex structure on the middle-dimensional cohomology space that is compatible with the natural symplectic form given by integrating the wedge product. In particular, when k=0 we get a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface, with monodromies along boundary components belonging to fixed conjugacy classes when the surface has boundary, that is compatible with the standard symplectic form on the moduli space.

In the present paper, we prove an improved Combes–Thomas estimate, viz. the Combes–Thomas estimate in trace-class norms, for magnetic Schrödinger operators under general assumptions. In particular, we allow for unbounded potentials. We also show that for any function in the Schwartz space on the reals the operator kernel decays, in trace-class norms, faster than any polynomial.

The purpose of this paper is to organize some results on the local geometry of CR singular real-analytic manifolds that are images of CR manifolds via a CR map that is a diffeomorphism onto its image. We find a necessary (sufficient in dimension 2) condition for the diffeomorphism to extend to a finite holomorphic map. The multiplicity of this map is a biholomorphic invariant that is precisely the Moser invariant of the image, when it is a Bishop surface with vanishing Bishop invariant. In higher dimensions, we study Levi-flat CR singular images and we prove that the set of CR singular points must be large, and in the case of codimension 2, necessarily Levi-flat or complex. We also show that there exist real-analytic CR functions on such images that satisfy the tangential CR conditions at the singular points, yet fail to extend to holomorphic functions in a neighborhood. We provide many examples to illustrate the phenomena that arise.

We consider real interpolation methods defined by means of slowly varying functions b and symmetric spaces E, for which we present extreme reiteration theorems. As an application we identify, for all possible values of θ∈[0,1], the interpolation spaces (L₁, LlogL)_{θ, b, E} and (L_exp, L_∞)_{θ, b, E}.