Smoothing theorems for Radon transforms over hypersurfaces and related operators

We extend the theorems of [M. Greenblatt, L^{p} Sobolev regularity of averaging operators over hypersurfaces and the Newton polyhedron, J. Funct. Anal. 276 2019, 5, 1510–1527] on {L^{p}} to {L^{p}_{s}} Sobolev improvement for translation invariant Radon and fractional singular Radon transforms over hypersurfaces, proving {L^{p}} to {L^{q}_{s}} boundedness results for such operators. Here {q\geq p} but s can be positive, negative, or zero. For many such operators we will have a triangle {Z\subset(0,1)\times(0,1)\times{\mathbb{R}}} such that one has {L^{p}} to {L^{q}_{s}} boundedness for {({1\over p},{1\over q},s)} beneath Z, and in the case of Radon transforms one does not have {L^{p}} to {L^{q}_{s}} boundedness for {({1\over p},{1\over q},s)} above the plane containing Z, thereby providing a Sobolev space improvement result which is sharp up to endpoints for {({1\over p},{1\over q})} below Z. This triangle Z intersects the plane {\{(x_{1},x_{2},x_{3}):x_{3}=0\}}, and therefore we also have an {L^{p}} to {L^{q}} improvement result that is also sharp up to endpoints for certain ranges of p and q.

Compactness criteria for real algebraic sets and Newton polyhedra

Let {f\colon\mathbb{R}^{n}\rightarrow\mathbb{R}} be a polynomial and {\mathcal{Z}(f)} its zero set. In this paper, in terms of the so-called Newton polyhedron of f, we present a necessary criterion and a sufficient condition for the compactness of {\mathcal{Z}(f)} . From this we derive necessary and sufficient criteria for the stable compactness of {\mathcal{Z}(f)} .

THE HOLOMORPHY CONJECTURE FOR NONDEGENERATE SURFACE SINGULARITIES

The holomorphy conjecture roughly states that Igusa’s zeta function associated to a hypersurface and a character is holomorphic on$\mathbb{C}$whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article, we prove the holomorphy conjecture for surface singularities that are nondegenerate over$\mathbb{C}$with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volumes (which appear in the formula of Varchenko for the zeta function of monodromy) of the faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast to the context of the trivial character, we here need to show fakeness of certain candidate poles other than those contributed by$B_{1}$-facets.

The Motivic Igusa Zeta Series of Some Hypersurfaces Non-Degenerated with Respect to their Newton Polyhedron

We describe some algorithms, without using resolution of singularities, that establish the rationality of the motivic Igusa zeta series of certain hypersurfaces that are non-degenerated with respect to their Newton polyhedron. This includes, in any characteristic, the motivic rationality for polydiagonal hypersurfaces, vertex singularities, binomial hypersurfaces, and Du Val singularities.

Reduction to Restriction Estimates near the Principal Root Jet

This chapter shows that one may reduce the desired Fourier restriction estimate to a piece Ssubscript Greek small letter psi of the surface S lying above a small, “horn-shaped” neighborhood Dsubscript Greek small letter psi of the principal root jet ψ‎, on which ∣x₂ − ψ‎(x₁)∣ ≤ ε‎xᵐ₁. Here, ε‎ > 0 can be chosen as small as one wishes. The proof then provides the opportunity to introduce some of the basic tools which will be applied frequently, such as dyadic domain decompositions, rescaling arguments based on the dilations associated to a given edge of the Newton polyhedron, in combination with Greenleaf's restriction and Littlewood–Paley theory, hence summing the estimates that have been obtained for the dyadic pieces.