arXiv subjects

Robert Schippa

Publications and source records attributed to Robert Schippa.

At least 19 recordsLinked to original sources

arXiv2026

Low-regularity well-posedness for dispersive equations with derivative nonlinearity and quasi-periodic initial data

We show low-regularity well-posedness of the Korteweg-de Vries equation with spatially quasi-periodic initial data. To this end, we employ frequency-dependent time localization and a bilinear version of the C\'ordoba--Fefferman square function estimate to show a bilinear Strichartz estimate for quasi-periodic functions. The solutions are proved to preserve the Sobolev regularity of the initial data, which was not the case in earlier works. The argument extends to other dispersion relations and higher order nonlinearities.

math.AP↗

arXiv2026

Local smoothing for rough wave equations

We prove local smoothing estimates for wave equations with $C^{r}$ coefficients, relying on bilinear restriction estimates associated with rough wave propagation. In two and three dimensions, for $C^{1,1}$ coefficients we recover the sharp $L^p$-$L^q$ local smoothing bounds previously established for smooth coefficients. These are implied by stronger $L^{p}$-$L^{q}$ decoupling inequalities with endpoint derivative loss. As a corollary, we obtain non-trivial $L^{p}$-$L^{p}$ local smoothing estimates as well.

math.AP↗

arXiv2026

Decoupling for complex curves and improved decoupling for the cubic moment curve

We prove sharp $\ell^2$-decoupling inequalities for non-degenerate complex curves via the bilinear argument due to Guo--Li--Yung--Zorin-Kranich, which in turn is inspired by the efficient congruencing argument of Wooley. Secondly, quantifying the iteration in the cubic case, we obtain a logarithmic refinement of the decoupling inequality for the cubic moment curve.

math.CA↗

arXiv2026

The Cauchy problem for the periodic Kadomtsev--Petviashvili--II equation below $L^2$

We extend Bourgain's $L^2$-wellposedness result for the KP-II equation on $\mathbb{T}^2$ to initial data with negative Sobolev regularity. The key ingredient is a new linear $L^4$-Strichartz estimate which is effective on frequency-dependent time scales. The $L^4$-Strichartz estimates follow from combining an $\ell^2$-decoupling inequality recently proved by Guth--Maldague--Oh with semiclassical Strichartz estimates. Moreover, we rely on a variant of Bourgain's bilinear Strichartz estimate on frequency-dependent times, which is proved via the Córdoba--Fefferman square function estimate.

math.AP↗

arXiv2025

Wave packet decompositions and sharp bilinear estimates for rough Hamiltonian flows

The goal of this paper is to prove bilinear $L^p$ estimates for rough dispersive evolutions satisfying non-degeneracy and transversality assumptions. The estimates generalize the sharp Fourier extension estimates for the cone and the paraboloid. To this end, we require a wave packet decomposition with localization properties in space-time and space-time frequencies. Secondly, we construct a refined wave packet parametrix for dispersive equations with $C^{1,1}$-coefficients by using the FBI transform. As a consequence, we obtain bilinear estimates for solutions to dispersive equations with $C^{1,1}$ coefficients provided that the solutions interact transversely.

math.AP↗

arXiv2025

Global results for weakly dispersive KP-II equations on the cylinder

We consider the dispersion-generalized KP-II equation on a partially periodic domain in the weakly dispersive regime. We use Fourier decoupling techniques to derive essentially sharp Strichartz estimates. With these at hand, we show global well-posedness of the quasilinear Cauchy problem in $L^2(\mathbb{R} \times \mathbb{T})$. Finally, we prove a long-time decay property of solutions with small mass by using the Kato smoothing effect in the fractional case.

math.AP↗

arXiv2025

Square function estimates for cones over quadratic manifolds

We extend the $L^4$-square function estimates for the parabola and the half-cone to quadratic manifolds in higher dimensions and their conical extensions. To this end, we require transversality for the tangent spaces of the quadratic manifolds at separated points. This allows us to show biorthogonality for the associated system of quadratic equations. For the conical extensions we obtain a wave envelope estimate by High-Low arguments.

math.CA↗

arXiv2024

Improved well-posedness for quasilinear and sharp local well-posedness for semilinear KP-I equations

We show new well-posedness results in anisotropic Sobolev spaces for dispersion-generalized KP-I equations with increased dispersion compared to the KP-I equation. We obtain the sharp dispersion rate, below which generalized KP-I equations on $\mathbb{R}^2$ and on $\mathbb{R} \times \mathbb{T}$ exhibit quasilinear behavior. In the quasilinear regime, we show improved well-posedness results relying on short-time Fourier restriction. In the semilinear regime, we show sharp well-posedness with analytic data-to-solution mapping. On $\mathbb{R}^2$ we cover the full subcritical range, whereas on $\mathbb{R} \times \mathbb{T}$ the sharp well-posedness is strictly subcritical. Nonlinear Loomis-Whitney inequalities are one ingredient. These are presently proved for Borel measures with growth condition reflecting the different geometries of the plane $\mathbb{R}^2$, the cylinder $\mathbb{R} \times \mathbb{T}$, and the torus $\mathbb{T}^2$. Finally, we point out that on tori $\mathbb{T}^2_γ$, KP-I equations are never semilinear.

math.AP↗

arXiv2024

Generalized square function estimates for curves and their conical extensions

We show sharp square function estimates for curves in the plane whose curvature degenerates at a point and estimates sharp up to endpoints for cones over these curves. To this end, for curves of finite type we extend the classical Córdoba--Fefferman biorthogonality. For cones over degenerate curves, we analyze wave envelope estimates proved via High-Low-decomposition. The arguments are subsequently extended to the cone over the complex parabola.

math.CA↗

arXiv2024

Strichartz estimates for quasi-periodic functions and applications

We show Strichartz estimates for quasi-periodic functions with decaying Fourier coefficients via $\ell^2$-decoupling. When we additionally average in time, further improvements can be obtained. Next, we apply multilinear refinements to show low regularity local well-posedness for nonlinear Schrödinger equations. For the cubic nonlinear Schrödinger equation the approach yields the sharp local well-posedness result.

math.AP↗

arXiv2024

Local smoothing for the Hermite wave equation

We show local smoothing estimates in $L^p$-spaces for solutions to the Hermite wave equation. For this purpose, we obtain a parametrix given by a Fourier Integral Operator, which we linearize. This leads us to analyze local smoothing estimates for solutions to Klein-Gordon equations. We show $\ell^2$-decoupling estimates adapted to the mass parameter to obtain local smoothing with essentially sharp derivative loss. In one dimension as consequence of square function estimates, we obtain estimates sharp up to endpoints. Finally, we elaborate on the implications of local smoothing estimates for Hermite Bochner--Riesz means.

math.AP↗

arXiv2024

Low regularity well-posedness for KP-I equations: the dispersion-generalized case

We prove new well-posedness results for dispersion-generalized Kadomtsev--Petviashvili I equations in $\mathbb{R}^2$, which family links the classical KP-I equation with the fifth order KP-I equation. For strong enough dispersion, we show global well-posedness in $L^2(\mathbb{R}^2)$. To this end, we combine resonance and transversality considerations with Strichartz estimates and a nonlinear Loomis--Whitney inequality. Moreover, we prove that for small dispersion, the equations cannot be solved via Picard iteration. In this case, we use an additional frequency dependent time localization.

math.AP↗

arXiv2023

Improved global well-posedness for mass-critical nonlinear Schrödinger equations on tori

We show new global well-posedness results for mass-critical nonlinear Schrödinger equations on tori in one and two dimensions. For the quintic nonlinear Schrödinger equation on the circle we show global well-posedness for initial data in $H^s(\mathbb{T})$ for $s>\frac{1}{3}$ and $\| u_0 \|_{L^2(\mathbb{T})} \ll 1$. In two dimensions we show global well-posedness on possibly irrational tori for $s>\frac{3}{5}$. In the focusing case we need a smallness condition for the mass, whereas in the defocusing case large mass is covered.

math.AP↗

arXiv2023

Refinements of Strichartz estimates on tori and applications

We show trilinear Strichartz estimates in one and two dimensions on frequency-dependent time intervals. These improve on the corresponding linear estimates of periodic solutions to the Schrödinger equation. The proof combines decoupling iterations with bilinear short-time Strichartz estimates. Secondly, we use decoupling to show new linear Strichartz estimates on frequency dependent time intervals. We apply these in case of the Airy propagator to obtain the sharp Sobolev regularity for the existence of solutions to the modified Korteweg-de Vries equation.

math.AP↗

arXiv2023

Oscillatory integral operators with homogeneous phase functions

Oscillatory integral operators with $1$-homogeneous phase functions satisfying a convexity condition are considered. For these we show the $L^p - L^p$-estimates for the Fourier extension operator of the cone due to Ou--Wang via polynomial partitioning. For this purpose, we combine the arguments of Ou--Wang with the analysis of Guth--Hickman--Iliopoulou, who previously showed sharp $L^p-L^p$-estimates for non-homogeneous phase functions with variable coefficients under a convexity assumption. Furthermore, we provide examples exhibiting Kakeya compression, which shows the estimates to be sharp. We apply the oscillatory integral estimates to show new local smoothing estimates for wave equations on compact Riemannian manifolds $(M,g)$ with $\dim M \geq 3$. This generalizes the argument for the Euclidean wave equation due to Gao--Liu--Miao--Xi.

math.CA↗

arXiv2023

Strichartz estimates for equations with structured Lipschitz coefficients

Sharp Strichartz estimates are proved for Schrödinger and wave equations with Lipschitz coefficients satisfying additional structural assumptions. We use Phillips functional calculus as a substitute for Fourier inversion, which shows how dispersive properties are inherited from the constant coefficient case. Global Strichartz estimates follow provided that the derivatives of the coefficients are integrable. The estimates extend to structured coefficients of bounded variations. As applications we derive Strichartz estimates with additional derivative loss for wave equations with Hölder-continuous coefficients and solve nonlinear Schrödinger equations. Finally, we record spectral multiplier estimates, which follow from the Strichartz estimates by well-known means.

math.AP↗

arXiv2023

Strichartz estimates for Maxwell equations on domains with perfectly conducting boundary conditions

We consider Maxwell equations on a smooth domain with perfectly conducting boundary conditions in isotropic media in two and three dimensions. In the charge-free case we recover Strichartz estimates due to Blair--Smith--Sogge for wave equations on domains up to endpoints. For the proof we suitably extend Maxwell equations over the boundary, which introduces coefficients on the full space with codimension-$1$ Lipschitz singularity. This system can be diagonalized to half-wave equations amenable to the results of Blair--Smith--Sogge. In two dimensions, we improve the local well-posedness of the Maxwell system with Kerr nonlinearity via Strichartz estimates.

math.AP↗

arXiv2022

Low regularity well-posedness of KP-I equations: the three-dimensional case

In this paper, low regularity local well-posedness results for the Kadomtsev--Petviashvili--I equation posed in spatial dimension $d =3$ are proved. Periodic, non-periodic and mixed settings as well as generalized dispersion relations are considered. In the weak dispersion regime, these initial value problems show a quasilinear behavior so that bilinear and energy estimates on frequency dependent time scales are used in the analysis.

math.AP↗