Searcharxiv⌕ Search

arXiv subjects

Jong-Guk Bak

Publications and source records attributed to Jong-Guk Bak.

6 recordsLinked to original sources
arXiv2017

Bilinear restriction estimates for surfaces of codimension bigger than one

In connection with the restriction problem in $\mathbb R^n$ for hypersurfaces including the sphere and paraboloid, the bilinear (adjoint) restriction estimates have been extensively studied. However, not much is known about such estimates for surfaces with codimension (and dimension) larger than one. In this paper we show sharp bilinear $L^2 \times L^2 \to L^q$ restriction estimates for general surfaces of higher codimension. In some special cases, we can apply these results to obtain the corresponding linear estimates.

math.CA↗
arXiv2013

Restriction of Fourier transforms to some complex curves

The purpose of this paper is to prove a Fourier restriction estimate for certain 2-dimensional surfaces in $\bbR^{2d}$, $d\ge 3$. These surfaces are defined by a complex curve $γ(z)$ of simple type, which is given by a mapping of the form % \[ z\mapsto γ(z) = \big(z, \, z^2,..., \, z^{d-1}, \, ϕ(z) \big) \] % where $ϕ(z)$ is an analytic function on a domain $Ω\subset \bbC$. This is regarded as a real mapping $z=(x,y) \mapsto γ(x,y)$ from $Ω\subset \bbR^2$ to $\bbR^{2d}$. Our results cover the case $ϕ(z) = z^N$ for any nonnegative integer $N$, in all dimensions $d\ge 3$. Furthermore, when $d=3$, we have a uniform estimate, where $ϕ(z)$ may be taken to be an arbitrary polynomial of degree at most $N$. These results are analogues of the uniform restricted strong type estimate in \cite{BOS3}, valid for polynomial curves of simple type and some other classes of curves in $\bbR^d$, $d\ge 3$.

math.CA↗
arXiv2010

Extensions of the Stein-Tomas theorem

We prove an endpoint version of the Stein-Tomas restriction theorem, for a general class of measures, and with a strengthened Lorentz space estimate. A similar improvement is obtained for Stein's estimate on oscillatory integrals of Carleson-Sjölin-Hörmander type and some spectral projection operators on compact manifolds, and for classes of oscillatory integral operators with one-sided fold singularities.

math.CA↗
arXiv2008

Restriction of Fourier transforms to curves, II: Some classes with vanishing torsion

We consider the Fourier restriction operators associated to certain degenerate curves in R^d for which the highest torsion vanishes. We prove estimates with respect to affine arclength and with respect to the Euclidean arclength measure on the curve. The estimates have certain uniform features, and the affine arclength results cover families of flat curves.

math.CA↗
arXiv2008

Restriction of Fourier transforms to curves and related oscillatory integrals

We prove sharp endpoint results for the Fourier restriction operator associated to nondegenerate curves in $\Bbb R^d$, $d\ge 3$, and related estimates for oscillatory integral operators. Moreover, for some larger classes of curves in $\Bbb R^d$ we obtain sharp uniform $L^p\to L^q$ bounds with respect to affine arclength measure, thereby resolving a problem of Drury and Marshall.

math.CA↗