Jong-Guk Bak

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-Sjlin-Hrmander type and some spectral projection operators on compact manifolds, and for classes of oscillatory integral operators with one-sided fold singularities.

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.

Abstract 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.

The Bochner-Riesz operator $T^{\alpha }$ on $\mathbf {R}^{n}$ of order $\alpha$ is defined by \begin{equation*}(T^{\alpha } f)\;\widehat {}\;(\xi ) = {\frac {(1-|\xi |^{2})_{+}^{\alpha } }{\Gamma (\alpha +1)}} \hat {f}(\xi ) \end{equation*} where $\;\widehat {}\;$ denotes the Fourier transform and $r_{+}^{\alpha } = r^{\alpha }$ if $r>0$, and $r_{+}^{\alpha }=0$ if $r\leq 0$. We determine all pairs $(p,q)$ such that $T^{\alpha }$ on $\mathbf {R}^{2}$ of negative order is bounded from $L^{p}(\mat

Abstract. Consider the Fourier restriction operators associated to curves in ℝ d , . We prove for various classes of curves the endpoint restricted strong type estimate with respect to affine arclength measure on the curve. An essential ingredient is an interpolation result for multilinear operators with symmetries acting on sequences of vector-valued functions.

We consider the oscillatory integral operator defined by <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript lamda Baseline f left-parenthesis x right-parenthesis equals integral Underscript double-struck upper R Endscripts e Superscript i lamda phi left-parenthesis x comma t right-parenthesis Baseline a left-parenthesis x comma t right-parenthesis f left-parenthesis t right-parenthesis d t"> <mml:semantics> <mml:mrow>

Abstract Theorems 1 and 2 are known results concerning L p – L q estimates for certain operators wherein the point (1/ p , 1/ q ) lies on the line of duality 1/ p + 1/ q = 1. In Theorems 1′ and 2′ we show that with mild additional hypotheses it is possible to prove L p - L q estimates for indices (1/ p , 1/ q ) off the line of duality. Applications to Bochner-Riesz means of negative order and uniform Sobolev inequalities are given.

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