곱함수 연산자의 르베그 공간 계측

Abstract

The generalized Riesz mean of order $\alpha$ associated with afunction $\rho$ is defined by\begin{equation*}\widehat{S^{\alpha}_tf}(\xi)=\Big(1-\frac{\rho(\xi)}t\Big)_+^\alpha\widehat{f}(\xi), \quad \xi = (\xi ', \xid)\in \mathbb R^d\times\mathbb R.\end{equation*} When $\rho(\xi)=

\xi

$, $S^\alpha_t$ is the standard Bochner-Rieszoperator in $\mathbb R^{d+1}$ (referred to later as spherical means,as the set $

=1$ is a sphere). When$\rho(\xi)=\frac{

\xi'

^2}{\xi_{d+1}^2}$ with a cutoff in$\xi_{d+1}$, it is called a cone multiplier operator in $\mathbbR^{d+1}$. There has been a lot of work devoted to the problem of$L^p$ and $L^p-L^q$ boundedness of the Bochner-Riesz and conemultiplier operators. They are generally known as difficult problemsin this field and there are still many open problems. In fact, forthe cone multiplier there are some additional difficulties ascompared with Bochner-Riesz problem.In this thesis, we consider $L^p$ and maximal $L^p$ estimates forthe generalized Riesz means which are associated with thecylindrical distance function $\rho(\xi) = \max \{

\xxi

,

\xid

\}$,$(\xxi,\xid)\in\mathbb R^{d}\times\mathbb R$. We prove theseestimates up to the currently known range of the sphericalBochner-Riesz and its maximal operators. This is done byestablishing implications between the corresponding estimates forthe spherical Bochner-Riesz and the cylindrical multiplieroperators.Secondly, we obtain some sharp $L^{p}-L^{q}$ estimates and therestricted weak-type endpoint estimates for the multiplier operatorof negative order associated with conic surfaces in $\mathbb{R}^{3}$which have finite type degeneracy.