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곱함수 연산자의 르베그 공간 계측

곱함수 연산자의 르베그 공간 계측
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The generalized Riesz mean of order $\alpha$ associated with a function $\rho$ 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)=
$, $S^\alpha_t$ is the standard Bochner-Riesz operator in $\mathbb R^{d+1}$ (referred to later as spherical means, as the set $
=1$ is a sphere). When $\rho(\xi)=\frac{
^2}{\xi_{d+1}^2}$ with a cutoff in $\xi_{d+1}$, it is called a cone multiplier operator in $\mathbb R^{d+1}$. 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 cone multiplier operators. They are generally known as difficult problems in this field and there are still many open problems. In fact, for the cone multiplier there are some additional difficulties as compared with Bochner-Riesz problem. In this thesis, we consider $L^p$ and maximal $L^p$ estimates for the generalized Riesz means which are associated with the cylindrical distance function $\rho(\xi) = \max \{
\}$, $(\xxi,\xid)\in\mathbb R^{d}\times\mathbb R$. We prove these estimates up to the currently known range of the spherical Bochner-Riesz and its maximal operators. This is done by establishing implications between the corresponding estimates for the spherical Bochner-Riesz and the cylindrical multiplier operators. Secondly, we obtain some sharp $L^{p}-L^{q}$ estimates and the restricted weak-type endpoint estimates for the multiplier operator of negative order associated with conic surfaces in $\mathbb{R}^{3}$ which have finite type degeneracy.
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