| This dissertation consists of two parts. (1) A review of properties of general homogeneous convex cones in $Rsp{n}$. (2) Norm inequalities for certain integral operators on such domains. For a special class of integral operators K, Kf(x) = $intsb{V}k(x,y)f(y)dy$, defined on a homogeneous convex con V, we obtain the following weighted ($Lsp{p}$, $Lsp{q}$) norm inequality: (1 $leq pleq qleqinfty$) $$left(intsb{V} Deltasbsp{V}{gamma - q} (x)(Kf(x))sp{q} dxright)sp{sp{1/q}}{le}cleft(intsb{V} fsp{p} (x) Deltasbsp{V}{beta p + (gamma + 1) p/q - 1} (x)dxright)sp{sp{1/p}}$$where $Deltasbsp{V}{delta}(x)$, the $Rsp{n}$ analogue of $xsp{delta}$, is a special weight function. We then proceed to apply the above result and others to some important special operators and their duals. These operators are: Riemann-Liouville's, Weyl's and Laplace's. Hardy's operator and its dual are special cases of Riemann-Liouville's and Weyl's. |
