Multidimensional extensions of Pólya-Knopp-type inequalities over spherical cones (Q2830461)

The authors introduce a new type of limit process to evaluate the modular-type operator norm of an integral operator.NEWLINENEWLINEIn particular, it is focused on the extensions of the Pólya-Knopp inequality NEWLINE\[CARRIAGE_RETURNNEWLINE\int^\infty_0\exp\left(\frac{1}{x}\int^x_0\log f(t)\,dt\right)dx\leq e\int^\infty_0f(x)\,dx,\ f\geq 0.CARRIAGE_RETURNNEWLINE\]NEWLINE These extensions are presented in the form NEWLINE\[CARRIAGE_RETURNNEWLINE\left\{\int_E(\Phi\circ\mathbb Kf(x))^q\,d\mu\right\}^{1/q}\leq C\left\{\int_E(\Phi\circ f(x))^p\,d\nu\right\}^{1/p},CARRIAGE_RETURNNEWLINE\]NEWLINE where \(E\) is a spherical cone in \(\mathbb R^n\), \(f\in D_{\mathbb K}\cap L^p_\Phi (d\nu)\), \(p,q\neq 0\), \(\mu\) and \(\nu\) are two \(\sigma\)-finite Borel measures on \(E\), \(\Phi\in CV^+(I)\), \(\Phi\circ f (x)=\Phi(f (x))\), and \(\mathbb Kf (x)\) is in one of the forms NEWLINE\[CARRIAGE_RETURNNEWLINE\mathbb Kf(x):=\int_{\tilde S_x}k(x,t)f(t)\,d\sigma(t),\ x\in E,CARRIAGE_RETURNNEWLINE\]NEWLINE or NEWLINE\[CARRIAGE_RETURNNEWLINE\tilde{\mathbb K}f(x):=\int_{E\backslash s_x}k(x,t)\,d\sigma(t),\ x\in E.CARRIAGE_RETURNNEWLINE\]NEWLINE Here, \(D_{\mathbb K}\) is the space of those \(f\) such that \(\mathbb Kf(x)\) is well-defined for \(\mu\) a.e.\ \(x\in E\) and \(L^p_\Phi(d\nu)\) is the set of all real-valued Borel measurable \(f\) with NEWLINE\[CARRIAGE_RETURNNEWLINE\| f\|_{\Phi,p,\nu}:=\left\{\int_E(\Phi\circ f(x))^p\,d\nu\right\}^{1/p}<\infty.CARRIAGE_RETURNNEWLINE\]NEWLINEWith the above notations, \(CV^+(I)\) is the set of nonnegative convex functions defined on an open interval \(I\) in \(\mathbb R\), \(\tilde S_x=\cup_{0<s\leq\| x\|}sA\), \(S_x=\tilde S_x\backslash \| x\| A\), \(k(x, t)\geq 0\) is locally integrable over \(E\times E\), and \(\sigma\) is a \(\sigma\)-finite Borel measure on \(E\). Then letting NEWLINE\[CARRIAGE_RETURNNEWLINE\| \mathbb K\|_\ast=\sup\limits_f\frac{\|\Phi\circ\mathbb Kf\|_{q,\mu}}{\|\Phi\circ f\|_{p,\nu}},\ f\in D_{\mathbb K}\cap L^p_\Phi(d\nu),\ \|\Phi\circ f\|_{p,\nu}\neq 0,CARRIAGE_RETURNNEWLINE\]NEWLINE the authors present their results in terms of \(\|\mathbb K\|_\ast\). Thus, under certain conditions, many estimates are found for \(\|\mathbb K\|_\ast\).NEWLINENEWLINEThe results are of interest to researchers in the field. Without being expert, however, the paper is not very easy to follow.