Orlicz boundedness for certain classical operators (Q2773349)

The authors study boundedness of the fractional maximal operator \(M^\alpha_\Omega\), \(0\leq\alpha<n\), associated to an open bounded set \(\Omega\) in \(\mathbb R^ n\), that is, boundedness of the operator NEWLINE\[NEWLINE M^\alpha_\Omega f(x):= \sup_{r>0}\frac1{| \Omega\cap B(x,r)| ^{1-\alpha/n}} \int_{B(x,r)}| f(y)| \,dy, \quad x\in\Omega, NEWLINE\]NEWLINE (where \(B(x,r):=\{y\in\mathbb R^ n; | x-y| <r\}\) and \(| E| \) denotes the Lebesgue measure of the set \(E\)) from the Orlicz space \(L^\Psi(\Omega)\) into the Orlicz space \(L^\Phi(\Omega)\). They find necessary and sufficient conditions for the functions \(\Phi\) and \(\Psi\) (defined on \([0,\infty)\), taking the value zero at zero and with non-negative continuous derivatives), such that, under very mild extra assumptions on \(\Phi\) and \(\Psi\), the boundedness holds.NEWLINENEWLINEFor functions \(\Phi\) of finite upper type these results can be extended to the Hilbert transform on the one-dimensional torus and to the fractional integral operator \(I^\alpha_\Omega\) (\(I^\alpha_\Omega f(x)= \int_\Omega (f(y)/| x-y| ^{n-\alpha})\,dy\), \(x\in\Omega\)), \(0<\alpha<n\). As an application the authors obtain some generalized Trudinger-type inequalities for the operator \(I^\alpha_\Omega\).