Extrapolation and weighted norm inequalities between Lebesgue and Lipschitz spaces in the variable exponent context
Several extrapolation results starting from weighted norm inequalities between Lebesgue and Lipschitz spaces given by \[ \sup_{B}\frac{\|w\chi_B\|_\infty}{|B|^{1+\delta/n}}\int_B|f(x)-m_B(f)|dx\leq C\|gw\|_s \] are obtained, where \(1<\beta<\infty\), \(0\leq\delta<1\), \(\delta/n=1/\beta-1/s\), \(f\) and \(g\) are two measurable functions and \(w\) belongs to a suitable class of weights. This hypothesis leads to a large class of inequalities for sublinear operators acting between weighted \(L^p\) and \(L^q\) spaces; weighted \(L^p\) and Lipschitz integral spaces; variable Lebesgue spaces \(L^{p(\cdot)}\) and \(L^{q(\cdot)}\); variable Lebesgue spaces \(L^{p(\cdot)}\) and variable versions of Lipschitz integral spaces. An important ingredient of the proofs is the new pointwise estimate \(f_\gamma^\#(x)\leq M_{\delta-\gamma}f_\delta^\#(x)\) for \(0\leq\gamma\leq\delta<1\), relating the fractional sharp maximal function \(f_\delta^\#\) and the fractional maximal function \(M_\delta f\).
scientific article; zbMATH DE number 6536926
scientific article