New Brezis–Van Schaftingen–Yung–Sobolev type inequalities connected with maximal inequalities and one parameter families of operators

Motivated by the recent characterization of Sobolev spaces due to Brezis–Van Schaftingen–Yung we prove new weak-type inequalities for one parameter families operators connected with mixed norm inequalities. The novelty our approach comes from fact that underlying measure space incorporates as a variable. We also show framework can be adapted treat related characterizations obtained earlier Bourgain–Nguyen. connection classical and fractional order is shown through use generalized Riesz potential Caffarelli–Silvestre extension principle. Higher are considered. indicate many examples applications PDE's different areas Analysis, suggesting vast future research. In direction, inspired methods originally Gagliardo Garsia, obtain maximal which combined applied type in context Calderón–Campanato spaces. particular, Log versions Gagliardo–Brezis–Van introduced compared corresponding limiting spaces, resulting sharpening Crippa–De Lellis Brué–Nguyen.