"fall under his responsibility" is grammatically correct and can be used in written English.
It means that something is the responsibility of a particular person or falls within their area of responsibility. Example: The finances of the company fall under his responsibility as the chief financial officer.
"The public deserve to know that everything has been done to ensure that Mr Robert has done no wrong, particularly since the allegations surrounding his involvement as a director of companies would fall under his responsibilities as a government minister," he said.
Men's lives, human lives, at issue and under his responsibility.
Nyien Nyang town, close to Leitchor refugee camp, is under his responsibility.
"Everyone seemed to fall under his spell".
Rural areas fall under the responsibility of the central government's ministry of health.
All the analysis, demonstration, and examination work commonly does not fall under the responsibility of the people from any single functional organization.
Previously these closely linked areas were split between two different ministers, now they both fall under the responsibility of one fully fledged minister of state.
However, customer service and customer care often fall under the responsibility of one department, while everyone in the organization should be invested in customer experience.
The UK government is currently in talks with Scottish ministers on devolving powers which it says are needed to make the referendum legally watertight, as constitutional issues fall under the responsibility of Westminster.
Particular focus is directed towards the three volcanoes that fall under the responsibility of the Institut de Physique du Globe of Paris, namely La Montagne Pelée in Martinique, La Soufrière in Guadeloupe, and the Soufrière Hills in Montserrat.
Due to the risks associated with MEC identification and removal, MEC removal and subsequent land management fall under the responsibility of multiple government agencies in concert with private companies.
