Database
[{title:"match1",start:"2022-01-01T00:00:00Z",end:"2022-02-02T00:00:00Z"},{title:"notMatch1",start:"2022-01-15T00:00:00Z",end:"2022-01-16T00:00:00Z"},{title:"notMatch2",start:"2022-01-01T00:00:00Z",end:"2022-01-16T00:00:00Z"}]
Query
db.collection.aggregate([{"$match":{$expr:{"$and":[{"$lte":[{$week:{"$toDate":"$start"}},{$week:{"$toDate":"2022-01-10"}}]},{"$gte":[{$week:{"$toDate":"$end"}},{$week:{"$toDate":"2022-01-20"}}]}]}}}])