When playing sheet music as written, you play the notes that are shown.
At the start of the passage you showed,
the F and A above middle C are written, and the D below middle C is written.
Therefore you should be playing those three notes at the start of that measure.
How you play the notes is your business. I would play (and hold) the two notes in the upper staff with my right hand and play the sequence of notes in the lower staff with my left hand.
But you could decide to play the two notes in the upper staff with your left hand and play the notes in the lower staff with your right hand.
Notice that you still play the F and A above middle C and the D below middle C, so if you decide to play the D with your right hand you have to cross one arm over the other in order to reach the notes.
In the edition of the Gershwin preludes I play from, part of the second prelude consists of chord-like notes in the upper staff and melody-like notes in the lower staff, somewhat like the music you have shown. (In the Gershwin prelude, however, the chords are repeated several times a measure rather than held.)
There is a note printed over this passage that says, "Optional Version: Reverse Hands". Here the written sheet music itself is suggesting you could choose to cross your arms one over the other to play the lower notes with your right hand.
I play that passage using my left hand for the lower notes and my right hand for the upper notes. I decided I would rather try to become good at playing a melody with my left hand than to play with my arms crossed over.
But that was a choice I made, not something dictated by the written music.