When giving an item using commands that has a colored name, when you scroll over the item, or hold it, the colored name does not show up over the hot bar. The 'bold' and 'italic' features and the others are not affected by this but the colored item names do not appear over the hot bar.

If you put this command into a command block or chat:

/give @p minecraft:apple{display:{Name:"

{\"text\":\"Apple\",\"color\":\"red\",\"bold\":true,\"italic\":false}

"}} 1

You will see that the name is still bold and non-italicized over the hot bar, but the name will show up as white, not red like specified in the command. The colored name does still show up if you hover over the item in your inventory.