$line_split[0] will only be undefined if $line doesn't contain any other character than \t (including being empty). And the fact it's undefined stems from @line_split not having any entries. So, you may do something like:

if (@line_split && $line_split[0] eq "") {
    # Has empty leading fields
}
[download]

or

if (!@line_split || $line_split[0] eq "") {
    # Has empty leading fields, or no fields at all
}
[download]

depending on what you want to do if there are no fields at all.