<<< tex
\newcommand\Div{\operatorname{div}}
\newcommand\hl{\operatorname*{\star-lim'sup}}
\DeclareMathOperator\curl{curl}
\DeclareMathOperator*\argmax{argmax}
\begin{cases}
\Div_x u \Div^x u\Div_x^y u \Div^y_x u & \text{operatorname}\\
\hl_{x\to0} u \hl^w u \hl_{x\to0}^w u \hl^w_{x\to0} u & \text{operatorname*}\\
\hl\nolimits_{x\to0} u \hl\nolimits^w u
\hl\nolimits_{x\to0}^w u \hl\nolimits^w_{x\to0} u & \text{operatorname*:nolimits}\\
\curl_x v \argmax_K u & \text{DeclareMathOperator}
\end{cases}

>>> native
[ EDelimited
    "{"
    ""
    [ Right
        (EArray
           [ AlignLeft , AlignLeft ]
           [ [ [ ESub (EMathOperator "div") (EIdentifier "x")
               , EIdentifier "u"
               , ESuper (EMathOperator "div") (EIdentifier "x")
               , EIdentifier "u"
               , ESubsup (EMathOperator "div") (EIdentifier "x") (EIdentifier "y")
               , EIdentifier "u"
               , ESubsup (EMathOperator "div") (EIdentifier "x") (EIdentifier "y")
               , EIdentifier "u"
               ]
             , [ EText TextNormal "operatorname" ]
             ]
           , [ [ EUnder
                   True
                   (EMathOperator "\8902-lim'sup")
                   (EGrouped [ EIdentifier "x" , ESymbol Rel "\8594" , ENumber "0" ])
               , EIdentifier "u"
               , EOver True (EMathOperator "\8902-lim'sup") (EIdentifier "w")
               , EIdentifier "u"
               , EUnderover
                   True
                   (EMathOperator "\8902-lim'sup")
                   (EGrouped [ EIdentifier "x" , ESymbol Rel "\8594" , ENumber "0" ])
                   (EIdentifier "w")
               , EIdentifier "u"
               , EUnderover
                   True
                   (EMathOperator "\8902-lim'sup")
                   (EGrouped [ EIdentifier "x" , ESymbol Rel "\8594" , ENumber "0" ])
                   (EIdentifier "w")
               , EIdentifier "u"
               ]
             , [ EText TextNormal "operatorname*" ]
             ]
           , [ [ ESub
                   (EMathOperator "\8902-lim'sup")
                   (EGrouped [ EIdentifier "x" , ESymbol Rel "\8594" , ENumber "0" ])
               , EIdentifier "u"
               , ESuper (EMathOperator "\8902-lim'sup") (EIdentifier "w")
               , EIdentifier "u"
               , ESubsup
                   (EMathOperator "\8902-lim'sup")
                   (EGrouped [ EIdentifier "x" , ESymbol Rel "\8594" , ENumber "0" ])
                   (EIdentifier "w")
               , EIdentifier "u"
               , ESubsup
                   (EMathOperator "\8902-lim'sup")
                   (EGrouped [ EIdentifier "x" , ESymbol Rel "\8594" , ENumber "0" ])
                   (EIdentifier "w")
               , EIdentifier "u"
               ]
             , [ EText TextNormal "operatorname*:nolimits" ]
             ]
           , [ [ ESub (EMathOperator "curl") (EIdentifier "x")
               , EIdentifier "v"
               , EUnder True (EMathOperator "argmax") (EIdentifier "K")
               , EIdentifier "u"
               ]
             , [ EText TextNormal "DeclareMathOperator" ]
             ]
           ])
    ]
]