%--Paired Delimiters-- $ \newcommand{\abs}[1]{\left\lvert #1 \right\rvert} \newcommand{\ang}[1]{\left\langle #1 \right\rangle} \newcommand{\bkt}[1]{\left\lbrack #1 \right\rbrack} \newcommand{\nor}[1]{\left\lVert #1 \right\rVert} \newcommand{\pth}[1]{\left( #1 \right)} \newcommand{\set}[1]{\left\lbrace #1 \right\rbrace} \newcommand{\abset}[1]{\abs{\set{#1}}} \newcommand{\ptset}[1]{\pth{\set{#1}}} $ %--Operators-- $ \newcommand{\grad}{\nabla} \newcommand{\La}{\Delta} \renewcommand{\div}{\operatorname{div}} \newcommand{\curl}{\operatorname{curl}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\Hess}{\operatorname{Hess}} \newcommand{\mm}{\mathcal{M}} \newcommand{\inv}{^{-1}} \newcommand{\tensor}{\otimes} \newcommand{\cross}{\times} \newcommand{\weak}[1]{\xrightharpoonup{#1}} $ %--Notations-- $ \newcommand{\Vol}{\mathrm{Vol}} \newcommand{\loc}{\mathrm{loc}} \renewcommand{\dim}{\mathrm{dim}} \newcommand{\Span}{\mathrm{Span}} \newcommand{\diam}{\mathrm{diam}} \newcommand{\Id}{\mathrm{Id}} \newcommand{\Lip}{\mathrm{Lip}} \newcommand{\dist}{\mathrm{dist}} \newcommand{\inv}{^{-1}} $ %--Algebra-- $ \newcommand{\hfsq}[1]{\frac{\abs{#1} ^2}{2}} $ %--Sets-- $ \newcommand{\R}{\mathbb{R}} \newcommand{\RR}[1]{\R ^{#1}} \newcommand{\Rd}{\RR d} \newcommand{\Rn}{\RR n} \renewcommand{\S}{\mathbb{S}} \renewcommand{\Z}{\mathbb{Z}} \newcommand{\T}{\mathbb{T}} \newcommand{\ssubset}{\subset \subset} \newcommand{\spt}{\operatorname{spt}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\diam}{\mathrm{diam}} \newcommand{\diag}{\operatorname{diag}} \renewcommand{\dim}{\mathrm{dim}} \newcommand{\dimH}{\mathrm{dim} _\mathcal{H}} \newcommand{\Sing}{\mathrm{Sing}} \newcommand{\Vol}{\mathrm{Vol}} \newcommand{\inds}[1]{\mathbf1_{\set{#1}}} \newcommand{\ind}[1]{\mathbf1_{#1}} $ %--Summations-- $ \newcommand{\SUM}[3]{\sum \cnt{#1}{#2}{#3}} \newcommand{\PROD}[3]{\prod \cnt{#1}{#2}{#3}} \newcommand{\cnt}[3]{ _{#1 = #2} ^{#3} } \newcommand{\seq}[2]{\set{#1 _{#2}} _{#2}} \newcommand{\seqi}[2]{\set{#1 _{#2}} \cnt{#2}1i} $ %--Superscript and Subscript $ \newcommand{\pp}[1]{^{(#1)}} \newcommand{\bp}[1]{^{(#1)}} \newcommand{\pps}[2][1]{^{#2 + #1}} \newcommand{\bps}[2][1]{_{#2 + #1}} \newcommand{\pms}[2][1]{^{#2 - #1}} \newcommand{\bms}[2][1]{_{#2 - #1}} $ %--Notations-- $ \newcommand{\loc}{\mathrm{loc}} \newcommand{\Span}{\operatorname{Span}} \newcommand{\argmin}{\operatorname{argmin}} \newcommand{\argmax}{\operatorname{argmax}} \newcommand{\osc}{\operatorname{osc}} \newcommand{\Id}{\mathrm{Id}} \newcommand{\Lip}{\operatorname{Lip}} \newcommand{\Leb}{\operatorname{Leb}} \newcommand{\PV}{\mathrm{P.V.}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\at}[1]{\bigr\rvert _{#1}} \newcommand{\At}[1]{\biggr\rvert _{#1}} \newcommand{\half}{\frac12} $ %--Text-- $ \newcommand{\inn}{\text{ in }} \newcommand{\onn}{\text{ on }} \renewcommand{\ae}{\text{ a.e. }} \newcommand{\st}{\text{ s.t. }} \newcommand{\forr}{\text{ for }} \newcommand{\as}{\text{ as }} $ %--Differential-- $ \newcommand{\d}{\mathop{\kern0pt\mathrm{d}}\!{}} \newcommand{\dt}{\d t} \newcommand{\dx}{\d x} \newcommand{\dy}{\d y} \newcommand{\ptil}{\partial} \newcommand{\pt}{\ptil _t} \newcommand{\pfr}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\dfr}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}} \newcommand{\ddt}{\dfr{}t} \newcommand{\pthf}[2]{\pth{\frac{#1}{#2}}} $ %--Integral-- $ \newcommand{\intR}{\int _0 ^\infty} \newcommand{\intRn}{\int _{\Rn}} \newcommand{\intRd}{\int _{\Rd}} \newcommand{\fint}{-\!\!\!\!\!\!\int} \newcommand{\intset}[1]{\int _{\set{#1}}} $ %--Greek-- $ \newcommand{\e}{\varepsilon} \newcommand{\vp}{\varphi} $ %--Norm-- $ \newcommand{\nmL}[2]{\nor{#2} _{L ^#1}} $