Minus sign too big in superscript inside subscript in math display mode

Question

The following code:

Inline mode $\|f\|_{H^{-1}}, \|f\|_{_{H^{-1}}}, \|f\|_{_{\scalebox{0.7}{$H^{-1}$}}},\|f\|_{_{\scalebox{0.5}{$H^{-1}$}}}$, 
display mode $$\|f\|_{H^{-1}}, \|f\|_{_{H^{-1}}}, \|f\|_{_{\scalebox{0.7}{$H^\{-1}$}}},\|f\|_{_{\scalebox{0.5}{$H^{-1}$}}}$$

Results in the following:

I want to have $H^{-1}$ as a sub-index, in the first case it is just too big to be a subindex.

I added a second subindex in the second, the H looks good, but the -1 is just too big, in particular the minus sign.

I added scalebox, in the last two cases, which seems to give a better looking result, and it actually scale correctly the term H^{-1}, but I hate to use it, I can see this failing once I have to modify this for other formats, but I'm not sure.

Is there a way to tell that minus sign to scale appropriately with respect to the sub-index level used?

---

It seems that one way to solve the problem is to correctly set the norms either by grouping or defining a norm command as perfectly shown in some answer.

I still found that the minus sign is too big in what follows:

\begin{equation}
  \begin{aligned}
    &f^{-1}(x) &&L_{f^{-1}}[\eta] \\
    &f(x)   &&L_{f}[\eta]
  \end{aligned}
\end{equation}

The "-1" sign takes even more width than the "f" in the first case, in the seconds it gets close to the width of the $L_f$.

I imagine I could use a different symbol, i.e. a minus in $1-2$ could be different that in $-1$.

Answer 1

(I revised this answer thoroughly after receiving additional comments from the OP.)

I think there are two issues that need to be addressed: One is the spacing around the - ("minus") sign when it occurs in scriptscript-style math mode, and the other is the vertical positioning of the subscript formulas that follow a "norm" (double vertical bar) symbol.

Regarding the first issue, I suggest you define some macros like this

\newcommand{\mym}{\mkern-1.5mu-\mkern-3mu 1}
\newcommand\finv{f^{\mym}}
\newcommand\Hinv{H^{\mym}}

and then write $L_{\finv}[\eta]$ instead of L_{f^{-1}}[\eta]. Note that this approach simply reduces the amount of whitespace around the scriptscript-style minus symbol. It does not reduce the size of either the minus symbol or the digit 1; I'm concerned that reducing the size of the scriptscript-style glyphs would also reduce their basic readability. If you think the compressed -1 still takes up too much space, you should probably coming up with new and space-saving notation to denote the inverse of a function. E.g., something like f* or \bar{f}...

Regarding the second issue: Note that the formula \|f\|_{H^{-1}} may be broken down into two sub-formulas: \|f\| and _{H^{-1}}. What you're encountering is that TeX has special rules for placing subscripts that follow a "math atom" such as \|: The subscripts (and superscripts) are set in a cramped mode. While this is OK in most settings, it's clearly not optimal for your use case. I can think of two remedies: Either change the first sub-formula from \|f\| to {\|f\|}, changing its type to "math-ordinary", or -- more LaTeX-ishly -- define a macro called \norm (say) using the machinery of the mathtools package, and then write \norm{f} instead of {\|f\|}.

Combining the solutions to the two issues, I think you should write \norm{f}_{\Hinv} instead of \|f\|_{H^{-1}}.

\documentclass{article}
\usepackage{mathtools} % for "\DeclarePairedDelimiter" macro
\DeclarePairedDelimiter\norm\lVert\rVert     % create a "\norm" macro
\newcommand{\mym}{\mkern-1.5mu-\mkern-3mu 1} % short for "my minus one"
\newcommand\finv{f^{\mym}}
\newcommand\Hinv{H^{\mym}}

\begin{document}
$L_{f^{-1}}[\eta]$ vs.\ $L_{\finv}[\eta]$

\bigskip
\emph{Original code}

\quad$\|f\|_{H^{-1}}\quad\|f\|_{\Hinv}$

\bigskip
\emph{Employ grouping}

\quad${\|f\|}_{H^{-1}}\quad{\|f\|}_{\Hinv}$

\bigskip
\emph{Dedicated \emph{\texttt{\textbackslash norm}} macro}

\quad$\norm{f}_{H^{-1}}\quad\norm{f}_{\Hinv}$
\end{document}

Answer 2

What you probably want is adding a (possibly empty) superscript, which will push down the subscript.

Here's an augmented version of \DeclarePairedDelimiter that does it:

\documentclass{article}

\usepackage{mathtools} % for \DeclarePairedDelimiter
\usepackage{xparse}    % for \NewDocumentCommand
\usepackage{booktabs}  % for the table

\DeclarePairedDelimiter{\normAux}{\lVert}{\rVert}
\NewDocumentCommand{\norm}{sO{}me{_^}}{%
  \IfBooleanTF{#1}
    {\normAux*{#3}\normS#4}
    {\normAux[#2]{#3}\normS#4}%
}
\NewDocumentCommand{\normS}{mm}{%
  \IfValueT{#1}{_{#1}}^{\IfValueT{#2}{#2}}%
}

\begin{document}

\renewcommand{\arraystretch}{1.5}

\begin{tabular}{lll}
\toprule
Style & New & Old \\
\midrule
Inline
  & $\norm{f}$ & $\|f\|$ \\
  & $\norm{f}^{2}$ & $\|f\|^2$ \\
  & $\norm{f}_{H^{-1}}$ & $\|f\|_{H^{-1}}$ \\
  & $\norm{f}_{H^{-1}}^{2}$ & $\|f\|_{H^{-1}}^{2}$ \\
\midrule
Display
  & $\displaystyle\norm{f}$ & $\displaystyle\|f\|$ \\
  & $\displaystyle\norm{f}^{2}$ & $\displaystyle\|f\|^2$ \\
  & $\displaystyle\norm{f}_{H^{-1}}$ & $\displaystyle\|f\|_{H^{-1}}$ \\
  & $\displaystyle\norm{f}_{H^{-1}}^{2}$ & $\displaystyle\|f\|_{H^{-1}}^{2}$ \\
\midrule
Sizes
  & \multicolumn{2}{l}{%
      $\norm[\big]{f}_{H^{-1}},\norm[\Big]{f}_{H^{-1}},\norm*{\dfrac{f}{2}}_{H^{-1}}$%
    } \\
\bottomrule
\end{tabular}

\end{document}

(Beware: the syntax for the e argument type might change.)

Answer 3

I would complete the answer from Miko. Why it works? Why ${\|f\|}_H$ behaves differently than $\|f\|_H$? The answer is in TeXbook, page 445, paragraphs 18a and 18b. Roughly speaking, when nucleus (i.e. the object which is subscripted) is single character, then the subscript is lowered by \fondimen16 independent of the depth of the nucleus. But if the nucleus includes more characters then the depth of nucleus is measured and the subscript is lowered by this depth plus \fondtimen19. The reason of this feature is our requirement that we need to have subscripts of $a_2, f_2$ in the same vertical position.