%Authors: L. Grafakos, S. Montgomery-Smith, and O. Motrunich %Title: A sharp estimate for the Hardy-Littlewood maximal function %Filename: grafakosmontsmithmotrunichhardy.tex %TeX: AMSTeX %Length: ????? %Received Date N/A %SubjectClass: 42B25 %Abstract: The best constant in the usual $L^p$ norm inequality for the centered % Hardy-Littlewood maximal function on $\Bbb R^1$ is obtained for the % class of all ``peak-shaped'' functions. A positive function % on the line is called ``peak-shaped'' if it is positive and % convex except at one point. % The techniques we use include convexity and % an adaptation of the standard Euler-Langrange variational % method. %Citation: preprint. %Special character check block %32 space 33 ! exclam. pt. 34 " double quote 35 # sharp %36 $ dollar 37 % percent 38 & ampersand 39 ' prime %40 ( left paren. 41 ) rt. paren. 42 * asterisk 43 + plus %44 , comma 45 - minus 46 . period 47 / divide %58 : colon 59 ; semi-colon 60 < less than 61 = equal %62 > greater than 63 ? question mark 64 @ at %91 [ left bracket 92 \ backslash 93 ] right bracket 94 ^ caret %95 _ underline 96 ` left single quote %123 { left brace 124 | vertical bar 125 } right brace 126 ~ tilda %Insert your TeX file starting here. \input amstex \documentstyle{amsppt} \magnification\magstep1 \NoRunningHeads %\hsize 6.5 truein \vsize 8.0 truein %general \define\k{\ref\key} \define\pp{\pages} \define\en{\endref} \define\nod{\noindent} \define\f{\frac} \define\tf{\tfrac} \define\q{\quad} \define\qq{\qquad} \define\qqq{\quad\qquad} \define\qqqq{\qquad\qquad} %blackboard bold \define\r1{\Bbb R^1} \define\rr{\Bbb R} \define\rp{\Bbb R^+} \define\rn{\Bbb R^n} \define\zz{\Bbb Z} \define\zp{\Bbb Z^+} \define\torus{\Bbb T^1} \define\tn{\Bbb T^n} \define\cc{\Bbb C^1} \define\cn{\Bbb C^n} \define\hn{\Bbb H^n} %calligraphic letters \define\cb{\Cal P} \define\cd{\Cal D} \define\ce{\Cal E} \define\cf{\Cal F} \define\cm{\Cal M} \define\cs{\Cal S} %math symbols \define\p{\partial} \define\ppr{p^\prime} \define\nf{\infty} \define\pnf{+\infty} \define\mnf{-\infty} \define\ci{C^\infty} \define\coi{C_0^\infty} \define\lo{ {L^1} } \define\lt{ {L^2} } \define\lp{ {L^p} } \define\li{ {L^\infty} } \define\half{\f{1}{2}} \define\dtt{\, \frac{dt}{t}} \define\dss{\, \frac{ds}{s}} \define\drr{\, \frac{dr}{r}} \define\pv{{\text{p.v.}}} \define\loc{ {\text{loc}} } \define\il{\int\limits} \define\ra{\rightarrow} %Greek \define\al{\alpha} \define\be{\beta} \define\ga{\gamma} \define\Ga{\Gamma} \define\de{\delta} \define\De{\Delta} \define\ve{\varepsilon} \define\ze{\zeta} \define\th{\theta} \define\Th{\Theta} \define\la{\lambda} \define\La{\Lambda} \define\si{\sigma} \define\Si{\Sigma} \define\vp{\varphi} \define\om{\omega} \define\Om{\Omega} \hsize=30truecc \baselineskip=16truept \topmatter \title A sharp estimate for the Hardy-Littlewood maximal function\endtitle \author Loukas Grafakos$^*$, Stephen Montgomery-Smith$^{**}$, and Olexei Motrunich$^{***}$\endauthor \affil University of Missouri, Columbia and Princeton University\endaffil \noindent \address Department of Mathematics, University of Missouri, Columbia, MO 65211, \newline Department of Mathematics, University of Missouri, Columbia, MO 65211, \newline Department of Physics, Princeton University, Princeton, NJ 08544 \endaddress \email \newline \noindent loukas\@math.missouri.edu, stephen\@math.missouri.edu, motrunich\@princeton.edu \endemail \thanks $\,\,\,\,\,\, ^*$Research partially supported by the University of Missouri Research Board. \endthanks \thanks $\,\,\, ^{**}$Research partially supported by the National Science Foundation. \endthanks \thanks $^{***}$Research partially supported by the University of Missouri-Columbia Research Council. \endthanks \abstract The best constant in the usual $L^p$ norm inequality for the centered Hardy-Littlewood maximal function on $\Bbb R^1$ is obtained for the class of all ``peak-shaped'' functions. A function on the line is called ``peak-shaped'' if it is positive and convex except at one point. The techniques we use include variational methods. AMS Classification (1991): 42B25 \endabstract \endtopmatter \document {\bf 0. Introduction.} \smallskip Let $$ (Mf)(x)= \sup_{\de >0} {1 \over 2 \de }\int_{x-\de}^{x+\de} |f(t)| \, dt\tag0.1 $$ be the centered Hardy-Littlewood maximal operator on the line. This paper grew out from our attempt to find the operator norm of $M$ on $\lp (\r1)$. Since $M$ is a positive operator, we may restrict our attention to positive functions. Let $\cb$ be the set of all positive functions $f$ on $\r1$, which are convex except at one point (where we also allow $f$ to be discontinuous). We call such functions ``peak-shaped.'' We were able to find the best constant in the inequality $$ \|Mf\|_{L^p} \le C(p) \|f\|_{L^p}\qqq \text{for $f$ in $\cb \cap \lp$.}\tag0.2 $$ for $1
1} {(\tau +1)^{p-1\over p} + (\tau -1)^{p-1\over p} \over 2\tau {p-1\over p}},\tag0.4 $$ that is, the constant in (0.3). \endproclaim One may ask the corresponding question when $p=1$. It was communicated to us by Jos\'e Barrionuevo [Ba] that the best constant $C_1$ in the weak type inequality $$ |\{x: (Mf)(x)>\la \}| \le C_1 { \|f\|_{L^1} \over \la } $$ for $f$ in $\cb \cap L^1$ is $C_1=1$. This result is sharp and is analogous to our result when $p=1$. (In fact, this result is valid for the wider class of positive functions that are increasing on $(-\infty,c)$, and decreasing on $(c,\infty)$ for some number $c$.) It is still a mystery what happens for general functions $f$. It is conjectured in [DGS] that $c_p$ is the operator norm of the Hardy-Littlewood maximal function on $L^p(\Bbb R^1)$. Our methods will not work for arbitrary functions and we will point out during the proof where they break down. For general functions $f \in L^1$, the conjecture used to be that $C_1=3/2$. However, it has recently been shown by Aldaz [Al] that $C_1$ lies between $3/2$ and n $2$. This result tends to suggest that the value $c_p$ given by our Theorem is not the best constant for general $f\in L^p$. The authors would like to thank the anonymous referee for many valuable comments, and for pointing out errors in the original version of the manuscript. \bigskip {\bf 1. Some preliminary Lemmas.} \smallskip Throughout this paper we fix a $p$ with $1
0$, and
$\xi_x(0) = f(x)$. It can be
seen that $\xi_x(t)$ is a $C^\nf$ function of $t> 0$
(except at $t = |x|$, where it is merely continuous)
and that it tends to zero
as $t\ra \nf$.
Furthermore, we see that
$$ \xi_x'(t) = {{1\over2}(f(x+t)+f(x-t)) - \xi_x(t) \over t} .$$
Convexity shows us that $\xi_x'(t) \ge 0$ for $t \in (0,|x|)$, and
the third condition on $f$ shows us that $\xi_x'(t) > 0$ for $t$ close
to $|x|$. Thus we see that
$\xi_x(t)$ is non-decreasing for $t$ in some open neighborhood
of $(0, |x|]$.
The global maximum of $\xi_x$ over $[0,\nf)$ is
equal to $(Mf)(x)$. This maximum
is attained on some set of real numbers $B_x=\{ t: \xi_x(t) =
\sup_{u\ge 0}\xi_x(u)\}$. Set $\de(x)= \max B_x$. Since $B_x$ is
a closed set, it contains $\de(x)$. Note that $\de(x) > |x|$ for
$x\ne0$.
Thus
$$
(Mf)(x)= {1 \over 2 \de(x) }\int_{x-\de(x)}^{x+\de(x)}
f(t) \, dt.\tag1.1
$$
Since $\de(x)$ is a critical point of $\xi_x$, it follows that
$\xi_x'(\de(x))=0$. A simple calculation and (1.1)
give formula (1.2) below.
Now fix $x_0 \ne 0$.
By the Implicit Function Theorem, the equation
$\xi_x'(\de)=0$ can be solved for $\de$ as a smooth
function of $x$ in the vicinity of any point $(x_0,\de(x_0))$,
as long as $\dfrac{ \p \xi_x'(\de)}{ \p \de} \neq 0$ at $(x_0,\de(x_0))$.
This condition is equivalent to
$f'(x_0+\de(x_0))\neq f'(x_0-\de(x_0))$, which follows from the fact
that $x_0+\de(x_0)$ and $x_0-\de(x_0)$ lie on opposite sides of
the origin and that
$f$ has different kind of monotonicity on each side.
Therefore $\de$ coincides with a smooth function in the
neighborhood of every point $x_0 \ne 0$, which implies that
$\de(x)$ is a smooth function of $x \ne 0$.
As a consequence $(Mf)(x)$ is also smooth for $x \ne 0$.
We notice that for sufficiently small $|x|$ that
$\de(x) = (1+\tau_{2p}) |x|$ for a fixed value $\tau_{2p}$, and
that $(Mf)(x) = c_{2p} |x|^{-1/2p}$.
Thus $\de(x)$ is a continuous function of $x$.
\proclaim{Lemma 1} For $x\ne 0$, we have
$$
(Mf)(x)= {f(x+\de(x))+f(x-\de(x))\over 2},\tag1.2
$$
and
$$
(Mf)'(x)= {f(x+\de(x))-f(x-\de(x))\over 2\de(x)}.\tag1.3
$$
\endproclaim
{\smc Proof.} $\,$ (1.2) is proved as indicated above.
To prove (1.3), differentiate the identity (1.1) and use (1.2).
This completes the proof of Lemma 1. $QED$.
\medskip
Formula (1.3) indicates that the points $x+\de(x)$ and
$x-\de(x)$ are the $x$-coordinates of some two points of
intersection of the graph of $f$ with the
tangent line to the graph of $Mf$ at $(x,f(x))$.
\proclaim{Lemma 2} If $x>0$ then $\delta'(x) > 1$, and if
$x<0$ then $\delta'(x) < -1$. Moreover $Mf$ is in $\cb$ with
its maximum at $0$.
\endproclaim
{\smc Proof.} $\,$
We begin by showing that $Mf$ has no inflection points away
from $0$. Differentiating (1.2) and (1.3) we obtain
that for $x\ne 0$, we have
$$
\align
(Mf)'(x) &= f'(x+\de(x)){ (1+\de'(x))\over 2} +
f'(x-\de(x)){ (1-\de'(x))\over 2 } \tag1.4 \\
(Mf)'(x)\de'(x) +\de(x)(Mf)''(x) &=
f'(x+\de(x)){ (1+\de'(x))\over 2} -
f'(x-\de(x)){ (1-\de'(x))\over 2 }. \tag1.5
\endalign
$$
If $q \ne 0$ were an inflection point, then $(Mf)''(q)=0$,
and by (1.4) and (1.5) it would follow that
$$\align
f'(q+\de(q)) (1+\de'(q))&=(1+\de'(q))(Mf)'(q)\\
\noalign{\noindent or}
f'(q-\de(q)) (1-\de'(q))&=(1-\de'(q))(Mf)'(q).
\endalign$$
Then $(Mf)'(q) $ would be equal to either
$f'(q+\de(q))$ or $f'(q-\de(q))$. By
Lemma 1, $ (Mf)'(q)$ is the slope of the line
segment that joins $(q-\de(q),f(q-\de(q)))$ to
$(q+\de(q),f(q+\de(q)))$. By the convexity conditions
on $f$, this line would then necessarily lie on
the graph of $f$. By (1.2), this would imply that
$(Mf)(q) \le f(q)$, a contradiction if condition~(3) is imposed
upon $f$.
Therefore $Mf$ has no inflection points away from $0$, hence it is either concave or
convex there.
Since $(Mf)(x)$ looks like ${1\over x}$ near $\pm \nf$, it follows that
$Mf$ is convex on $(-\infty,0)$ and on $(0,+\infty)$.
We now show that if $x < 0$, then $\de'(x) < -1$.
Let $x_1