# Factorisation in Fourier Restriction Theory and near Extremisers

Stefan Buschenhenke  Stefan Buschenhenke: Mathematisches Seminar,
C.A.-Universität Kiel, Heinrich-Hecht-Platz 6, D-24118 Kiel, Germany
<EMAIL_ADDRESS>http://www.math.uni-
kiel.de/analysis/de/buschenhenke

###### Abstract.

We give an alternative argument to the application of the so-called Maurey-
Nikishin-Pisier factorisation in Fourier restriction theory. Based on an
induction-on-scales argument, our comparably simple method applies to any
compact quadratic surface, in particular compact parts of the paraboloid and
the hyperbolic paraboloid. This is achieved by constructing near extremisers
with big ”mass”, which itself might be of interest.

##  1\. Introduction

###  1.1. Historical background

The restriction problem for the Fourier transform, introduced by Stein [St86]
in the seventies, asks for which values of $1\leq r,s\leq\infty$ the
restriction of the Fouriertransform $Rf=\hat{f}|_{S}$ to a given submanifold
$S\subset{\mathbb{R}}^{n+1}$ gives a bounded operator, i.e., for which values
of $1\leq r,s\leq\infty$ is

$\left(\int_{S}|\hat{f}|^{r}\mathrm{d}\sigma\right)^{1/r}\leq
C\|f\|_{L^{s}({\mathbb{R}}^{n+1})}$

for all $f\in\mathcal{S}({\mathbb{R}}^{n+1})$? Here, $\sigma$ denotes the
surface measure of $S$, and for $r=\infty$, the left-hand-side has to be
modified the obvious way.
In restriction theory, it is often useful to study the adjoint restriction
operator associated to $S$, the _extension operator_

${\mathcal{E}}(f)(x,t)=\widehat{f\mathrm{d}\sigma}(x,t).$

For $S$ being a compact n-dimensional hypersurface with non-vanishing Gaussian
curvature, the famous restriction conjecture proposed by Stein in the 70’s is
one of the hardest problems in harmonic analysis and has inspired many
groundbreaking work. In the language of the adjoint setting, the conjecture
states that the extension operator is bounded from $L^{q}(S)$ to
$L^{p}({\mathbb{R}}^{n+1})$ if and only if $p>\frac{2n+2}{n}$ and
$\frac{1}{q^{\prime}}\geq\frac{n+2}{p}$ (Fig. 1). Testing on the constant
function, or respectively on a characteristic function of a small ball shows
that both conditions are indeed necessary.

Figure 1. Conjectured range for boundedness of the extension operator

For the important Hilbert space case $q=2$, the conjecture was confirmed Stein
and Tomas [To75], [St86]. Further important contributions were due to J.
Bourgain in the nineties ([B91], [B95a], [B95b]), and the introduction of the
so-called bilinear method by Tao, Moyu, Vargas, Vega and Wulff ([MVV96],
[MVV99], [TVV98] [TVI00], [TVII00], [W01], [T03]). A new approach to the
problem has been the study of multilinear variants of the extension operator
in [BCT06], which in turn imply estimates for the ”standard” linear extension
operator [BG11]. Recently, a major step forward was achieved by the polynomial
partitioning method, developed by L. Guth and N.H. Katz ([GK15], [G16], [G17])
(see also [HR19] and [Wa18] for recent improvements).
Observe that the adjoint restriction estimate for $q=1$ and $p=\infty$ is
trivial, and a $L^{q}(S)$-$L^{p}({\mathbb{R}}^{n+1})$-estimate implies a
$L^{\tilde{q}}(S)$-$L^{p}({\mathbb{R}}^{n+1})$-estimate for any $\tilde{q}>q$
by Hölders inequality, since $S$ has finite measure. On the other hand, the
two critical lines $p=\frac{2n+2}{n}$ and $\frac{1}{q^{\prime}}=\frac{n+2}{p}$
intersect on the diagonal at $q=p=\frac{2n+2}{n}$, and proving a estimate for
$q=p$ arbitrary close to the critical value $\frac{2n+2}{n}$ is arguably the
hardest case, as one then obtains
$L^{q}(S)$-$L^{p}({\mathbb{R}}^{n+1})$-boundedness for the open range
$p>\frac{2n+2}{n}$ and $\frac{1}{q^{\prime}}>\frac{n+2}{p}$ by interpolating
with the trivial $L^{1}(S)$ to $L^{\infty}({\mathbb{R}}^{n+1})$ estimate and
by Hölder. It is even possible to obtain estimates for the endpoint line
$\frac{1}{q^{\prime}}=\frac{n+2}{p}$ by switching to a bilinear regime and
interpolating with any bilinear estimate above the critical line, though we
will not explain this in detail.
Now, to prove extension estimates on the diagonal, at least for $S$ being the
sphere, it is indeed sufficient to prove
$L^{\infty}(S)$-$L^{p}({\mathbb{R}}^{n+1})$-estimates by using the Nikishin-
Maurey-Pisier factorisation argument. To be more precise, for any fixed $p\geq
2$, the following weak type estimate for the restriction operator $R$

(1.1)
$\displaystyle\|Rf\|_{L^{1,\infty}(S)}\lesssim\|f\|_{L^{p^{\prime}}({\mathbb{R}}^{n+1})}$

is equivalent to

(1.2)
$\displaystyle\|Rf\|_{L^{p^{\prime},\infty}(S)}\lesssim\|f\|_{L^{p^{\prime}}({\mathbb{R}}^{n+1})}.$

In other words, $R$ factors through $L^{p^{\prime},\infty}(S)$, via
composition with the embedding $L^{p^{\prime},\infty}(S)\hookrightarrow
L^{1,\infty}(S)$. That (1.2) implies (1.1) is trivial by embedding of Lorentz
spaces, the converse is far from being obvious; this is what we call the
factorisation argument (cf. Fig. 1) and for the sphere is due to Bourgain
[B91].
Bascially, this factorisation argument consist of two ingredients, the actual,
general Nikishin-Maurey-Pisier factorisation, plus a invariance consideration
as final step.

The main part, the Nikishin-Maurey-Pisier factorisation argument, indeed works
in a much more general setting, namely sublinear operators on Banach spaces of
Rademacher type $p^{\prime}$, but we will not dwell on that. For an excellent
survey on this argument, we refer the interested reader to the book of Garcia-
Cuerva and Rubio de Francia [GR85], which builds upon the work of Nikishin
[N70, N72] and Maurey [M73, M74].
The Nikishin-Maurey-Pisier factorisation as in Theorem 1.7 and 2.4 in Chapter
VI in [GR85], simplified for our purposes, states that (1.1) implies the
following so-called Nikishin condition:
For any $\varepsilon>0$, there exist a constant $C_{\varepsilon}$ and a set
$E\subset S$ such that

$\displaystyle\sigma(S\backslash E)<$
$\displaystyle\varepsilon\qquad\text{and}$ $\displaystyle\sigma(\xi\in
E:|Rf(\xi)|>\lambda)\leq$ $\displaystyle
C_{\varepsilon}\left(\frac{\|f\|_{p}}{\lambda}\right)^{p}$

for all $f\in L^{p}({\mathbb{R}}^{n+1})$ and $\lambda>0$. This is almost a
weak type $p$ estimate, but for the exceptional set $S\backslash E$, which is
small. So far the argument works even for the aforementioned more general
operators.
The proof of the Nikishin-Maurey-Pisier factorisation in [GR85] uses as two
major tools: Firstly Khintchine’s inequality, and secondly Zorn’s Lemma. Both
will in some sense feature in our proof as well, but only in a very simplified
form. The complexity of our ”Khintchine inequality” will be at the level of
the parallelogram identity, and instead of the Lemma of Zorn or the equivalent
axiom of choice, we will simply have to choose a representative from a set of
at most two elements. The reason for this perceived simplification is that we
apply them repeatedly, as we will use an inductive argument. We believe that
this argument is arguably easier accessible and more natural, at least in the
Fourier restriction community, where induction on scales methods are standard
by now.

Apart from this Nikishin-Maurey-Pisier factorisation, to deal with the above
exceptional set, an extra invariance argument is needed. For the surface in
question being the sphere, this crucial final step then relies on the
rotational invariance of spheres, see Bourgain [B91]. According to Bourgain in
the same paper, this generalises to the paraboloid, although he does not give
a proof and it seems less obvious to the author. The unbounded paraboloid
comes with invariance under affine transformations, but restriction estimate
(1.1) is known to fail for the unbounded paraboloid due to scaling
considerations. A compact part of the paraboloid, on the other hand, is not
invariant under affine transformations.
However, our approach is somewhat different, and we will be able to adapt the
invariance argument to a quadratic surface setting to meet our purposes in
Section 2.2.

###  1.2. Localisation

It is a by now standard to consider local estimates, that is, instead of
attacking restriction estimates

(1.3) $\displaystyle\|{\mathcal{E}}f\|_{L^{p}({\mathbb{R}}^{n})}\leq
C\|f\|_{L^{q}(S)},$

directly, one deals with the localised versions

(1.4) $\displaystyle\|{\mathcal{E}}f\|_{L^{p}(B_{R})}\leq
C_{\varepsilon}R^{\varepsilon}\|f\|_{L^{q}(S)},$

where $B_{R}$ is a ball of radius $R$. Note that the estimate is invariant in
the centre of $B_{R}$, as a translation of the centre only amounts to a
harmless modulation of $f$. Indeed, by certain $\varepsilon$-removal
techniques, (1.3) essentially (up to an endpoint) reduces to proving (1.4) for
arbitrary small $\varepsilon$ , see for instance Theorem 1.2 in [T99] or
Theorem 5.3 in [K17] for a later version.
At the first glance stronger, but indeed equivalent to the local restriction
estimate is

(1.5) $\displaystyle\|{\mathcal{E}}f\|_{L^{p}(T_{R})}\leq
C_{\varepsilon}R^{\varepsilon}\|f\|_{L^{q}(S)},$

where $T_{R}$ now is the horizontal plate $T_{R}={\mathbb{R}}^{n}\times[-R,R]$
of thickness $2R$. Clearly (1.5) implies (1.4), while the converse uses
certain orthogonality arguments. A detailed proof can be found in Lemma 5.1 in
[BMV20].

###  1.3. The main result

In order to present our main result, let us introduce some notation. Let
$\Omega=[0,1]^{n}$ and $S=\\{(\xi,Q(\xi))|\xi\in\Omega\\}$, where $Q$ is any
quadratic form on ${\mathbb{R}}^{n}$. There is no restriction on the signature
of $Q$, nor do we require any bounds on the eigenvalues of $Q$.
We will call the surface $S$ a quadric and frequently identify a function
$f\in L^{1}(S)$ with the function $\xi\to f(\xi,Q(\xi))$ defined on $\Omega$,
and write $f(\xi)$ when we mean $f(\xi,Q(\xi))$. With the same understanding,
we will also write $L^{q}(S)$ where we mean $L^{q}(\Omega)$. Let
${\mathcal{E}}$ be the adjoint restriction or extension operator associated to
$S$, that is

${\mathcal{E}}(f)(x,t)=\int_{\Omega}f(\xi)e^{ix\cdot\xi+tQ(\xi)}\mathrm{d}\sigma(\xi).$

For convenience, we will denote the adjoint restriction operator for the whole
quadric
$\\{(\xi,Q(\xi))|\xi\in{\mathbb{R}}^{n}\\}$ by ${\mathcal{E}}$ as well
whenever that does not lead to any ambiguity.

Since we are aiming at a restricted weak type estimate, we will restrict
ourselves to a sufficiently rich subset of $L^{q}(S)$-functions, including all
characteristic functions. However, in order to allow more flexibility, we will
also allow signs. We define the set of signed characteristic functions:

$D:=\\{f:\Omega\to\\{0\\}\cup\mathbb{S}^{1}|f\text{ is measurable}\\},$

where $\mathbb{S}^{1}:=\\{z\in{\mathbb{C}}:|z|=1\\}$.

For $f\in D$, define the ”carrier” $\text{\rm car\,}f:=\\{f\neq
0\\}=\\{|f|=1\\}$ of $f\in D$. We will refer to $\text{\rm
mass\,}(f):=|\text{\rm car\,}f|$ as the ”mass” of $f$

Note that $D$ is not a vector space, as for $f,g\in D$, in general $f+g\not\in
D$, since their carriers might overlap. However, still $(f+g)\chi_{(\text{\rm
car\,}f\cap\text{\rm car\,}g)^{c}}\in D$, and that will suffice for our
considerations. Some further simple remarks about the set $D$:

######  Remark 1.1.

Let $f,g\in D$, $E\subset\Omega$ measurable, and $\psi$ a measurable and real-
valued function on $\Omega$. Then

  1. (i)

$fg\in D$

  2. (ii)

$-f\in D$

  3. (iii)

$e^{i\psi(\cdot)}f\in D$

  4. (iv)

$f\chi_{E}\in D$.

We fix $p$. For $1\leq q\leq\infty$ and $T\subset{\mathbb{R}}^{n+1}$
measurable, let ${\mathcal{C}}_{q}(T)$ denote the best constant $C$ such that

$\|{\mathcal{E}}f\|_{L^{p}(T)}\leq C\|f\|_{L^{q}(S)}$

for all $f\in D$. Here, we will consider horizontal plates

$T_{R}={\mathbb{R}}^{n}\times[-R,R]$

of thickness $2R$. Abusing notation, we will write ${\mathcal{C}}_{q}(R)$ when
we mean ${\mathcal{C}}_{q}(T_{R})$.
By trivial considerations, ${\mathcal{C}}_{q}(R)\lesssim R^{1/p}<\infty$ for
all $q\geq p^{\prime}$.

Note that if $I$ is any interval of length $|I|=2R$, then

${\mathcal{C}}_{q}({\mathbb{R}}^{n}\times
I)={\mathcal{C}}_{q}({\mathbb{R}}^{n}\times[-R,R])$

since ${\mathcal{E}}f(x,t+s)={\mathcal{E}}(e^{isQ}f)$ and $D$ is closed under
modulations (iii).

Another small but important observation is: Since $[-2R,2R]$ is the union of
the two intervals $[-2R,0]$ and $[0,2R]$ of length $2R$, we have the following
doubling property:

######  Remark 1.2.

(1.6) $\displaystyle{\mathcal{C}}_{q}(2R)\leq
2^{\frac{1}{p}}{\mathcal{C}}_{q}(R).$

We now present our main result.

######  Theorem 1.3.

For any $p<q\leq\infty$ there exists a constant $C_{1}>1$ such that for all
$R>1$

(1.7) $\displaystyle{\mathcal{C}}_{\infty}(R)\leq{\mathcal{C}}_{q}(R)\leq
C_{1}{\mathcal{C}}_{\infty}(R).$

For convenience, we formulate a version that is more along the lines of the
usual language in restriction theory:

######  Corollary 1.4.

Let $p_{0}>(2n+2)/n$. Assume we have the restriction estimate

$\|{\mathcal{E}}g\|_{L^{p_{0}}({\mathbb{R}}^{n+1})}\leq C\|g\|_{\infty}$

for all $g\in D\subset L^{\infty}(S)$. Then for all $p>p_{0}$, and $g\in
L^{p}(S)$,

$\|{\mathcal{E}}g\|_{L^{p}({\mathbb{R}}^{n+1})}\leq C^{\prime}\|g\|_{p}.$

###### Proof.

The assumptions of the corollary imply in particular

${\mathcal{C}}_{\infty}(T_{R})\leq C$

for all $R>1$. But then

${\mathcal{C}}_{p}(T_{R})\leq CC_{1}^{;}$

for all $p>p_{0}$, and since $D$ contains all characteristic functions
$\chi_{E}$ of measurable sets $E$, we obtain for $R\to\infty$

$\|{\mathcal{E}}\chi_{E}\|_{L^{p_{0}}({\mathbb{R}}^{n+1})}\leq
CC_{1}|E|^{1/p}.$

Marcinkiewicz interpolation with the trivial $L^{1}(S)\to
L^{\infty}({\mathbb{R}}^{n+1})$ result gives the claim. ∎

######  Remark 1.5.

Though we discuss the proof in the setting of our quadrics, an analogous
argument applies for the sphere, giving an alternative proof for the
factorisation argument for Fourier restriction to the sphere. Actually, for
the sphere, our argument can be simplified compared to the case for quadrics.
Since Lemma 2.3 does not require the extra parameter $\delta$ from Lemma 2.4,
there is no need for the rescaling argument.

##  2\. Further remarks and Preliminaries

###  2.1. Limitations of factorisation arguments in restriction theory

The proof we present works for any quadratic phase function, regardless of its
signature. Perturbations are a different matter. Although some kind of
factorisation theorem might be conjectured to hold for perturbations of
quadratic forms, i.e., compact surfaces with non-vanishing curvature, a proof
would require further ideas, as they are not affine invariant.
Going beyond surfaces with non-vanishing curvature however, factorisation may
fail. Müller, Vargas and the author studied restriction estimates for surfaces
of finite type, the prototypical example being the graph $S_{m,l}$ of the
phase function $\psi_{m,l}(x,y)=x^{m}+y^{l}$, $x,y\in[0,1]^{2}$. The range of
admissible exponents can be described in terms of the so-called _height_ $h$,
in this case given by $\frac{1}{h}=\frac{1}{m}+\frac{1}{l}$ and
$M=\max\\{m,l\\}$. The following holds true [BMV17]:

######  Theorem 2.1 (Buschenhenke, Müller, Vargas).

Let $p>\frac{10}{3}$. Then the extension operator associated to $\psi_{m,l}$
is bounded from $L^{q}(S_{m,l})$ to $L^{p}({\mathbb{R}}^{n+1})$ if and only if
$p>h+1$, $\frac{1}{q^{\prime}}\geq\frac{h+1}{p}$ and
$\frac{1}{q}+\frac{2M+1}{p}<\frac{M+2}{2}$.

The condition $p>\frac{10}{3}$ is not sharp, but the natural limit for the
bilinear method used in [BMV17]. We emphasize that the condition
$\frac{1}{q}+\frac{2M+1}{p}<\frac{M+2}{2}$ does not allow a factorisation
theorem. We give a concrete example where indeed (1.1) holds for certain
values of $p$, but (1.2) does not:

######  Example 2.2.

Let $l=2$ and $m=M=7$. We see that we have boundedness from
$L^{\infty}(S_{m,l})$ to $L^{p}({\mathbb{R}}^{n+1})$ for all $p>\frac{10}{3}$;
however, we have no boundedness from $L^{p}(S_{m,l})$ to
$L^{p}({\mathbb{R}}^{n+1})$ unless even $p>\frac{32}{9}>\frac{10}{3}:$
Here, the height is $h=\frac{14}{9}$, so we have $p>\frac{10}{3}>h+1$ in any
case. The critical condition is $\frac{1}{q}+\frac{15}{p}<\frac{9}{2}$, which
for $q=\infty$ gives $p>\frac{10}{3}$, but for $q=p$, necessarily
$p>\frac{32}{9}$.

###  2.2. Invariance considerations

The factorisation argument in [BG11] relies on the rotation invariance of the
sphere. This rotation invariance can be exploited in the following lemma:

######  Lemma 2.3.

Let $E,F\subset\mathbb{S}^{n-1}$ measurable. Then there exist a rotation
$\rho\in\mathcal{O}(n)$ such that

$\sigma(E\cap\rho(F))=\sigma(E)\,\sigma(F).$

Since we will not directly apply this lemma, but use a variation adapted to
our quadric, namely Lemma 2.4, we will leave the (very similar) proof to the
reader.

Our quadric does not come with such rotation invariance; instead, we will make
use of translations in parameter space, amounting to affine transformations of
the surface. The obvious problem is that while ${\mathbb{R}}^{n}$ is
translation-invariant, the compact subset $\Omega\subset{\mathbb{R}}^{n}$ is
not. The following Lemma will meet our purposes:

######  Lemma 2.4.

Let $\delta>0$ and $E,F\subset{\mathbb{R}}^{n}$ measurable. Then there exist a
$\lambda\in[0,\delta]^{n}$ such that

$|E\cap(F+\lambda)|\leq\delta^{-n}|E|\,|F|.$

######  Remark 2.5.

We will apply the lemma for $E=F\subset\Omega$ with $|E|=|F|\ll 1$. Observe
that in general $F+\lambda$ is not a subset of $\Omega$ anymore, but we only
know that $F+\lambda$ is contained in $(1+\delta)\Omega$, as is $E$. We will
deal with these issue by rescaling. Furthermore, we have to account for the
parameter $\delta$ and accommodate a blow-up by $\delta^{-n}\gg 1$ on the
right-hand-side. This has to be balanced against the smallness of $|E|=|F|\ll
1$. We will give more details later in the proofs.

###### Proof.

We have

(2.8) $\displaystyle|E|\,|F|=$
$\displaystyle\int_{{\mathbb{R}}^{n}}\int_{{\mathbb{R}}^{n}}\chi_{E}(x)\chi_{F}(x-y)\mathrm{d}x\mathrm{d}y$
(2.9) $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{n}}|E\cap(F+y)|\mathrm{d}y\geq\int_{[0,\delta]^{n}}|E\cap(F+y)|\mathrm{d}y.$

By pigeonholing, there exist a $\lambda\in[0,\delta]^{n}$ such that

$|E|\,|F|\geq\delta^{n}|E\cap(F+\lambda)|,$

as we claimed. ∎

###  2.3. Clarkson’s inequality

Another tool we are going to use is the so-called Clarkson’s inequality [C36].

######  Lemma 2.6.

Let $(X,\mu)$ be a measure space, and $2\leq p<\infty$. Then for all $F,G\in
L^{p}(X)$, we have for $p<\infty$

(2.10)
$\displaystyle\|F\|_{L^{p}(X)}^{p}+\|G\|_{L^{p}(X)}^{p}\leq\frac{\|F+G\|_{L^{p}(X)}^{p}+\|F-G\|_{L^{p}(X)}^{p}}{2}.$

######  Remark 2.7.

There is a analogue version for the case $p=\infty$, and Clarkson has a
similar, though slightly more complicated formula for $p<2$, but since we are
only interested in the case $2\leq p<\infty$, we refer the reader to [C36] or
[LT96] for further information.

For sake of completeness, we sketch the short proof.

###### Proof.

We begin by taking the root of the well-known parallelogram identity: For
$z,w\in{\mathbb{C}}$ we have

(2.11)
$\displaystyle(|z|^{2}+|w|^{2})^{\frac{1}{2}}=\left(\frac{|z+w|^{2}+|z-w|^{2}}{2}\right)^{\frac{1}{2}}.$

Since $p\geq 2$, we can use the embeddings of $\ell^{p}$-spaces, for the left-
hand side with the counting measure, for the right-hand side with probability
measure to conclude that

$\displaystyle(|z|^{p}+|w|^{p})^{\frac{1}{p}}\leq\left(\frac{|z+w|^{p}+|z-w|^{p}}{2}\right)^{\frac{1}{p}}.$

Application to $z=F(x)$ and $w=G(x)$ gives

$\displaystyle\|F\|_{L^{p}(X)}^{p}+\|G\|_{L^{p}(X)}^{p}=$
$\displaystyle\int_{X}|F(x)|^{p}+|G(x)|^{p}d\mu(x)$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{X}|F(x)+G(x)|^{p}+|F(x)-G(x)|^{p}d\mu(x)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\|F+G\|_{L^{p}(X)}^{p}+\frac{1}{2}\|F-G\|_{L^{p}(X)}^{p},$

which is what we claimed. ∎

##  3\. Induction on scales

###  3.1. Near extremisers

Given an error $\varepsilon>0$, we will call a function $f\in D$ an
$\varepsilon$-near extremiser for ${\mathcal{C}}_{q}(R)$, if

(3.12)
$\displaystyle\|{\mathcal{E}}f\|_{L^{p}(T_{R})}\geq(1-\varepsilon){\mathcal{C}}_{q}(R)\|f\|_{L^{q}(S)}.$

Clearly, $\varepsilon$-near extremisers always exist.

Note that if $f\in D$ is such an $\varepsilon$-near extremiser for
${\mathcal{C}}_{q}$, then for any $1\leq\tilde{q}\leq\infty$, we have

$(1-\varepsilon){\mathcal{C}}_{q}(R)\|f\|_{L^{q}(S)}\leq\|{\mathcal{E}}f\|_{L^{p}(T_{R})}\leq{\mathcal{C}}_{\tilde{q}}(R)\|f\|_{L^{\tilde{q}}(S)}.$

If there would be an $\varepsilon$-near extremiser $f$ for
${\mathcal{C}}_{q}(R)$ with $|\text{\rm car\,}f|=1$, then
${\mathcal{C}}_{q}(R)$ and ${\mathcal{C}}_{\tilde{q}}(R)$ would be comparable
up to a factor $(1-\varepsilon)$. Thus our aim will be to construct near
extremisers with large mass $|\text{\rm car\,}f|$.

Furthermore, for $\varepsilon>0,\alpha\geq 1$, we will call $f\in D$ a
$(\varepsilon,\alpha)$-near extremiser for ${\mathcal{C}}_{q}(R)$, if

(3.13) $\displaystyle\|{\mathcal{E}}f\|_{L^{p}(T_{\alpha
R})}\geq(1-\varepsilon){\mathcal{C}}_{q}(R)\|f\|_{L^{q}(S)}.$

Clearly, any $\varepsilon$-near extremiser is a $(\varepsilon,\alpha)$-near
extremiser.

Our key result is the following lemma, which essentially allows us to increase
the mass of a near extremiser:

######  Lemma 3.1.

Let $2\leq p,q\leq\infty$, $\delta>0$, $0<\kappa<1<\alpha<2$. Assume that
$f\in D$ is a $(1-\kappa,\alpha)$-near extremiser for ${\mathcal{C}}_{q}(R)$,
that is

(3.14) $\displaystyle\|{\mathcal{E}}f\|_{L^{p}(T_{\alpha
R})}\geq\kappa{\mathcal{C}}_{q}(R)\|f\|_{L^{q}(S)}.$

Then there exists a $f_{1}\in D$ such that

(3.15) $\displaystyle 2\,\text{\rm mass\,}(f)\geq\text{\rm mass\,}(f_{1})\geq
2\,\text{\rm mass\,}(f)\left(1-\frac{\text{\rm
mass\,}(f)}{2\delta^{n}}\right)\frac{1}{(1+\delta)^{n}}$

and

(3.16) $\displaystyle\|{\mathcal{E}}f_{1}\|_{L^{p}(T_{(1+\delta)^{2}\alpha
R})}\geq
2^{\frac{1}{p}-\frac{1}{q}}(1+\delta)^{\frac{n+2}{p}-\frac{n}{q^{\prime}}}\left(\kappa-\left[\frac{\text{\rm
mass\,}(f)}{\delta^{n}}\right]^{\frac{1}{q}}\right){\mathcal{C}}_{q}(R)\|f_{1}\|_{L^{q}(S)}.$

The notation clears up if we restrict ourselves to the regime $p<q$. Then for
sufficiently small $\delta>0$,

(3.17) $\displaystyle
2^{\frac{1}{p}-\frac{1}{q}}(1+\delta)^{\frac{n+2}{p}-\frac{n}{q^{\prime}}}\geq
2^{\frac{1}{p}-\frac{1}{q}}(1+\delta)^{-n}\geq 1.$

######  Lemma 3.2.

For $p<q\leq\infty$ exist a $\delta_{0}>$ so that for all
$0<\delta<\delta_{0}$and $0<\kappa<1<\alpha<2$, the following holds true.
Assume that $f\in D$ is a $(1-\kappa,\alpha)$-near extremiser for
${\mathcal{C}}_{q}(R)$, that is

(3.18) $\displaystyle\|{\mathcal{E}}f\|_{L^{p}(T_{\alpha
R})}\geq\kappa{\mathcal{C}}_{q}(R)\|f\|_{L^{q}(S)}.$

Then there exists an $f_{1}\in D$ such that

(3.19) $\displaystyle 2\,\text{\rm mass\,}(f)\geq\text{\rm mass\,}(f_{1})\geq
2\,\text{\rm mass\,}(f)\left(1-\frac{\text{\rm
mass\,}(f)}{2\delta^{n}}\right)\frac{1}{(1+\delta)^{n}}$

and

(3.20) $\displaystyle\|{\mathcal{E}}f_{1}\|_{L^{p}(T_{(1+\delta)^{2}\alpha
R})}\geq\left(\kappa-\left[\frac{\text{\rm
mass\,}(f)}{\delta^{n}}\right]^{\frac{1}{q}}\right){\mathcal{C}}_{q}(R)\|f_{1}\|_{L^{q}(S)}.$

A number of comments are in order:
First of all, we use the notion of $(\varepsilon,\alpha)$-near extremisers,
where we only assume a lower bound for the integral on the bigger plate
$T_{rR}$ of thickness $2rR$, instead of the more natural $T_{R}$. This is
because when passing to $f_{1}$, we have to increase the thickness by
$(1+\delta)^{2}$ due to a rescaling. To allow an inductive application of the
lemma, we make use of the parameter $\alpha>1$.
Secondly, one should think of $\delta$ being small, and take into account that
the lemma is only effective if $\text{\rm mass\,}(f)\ll\delta^{n}$, so that
$\kappa$ is shrinked not to much in the inductive process, while the mass of
$f_{1}$ almost doubles compared to the mass of $f$.

Figure 2. Setup for the proof of Lemma 3.1

Lemma 3.2 follows from Lemma 3.1, which we prove now.

###### Proof.

Let denote $E:=\text{\rm car\,}f$. By Lemma 2.4, we find some
$\lambda\in[0,\delta]^{n}$ such that for $E_{\cap}:=E\cap(\lambda+E)$ we have

(3.21) $\displaystyle|E_{\cap}|\leq\frac{|E|^{2}}{\delta^{n}}.$

Now let $f_{\lambda}:=f(\cdot-\lambda)$ and

(3.22) $\displaystyle f_{\pm}:=f\pm f_{\lambda}(1-\chi_{E_{\cap}}).$

Note that $f_{\pm}$ may not be supported in $\Omega$, but only in
$(1+\delta)\Omega$ (cf. Fig. 2). Observe that

(3.23) $\displaystyle|\text{\rm car\,}f_{\pm}|=|E|+|(\lambda+E)\backslash
E_{\cap}|\geq 2|E|-\frac{|E|^{2}}{\delta^{n}}\geq
2|E|\left(1-\frac{|E|}{2\delta^{n}}\right)$

and

(3.24) $\displaystyle|\text{\rm car\,}f_{\pm}|\leq 2|E|.$

Furthermore, if $Q(\xi)=\xi^{t}A\xi$, then

(3.25) $\displaystyle|{\mathcal{E}}f_{\lambda}(x,t)|=|\int
f(\xi-\lambda)e^{i(x\cdot\xi+tQ(\xi))}\mathrm{d}\xi|=|\int
f(\xi)e^{i([x+t(A+A^{t})\lambda]\cdot\xi+tQ(\xi))}\mathrm{d}\xi|=|{\mathcal{E}}f(x+t(A+A^{t})\lambda,t)|,$

Since $T_{\alpha R}$ is invariant with respect to the map
$(x,t)\to(x+t(A+A^{t})\lambda,t)$, we see that

$\|{\mathcal{E}}f_{\lambda}\|_{L^{p}(T_{\alpha
R})}=\|{\mathcal{E}}f\|_{L^{p}(T_{\alpha R})}.$

Using the $(1-\kappa,\alpha)$-near extremiser property and Clarkson’s
inequality (Lemma 2.6), we conclude that

$\displaystyle 2^{\frac{1}{p}}{\mathcal{C}}_{q}(R)\kappa|E|^{\frac{1}{q}}\leq$
$\displaystyle 2^{\frac{1}{p}}\|{\mathcal{E}}f\|_{L^{p}(T_{\alpha R})}$
$\displaystyle=$ $\displaystyle\left(\|{\mathcal{E}}f\|^{p}_{L^{p}(T_{\alpha
R})}+\|{\mathcal{E}}f_{\lambda}\|^{p}_{L^{p}(T_{\alpha
R})}\right)^{\frac{1}{p}}$ $\displaystyle\leq$
$\displaystyle\left(\frac{1}{2}\sum_{\pm}\|{\mathcal{E}}(f\pm
f_{\lambda})\|^{p}_{L^{p}(T_{\alpha R})}\right)^{\frac{1}{p}}.$

Recalling that $f_{\pm}=f\pm f_{\lambda}(1-\chi_{E_{\cap}})$, we see that

$\displaystyle 2^{\frac{1}{p}}{\mathcal{C}}_{q}(R)\kappa|E|^{\frac{1}{q}}\leq$
$\displaystyle\left(\frac{1}{2}\sum_{\pm}\left[\|{\mathcal{E}}f_{\pm}\|_{L^{p}(T_{\alpha
R})}+\|{\mathcal{E}}(f_{\lambda}\chi_{E_{\cap}})\|_{L^{p}(T_{\alpha
R})}\right]^{p}\right)^{\frac{1}{p}}$ $\displaystyle\leq$
$\displaystyle\left(\frac{1}{2}\sum_{\pm}\|{\mathcal{E}}f_{\pm}\|^{p}_{L^{p}(T_{\alpha
R})}\right)^{\frac{1}{p}}+\|{\mathcal{E}}(f_{\lambda}\chi_{E_{\cap}})\|_{L^{p}(T_{\alpha
R})}.$

Here, we used the triangle inquality for both the $L^{p}$-norm on $T_{\alpha
R}$ and the outer $\ell_{p}$-norm.
We now bootstrap the error term
$\|{\mathcal{E}}(f_{\lambda}\chi_{E_{\cap}})\|_{L^{p}(T_{\alpha R})}$. First
observe that

$\text{\rm car\,}(f_{\lambda}\chi_{E_{\cap}})\subset E_{\cap}\subset
E\subset\Omega,$

securing that $f_{\lambda}\chi_{E_{\cap}}\in D$. Therefore by Remark 1.2

(3.26)
$\displaystyle\|{\mathcal{E}}(f_{\lambda}\chi_{E_{\cap}})\|_{L^{p}(T_{\alpha
R})}\leq\|{\mathcal{E}}(f_{\lambda}\chi_{E_{\cap}})\|_{L^{p}(T_{2R})}\leq
2^{\frac{1}{p}}{\mathcal{C}}_{q}(R)|E_{\cap}|^{1/q}\leq
2^{\frac{1}{p}}{\mathcal{C}}_{q}(R)({\delta}^{-n}|E|^{2})^{1/q},$

and hence

(3.27) $\displaystyle
2^{\frac{1}{p}}{\mathcal{C}}_{q}(R)|E|^{\frac{1}{q}}(\kappa-\delta^{-n/q}|E|^{1/q})\leq\left(\frac{1}{2}\sum_{\pm}\|{\mathcal{E}}f_{\pm}\|^{p}_{L^{p}(T_{\alpha
R})}\right)^{\frac{1}{p}}.$

Since the left-hand side is a lower bound for the average of
$\|{\mathcal{E}}f_{\pm}\|_{L^{p}(T_{\alpha R})}$ over the two signs $\pm$, it
must be a lower bound for at least one of these terms.
Therefore, for either $\tilde{f}=f_{+}$ or $\tilde{f}=f_{-}$, we have

(3.28) $\displaystyle
2^{\frac{1}{p}}{\mathcal{C}}_{q}(R)|E|^{\frac{1}{q}}(\kappa-\delta^{-n/q}|E|^{1/q})\leq\|{\mathcal{E}}\tilde{f}\|_{L^{p}(T_{\alpha
R})},$

and since $\|\tilde{f}\|_{q}\leq(2|E|)^{\frac{1}{q}}$, the claim follows by a
scaling argument. Note that for

$f_{1}(\xi):=\tilde{f}((1+\delta)\xi),$

while $\text{\rm car\,}\tilde{f}\subset(1+\delta)\Omega$, we have $\text{\rm
car\,}f_{1}\subset\Omega$.

∎

We now proceed by iterating Lemma 3.2, increasing the mass of the near
extremiser in every step:

######  Lemma 3.3.

Let $p<q\leq\infty$. For any $\varepsilon>0$, there exists a $\gamma>0$ such
that for all $R>0$ the following holds: If $f_{0}$ is a
$\varepsilon/2$-extremiser for ${\mathcal{C}}_{q}(T)$ and $\text{\rm
mass\,}f_{0}<\gamma$, then there exists an $f\in D$ such that $\text{\rm
mass\,}f\geq\gamma$ and $f$ is an $(\varepsilon,1+\varepsilon)$-near
extremiser, that is,

(3.29)
$\displaystyle\|{\mathcal{E}}(f)\|_{L^{p}(T_{(1+\varepsilon)R})}\geq(1-\varepsilon){\mathcal{C}}_{q}(R)\|f\|_{L^{q}(S)}.$

###### Proof.

In the proof, we will choose a parameter $\beta>0$, $\beta\ll 1$, fulfilling
several smallness conditions, that may depend on $\varepsilon$, but not on
$R$, and then choose $\gamma:=\beta^{4n+2}$.
We assume that $f_{0}$ is an $\varepsilon/2$-near extremiser for
${\mathcal{C}}_{q}(R)$ and $|\text{\rm car\,}f_{0}|\leq\beta^{4n+2}$.
Inductively, we will construct sequences of functions $f_{j}\in D$,
$j=0,\ldots,J$ such that for $E_{j}=\text{\rm car\,}f_{j}$ and the parameters
$\delta_{j}$, defined by the formula

(3.30) $\displaystyle|E_{j}|^{1/2}=\beta\delta_{j}^{n}$

we have

(3.31)
$\displaystyle\|{\mathcal{E}}f_{J}\|_{L^{p}(\prod_{j=0}^{J-1}(1+\delta_{j})^{2}\cdot
T)}\geq\left(1-\frac{\varepsilon}{2}-\sum_{j=0}^{J-1}\left[\frac{|E_{j}|}{\delta_{j}^{n}}\right]^{1/q}\right){\mathcal{C}}_{q}(T)\|f_{j}\|_{q},$
(3.32) $\displaystyle|E_{j+1}|\geq\frac{3}{2}|E_{j}|,\qquad\forall 0\leq j<J$

and

(3.33) $\displaystyle|E_{j}|<\beta^{4n+2}\qquad\forall 0\leq j<J.$

The aim is to construct such a sequence with

(3.34) $\displaystyle|E_{J}|\geq\beta^{4n+2}.$

For $J=0$, (3.31) is clear, and (3.32) and (3.33) are void. So, assume we
constructed a sequence $f_{0},\ldots,f_{J}$ so that (3.31), (3.32) and (3.33)
hold true. If $|E_{J}|\geq\beta^{4n+2}$, we are done. If
$|E_{J}|<\beta^{4n+2}$, we iterate. Observe first that since
$|E_{j}|^{1/2}\beta^{-1}\leq\beta^{2n},$ we have

(3.35) $\displaystyle\delta_{j}\leq\beta^{2}<\delta_{0}$

if $\beta$ is sufficiently small. Here, $\delta_{0}$ is the constant from
Lemma 3.2. We seek to apply Lemma 3.2 to $f=f_{J}$ and the parameter
$\delta=\delta_{J}$, with
$\kappa=\kappa_{J}=1-\frac{\varepsilon}{2}-\sum\limits_{i=0}^{J-1}\left[\frac{|E_{i}|}{\delta_{i}^{n}}\right]^{1/q}$
and $\alpha=\alpha_{J}=\prod\limits_{i=0}^{J-1}(1+\delta_{i})^{2}$. Therefore,
we have to ensure $\kappa_{J}>0$ and $\alpha_{J}\leq 2$.

In fact, we will show at the end of the proof that

(3.36) $\displaystyle\alpha_{J}\leq$ $\displaystyle 1+\varepsilon,$ (3.37)
$\displaystyle\kappa_{J}\geq$ $\displaystyle 1-\varepsilon.$

provided (3.32) and (3.33) hold true. Taking that for granted for now, Lemma
3.2 provides us with an $f_{J+1}\in D$ such that

(3.38) $\displaystyle|E_{J+1}|\geq
2|E_{J}|\left(1-\frac{|E_{J}|}{2\delta_{J}^{n}}\right)\frac{1}{(1+\delta_{J})^{n}}$

and

(3.39)
$\displaystyle\|{\mathcal{E}}f_{J}\|_{L^{p}((1+\delta_{J})^{2}\alpha_{J}T_{R})}\geq\left(\kappa-\left[\frac{|E_{J}|}{\delta_{J}^{n}}\right]^{1/q}\right){\mathcal{C}}_{q}(T)\|f_{J}\|_{q}.$

We check that the sequence $f_{0},\ldots,f_{J+1}$ fulfills (3.31), (3.32) and
(3.33): (3.31) is immediate, (3.33) by assumption. For (3.32),

since $\delta_{j}\leq\beta^{2}$, (3.38) implies

(3.40) $\displaystyle|E_{j+1}|\geq$
$\displaystyle|E_{j}|(2-\beta|E_{j}|^{1/2})(1+\delta_{j})^{-n}$ (3.41)
$\displaystyle\geq$
$\displaystyle|E_{j}|(2-\beta)(1+\beta^{2})^{-n}\geq\frac{3}{2}|E_{j}|$

provided $(2-\beta)(1+\beta^{2})^{-n}\geq 3/2$.

We claim that the procedure stops after finitely many steps. If not, we would
obtain an infinite sequence of $E_{j}$, which according to (3.33) would be
bounded, contradicting (3.32).

Hence there exist a finite $J$ such that (3.31) holds and
$|E_{J}|\geq\beta^{4n+2}$. Using (3.36) and (3.37), we see that (3.31) implies
that $f_{J+1}$ is an $(\varepsilon,1+\varepsilon)$-near extremiser.

It remains to show that

$\displaystyle\alpha_{J}\leq$ $\displaystyle 1+\varepsilon,$
$\displaystyle\kappa_{J}\geq$ $\displaystyle 1-\varepsilon$

provided (3.32) and (3.33) hold true. Since by (3.32),
$|E_{j}|\leq\left(\frac{2}{3}\right)^{J-1-j}|E_{J-1}|$, for any exponent
$r>0$, we have

(3.42) $\displaystyle\sum_{j=0}^{J-1}|E_{j}|^{r}\leq$
$\displaystyle\sum_{j=0}^{J-1}\left(\frac{2}{3}\right)^{r(J-1-j)}|E_{J-1}|^{r}\leq
c_{r}\beta^{(4n+2)r}.$

For $r=1/(2q)$, we obtain

(3.43)
$\displaystyle\sum_{j=0}^{J-1}\left[\frac{|E_{j}|}{\delta_{j}^{n}}\right]^{1/q}=\beta^{1/q}\sum_{j=0}^{J-1}|E_{j}|^{1/(2q)}\leq
c_{\frac{1}{2q}}\beta^{(2n+2)/q}\leq\varepsilon/2,$

provided $\beta$ is sufficiently small, and hence (3.37). Furthermore,

$\displaystyle\log(\prod_{j=0}^{J-1}(1+\delta_{j})^{2})\leq$ $\displaystyle
2\sum_{j=0}^{J-1}\delta_{j}$ $\displaystyle=$ $\displaystyle
2\beta^{-1/n}\sum_{j=0}^{J-1}|E_{j}|^{1/(2n)}$ $\displaystyle\leq$
$\displaystyle 2c_{\frac{1}{2n}}\beta^{\frac{4n+2}{2n}-\frac{1}{n}}$
$\displaystyle=$ $\displaystyle 2c_{\frac{1}{2n}}\beta^{2}\leq\beta,$

again for sufficiently small $\beta$. Then

(3.44)
$\displaystyle\alpha_{J}=\prod_{j=0}^{J-1}(1+\delta_{j})^{2}\leq\exp(\beta)\leq
1+\varepsilon,$

if $\beta$ is small enough with respect to $\varepsilon$. ∎

As an immediate consequence, we can deduce the existence of near-extremisers
with large mass:

######  Corollary 3.4.

Let $\frac{2n+2}{n}\leq p\leq q<\infty$. For any $\varepsilon>0$, there exists
a $\gamma>0$ such that the following holds: For all $R\geq 1$ there exists an
$f\in D$ such that $|\text{\rm car\,}f|\geq\gamma$ and

(3.45)
$\displaystyle\|{\mathcal{E}}f\|_{L^{p}((1+\varepsilon)T_{R})}\geq(1-\varepsilon){\mathcal{C}}_{q}(R)\|f\|_{L^{q}(S)}.$

###### Proof.

Given $\varepsilon>0$, we choose $\gamma>0$ as in Lemma (3.3). Let $f_{0}$ be
a $1-\varepsilon/2$-near extremiser of ${\mathcal{C}}_{q}(R)$. If $|\text{\rm
car\,}f_{0}|\geq\gamma$, we are done by choosing $f=f_{0}$. If $|\text{\rm
car\,}f_{0}|<\gamma$, the aforementioned lemma applies.

∎

We can now prove our main result, Theorem 1.3. We have to show that

(3.46) $\displaystyle{\mathcal{C}}_{\infty}(R)\leq{\mathcal{C}}_{q}(R)\leq
C_{1}^{1/q}{\mathcal{C}}_{\infty}(R).$

###### Proof.

The first inequality is a simple consequence of Hölders inequality. The second
inequality is trivial for the case $q=\infty$, and for $q<\infty$, by an
interpolation argument, it is enough to consider the case $q=p$.
Choose $\gamma>0$ according to Corollary (3.4), for $\varepsilon=1/2$ and
$q=p$. Then for all $R>0$, there exist an $f\in D$ such that

(3.47) $\displaystyle\gamma^{1/p}{\mathcal{C}}_{p}(R)\leq\|f\|_{p}\
{\mathcal{C}}_{p}(R)\leq 2\|{\mathcal{E}}f\|_{L^{p}(T_{2R})}.$

On the other hand,

(3.48) $\displaystyle\gamma^{1/p}{\mathcal{C}}_{p}(R)\leq
2\|{\mathcal{E}}f\|_{L^{p}(T_{2R})}\leq 2{\mathcal{C}}_{\infty}(2R)\leq
2^{1+1/p}{\mathcal{C}}_{\infty}(T),$

hence we can choose $C_{1}=\dfrac{2^{p+1}}{\gamma}$. ∎

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# The cohomology of the connective spectra for K-theory revisited

Donald M. Davis  Department of Mathematics, Lehigh University
Bethlehem, PA 18015, USA<EMAIL_ADDRESS>and  W. Stephen Wilson  Department
of Mathematics, Johns Hopkins University
Baltimore, MD 01220, USA<EMAIL_ADDRESS>
###### Abstract.

The stable mod 2 cohomologies of the spectra for connective real and complex
K-theories are well known and easy to work with. However, the known bases are
in terms of the anti-automorphism of Milnor basis elements. We offer simple
bases in terms of admissible sequences of Steenrod operations that come from
the Adem relations. In particular, the basis for the complex case is that you
don’t use any Steenrod operations in degree one or $2^{n}+1$, $n>0$.

###### Key words and phrases:

Adem relations, connective $K$-theory, Steenrod algebra

###### 2020 Mathematics Subject Classification:

55S10, 55R45, 55N15

Without Maple software, the Induction step 4.2 would never have been found.

##  1\. Introduction

Our goal is to give simple bases for the mod 2 cohomologies, $H^{*}(bo)$ and
$H^{*}(bu)$, for the connective real and complex K-theory spectra
respectively.

Let $I=(i_{1},i_{2},\ldots,i_{k})$. We let $Sq^{I}=Sq^{i_{1}}Sq^{i_{2}}\cdots
Sq^{i_{k}}$ be a composition of Steenrod squares. We have the length of $I$
$\ell(I)=k$ and the degree of $I$ $|I|=|Sq^{I}|=\sum i_{s}$. We say $I$ is
admissible if $i_{s}\geq 2i_{s+1}$ for all $s$. The admissible $Sq^{I}$ form
the Serre-Cartan basis for the mod 2 Steenrod algebra, $\mathcal{A}$. Let
$\mathcal{A}_{1}$ be the sub-algebra generated by $Sq^{1}$ and $Sq^{2}$. Let
$E_{1}$ be the sub-algebra generated by $Q_{0}=Sq^{1}$ and
$Q_{1}=Sq^{1}Sq^{2}+Sq^{2}Sq^{1}$.

Let ${\mathbf{Z}}_{2}$ be the integers mod 2. It has been known for a long
time ([Sto63]) that
$H^{*}(bo)=\mathcal{A}\otimes_{\mathcal{A}_{1}}{\mathbf{Z}}_{2}=\mathcal{A}/\\!\\!/\mathcal{A}_{1}$
and
$H^{*}(bu)=\mathcal{A}\otimes_{E_{1}}{\mathbf{Z}}_{2}=\mathcal{A}/\\!\\!/E_{1}$.
The usual basis for $H^{*}(bo)$ involves applying the anti-automorphism to
Milnor basis ([Mil58]) elements $Sq(R)$ with $R=(4r_{1},2r_{2},r_{3},\ldots)$.
One can also extract an exotic basis for $H^{*}(bo)$ from [Mor07] that is
probably related to the spaces in the Omega spectrum that the elements are
created on.

We can now state our main theorem:

######  Theorem 1.1.

Bases for $H^{*}(bo)$ and $H^{*}(bu)$ are given by all $Sq^{I}$ with $I$
admissible, no $i_{s}=2^{n}+1$ for $n\geq 1$ and $i_{k}\geq 4$ and $i_{k}\geq
2$ respectively.

Along the way we ran across some things about the Steenrod algebra that may be
of independent interest.

######  Definition 1.2.

Let $T_{b}\subset\mathcal{A}$ be the span of all admissible $Sq^{I}$ with
$i_{1}\leq b$.

######  Proposition 1.3.

$Sq^{a}\,T_{b}\subset T_{n}$ where

$n=\begin{cases}a&\text{if\ \ }a\geq b\\\ 2b-1&\text{if\ \ }2b>a\geq b\\\
a+b&\text{if\ \ }b>a>0.\end{cases}$

######  Remark 1.4.

Note that it is always true that $Sq^{a}T_{b}\subset T_{a+b}$.

The subalgebra $\mathcal{A}_{n}$ of $\mathcal{A}$ is generated by
$Sq^{1},Sq^{2},Sq^{4},\cdots,Sq^{2^{n}}$. As usual, let $\alpha(n)$ be the
number of ones in the binary expansion of $n$. We toyed with a conjecture for
a basis for $\mathcal{A}_{n}$ but it was false already in degree 49 for
$\mathcal{A}_{2}$. One observation survived:

######  Proposition 1.5.

In $\mathcal{A}/\\!\\!/\mathcal{A}_{n}$, if $\alpha(m)\leq n$, then
$Sq^{m+1}\in T_{m}$.

We first prove the results about the Steenrod algebra. Then we apply these
results to prove Theorem 1.1.

##  2\. Results on the Steenrod algebra

We will make constant use of the Thom spectrum, $MO$, for the unoriented
cobordism case. From [Thom54] we know that $H^{*}(MO)$ is free over
$\mathcal{A}$ and that one copy of $\mathcal{A}$ sits on the Thom class $U\in
H^{0}(MO)$. We need the Stiefel-Whitney (S-W) classes, $w_{i}\in H^{i}(BO)$,
and the Thom isomorphism $H^{*}(BO)\cong H^{*}(MO)$ that takes $w_{i}$ to
$Uw_{i}$. We need the connection between the S-W classes and the Steenrod
algebra given by $Sq^{n}U=Uw_{n}$ and

$Sq^{i}(w_{j})={\sum}_{t=0}^{i}\binom{j+t-i-1}{t}w_{i-t}w_{j+t}.$

Keep in mind that $Sq^{n}w_{n}=w_{n}^{2}$ and $Sq^{i}w_{n}=0$ when $i>n$.

The cohomology, $H^{*}(BO)$, is a polynomial algebra on the S-W classes,
[MS74]. We put an order on the monomials. We have $M<M^{\prime}$ if the degree
of $M^{\prime}$ is greater than that of $M$. Next, if they have the same
degree, the one with the largest $w_{n}$ is greater. If they have the same
largest $w_{n}$, we go to the next largest and so on. To use Thom’s examples
from his paper: $w_{4}<w_{4}w_{1}^{2}<w_{4}w_{2}w_{1}<w_{4}w_{3}$.

######  Lemma 2.1 (Thom, in the proof of II.8, [Thom54]).

For $I=(i_{1},i_{2},\ldots,i_{k})$ admissible, in $H^{*}(MO)$,

$Sq^{I}(U)=U(w_{i_{1}}w_{i_{2}}\cdots w_{i_{k}}+\Delta)$

where $\Delta$ is a sum of monomials of lower order.

######  Remark 2.2.

This result allows us to distinguish between admissible $Sq^{I}$ using the S-W
classes in the Thom spectrum. Because it was 1954, Thom worked in the stable
range of $MO(n)$ where his Thom class was $w_{n}$.

###### Proof of Proposition 1.3..

It is enough to consider the case when $i_{1}=b$. When $a\geq 2b$, there is
nothing to prove because $Sq^{a}Sq^{I}$ is already admissible. When $2b>a\geq
b$, this is no longer the case. If $Sq^{a}Sq^{I}$ is written in terms of
admissible $Sq^{J}$, we need to determine what the maximum possibility is for
$j_{1}$. We look at

$\displaystyle Sq^{a}Sq^{I}(U)$ $\displaystyle=$ $\displaystyle
Sq^{a}\Big{(}U(w_{b}w_{i_{2}}\cdots w_{i_{k}}+\Delta)\Big{)}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{a}Sq^{a-j}(U)Sq^{j}(w_{b}w_{i_{2}}\cdots
w_{i_{k}}+\Delta)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{a}Uw_{a-j}Sq^{j}(w_{b}w_{i_{2}}\cdots
w_{i_{k}}+\Delta).$

Since $w_{b}$ is the largest S-W class, the largest possible new S-W class is
given by $Sq^{j}(w_{b})$, but the largest this can be is
$Sq^{b-1}(w_{b})=w_{2b-1}$ plus other terms with products. Similarly, if
$a=2b-1$, we could get $w_{2b-1}$ when $j=0$ in the formula. Not only is
$n=2b-1$ the largest possible, but it is realized.

Using the same formula when $b>a>0$, the largest possible $w_{n}$ is when
$Sq^{a}w_{b}$ includes $w_{a+b}$ and that is only realized when
$\binom{b-1}{a}=1$ (mod 2). This concludes the proof. ∎

It is time to introduce one of our key tools, the Adem relations:

$Sq^{a}Sq^{b}=\sum_{i}^{[a/2]}\binom{b-1-i}{a-2i}Sq^{a+b-i}Sq^{i}.$

These apply when $a<2b$, that is, when $Sq^{a}Sq^{b}$ is not admissible. The
resulting terms are admissible. The sum is from the maximum of $0$ or $a-b+1$.

###### Proof of Proposition 1.5.

We induct on $m$. Let $m=2^{k_{1}}+\cdots+2^{k_{\ell}}$ with
$k_{1}>\cdots>k_{\ell}$ and $\ell\leq n$. Let $s=\min\\{i:k_{i}>k_{i+1}+1\\}$.
If no such $s$, let $s=\ell$. If $s=\ell$ and $k_{\ell}=0$, then
$m+1=2^{\ell}$, and we are done since $Sq^{2^{\ell}}=0$ in
$\mathcal{A}/\\!\\!/\mathcal{A}_{n}$. Otherwise, write $m=2a+b$ with

$a=2^{k_{1}-1}+\cdots+2^{k_{s}-1}\text{ and
}b=2^{k_{s+1}}+\cdots+2^{k_{\ell}}.$

Note that if $s=\ell$, then $b=0$ and $m$ is even, having already done the odd
case. Then $Sq^{a+b+1}\in T_{a+b}$ by the induction hypothesis. Proposition
1.3 says $Sq^{a}T_{a+b}\subset T_{2a+b}$, so we have $Sq^{a}Sq^{a+b+1}\in
T_{2a+b}$. Since $\binom{a+b}{a}=1$, (we are always working mod 2), the Adem
relation gives

$Sq^{a}Sq^{a+b+1}=\sum\binom{a+b-i}{a-2i}Sq^{2a+b+1-i}Sq^{i}=Sq^{2a+b+1}+\Delta.$

Here $\Delta\in T_{2a+b}$ and so is the left hand side, so
$Sq^{m+1}=Sq^{2a+b+1}\in T_{2a+b}=T_{m}$. ∎

We now consider the obvious homomorphism from the span of admissible monomials
described in Theorem 1.1 into $\mathcal{A}/\\!\\!/\mathcal{A}_{1}$. In Section
5, we do the deduction for $\mathcal{A}/\\!\\!/E_{1}$.

##  3\. Injectivity

We wish to show that the $Sq^{I}$ of Theorem 1.1 are linearly independent.
This follows directly from Lemma 2.1 once the background is set up. For that
we need the polynomial algebra from [Thomas62]

$H^{*}(BSpin)=P[w_{i}]\qquad i\geq 4\qquad i\neq 2^{n}+1.$

We give a quick proof of this because it involves techniques we need anyway.

Mod decomposables, we have the following easily verified formulas:

$Sq^{2^{k}}(w_{2^{k}+1})\equiv w_{2^{k+1}+1}\qquad
Sq^{(2^{n},2^{n-1},\ldots,4,2,1)}(w_{2})\equiv w_{2^{n+1}+1}.$

The second follows immediately from the first. Because $w_{2}=0\in
H^{*}(BSpin)$ by definition, any Steenrod operations on it are zero as well.
The formula tells us that $w_{2^{n+1}+1}$ is decomposable in $H^{*}(BSpin)$.
Most are non-trivial, but we do have $w_{3}=w_{5}=w_{9}=0$.

We know $H^{*}(BSO)=P[w_{i}]$ with $i>1$. We have a fibration
$BSpin\rightarrow BSO\rightarrow K_{2}=K({\mathbf{Z}}_{2},2)$, where the last
map is given by $w_{2}$. Note that $H^{*}(K_{2})$ is a polynomial algebra on
the $Sq^{(2^{n},2^{n-1},\ldots,4,2,1)}(\iota_{2})$. The above computation
shows the map $H^{*}(K_{2})\longrightarrow H^{*}(BSO)$ is an injection giving
us a short exact sequence of Hopf algebras $H^{*}(K_{2})\longrightarrow
H^{*}(BSO)\longrightarrow H^{*}(BSpin)$ from the Eilenberg-Moore (or Serre)
spectral sequence. This gives $H^{*}(BSpin)$ and the decomposability of the
$w_{2^{n}+1}$.

We are also going to look at the Thom spectrum, $MSpin$. Let $U$ be the Thom
class in $H^{0}(MSpin)$. The reason we are looking at the Thom spectrum is
because as a module over the Steenrod algebra, $H^{*}(MSpin)$ is a sum of
cyclic modules and the module generated by $U$ is precisely
$\mathcal{A}/\\!\\!/\mathcal{A}_{1}$, [ABP67].

###### Proof of injectivity for Theorem 1.1..

If there were a relation in $\mathcal{A}/\\!\\!/\mathcal{A}_{1}$ among the
admissible $Sq^{I}$ with no $i_{s}=2^{n}+1$ and $i_{k}\geq 4$, Lemma 2.1 would
imply a similar relations among the S-W classes in $H^{*}(MSpin)$. But because
we are not using the $w_{2^{n}+1}$, these are linearly independent. ∎

##  4\. Surjectivity

All that is left to do with our Theorem 1.1 is to show that any admissible
$Sq^{I}$ with some $i_{s}=2^{n}+1$ can be written in terms of admissible
$Sq^{J}$ with no $i_{s}=2^{t}+1$.

We specialize Proposition 1.3 to $Sq^{2^{n}+j}T_{2^{n}}\subset T_{2^{n+1}}$
when $2^{n}>j\geq 0$. Proposition 1.3 actually tells us $T_{2^{n+1}-1}$ but we
don’t need that little extra bit.

######  Lemma 4.1.

In $\mathcal{A}/\\!\\!/\mathcal{A}_{1}$, if $J=(2^{n+1}+1,i_{0},\ldots,i_{k})$
is admissible, then $Sq^{J}\in T_{2^{n+1}}$.

###### Proof of Theorem 1.1 from Lemma 4.1..

Take $Sq^{I}$ with $I$ admissible and with $I$ ending in the shortest $J$ as
in the Lemma. Since $Sq^{J}$ can be written in terms of $Sq^{K}$ admissible
with $k_{1}<2^{n+1}+1$, when this sum replaces $Sq^{J}$ in $Sq^{I}$, it is
still admissible. By induction, we do not have to worry about smaller $J$ like
this showing up. Since $I$ is finite, this process of replacement is also
finite. We have shown that every $Sq^{I}$, $I$ admissible, can be replaced
with one of the desired form, and we have shown that the $Sq^{I}$ of this form
are linearly independent. This concludes the proof of Theorem 1.1 from Lemma
4.1. ∎

###### Proof of Lemma 4.1..

When $J=(2^{n+1}+1)$, we can use the $\mathcal{A}/\\!\\!/\mathcal{A}_{1}$ case
of Proposition 1.5.

To start our induction on $k$, we need the $J=(2^{n+1}+1,i_{0})$ case. We
begin with $Sq^{2^{n}+1}=\Delta\in T_{2^{n}}$ from Proposition 1.5 and apply
$Sq^{2^{n}+i_{0}}$. For $J$ to be admissible, we have $i_{0}\leq 2^{n}$. From
Proposition 1.3, $Sq^{2^{n}+i_{0}}T_{2^{n}}\subset T_{2^{n+1}}$ so
$Sq^{2^{n}+i_{0}}Sq^{2^{n}+1}\in T_{2^{n+1}}$ since $Sq^{2^{n}+i_{0}}\Delta\in
T_{2^{n+1}}$. All we need now is:

$\displaystyle Sq^{2^{n}+i_{0}}Sq^{2^{n}+1}$ $\displaystyle=$
$\displaystyle\sum_{s\geq
i_{0}}\binom{2^{n}-s}{2^{n}+i_{0}-2s}Sq^{2^{n+1}+1+i_{0}-s}Sq^{s}$
$\displaystyle=$ $\displaystyle
Sq^{2^{n+1}+1}Sq^{i_{0}}+\sum_{s>i_{0}}\binom{2^{n}-s}{2^{n}+i_{0}-2s}Sq^{2^{n+1}+1+i_{0}-s}Sq^{s}.$

The terms in the sum are all also in $T_{2^{n+1}}$ so the same is true for
$Sq^{2^{n+1}+1}Sq^{i_{0}}$.

The following induction proves our Lemma 4.1 because the two terms in
$T_{2^{n+1}}$ force the third term to be there as well. The induction is
started above as it can be rephrased in the format of our induction below as
the $k=0$ case.

######  Induction 4.2.

In $\mathcal{A}/\\!\\!/\mathcal{A}_{1}$, with $(2^{n+1}+1,i_{0},\ldots,i_{k})$
admissible,

$Sq^{2^{n}+i_{0}}Sq^{2^{n-1}+i_{1}}\cdots Sq^{2^{n-k}+i_{k}}Sq^{2^{n-k}+1}\in
T_{2^{n+1}}$

and is equal in $\mathcal{A}/\\!\\!/\mathcal{A}_{1}$ to

$Sq^{2^{n+1}+1}Sq^{i_{0}}Sq^{i_{1}}\cdots Sq^{i_{k}}+\Delta_{n+1}\text{ with
}\Delta_{n+1}\in T_{2^{n+1}}.$

###### Proof of our Induction 4.2.

By induction on $k$, we can write

$Sq^{2^{n-1}+i_{1}}\cdots Sq^{2^{n-k}+i_{k}}Sq^{2^{n-k}+1}\in T_{2^{n}}$

and it is equal to

$Sq^{2^{n}+1}Sq^{i_{1}}\cdots Sq^{i_{k}}+\Delta_{n}\text{ with }\Delta_{n}\in
T_{2^{n}}.$

Now we take $Sq^{2^{n}+i_{0}}$ times everything. Since
$Sq^{2^{n}+i_{0}}T_{2^{n}}\subset T_{2^{n+1}}$, we have
$Sq^{2^{n}+i_{0}}\Delta_{n}=\Delta_{n+1}\in T_{2^{n+1}}$ and

$Sq^{2^{n}+i_{0}}Sq^{2^{n-1}+i_{1}}\cdots Sq^{2^{n-k}+i_{k}}Sq^{2^{n-k}+1}\in
T_{2^{n+1}}$

and is equal in $\mathcal{A}/\\!\\!/\mathcal{A}_{1}$ to

$Sq^{2^{n}+i_{0}}Sq^{2^{n}+1}Sq^{i_{1}}\cdots Sq^{i_{k}}+\Delta_{n+1}.$

All we need is the term of Lemma 4.1.

$\displaystyle Sq^{2^{n}+i_{0}}Sq^{2^{n}+1}Sq^{i_{1}}\cdots Sq^{i_{k}}$
$\displaystyle=$ $\displaystyle\Big{(}\sum_{s\geq
i_{0}}\binom{2^{n}-s}{2^{n}+i_{0}-2s}Sq^{2^{n+1}+1+i_{0}-s}Sq^{s}\Big{)}Sq^{i_{1}}\cdots
Sq^{i_{k}}$ $\displaystyle=$ $\displaystyle
Sq^{2^{n+1}+1}Sq^{i_{0}}Sq^{i_{1}}\cdots Sq^{i_{k}}$
$\displaystyle+\Big{(}\sum_{s>i_{0}}\binom{2^{n}-s}{2^{n}+i_{0}-2s}Sq^{2^{n+1}+1+i_{0}-s}Sq^{s}\Big{)}Sq^{i_{1}}\cdots
Sq^{i_{k}}.$

Since $s>i_{0}$, the elements in the sum are admissible and in $T_{2^{n+1}}$.
∎

Lemma 4.1 follows. ∎

##  5\. $H^{*}(bu)$

We give a quick derivation of $H^{*}(bu)$ from $H^{*}(bo)$. We have the
standard fibration

$bo\longrightarrow bu\longrightarrow\Sigma^{2}bo.$

This gives a short exact sequence of $\mathcal{A}$ modules. The map from
$H^{*}(\Sigma^{2}bo)$ takes $1$ to $Sq^{2}$ and is injective so must hit all
$Sq^{I}Sq^{2}$ with $Sq^{I}$ a basis for $H^{*}(bo)$. The surjection
$H^{*}(bu)\longrightarrow H^{*}(bo)$ must hit the $Sq^{I}$ for a basis for
$H^{*}(bo)$. This is the stated answer for $H^{*}(bu)$ in Theorem 1.1.

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