# Quantitative affine approximation for UMD targets

Tuomas Hytönen  University of Helsinki, Helsinki, Finland
<EMAIL_ADDRESS>,  Sean Li  University of Chicago, Chicago IL, USA
<EMAIL_ADDRESS>and  Assaf Naor  Princeton University, Princeton NJ,
USA<EMAIL_ADDRESS>
###### Abstract.

It is shown here that if $(Y,\|\cdot\|_{Y})$ is a Banach space in which
martingale differences are unconditional (a UMD Banach space) then there
exists $c=c(Y)\in(0,\infty)$ with the following property. For every
$n\in\mathbb{N}$ and $\varepsilon\in(0,1/2]$, if $(X,\|\cdot\|_{X})$ is an
$n$-dimensional normed space with unit ball $B_{X}$ and $f:B_{X}\to Y$ is a
$1$-Lipschitz function then there exists an affine mapping $\Lambda:X\to Y$
and a sub-ball $B^{*}=y+\rho B_{X}\subseteq B_{X}$ of radius
$\rho\geqslant\exp(-(1/\varepsilon)^{cn})$ such that
$\|f(x)-\Lambda(x)\|_{Y}\leqslant\varepsilon\rho$ for all $x\in B^{*}$. This
estimate on the macroscopic scale of affine approximability of vector-valued
Lipschitz functions is an asymptotic improvement (as $n\to\infty$) over the
best previously known bound even when $X$ is $\mathbb{R}^{n}$ equipped with
the Euclidean norm and $Y$ is a Hilbert space.

T. H. was supported by ERC grant ANPROB and by the Academy of Finland through
projects 130166, 133264 and CoE in Analysis and Dynamics. S. L. was supported
by NSF grants CCF-0832795 and DMS-1303910. A. N. was supported by NSF grant
CCF-0832795, BSF grant 2010021, the Packard Foundation and the Simons
Foundation.

##  1\. Introduction

In what follows, the unit ball of a normed space $(X,\|\cdot\|_{X})$ is
denoted $B_{X}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x\in X:\
\|x\|_{X}<1\\}$. For $p\in[1,\infty]$ and $n\in\mathbb{N}$, the space
$\mathbb{R}^{n}$ equipped with the $\ell_{p}$ norm is denoted as usual by
$\ell_{p}^{n}$. Given two metric spaces $(U,d_{U})$ and $(V,d_{V})$, the
Lipschitz constant of a mapping $f:U\to V$ is denoted $\|f\|_{\mathrm{Lip}}$.
Throughout this article, given $a,b\in(0,\infty)$, the notations $a\lesssim b$
and $b\gtrsim a$ mean that $a\leqslant cb$ for some universal constant
$c\in(0,\infty)$. The notation $a\asymp b$ stands for $(a\lesssim
b)\wedge(b\lesssim a)$.

For $n\in\mathbb{N}$ and $\varepsilon\in(0,1)$ let
$r_{n}(\varepsilon)\geqslant 0$ be the supremum over those $r\in[0,1)$ such
that for every Lipschitz function $f:B_{\ell_{2}^{n}}\to\ell_{2}$ there exists
a linear mapping $T:\ell_{2}^{n}\to\ell_{2}$, a vector $a\in\ell_{2}$, and a
sub-ball $B^{*}=x_{0}+\rho B_{\ell_{2}^{n}}\subseteq B_{\ell_{2}^{n}}$ of
radius $\rho\geqslant r$, such that

$\forall\ x\in
B^{*},\qquad\frac{\|f(x)-(a+Tx)\|_{\ell_{2}}}{\rho}\leqslant\varepsilon\|f\|_{\mathrm{Lip}}.$
(1)

Thus, all the Hilbert space-valued $1$-Lipschitz functions on the Euclidean
unit ball of $\mathbb{R}^{n}$ are guaranteed to be $\varepsilon$-close to some
affine function on some sub-ball of radius at least $r_{n}(\varepsilon)$,
where $\varepsilon$-closeness is measured relative to the scale of the sub-
ball. A lower bound on $r_{n}(\varepsilon)$ corresponds to a differentiation-
type theorem asserting that any such function is macroscopically close to
being affine rather than being infinitesimally affine. Crucially, the
macroscopic lower bound on the scale of affine approximability is independent
of the given function.

The basic question in which we are interested is that of determining the
asymptotic behavior of $r_{n}(\varepsilon)$ as $n\to\infty$. Qualitatively, we
ask for an asymptotic understanding of those Hilbert space-valued Lipschitz
functions on $B_{\ell_{2}^{n}}$ that are hardest to approximate by affine
functions.

There is a big gap between the known upper and lower bounds on
$r_{n}(\varepsilon)$. We have $r_{n}(\varepsilon)\leqslant
e^{-cn/\varepsilon^{2}}$, where $c\in(0,\infty)$ is a universal constant; see
Section 2.2 below. The only known lower bound [57] on $r_{n}(\varepsilon)$ is
$r_{n}(\varepsilon)\geqslant e^{-(n/\varepsilon)^{Cn}}$, where
$C\in(0,\infty)$ is a universal constant. For concreteness, by choosing, say,
$\varepsilon=\frac{1}{4}$ and denoting $r_{n}=r_{n}(\frac{1}{4})$, the best
known bounds on $r_{n}$ become

$e^{-n^{Kn}}\leqslant r_{n}\leqslant e^{-\kappa n},$ (2)

for some universal constants $\kappa,K\in(0,\infty)$. An illustrative special
case of the main result that is obtained here (to be described in full below)
is the following asymptotic improvement over the leftmost inequality (2),
which holds for every $n\in\mathbb{N}$ and for some universal constant
$K\in(0,\infty)$.

$r_{n}\geqslant e^{-e^{Kn}}.$ (3)

###  1.1. The modulus of $L_{p}$ affine approximabilty

Despite the fact that the above question was phrased in the context of Hilbert
spaces, a setting which arguably best highlights its fundamental nature, it is
important to study it in the context of mappings between more general normed
spaces; it is in this setting, for example, that it becomes relevant to
Bourgain’s discretization problem [13, 35], as explained in [57, Section 1.1]
(see Remark 3 below).

######  Definition 1.

Fix $n\in\mathbb{N}$ and let $(X=\mathbb{R}^{n},\|\cdot\|_{X})$ be an
$n$-dimensional Banach space. Also, let $(Y,\|\cdot\|_{Y})$ be an inifinite
dimensional Banach space. For $p\in(0,\infty]$ and $\varepsilon\in(0,1)$
define $r_{p}^{X\to Y}(\varepsilon)$ to be the supremum over those $r\geqslant
0$ with the following property. For every Lipschitz function $f:B_{X}\to Y$
there exists $y\in X$ and $\rho\in[r,\infty)$ such that $y+\rho B_{X}\subseteq
B_{X}$, and there exists $a\in Y$ and a linear operator $T:X\to Y$ whose
operator norm satisfies $\|T\|_{X\to Y}\leqslant 3\|f\|_{\mathrm{Lip}}$, such
that

$\left(\frac{1}{\mathrm{vol}(x+\rho B_{X})}\int_{y+\rho
B_{X}}\left\|f(z)-(a+Tz)\right\|_{Y}^{p}\mathrm{d}z\right)^{\frac{1}{p}}\leqslant\varepsilon\rho\|f\|_{\mathrm{Lip}}.$
(4)

We call $r_{p}^{X\to Y}(\cdot)$ the modulus of $L_{p}$ affine approximability
corresponding to $X$ and $Y$.

Using the notation of Definition 1, the quantity $r_{n}(\varepsilon)$ that we
defined above can be written as

$r_{n}(\varepsilon)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}r_{\infty}^{\ell_{2}^{n}\to\ell_{2}}(\varepsilon).$

Indeed, the $L_{\infty}$ requirement (1) implies that
$\|T\|_{\ell_{2}^{n}\to\ell_{2}}\leqslant(1+2\varepsilon)\|f\|_{\mathrm{Lip}}\leqslant
3\|f\|_{\mathrm{Lip}}$. For finite $p$, the $L_{p}$ bound (4) does not
automatically imply a bound on $\|T\|_{X\to Y}$, which is the reason why we
added the requirement $\|T\|_{X\to Y}\leqslant 3\|f\|_{\mathrm{Lip}}$ as part
of the definition of $r_{p}^{X\to Y}(\varepsilon)$.

The case $p=\infty$ of Definition 1, for which we shall use below the simpler
notation

$r^{X\to
Y}(\varepsilon)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}r_{\infty}^{X\to
Y}(\varepsilon),$

was introduced by Bates, Johnson, Lindenstrauss, Preiss and Schechtman [6],
who proved that $r^{X\to Y}(\varepsilon)>0$ for all $\varepsilon\in(0,1)$ if
and only if $Y$ admits an equivalent uniformly convex norm; see [6] for
beautiful geometric applications of this result. The best known lower bound on
$r^{X\to Y}(\varepsilon)$ (in terms of $n$, $\varepsilon$ and the modulus of
uniform convexity of $Y$) was obtained in [57]. This bound is

$r^{X\to Y}(\varepsilon)\geqslant e^{-(n/\varepsilon)^{c(Y)n}},$ (5)

where $c(Y)\in(0,\infty)$ depends only the modulus of uniform convexity of the
target Banach space $(Y,\|\cdot\|_{Y})$. An explicit estimate on $c(Y)$
appears in [57, Thm. 1.1]. Here we obtain an estimate that is asymptotically
better than (5) as $n\to\infty$ provided that $Y$-valued martingale
differences are unconditional ($Y$ is a UMD Banach space). Note that all the
classical reflexive Banach spaces have this property, but one can construct
[67] uniformly convex Banach spaces that are not UMD.

Formally, a Banach space $(Y,\|\cdot\|_{Y})$ is said to be a UMD Banach space
if there exists $\beta\in(1,\infty)$ such that if $\\{M_{j}\\}_{j=0}^{\infty}$
is a $Y$-valued square-integrable martingale defined on some probability space
$(\Omega,\mathbb{P})$ then for every $n\in\mathbb{N}$ and every choice of
signs $\varepsilon_{1},\ldots,\varepsilon_{n}\in\\{-1,1\\}$ we have

$\int_{\Omega}\Big{\|}M_{0}+\sum_{j=1}^{n}\varepsilon_{n}\left(M_{j}-M_{j-1}\right)\Big{\|}_{Y}^{2}\mathrm{d}\mathbb{P}\leqslant\beta^{2}\int_{\Omega}\left\|M_{n}\right\|_{Y}^{2}\mathrm{d}\mathbb{P}.$
(6)

If $Y$ is a UMD Banach space then the infimum over those $\beta\in(1,\infty)$
for which (6) holds true for every square-integrable $Y$-valued martingale
$\\{M_{j}\\}_{j=0}^{\infty}$ is denoted below by $\beta(Y)$. Examples of UMD
Banach spaces include all $L_{p}(\mu)$ spaces when $p\in(1,\infty)$, in which
case $\beta(L_{p}(\mu))\asymp p^{2}/(p-1)$. See [14] and the references
therein for more information on UMD spaces.

Theorem 2 below asserts an improved lower bound on the modulus of affine
approximability $r^{X\to Y}(\varepsilon)$, provided that $Y$ is a UMD space.

######  Theorem 2.

There is a universal constant $c\in[1,\infty)$ such that for every
$n\in\mathbb{N}$ and $\beta\in[2,\infty)$, if $(X,\|\cdot\|_{X})$ is an
$n$-dimensional normed space and $(Y,\|\cdot\|_{Y})$ is a UMD Banach space
satisfying $\beta(Y)\leqslant\beta$ then for every $\varepsilon\in(0,1/2)$ we
have

$r^{X\to Y}(\varepsilon)\geqslant\exp\left(-\frac{(\beta
n)^{c\beta}}{\varepsilon^{c(n+\beta)}}\right).$

######  Remark 3.

By substituting Theorem 2 into equation (12) of [57] one obtains a bound on
Bourgain’s discretization modulus in the special case of UMD targets that
improves over the bound that was deduced in [57, Section 1.1] and matches
Bourgain’s original bound [13]. Specifically, one obtains the refined estimate
that appears in equation (2) of [35]. This yields a new proof of the best
known general bound in Bourgain’s discretization problem via an approach that
is entirely different from Bourgain’s method, albeit in the special (though
still very general) case of UMD targets. We note that due to the recent
progress in [35] a stronger bound is available here when the target is
$L_{p}$.

The main reason why we study here the modulus of $L_{p}$ affine
approximability $r_{p}^{X\to Y}(\varepsilon)$ is that it relates to $r^{X\to
Y}(\varepsilon)$ through Lemma 4 below, whose simple proof appears in Section
2.1. The advantage of working with finite $p$ is that it allows us to use a
variety of analytic tools, such as vector-valued Littlewood–Paley theory and
complex interpolation.

######  Lemma 4.

Fix $n\in\mathbb{N}$ and $p\in[1,\infty)$. Suppose that $(X,\|\cdot\|_{X})$
and $(Y,\|\cdot\|_{Y})$ are Banach spaces with $\dim(X)=n$. Then for every
$\varepsilon\in(0,1)$ we have

$r^{X\to Y}(\varepsilon)\geqslant r_{p}^{X\to
Y}\left(\left(\frac{\varepsilon}{9}\right)^{1+\frac{n}{p}}\right).$

Due to Lemma 4, Theorem 2 is a consequence of Theorem 5 below, which is our
main result. Its proof is based on a vector-valued variant of an argument of
Dorronsoro [26], combined with a wide variety of additional analytic and
geometric ingredients of independent interest.

######  Theorem 5.

There exist universal constants $\kappa,C\in[1,\infty)$ such that for every
$\beta\in[2,\infty)$ and $n\in\mathbb{N}$, if $(X,\|\cdot\|_{X})$ is an
$n$-dimensional normed space and $(Y,\|\cdot\|_{Y})$ is a UMD Banach space
satisfying $\beta(Y)\leqslant\beta$, then for every $\varepsilon\in(0,1)$ we
have

$r_{\kappa\beta}^{X\to Y}(\varepsilon)\geqslant\exp\left(-\frac{(\beta
n)^{C\kappa\beta}}{\varepsilon^{\kappa\beta}}\right).$

###  1.2. Previous work

Over the past several decades, research on quantitative differentiation has
proceeded roughly along two lines of inquiry, one of which arising from
functional analysis and metric geometry and the other arising from
rectifiability questions in harmonic analysis. The present work belongs to the
former direction, but its main contribution is the use of methods from the
latter direction in this new context while incorporating various Banach space
theoretic tools.

Bates, Johnson, Lindenstrauss, Preiss and Schechtman studied [6] quantitative
differentiation in order to prove the rigidity of certain classes of Banach
spaces under nonlinear quotients. The same notion was used in [57] for metric
embeddings, namely as an alternative approach to Bourgain’s discretization
problem [13] when the target Banach space is uniformly convex. The methods in
this context fall under the category of (extensions of) “approximate midpoint
arguments,” as initiated by Enflo [7] to prove that $L_{1}$ is not uniformly
homeomorphic to $\ell_{1}$, and further developed in [13, 27, 46, 65] (see
also Chapter 10 of [8]). As examples of the many related applications to
quantitative embedding theory and rigidity questions, see also [60, 54, 20,
27, 19, 64, 55, 56].

Parallel developments of a different nature arose in harmonic analysis, as
part of the quest to develop a quantitative theory of rectifiability, with
applications to singular integrals. Notable contributions along these lines
include classical works of Stein (see the monograph [72]), through the works
of Dorronsoro [26], Jones [48, 49], David and Semmes [22, 23, 24], as well as
the more recent work of Azzam and Schul [2]. The work of Dorronsoro [26]
directly influenced the present article, and we were also greatly inspired by
the works of David and Semmes [22, 23, 24]. These works introduced and studied
quantities that correspond to Definition 1 when $X$ is the Euclidean space
$\mathbb{R}^{n}$ and $Y$ is the real line. Such methods also yield results for
mappings from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$, but with statements that
include implicit parameters that are allowed to depend on $m,n$. In [24] David
and Semmes compare their work to that of Bates, Johnson, Lindenstrauss, Preiss
and Schechtman [6], noting that the latter methods are different, and even
yield results for infinite dimensional spaces.

Our contribution here follows the harmonic-analytic methodology, while
overcoming several difficulties. Firstly, the literature on quantitative
rectifiability ignores the dependence on the dimension $n$, while this
dependence is the main topic of interest in the present context. In fact, a
direct examination of the dependence on $n$ that is implicit in the above
cited works reveals that it is insufficient for the purpose of obtaining
improved bounds on $r^{X\to Y}(\varepsilon)$. Secondly, in our setting the
domain $X$ is a general $n$-dimensional normed space $X$ rather than a
Euclidean space, and our arguments address this point. A final important
difference is that we treat infinite dimensional Banach spaces $Y$ as targets.
Overcoming this requires substantial effort, because the infinite dimensional
arguments of Bates, Johnson, Lindenstrauss, Preiss and Schechtman [6] do not
seem to be applicable in our setting. The present work yields an infinite
dimensional version of an inequality that Dorronsoro obtained [26] for real-
valued functions. Such an infinite dimensional extension of Dorronsoro’s work
is not routine, and in particular it does not hold true for arbitrary infinite
dimensional Banach space targets $Y$. In fact, the geometry of $Y$ influences
the structure of the inequality thus obtained, while stronger inequalities
hold true for real-valued functions. The assumption that $Y$ is UMD is used
several times in our argument through a rich UMD-valued Fourier-analytic
toolkit that has been developed by many authors over the past four decades.

###  1.3. Open questions

We list below some open questions that arise naturally from our work.

######  Question 6 (Asymptotics of $r^{X\to Y}(\varepsilon)$).

Obviously, the most tantalizing open question in the present context is to
determine the rate at which $r_{n}$ tends to $0$ as $n\to\infty$, even
roughly: say, is this rate exponential, doubly exponential, or of some
intermediate behavior? More importantly for potential applications, it remains
open to obtain sharp bounds on the quantity $r^{X\to Y}(\varepsilon)$ when $X$
is a finite dimensional Banach space, $Y$ is a uniformly convex Banach space
or belongs to some important class of Banach spaces (e.g. UMD spaces or
uniformly convex lattices), and $\varepsilon\in(0,1)$.

######  Question 7 (Infinite dimensional domains).

The question of characterizing those pairs of Banach space $X,Y$ for which
$r^{X\to Y}(\varepsilon)>0$ for every $\varepsilon\in(0,1)$ was solved in [6]
also when $\dim(X)=\infty$ and $\dim(Y)<\infty$: this happens if and only if
$X$ admits an equivalent uniformly smooth norm. It remains open to understand
the asymptotic behavior of $r^{X\to Y}(\varepsilon)$ in the setting of
uniformly smooth infinite dimensional domains and finite dimensional ranges.
In particular, the rate at which $r^{\ell_{2}\to\ell_{2}^{n}}(\frac{1}{4})$
tends to $0$ as $n\to\infty$ is unknown. An explicit lower bound on $r^{X\to
Y}(\varepsilon)$ in terms of the modulus of uniform smoothness of $X$ and
$\dim(Y)$ can be deduced from an examination of the proof in [6], but we
believe that this bound is far from being optimal.

######  Question 8 (Uniformly convex targets).

As stated earlier, by [6] we know that if $\dim(X)=n<\infty$ and
$\dim(Y)=\infty$ then $r^{X\to Y}(\varepsilon)>0$ for every
$\varepsilon\in(0,1)$ if and only if $Y$ admits and equivalent uniformly
convex norm. However, the best known lower bounds on $r^{X\to Y}(\varepsilon)$
in this (maximal) generality remain those of [57], and these bounds are weaker
than Theorem 2 (as $n\to\infty$), which does not cover all uniformly convex
ranges $Y$ because Pisier [67] proved that there exist uniformly convex Banach
spaces that are not UMD (even such uniformly convex Banach lattices exist, as
shown by Bourgain in [11]; see also the recent example by Qiu [69]). It would
be interesting to obtain an improved bound as in Theorem 2 under the weaker
assumption that $Y$ is uniformly convex. Our proof of Theorem 2 definitely
uses properties of $Y$ that imply the UMD property (e.g., we rely on the
boundedness of the $Y$-valued Hilbert transform, which was shown by Bourgain
[11] to imply that $Y$ is UMD). However, the conclusion of Theorem 2, or even
Theorem 5, may be valid when $Y$ is uniformly convex, and the same holds true
for some of our other results, such as Theorem 19 below. Certain aspects of
vector-valued Littelwood–Paley theory are known to hold true for uniformly
convex targets (see e.g. [59]), so it would be interesting to investigate the
extent to which the UMD property is needed for our results. If, on the other
hand, Theorem 5 or Theorem 19 imply the UMD property then this would be a new
characterization of UMD spaces.

######  Question 9 (Asymptotics of $r^{X\to Y}_{p}(\varepsilon)$ for finite
$p$).

It would be interesting to understand the asymptotic behavior of the modulus
of $L_{p}$ affine approximability $r_{p}^{X\to Y}(\varepsilon)$, even in the
special case when $X$ and $Y$ are both Hilbert spaces, $p=2$ and, say,
$\varepsilon=1/2$. A careful examination of the proof of Theorem 5 in this
Hilbertian setting (in which case some of the steps that we perform below are
not needed, and several estimates can be easily improved) reveals that for
some $c\in(0,\infty)$,

$\forall\,n\in\mathbb{N},\ \forall\,\varepsilon\in(0,1),\qquad
r_{2}^{\ell_{2}^{n}\to\ell_{2}}(\varepsilon)\gtrsim e^{-\frac{c(n\log
n)^{2}}{\varepsilon^{2}}}.$ (7)

We do not know the extent to which (7) is best possible; it seems plausible
that with more work one could improve the dependence on $n$ in the exponent,
but we do not presently have an upper bound that comes close to the lower
bound in (7). Note that while the moduli $r_{p}^{X\to Y}(\varepsilon)$ are
interesting in their own right, we do not have geometric applications of them
as in the case $p=\infty$. So, as a more amorphous research direction, it
would be interesting to find geometric applications of bounds on $r_{p}^{X\to
Y}(\varepsilon)$ (other than as a tool to bound $r_{\infty}^{X\to
Y}(\varepsilon)$, which is the application that we present here).

##  2\. Geometric preliminaries

Fix from now on an integer $n$ and an $n$-dimensional normed space
$(X,\|\cdot\|_{X})$. We shall also fix a normed space $(Y,\|\cdot\|_{Y})$. In
later sections we will need $Y$ to be a UMD Banach space, but the statements
of the present section hold true when $Y$ is a general normed space.

By John’s theorem [45] there exists a scalar product
$\langle\cdot,\cdot\rangle$ on $X$ with respect to which we can identify $X$
with $\mathbb{R}^{n}$ and we have

$\forall\,x\in\mathbb{R}^{n},\qquad\|x\|_{2}\leqslant\|x\|_{X}\leqslant\sqrt{n}\cdot\|x\|_{2},$
(8)

where $\|\cdot\|_{2}=\|\cdot\|_{\ell_{2}^{n}}$. We shall also use the standard
notation $B^{n}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}B_{\ell_{2}^{n}}$ and
$S^{n-1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\partial B^{n}$. This
Euclidean structure will be fixed from now on. Despite the fact that $X$ is
now endowed with two metrics (those induced by $\|\cdot\|_{X}$ and
$\|\cdot\|_{2}$), we shall tacitly maintain throughout the ensuing discussion
the convention that whenever $\Omega\subseteq\mathbb{R}^{n}$ and $f:\Omega\to
Y$ then $\|f\|_{\mathrm{Lip}(\Omega)}$ denotes the Lipschitz constant of $f$
with respect to the metric induced by $\|\cdot\|_{X}$, i.e.,

$\|f\|_{\mathrm{Lip}(\Omega)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sup_{\begin{subarray}{c}x,y\in\Omega\\\
x\neq y\end{subarray}}\frac{\|f(x)-f(y)\|_{Y}}{\|x-y\|_{X}}.$

When $\Omega=\mathbb{R}^{n}$ we shall also use the shorter notation
$\|f\|_{\mathrm{Lip}}=\|f\|_{\mathrm{Lip}(\mathbb{R}^{n})}$.

We shall use standard notation for vector-valued $L_{p}$ spaces. Specifically,
for every measurable subset $\Omega\subseteq\mathbb{R}^{n}$ of positive
Lebesgue measure and $p\in[1,\infty]$, we let $L_{p}(\Omega,Y)$ denote the
space of all measurable functions $f:\Omega\to Y$ such that

$\|f\|_{L_{p}(\Omega,Y)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left(\int_{\Omega}\|f(x)\|_{Y}^{p}\mathrm{d}x\right)^{\frac{1}{p}}<\infty.$

Given $f:\Omega\to Y$ and $x\in\mathbb{R}^{n}$ we denote by $f^{x}:\Omega-x\to
Y$ the translate of $f$ by $x$, i.e.,

$\forall\,y\in\Omega-x,\qquad
f^{x}(y)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f(x+y).$ (9)

For $u\in(0,\infty)$ and $f\in L_{1}(uB^{n},Y)$ let $T_{u}f:\mathbb{R}^{n}\to
Y$ be the linear operator defined by

$\forall\,w\in\mathbb{R}^{n},\qquad
T_{u}f(w)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{n+2}{{V}_{n}u}\int_{B^{n}}\langle
z,w\rangle f(uz)\mathrm{d}z,$ (10)

where

$V_{n}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathrm{vol}(B^{n})=\frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2}+1\right)}.$

The operator norm of $T_{u}f$ can be bounded in terms of the Lipschitz
constant of $f$ as follows.

######  Lemma 10.

Fix $u\in(0,\infty)$ and a Lipschitz map $f:uB^{n}\to Y$. Then

$\|T_{u}f\|_{X\to Y}\leqslant\|f\|_{\mathrm{Lip}(uB^{n})}.$

###### Proof.

By rescaling we may assume that $u=1$. Take
$w\in\mathbb{R}^{n}\smallsetminus\\{0\\}$. For every $t\in\mathbb{R}$ consider
the following affine hyperplane.

$H_{t}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\\{y\in\mathbb{R}^{n}:\
\left\langle
y,\frac{w}{\|w\|_{2}}\right\rangle=t\right\\}\subseteq\mathbb{R}^{n}.$

Then by the definition (10) and Fubini’s theorem we have

$\displaystyle T_{1}f(w)$
$\displaystyle=\frac{(n+2)\|w\|_{2}}{V_{n}}\int_{-\infty}^{\infty}t\int_{H_{t}\cap
B^{n}}f(z)\mathrm{d}z\mathrm{d}t$
$\displaystyle=\frac{(n+2)\|w\|_{2}}{V_{n}}\int_{0}^{\infty}t\int_{H_{t}\cap
B^{n}}\left(f(z)-f\left(z-\frac{2tw}{\|w\|_{2}}\right)\right)\mathrm{d}z\mathrm{d}t.$

Consequently,

$\displaystyle\|T_{1}f(w)\|_{Y}$
$\displaystyle\leqslant\frac{(n+2)\|w\|_{2}}{V_{n}}\int_{0}^{\infty}t\int_{H_{t}\cap
B^{n}}\left\|f(z)-f\left(z-\frac{2tw}{\|w\|_{2}}\right)\right\|_{Y}\mathrm{d}z\mathrm{d}t$
$\displaystyle\leqslant\frac{2(n+2)\|f\|_{\mathrm{Lip}(B^{n})}\|w\|_{X}}{V_{n}}\int_{0}^{\infty}t^{2}\mathrm{vol}_{n-1}\left(H_{t}\cap
B^{n}\right)\mathrm{d}t$
$\displaystyle=\frac{(n+2)\|f\|_{\mathrm{Lip}(B^{n})}\|w\|_{X}}{V_{n}}\int_{B^{n}}x_{1}^{2}\mathrm{d}x$
$\displaystyle=\|f\|_{\mathrm{Lip}(B^{n})}\|w\|_{X}.\qed$

We also record for future use the following simple estimate.

######  Lemma 11.

For every $u\in(0,\infty)$ and $p\in[1,\infty]$ we have the following operator
norm bound.

$\|T_{u}\|_{L_{p}(uB^{n},Y)\to
L_{p}(uB^{n},Y)}\lesssim\min\left\\{\sqrt{pn},n\right\\}.$

###### Proof.

By rescaling we may assume that $u=1$. If $f\in L_{p}(B^{n},Y)$ then

$\displaystyle\|T_{1}f\|_{L_{p}(B^{n},Y)}$
$\displaystyle\leqslant\frac{n+2}{V_{n}}\int_{B^{n}}\left\|w\mapsto\langle
z,w\rangle f(z)\right\|_{L_{p}(B^{n},Y)}\mathrm{d}z$ (11)
$\displaystyle=\frac{n+2}{V_{n}}\bigg{(}\int_{B^{n}}|w_{1}|^{p}dw\bigg{)}^{\frac{1}{p}}\int_{B^{n}}\|z\|_{2}\cdot\|f(z)\|_{Y}\mathrm{d}z$
(12)
$\displaystyle\lesssim\min\left\\{\sqrt{pn},n\right\\}\|f\|_{L_{p}(B^{n},Y)},$
(13)

where in (11) we used the definition (10) and the triangle inequality in
$L_{p}(B^{n},Y)$, in (12) we used rotation invariance, and (13) follows from
Hölder’s inequality combined with the following fact, which can be verified by
a direct computation and is also, say, a special case of inequality (4) in
[5].

$\bigg{(}\frac{1}{V_{n}}\int_{B^{n}}|w_{1}|^{p}\mathrm{d}w\bigg{)}^{\frac{1}{p}}\asymp\min\left\\{\sqrt{\frac{p}{n}},1\right\\}.\qed$

For $u\in(0,\infty)$ and $f\in L_{1}(uB^{n},Y)$ define

$P^{0}_{u}f\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{V_{n}}\int_{B^{n}}f(uz)\mathrm{d}z\in
Y.$ (14)

Thus, for every $\Omega\subseteq\mathbb{R}^{n}$ and $f\in L_{1}(\Omega,Y)$, if
$x\in\mathbb{R}^{n}$ and $u\in(0,\infty)$ satisfy $x+uB^{n}\subseteq\Omega$
then the vector $P^{0}_{u}f^{x}\in Y$ is the mean of $f$ over $x+uB^{n}$. The
following simple estimate will be used later.

######  Lemma 12.

Fix $p\in[1,\infty)$, $q\in(0,\infty)$ and $x\in\mathbb{R}^{n}$. Every
measurable $f:\mathbb{R}^{n}\to Y$ satisfies

$\int_{0}^{\infty}\int_{B^{n}}\frac{\left\|f^{x}(uy)-P^{0}_{u}f^{x}\right\|_{Y}^{p}}{u^{q+1}}\mathrm{d}y\mathrm{d}u\leqslant\frac{2^{p}}{n+q}\int_{\mathbb{R}^{n}}\frac{\|f^{x}(y)-f(x)\|_{Y}^{p}}{\|y\|_{2}^{n+q}}\mathrm{d}y.$

###### Proof.

Recalling (14), it follows from the triangle inequality in $L_{p}(B^{n},Y)$
and convexity that

$\forall\,v\in
Y,\qquad\int_{B^{n}}\left\|f^{x}(uy)-P^{0}_{u}f^{x}\right\|_{Y}^{p}\mathrm{d}y\leqslant
2^{p}\int_{B^{n}}\left\|f^{x}(uy)-v\right\|_{Y}^{p}\mathrm{d}y.$

By choosing here $v=f(x)$, we see that for every $u\in(0,\infty)$ we have

$\int_{B^{n}}\left\|f^{x}(uy)-P^{0}_{u}f^{x}\right\|_{Y}^{p}\mathrm{d}y\leqslant
2^{p}\int_{B^{n}}\left\|f^{x}(uy)-f(x)\right\|_{Y}^{p}\mathrm{d}y.$

Hence, denoting the surface measure on $S^{n-1}$ by $\sigma$, by integrating
in polar coordinates we get

$\displaystyle\int_{0}^{\infty}\int_{B^{n}}\frac{\left\|f^{x}(uy)-P^{0}_{u}f^{x}\right\|_{Y}^{p}}{u^{q+1}}\mathrm{d}y\mathrm{d}u$
$\displaystyle\leqslant
2^{p}\int_{0}^{\infty}\int_{B^{n}}\frac{\left\|f^{x}(uy)-f(x)\right\|_{Y}^{p}}{u^{q+1}}\mathrm{d}y\mathrm{d}u$
$\displaystyle=2^{p}\int_{0}^{\infty}\int_{0}^{1}\int_{S^{n-1}}r^{n-1}\frac{\left\|f^{x}(urw)-f(x)\right\|_{Y}^{p}}{u^{q+1}}\mathrm{d}\sigma(w)\mathrm{d}r\mathrm{d}u$
$\displaystyle=2^{p}\int_{0}^{1}\int_{0}^{\infty}\int_{S^{n-1}}r^{n-1}\frac{\left\|f^{x}(sw)-f(x)\right\|_{Y}^{p}}{(s/r)^{q+1}}\mathrm{d}\sigma(w)\frac{\mathrm{d}s}{r}\mathrm{d}r$
$\displaystyle=\frac{2^{p}}{n+q}\int_{\mathbb{R}^{n}}\frac{\|f^{x}(y)-f(x)\|_{Y}^{p}}{\|y\|_{2}^{n+q}}\mathrm{d}y.$
∎

Define an affine mapping $P^{1}_{u}f:\mathbb{R}^{n}\to Y$ by

$P^{1}_{u}f\stackrel{{\scriptstyle\mathrm{def}}}{{=}}P^{0}_{u}f+T_{u}f.$ (15)

By a simple change of variable, for every $y\in\mathbb{R}^{n}$ we have

$\left(P^{1}_{u}f^{x}\right)^{-x}(y)=P^{0}_{u}f^{x}+\sum_{j=1}^{n}\frac{y_{j}-x_{j}}{\int_{x+uB}(w_{j}-x_{j})^{2}\mathrm{d}w}\int_{x+uB}(z_{j}-x_{j})f(z)\mathrm{d}z.$
(16)

Consequently, if $f$ were a real-valued function then $(P^{1}_{u}f^{x})^{-x}$
would be the orthogonal projection in $L_{2}(x+uB^{n})$ of $f$ onto the
subspace consisting of all the affine mappings. Lemma 13 below shows that for
every $p\in[1,\infty]$, if $f\in L_{p}(x+uB^{n},Y)$ then the distance between
$f$ and $(P^{1}_{u}f^{x})^{-x}$ in $L_{p}(x+uB^{n},Y)$ is controlled by the
distance of $f$ to the subspace of $L_{p}(x+uB^{n},Y)$ consisting of all the
affine mappings. Such a statement was previously proved in [26], but since the
dependence on $n$ and $p$ is important in the present context, and is only
implicit in [26], we include its proof.

######  Lemma 13.

Fix $p\in[1,\infty]$, $u\in(0,\infty)$ and $x\in\mathbb{R}^{n}$. Suppose that
$f\in L_{p}(x+uB^{n},Y)$ and that $\Lambda:\mathbb{R}^{n}\to Y$ is affine.
Then

$\left\|f^{x}-P^{1}_{u}f^{x}\right\|_{L_{p}(uB^{n},Y)}=\left\|f-\left(P^{1}_{u}f^{x}\right)^{-x}\right\|_{L_{p}(x+uB^{n},Y)}\lesssim\min\left\\{\sqrt{pn},n\right\\}\cdot\left\|f-\Lambda\right\|_{L_{p}(x+uB^{n},Y)}.$

###### Proof.

By translation and a rescaling we may assume that $x=0$ and $u=1$. Since
$P_{1}^{1}\Lambda=\Lambda$,

$\displaystyle\left\|f-P^{1}_{1}f\right\|_{L_{p}(B^{n},Y)}$
$\displaystyle\leqslant$
$\displaystyle\left\|f-\Lambda\right\|_{L_{p}(B^{n},Y)}+\left\|P^{1}_{1}(f-\Lambda)\right\|_{L_{p}(B^{n},Y)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:def P0}}}{{\leqslant}}$
$\displaystyle\left\|f-\Lambda\right\|_{L_{p}(B^{n},Y)}+\left\|P^{0}_{1}(f-\Lambda)\right\|_{L_{p}(B^{n},Y)}+\left\|T_{1}(f-\Lambda)\right\|_{L_{p}(B^{n},Y)}.$

It remains to note that
$\|P^{0}_{1}(f-\Lambda)\|_{L_{p}(B^{n},Y)}\leqslant\|f-\Lambda\|_{L_{p}(B^{n},Y)}$
since $P_{1}^{0}$ is an averaging operator, and to apply Lemma 11. ∎

We end this section by recording for ease of later reference two consequences
of Lemma 13.

######  Corollary 14.

Fix $p\in[1,\infty)$, $q\in(0,\infty)$ and $x\in\mathbb{R}^{n}$. Every
measurable $f:\mathbb{R}^{n}\to Y$ satisfies

$\left(\int_{0}^{\infty}\int_{B^{n}}\frac{\|f^{x}(uy)-P^{1}_{u}f^{x}(uy)\|_{Y}^{p}}{u^{q+1}}\mathrm{d}y\mathrm{d}u\right)^{\frac{1}{p}}\lesssim\frac{\min\left\\{\sqrt{pn},n\right\\}}{(q+n)^{\frac{1}{p}}}\bigg{(}\int_{\mathbb{R}^{n}}\frac{\|f^{x}(y)-f(x)\|_{Y}^{p}}{\|y\|_{2}^{n+q}}\mathrm{d}y\bigg{)}^{\frac{1}{p}}.$

###### Proof.

Fix $u\in(0,\infty)$. Lemma 13 implies that every affine mapping
$\Lambda:\mathbb{R}^{n}\to Y$ satisfies

$\left(\int_{B^{n}}\left\|f^{x}(uy)-P_{u}^{1}f^{x}(uy)\right\|_{Y}^{p}\mathrm{d}y\right)^{\frac{1}{p}}\lesssim\min\left\\{\sqrt{pn},n\right\\}\left(\int_{B^{n}}\left\|f^{x}(uy)-\Lambda(uy)\right\|_{Y}^{p}\mathrm{d}y\right)^{\frac{1}{p}}.$
(17)

An application of (17) when $\Lambda$ is the constant $P^{0}_{u}f^{x}$ shows
that for every $u\in(0,\infty)$ we have

$\left(\int_{B^{n}}\|f^{x}(uy)-P^{1}_{u}f^{x}(uy)\|_{Y}^{p}\mathrm{d}y\right)^{\frac{1}{p}}\lesssim\min\left\\{\sqrt{pn},n\right\\}\left(\int_{B^{n}}\left\|f^{x}(uy)-P^{0}_{u}f^{x}\right\|_{Y}^{p}\mathrm{d}y\right)^{\frac{1}{p}}.$

This implies the desired estimate due to Lemma 12.∎

######  Corollary 15.

Fix $p\in[1,\infty)$, $q\in(p,\infty)$ and $x\in\mathbb{R}^{n}$. Suppose that
$f:\mathbb{R}^{n}\to Y$ is smooth. Then

$\left(\int_{0}^{\infty}\int_{B^{n}}\frac{\|f^{x}(uy)-P^{1}_{u}f^{x}(uy)\|_{Y}^{p}}{u^{q+1}}\mathrm{d}y\mathrm{d}u\right)^{\frac{1}{p}}\\\
\lesssim\frac{p\min\left\\{\sqrt{pn},n\right\\}}{q(n+q-p)^{\frac{1}{p}}}\sum_{j=1}^{n}\bigg{(}\int_{\mathbb{R}^{n}}\left\|\frac{\partial
f}{\partial x_{j}}(x+y)-\frac{\partial f}{\partial
x_{j}}(x)\right\|_{Y}^{p}\frac{\mathrm{d}y}{\|y\|_{2}^{n+q-p}}\bigg{)}^{\frac{1}{p}}.$
(18)

###### Proof.

Suppose that $x\in\mathbb{R}^{n}$ and $u\in(0,\infty)$. For every
$j\in\\{1,\ldots,n\\}$ define

$a_{j}(u,x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\int_{0}^{1}P^{0}_{su}f_{j}^{x}\mathrm{d}s\in
Y,\qquad\mathrm{where}\qquad
f_{j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\partial f}{\partial
x_{j}}.$ (19)

Also, define an affine function $\Lambda_{x,u}:\mathbb{R}^{n}\to Y$ by setting

$\forall\,y\in\mathbb{R}^{n},\qquad\Lambda_{x,u}(y)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f(x)+\sum_{j=1}^{n}y_{j}a_{j}(u,x).$
(20)

An application of (17) with $\Lambda=\Lambda_{x,u}$ shows that

$\left(\int_{0}^{\infty}\int_{B^{n}}\frac{\|f^{x}(uy)-P^{1}_{u}f^{x}(uy)\|_{Y}^{p}}{u^{q+1}}\mathrm{d}y\mathrm{d}u\right)^{\frac{1}{p}}\\\
\lesssim\min\\{\sqrt{pn},n\\}\left(\int_{0}^{\infty}\int_{B^{n}}\frac{\left\|f^{x}(uy)-\Lambda_{x,u}(uy)\right\|_{Y}^{p}}{u^{q+1}}\mathrm{d}y\mathrm{d}u\right)^{\frac{1}{p}}.$
(21)

Observe that

$\displaystyle f^{x}(uy)-\Lambda_{x,u}(uy)\stackrel{{\scriptstyle\eqref{eq:def
Lambda
choice}}}{{=}}\int_{0}^{1}\frac{d}{ds}f^{x}(suy)\mathrm{d}s-\sum_{j=1}^{n}uy_{j}a_{j}(u,x)\stackrel{{\scriptstyle\eqref{eq:def
aj}}}{{=}}\sum_{j=1}^{n}\int_{0}^{1}uy_{j}\big{(}f_{j}^{x}(suy)-P^{0}_{su}f_{j}^{x}\big{)}\mathrm{d}s.$

By the triangle inequality in $L_{p}(B^{n},Y)$, this implies that for every
$u\in(0,\infty)$,

$\left(\int_{B^{n}}\left\|f^{x}(uy)-\Lambda_{x,u}(uy)\right\|_{Y}^{p}\mathrm{d}y\right)^{\frac{1}{p}}\leqslant\sum_{j=1}^{n}\int_{0}^{1}u\bigg{(}\int_{B^{n}}|y_{j}|^{p}\left\|f_{j}^{x}(suy)-P^{0}_{su}f_{j}^{x}\right\|_{Y}^{p}\mathrm{d}y\bigg{)}^{\frac{1}{p}}\mathrm{d}s\\\
\leqslant\sum_{j=1}^{n}\int_{0}^{u}\bigg{(}\int_{B^{n}}\left\|f_{j}^{x}(ty)-P^{0}_{t}f_{j}^{x}\right\|_{Y}^{p}\mathrm{d}y\bigg{)}^{\frac{1}{p}}\mathrm{d}t=\sum_{j=1}^{n}\int_{0}^{u}h_{j}(t)\mathrm{d}t,$
(22)

where we used the crude estimate $\max_{y\in
B^{n}}\max_{j\in\\{1,\ldots,n\\}}|y_{j}|\leqslant 1$ and introduced the
notation

$\forall\,(j,t)\in\\{1,\ldots,n\\}\times(0,\infty),\qquad
h_{j}(t)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigg{(}\int_{B^{n}}\left\|f_{j}^{x}(ty)-P^{0}_{t}f_{j}^{x}\right\|_{Y}^{p}\mathrm{d}y\bigg{)}^{\frac{1}{p}}.$
(23)

A combination of (22) with the triangle inequality and Hardy’s inequality [72,
Section A.4] yields

$\left(\int_{0}^{\infty}\int_{B^{n}}\frac{\left\|f^{x}(uy)-\Lambda_{x,u}(uy)\right\|_{Y}^{p}}{u^{q+1}}\mathrm{d}y\mathrm{d}u\right)^{\frac{1}{p}}\leqslant\sum_{j=1}^{n}\bigg{(}\int_{0}^{\infty}\bigg{(}\int_{0}^{u}h_{j}(t)\mathrm{d}t\bigg{)}^{p}\frac{\mathrm{d}u}{u^{q+1}}\bigg{)}^{\frac{1}{p}}\\\
\leqslant\sum_{j=1}^{n}\frac{p}{q}\left(\int_{0}^{\infty}\frac{h_{j}(u)^{p}}{u^{q-p+1}}\mathrm{d}u\right)^{\frac{1}{p}}\stackrel{{\scriptstyle\eqref{eq:def
hj}}}{{=}}\frac{p}{q}\sum_{j=1}^{n}\bigg{(}\int_{0}^{\infty}\int_{B^{n}}\left\|f_{j}^{x}(ty)-P^{0}_{t}f_{j}^{x}\right\|_{Y}^{p}\frac{\mathrm{d}u}{u^{q-p+1}}\bigg{)}^{\frac{1}{p}}.$
(24)

By applying Lemma 12 to each of the functions $\\{h_{j}\\}_{j=1}^{n}$ (with
$q$ replaced by $q-p$), we conclude from (24) that the following estimate
holds true.

$\left(\int_{0}^{\infty}\int_{B^{n}}\frac{\left\|f^{x}(uy)-\Lambda_{x,u}(uy)\right\|_{Y}^{p}}{u^{q+1}}\mathrm{d}y\mathrm{d}u\right)^{\frac{1}{p}}\lesssim\frac{p}{q(n+q-p)^{\frac{1}{p}}}\sum_{j=1}^{n}\bigg{(}\int_{\mathbb{R}^{n}}\frac{\|f_{j}^{x}(y)-f_{j}(x)\|_{Y}^{p}}{\|y\|_{2}^{n+q-p}}\mathrm{d}y\bigg{)}^{\frac{1}{p}}.$

Recalling the definition of $f_{j}$ in (19), the desired estimate (18) now
follows from (21). ∎

###  2.1. Proof of Lemma 4

Fix $\varepsilon\in(0,1)$ and denote
$\delta\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(\varepsilon/9)^{1+n/p}$. Fix
also $0<r<r_{p}^{X\to Y}(\delta)$ and a $1$-Lipschitz function $f:B_{X}\to Y$.
By the definition of $r_{p}^{X\to Y}(\delta)$, we can find $y\in X$ and
$\rho\geqslant r$ such that $y+\rho B_{X}\subseteq B_{X}$, and there exists an
affine mapping $\Lambda:X\to Y$ with $\|\Lambda\|_{\mathrm{Lip}}\leqslant 3$,
such that

$\frac{1}{\mathrm{vol}(\rho B_{X})}\int_{y+\rho
B_{X}}\left\|f(u)-\Lambda(u)\right\|_{Y}^{p}\mathrm{d}u\leqslant\delta^{p}\rho^{p}=\frac{\varepsilon^{p+n}\rho^{p}}{9^{p+n}}.$
(25)

We claim that (25), combined with the fact that
$\|f-\Lambda\|_{\mathrm{Lip}}\leqslant\|f\|_{\mathrm{Lip}}+\|\Lambda\|_{\mathrm{Lip}}\leqslant
4$, yields

$\forall\,z\in y+\rho
B_{X},\qquad\left\|f(z)-\Lambda(z)\right\|_{Y}\leqslant\varepsilon\rho,$ (26)

thus completing the proof of Lemma 4.

Assume for the sake of obtaining a contradiction that (26) fails. Then there
exists $z\in y+\rho B_{x}$ such that
$\left\|f(z)-\Lambda(z)\right\|_{Y}>\varepsilon\rho$. Write
$\lambda=\varepsilon/9$ and define $w=\lambda y+(1-\lambda)z$. Observe that we
have $\|w-z\|_{X}=\lambda\|y-z\|_{X}\leqslant\lambda\rho$. Supposing that
$u\in X$ satisfies $\|u-w\|_{X}\leqslant\lambda\rho$, we therefore obtain

$\|u-y\|_{X}=\left\|(u-w)-(1-\lambda)(y-z)\right\|_{X}\leqslant\|u-w\|+(1-\lambda)\|y-z\|_{X}\leqslant\rho.$
(27)

Moreover, since $\|f-\Lambda\|_{\mathrm{Lip}}\leqslant 4$ we have

$\|f(u)-\Lambda(u)\|_{Y}\geqslant\|f(z)-\Lambda(z)\|_{Y}-4\|u-z\|_{X}\\\
>\varepsilon\rho-4\|u-w\|_{X}-4\|w-z\|_{X}\geqslant\varepsilon\rho-8\lambda\rho=\lambda\rho.$
(28)

It follows from (27) that $w+\lambda\rho B_{X}\subseteq y+\rho B_{X}$, and it
follows from (28) that for $u\in w+\lambda\rho B_{X}$ the integrand in the
left hand side of (25) is strictly larger than $\lambda\rho$. Hence, (25)
yields the following contradiction.

$\lambda^{p+n}\rho^{p}=\frac{\varepsilon^{p+n}\rho^{p}}{9^{p+n}}>\frac{\mathrm{vol}(\lambda\rho
B_{X})}{\mathrm{vol}(\rho B_{X})}\lambda^{p}\rho^{p}=\lambda^{p+n}\rho^{p}.$ ∎

###  2.2. An upper bound on the modulus of affine approximability

The example that is constructed below was obtained in collaboration with
Charles Fefferman; we thank him for agreeing that we include it here. A simple
construction from [57] shows that if $p\in[2,\infty)$ and
$\varepsilon\in(0,1)$ then

$r^{\ell_{2}^{n}\to\ell_{2}(\ell_{p})}(\varepsilon)\lesssim\frac{e^{-1/(c\varepsilon)^{p}}}{\sqrt{n}},$
(29)

where $c\in(0,\infty)$ is a universal constant. We shall now show how the
example of [57] can be tensorized so as to yield an improved dependence on
$n$, and we shall also briefly discuss the problem of bounding $r_{q}^{X\to
Y}(\varepsilon)$ for finite $q\geqslant 1$. The following lemma is an
$L_{q}$-variant of Lemma 4.1 of [57].

######  Lemma 16.

There exists a universal constant $C\in(0,\infty)$ with the following
property. For every $\varepsilon\in(0,1)$ and $p\in[1,\infty)$ there exists a
$1$-Lipschitz function $f:\mathbb{R}\to\ell_{p}$ such that for every
$q\in[1,\infty]$ and every affine mapping $\Lambda:\mathbb{R}\to\ell_{p}$, if
$a,b\in\mathbb{R}$ satisfy $-1\leqslant a<b\leqslant 1$ then

$\frac{b-a}{2}\geqslant
4e^{-(C/\varepsilon)^{p}}\implies\left(\frac{1}{b-a}\int_{a}^{b}\left\|f(x)-\Lambda(x)\right\|_{p}^{q}\mathrm{d}x\right)^{\frac{1}{q}}>\varepsilon\cdot\frac{b-a}{2}.$
(30)

Consequently,

$r_{q}^{\mathbb{R}\to\ell_{p}}(\varepsilon)\lesssim e^{-(C/\varepsilon)^{p}}.$

###### Proof.

Let $\varphi:\mathbb{R}\to\mathbb{R}$ be the piecewise affine (“sawtooth”)
function defined by $\varphi(2{{\mathbb{Z}}})=\\{0\\}$ and
$\varphi(1+2{{\mathbb{Z}}})=\\{1\\}$. Fix $m\in\mathbb{N}$ that will be
determined later. Denoting the standard basis of $\ell_{p}^{m}$ by
$\\{e_{j}\\}_{j=1}^{m}$, define $f:\mathbb{R}\to\ell_{p}^{m}$ by setting for
every $x\in\mathbb{R}$,

$f(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{m^{1/p}}\sum_{k=1}^{m}\frac{\varphi\left(2^{k}x\right)}{2^{k}}e_{k}.$
(31)

Since $\varphi$ is $1$-Lipschitz, it follows from (31) that also $f$ is
$1$-Lipschitz.

Fix $a,b\in\mathbb{R}$ satisfying $-1\leqslant a\leqslant b\leqslant 1$ and
$b-a\geqslant 4/2^{m}$. There exists $k\in\\{1,\ldots,m\\}$ such that
$4/2^{k}\leqslant b-a\leqslant 8/2^{k}$. Hence there is
$j\in\\{0,\ldots,2^{k-1}-1\\}$ for which
$[j/2^{k-1},(j+1)/2^{k-1}]\subseteq[a,b]$. Then, since $b-a\leqslant 8/2^{k}$,
for every affine mapping $\Lambda:\mathbb{R}\to\ell_{p}^{m}$ we have

$\frac{1}{b-a}\int_{a}^{b}\left\|f(x)-\Lambda(x)\right\|_{p}^{q}\mathrm{d}x\gtrsim
2^{k}\int_{j/2^{k-1}}^{(j+1)/2^{k-1}}\left\|f(x)-\Lambda(x)\right\|_{p}^{q}\mathrm{d}x.$
(32)

Writing $\Lambda=(\Lambda_{1},\ldots,\Lambda_{m})$, where
$\Lambda_{1},\ldots,\Lambda_{m}:\mathbb{R}\to\mathbb{R}$ are affine, it
follows from (31) and (32) that

$\displaystyle\frac{1}{b-a}\int_{a}^{b}\left\|f(x)-\Lambda(x)\right\|_{p}^{q}\mathrm{d}x$
$\displaystyle\gtrsim
2^{k}\int_{j/2^{k-1}}^{(j+1)/2^{k-1}}\left|\frac{\varphi\left(2^{k}x\right)}{m^{1/q}2^{k}}-\Lambda_{k}(x)\right|^{q}\mathrm{d}x$
$\displaystyle=\frac{1}{m^{q/p}2^{qk}}\int_{-1}^{1}\left|\varphi(y)-\mathscr{L}(y)\right|^{q}\mathrm{d}y,$
(33)

where $\mathscr{L}:\mathbb{R}\to\mathbb{R}$ is the affine function given for
every $y\in\mathbb{R}$ by
$\mathscr{L}(y)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}m^{1/p}2^{k}\Lambda_{k}((y+2j+1)/2^{k})$.
Recalling (14) and (16), we have $P_{1}^{1}\varphi\equiv 1/2$. Hence, by Lemma
13,

$\left(\int_{-1}^{1}\left|\varphi(y)-\mathscr{L}(y)\right|^{q}\mathrm{d}y\right)^{\frac{1}{q}}\gtrsim\left(\int_{-1}^{1}\left|\varphi(y)-\frac{1}{2}\right|^{q}\mathrm{d}y\right)^{\frac{1}{q}}\gtrsim
1.$ (34)

Since $b-a\asymp 2^{-k}$, by combining (2.2) and (34) we see that

$\left(\frac{1}{b-a}\int_{a}^{b}\left\|f(x)-\Lambda(x)\right\|_{p}^{q}\mathrm{d}x\right)^{\frac{1}{q}}\geqslant\frac{\eta}{m^{1/p}}\cdot\frac{b-a}{2},$
(35)

where $\eta\in(0,\infty)$ is a universal constant. Suppose that
$\varepsilon<\eta$ and choose $m$ to be the largest positive integer such that
$m<(\eta/\varepsilon)^{p}$. Then (35) implies the conclusion of (30). The
requirement here is that $b-a\geqslant 4/2^{m}$, and since
$m+1\geqslant(\eta/\varepsilon)^{p}$, this requirement is satisfied if
$b-a\geqslant 8/2^{(\eta/\varepsilon)^{p}}$, which follows from
$(b-a)/2\geqslant 4e^{-(\eta/(2\varepsilon))^{p}}$. Thus (30) holds true with
$C=\eta/2$. Note that, with this choice of $C$, the implication (30) holds
vacuously when $\varepsilon\geqslant\eta$. ∎

Lemma 17 below tensorizes Lemma 16 to improve over (29), obtaining exponential
decay.

######  Lemma 17.

There exists universal constants $K,\varepsilon_{0}\in(0,1)$ such that for
every $\varepsilon\in(0,\varepsilon_{0}]$, every $p\in[1,\infty)$ and every
$n\in\mathbb{N}$ we have

$r^{\ell_{2}^{n}\to\ell_{2}(\ell_{p})}(\varepsilon)\leqslant
e^{-n(K/\varepsilon)^{p}}.$ (36)

###### Proof.

By Lemma 16 (with $q=\infty$) we can fix $K,\varepsilon_{0}\in(0,1)$ such that
for every $\varepsilon\in(0,\varepsilon_{0}]$ and $p\in[1,\infty)$ there
exists a $1$-Lipschitz mapping $f:[-1,1]\to\ell_{p}$ such that for every
interval $[a,b]\subseteq[-1,1]$ with $(b-a)/2>e^{-(K/\varepsilon)^{p}}$ and
for every affine mapping $\Lambda:\mathbb{R}\to\ell_{p}$ there exists
$x\in[a,b]$ such that $\|f(x)-\Lambda(x)\|_{p}>\varepsilon(b-a)/2$.

Define $F:\mathbb{R}^{n}\to\ell_{2}^{n}(\ell_{p})$ by setting for every
$x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$,

$F(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left(f\left(x_{1}\right),\frac{f\left(e^{(K/\varepsilon)^{p}}x_{2}\right)}{e^{(K/\varepsilon)^{p}}},\ldots,\frac{f\left(e^{(n-1)(K/\varepsilon)^{p}}x_{n}\right)}{e^{(n-1)(K/\varepsilon)^{p}}}\right)\in\ell_{2}^{n}(\ell_{p}).$

By applying the $1$-Lipschitz condition for $f$ coordinate-wise, we see that
$F$ is $1$-Lipschitz as a mapping from $\ell_{2}^{n}$ to
$\ell_{2}^{n}(\ell_{p})$. Fix $x\in B^{n}$ and $r\in(0,1)$ such that
$x+rB^{n}\subseteq B^{n}$. Suppose from now on that
$r>e^{-n(K/\varepsilon)^{p}}$ and let $j\in\mathbb{N}$ be the largest integer
for which $r\leqslant e^{-(j-1)(K/\varepsilon)^{p}}$. Then
$j\in\\{1,\ldots,n\\}$ and
$e^{(j-1)(K/\varepsilon)^{p}}r>e^{-(K/\varepsilon)^{p}}$. Let
$\Lambda:\mathbb{R}^{n}\to\ell_{2}^{n}(\ell_{p})$ be an affine mapping, and
write $\Lambda=(\Lambda_{1},\ldots,\Lambda_{n})$, where
$\Lambda_{1},\ldots,\Lambda_{n}:\mathbb{R}^{n}\to\ell_{p}$ are affine.
Consider the affine mapping $\Lambda_{j}^{\prime}:\mathbb{R}\to\ell_{p}$ given
by
$\Lambda_{j}^{\prime}(y)=\Lambda_{j}(x_{1},\ldots,x_{j-1},y,x_{j+1},\ldots,x_{n})$.
Set $a=x_{j}e^{(j-1)(K/\varepsilon)^{p}}-re^{(j-1)(K/\varepsilon)^{p}}$ and
$b=x_{j}e^{(j-1)(K/\varepsilon)^{p}}+re^{(j-1)(K/\varepsilon)^{p}}$. Then
$(b-a)/2=re^{(j-1)(K/\varepsilon)^{p}}>e^{-(K/\varepsilon)^{p}}$, so by our
assumption on $f$ there exists
$w\in\left[x_{j}e^{(j-1)(K/\varepsilon)^{p}}-re^{(j-1)(K/\varepsilon)^{p}},x_{j}e^{(j-1)(K/\varepsilon)^{p}}+re^{(j-1)(K/\varepsilon)^{p}}\right]$
with

$\left\|f(w)-e^{(j-1)(K/\varepsilon)^{p}}\Lambda_{j}^{\prime}\left(we^{-(j-1)(K/\varepsilon)^{p}}\right)\right\|_{p}>\varepsilon
re^{(j-1)(K/\varepsilon)^{p}}.$

Setting $y=we^{-(j-1)(K/\varepsilon)^{p}}$, this is the same as asserting that
there exists $y\in[x_{j}-r,x_{j}+r]$ with

$\left\|e^{-(j-1)(K/\varepsilon)^{p}}f\left(e^{(j-1)(K/\varepsilon)^{p}}y\right)-\Lambda^{\prime}_{j}(y)\right\|_{p}>\varepsilon
r.$

Hence, writing $z=(x_{1},\ldots,x_{j-1},y,x_{j+1},\ldots,x_{n})$, we have that
$z\in x+rB^{n}$ and

$\left\|F(z)-\Lambda(z)\right\|_{\ell_{2}^{n}(\ell_{p})}\geqslant\left\|e^{-(j-1)(K/\varepsilon)^{p}}f\left(e^{(j-1)(K/\varepsilon)^{p}}y\right)-\Lambda^{\prime}_{j}(y)\right\|_{p}>\varepsilon
r,$ (37)

Since (37) holds true for every affine mapping
$\Lambda:\mathbb{R}^{n}\to\ell_{2}^{n}(\ell_{p})$ whenever $x+rB^{n}\subseteq
B^{n}$ and $r>e^{-n(K/\varepsilon)^{p}}$, the proof of (36) is complete. ∎

######  Remark 18.

An upper bound on $r_{p}^{\ell_{2}^{n}\to\ell_{2}(\ell_{p})}(\varepsilon)$ is
a consequence of Lemma 17 and Lemma 4. It seems likely that a significantly
stronger upper bound holds true, but the above tensorization procedure does
not seem to yield such an improvement. We leave the investigation of upper
bounds on the modulus of $L_{p}$ affine approximability as an interesting
question for future research.

##  3\. A UMD-valued Dorronsoro-type estimate

The main ingredient of the proof of Theorem 5 (hence also Theorem 2) is the
following result.

######  Theorem 19.

There exists a universal constant $\kappa\in[2,\infty)$ with the following
property. Suppose that $(Y,\|\cdot\|_{Y})$ is a UMD Banach space and write
$\beta=\beta(Y)$. Suppose that $f:\mathbb{R}^{n}\to Y$ is a Lipschitz and
compactly supported function. Then

$\bigg{(}\frac{1}{V_{n}}\iiint_{\mathbb{R}^{n}\times
B^{n}\times(0,\infty)}\frac{\left\|f^{x}(uy)-P_{u}^{1}f^{x}(uy)\right\|_{Y}^{\kappa\beta}}{u^{1+\kappa\beta}}\mathrm{d}x\mathrm{d}y\mathrm{d}u\bigg{)}^{\frac{1}{\kappa\beta}}\lesssim\beta^{15}n^{\frac{5}{2}}\big{(}\mathrm{vol}(\operatorname{supp}(f))\big{)}^{\frac{1}{\kappa\beta}}\|f\|_{\mathrm{Lip}}.$
(38)

In the case of real-valued functions on $\mathbb{R}^{n}$, such a statement was
first proved by Dorronsoro [26], with an implicit dependence on the dimension
$n$. Extensions and variants of Dorronsoro’s theorem have been further studied
within the theory of functions spaces, where norms like the one that appears
on the left of (38) define what is called _local approximation spaces_. See
[75, Section 1.7] for some discussion of the subsequent history and [75,
Section 3.5.1] for a different proof of a similar statement. There, the
dependence on $n$ is perhaps even more implicit, since it also depends on a
non-canonical choice of resolutions of identity used to define some general
function spaces.

This dependence on dimension is crucial for us here; specifically, we desire
polynomial growth in $n$ in the right hand side of (38), while a naïve
examination of the proof in [26] reveals that it yields a much worse (super-
exponential) dependence on $n$. Note also that in [26] the $L_{\kappa\beta}$
norm that appears in (38) can be replaced by an $L_{p}$ norm for any
$p\in(1,\infty)$, while in the present vector-valued setting the geometry of
the target space $Y$ influences the value of $p$. In fact, we shall prove a
more refined (and stronger) version of Theorem 19; see Theorem 40 below, in
which the $L_{\kappa\beta}$ norm that appears in (38) can be replaced by an
$L_{p}$ norm provided $Y$ is a UMD Banach space of Rademacher cotype $p$ (see
Section 4.4 below). These refinements are not important for our purposes,
i.e., for proving Theorem 5, but they do imply sharper results, e.g. when $Y$
is itself an $L_{q}(\mu)$ space. In the same vein, the exponent $15$ of
$\beta$ in (38) is not sharp (for the purpose of Theorem 5 we only desire a
polynomial dependence on $\beta$).

The bulk of the ensuing discussion is devoted to the proof of Theorem 19. Our
argument roughly follows the strategy of Dorronsoro in [26], combined with
substantial additions and modifications in order to obtain good dependence on
$n$ and also overcome difficulties that arise in the vector-valued setting and
are not present in the real-valued setting of [26]. As explained in the
Introduction, these complications reflect a genuine difference between the
vector-valued setting and the real-valued setting, as such results do not hold
true for general Banach space targets $Y$, so the geometry of $Y$ must somehow
enter into the argument. We did not investigate the extent to which Theorem 19
(and Theorem 40 below) are sharp in terms of the assumptions that are required
from $Y$ and the $L_{p}$ norm that could appear in (38) (or (137) below).

###  3.1. Proof of Theorem 5 assuming the validity of Theorem 19

Here we shall prove Theorem 5 while using Theorem 19. This will allow us to
focus later on Theorem 19 itself in order complete the justification of
Theorem 5, and hence, by Lemma 4, also (3) and Theorem 2.

Recalling our setting, we are given a $n$-dimensional normed space
$(X=\mathbb{R}^{n},\|\cdot\|_{X})$ such that (8) holds true. We are also given
a normed space $(Y,\|\cdot\|_{Y})$ that satisfies the assumptions of Theorem
19. Suppose that $f:B_{X}\to Y$ is $1$-Lipschitz. Without loss of generality
assume also that $f(0)=0$.

Define $\phi:[0,\infty)\to[0,\infty)$ by

$\forall\,u\in[0,\infty),\qquad\phi(u)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\\{\begin{array}[]{ll}1&\mathrm{if}\
u\leqslant\frac{1}{\sqrt{n}},\\\ n+1-n^{\frac{3}{2}}u&\mathrm{if}\
\frac{1}{\sqrt{n}}<u\leqslant\left(1+\frac{1}{n}\right)\frac{1}{\sqrt{n}},\\\
0&\mathrm{if}\
u>\left(1+\frac{1}{n}\right)\frac{1}{\sqrt{n}}.\end{array}\right.$

Then $\phi$ is supported in $[0,(1+1/n)/\sqrt{n}]$ and it is elementary to
verify the following inequalities.

$\forall\,u\in(0,\infty),\qquad\max\big{\\{}\phi(u),\sqrt{n}u\phi(u)\big{\\}}\leqslant
1.$ (39)

and

$\forall\,u,v\in(0,\infty),\qquad\big{|}\phi(u)-\phi(v)\big{|}\cdot\min\\{u,v\\}\leqslant(n+1)|u-v|.$
(40)

Since by (8) we have $\frac{1}{\sqrt{n}}B^{n}\subseteq B_{X}$, by (39) we know
that every $x\in\mathbb{R}^{n}$ satisfies $\phi(\|x\|_{2})x\in B_{X}$. We can
therefore define $F:\mathbb{R}^{n}\to Y$ by $F(x)=f(\phi(\|x\|_{2})x)$. Then,
$F(x)=f(x)$ if $\|x\|_{2}\leqslant 1/\sqrt{n}$, and $F(x)=0$ if
$\|x\|_{2}\geqslant(1+1/n)/\sqrt{n}$. Also, every $x,y\in\mathbb{R}^{n}$ with
$\|y\|_{X}\leqslant\|x\|_{X}$ satisfy

$\displaystyle\|F(x)-F(y)\|_{Y}$ $\displaystyle\leqslant$
$\displaystyle\|f\|_{\mathrm{Lip}(B_{X})}\big{\|}\phi(\|x\|_{2})x-\phi(\|y\|_{2})y\big{\|}_{X}$
$\displaystyle\leqslant$
$\displaystyle\phi(\|x\|_{2})\|x-y\|_{X}+\big{|}\phi(\|x\|_{2})-\phi(\|y\|_{2})\big{|}\|y\|_{X}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:john
posiiton}\wedge\eqref{eq:phi upper}}}{{\leqslant}}$
$\displaystyle\|x-y\|_{X}+\big{|}\phi(\|x\|_{2})-\|\phi(\|y\|_{2})\big{|}\sqrt{n}\|y\|_{2}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:phi lip}}}{{\leqslant}}$
$\displaystyle\|x-y\|_{X}+\sqrt{n}(n+1)\|x-y\|_{2}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:john posiiton}}}{{\leqslant}}$
$\displaystyle\left(n^{\frac{3}{2}}+\sqrt{n}+1\right)\|x-y\|_{X}.$

Thus $\|F\|_{\mathrm{Lip}(\mathbb{R}^{n})}\lesssim n^{\frac{3}{2}}$. Since
$\operatorname{supp}(F)\subseteq\left(1+\frac{1}{n}\right)\frac{1}{\sqrt{n}}B^{n}$,
it therefore follows from Theorem 19 that

$\displaystyle\bigg{(}\iiint_{\mathbb{R}^{n}\times
B^{n}\times(0,\infty)}\frac{\left\|F^{x}(uy)-P_{u}^{1}F^{x}(uy)\right\|_{Y}^{\kappa\beta}}{V_{n}u^{1+\kappa\beta}}\mathrm{d}x\mathrm{d}y\mathrm{d}u\bigg{)}^{\frac{1}{\kappa\beta}}$
$\displaystyle\leqslant
K\beta^{15}n^{\frac{5}{2}}\left(1+\frac{1}{n}\right)^{\frac{n}{\kappa\beta}}\left(\frac{V_{n}}{n^{\frac{n}{2}}}\right)^{\frac{1}{\kappa\beta}}n^{\frac{3}{2}}$
$\displaystyle<eK\beta^{15}n^{4}\left(\frac{V_{n}}{n^{\frac{n}{2}}}\right)^{\frac{1}{\kappa\beta}},$
(41)

where $K\in(0,\infty)$ is a universal constant. Note that

$(x,u)\in\left(\left(1-\frac{1}{n}\right)\frac{1}{\sqrt{n}}B^{n}\right)\times\left(0,\frac{1}{n^{3/2}}\right)\implies
x+uB^{n}\subseteq\frac{1}{\sqrt{n}}B^{n}.$

Since $F$ coincides with $f$ on $\frac{1}{\sqrt{n}}B^{n}$, it follows that

$(x,y,u)\in\left(\left(1-\frac{1}{n}\right)\frac{1}{\sqrt{n}}B^{n}\right)\times
B^{n}\times\left(0,\frac{1}{n^{3/2}}\right)\implies
P^{1}_{u}F^{x}=P^{1}_{u}f^{x}\quad\mathrm{and}\quad F^{x}(uy)=f^{x}(uy).$

These observations in combination with (3.1) imply that

$\bigg{(}\iiint_{\left(\left(1-\frac{1}{n}\right)\frac{1}{\sqrt{n}}B^{n}\right)\times
B^{n}\times\left(0,\frac{1}{n^{3/2}}\right)}\frac{\left\|f^{x}(uy)-P_{u}^{1}f^{x}(uy)\right\|_{Y}^{\kappa\beta}}{V_{n}u^{1+\kappa\beta}}\mathrm{d}x\mathrm{d}y\mathrm{d}u\bigg{)}^{\frac{1}{\kappa\beta}}<K\beta^{15}n^{4}\left(\frac{V_{n}}{n^{\frac{n}{2}}}\right)^{\frac{1}{\kappa\beta}}.$
(42)

Fix $M\in(1,\infty)$ whose precise value will be specified later. It follows
from (42) that there exist

$u\in\left(\frac{1}{Mn^{2/3}},\frac{1}{n^{3/2}}\right)\qquad\mathrm{and}\qquad
x\in\left(1-\frac{1}{n}\right)\frac{1}{\sqrt{n}}B^{n},$ (43)

such that

$\bigg{(}\frac{1}{V_{n}}\int_{B^{n}}\left\|f^{x}(uy)-P_{u}^{1}f^{x}(uy)\right\|_{Y}^{\kappa\beta}\mathrm{d}y\bigg{)}^{\frac{1}{\kappa\beta}}\leqslant\frac{2eK\beta^{15}n^{4}u}{\left(\log
M\right)^{\frac{1}{\kappa\beta}}}.$ (44)

Indeed, we would otherwise have

$\bigg{(}\iiint_{\left(\left(1-\frac{1}{n}\right)\frac{1}{\sqrt{n}}B^{n}\right)\times
B^{n}\times\left(0,\frac{1}{n^{3/2}}\right)}\frac{\left\|f^{x}(uy)-P_{u}^{1}f^{x}(uy)\right\|_{Y}^{\kappa\beta}}{V_{n}u^{1+\kappa\beta}}\mathrm{d}x\mathrm{d}y\mathrm{d}u\bigg{)}^{\frac{1}{\kappa\beta}}\\\
>\left(1-\frac{1}{n}\right)^{\frac{n}{\kappa\beta}}\left(\frac{V_{n}}{n^{\frac{n}{2}}}\right)^{\frac{1}{\kappa\beta}}\left(\int_{\frac{n^{-3/2}}{M}}^{n^{-3/2}}\frac{\mathrm{d}u}{u}\right)^{\frac{1}{\kappa\beta}}\frac{2eK\beta^{15}n^{4}}{\left(\log
M\right)^{\frac{1}{\kappa\beta}}}\geqslant
eK\beta^{15}n^{4}\left(\frac{V_{n}}{n^{\frac{n}{2}}}\right)^{\frac{1}{\kappa\beta}},$

thus contradicting (42). Observe that by Lemma 10 we have

$\|P_{u}^{1}f^{x}\|_{\mathrm{Lip}(\mathbb{R}^{n})}\stackrel{{\scriptstyle\eqref{eq:def
P1}}}{{=}}\|T_{u}f^{x}\|_{X\to
Y}\leqslant\|f^{x}\|_{\mathrm{Lip}(\mathbb{R}^{n})}\leqslant 1.$

Hence, if we set
$\Lambda\stackrel{{\scriptstyle\mathrm{def}}}{{=}}P^{1}_{x}f^{x}$ then
$\Lambda:\mathbb{R}^{n}\to Y$ is affine,
$\|\Lambda\|_{\mathrm{Lip}(\mathbb{R}^{n})}\leqslant 1$, and by a change of
variable one can rewrite (44) as follows.

$\bigg{(}\frac{1}{\mathrm{vol}(x+uB^{n})}\int_{x+uB^{n}}\left\|f(z)-\Lambda(z)\right\|_{Y}^{\kappa\beta}\mathrm{d}z\bigg{)}^{\frac{1}{\kappa\beta}}\leqslant\frac{2eK\beta^{15}n^{4}u}{\left(\log
M\right)^{\frac{1}{\kappa\beta}}}.$ (45)

This is not quite the type of conclusion that we desire, because the averaging
in (45) occurs over a Euclidean ball rather than a ball in
$(\mathbb{R}^{n},\|\cdot\|_{X})$. We overcome this via another averaging step.

Observe that

$x+\left(1-\frac{1}{n}\right)uB^{n}+\frac{u}{n}B_{X}\subseteq
x+uB^{n}\subseteq\frac{1}{\sqrt{n}}B^{n}\stackrel{{\scriptstyle\eqref{eq:john
posiiton}}}{{\subseteq}}B_{X}.$ (46)

Indeed, if $a,b\in\mathbb{R}^{n}$ satisfy $\|a\|_{2}\leqslant(1-1/n)u$ and
$\|b\|_{X}\leqslant u/n$ then,

$\|a+b\|_{2}\leqslant\|a\|_{2}+\|b\|_{2}\stackrel{{\scriptstyle\eqref{eq:john
posiiton}}}{{\leqslant}}\|a\|_{2}+\|b\|_{X}\leqslant\left(1-\frac{1}{n}\right)u+\frac{u}{n}=u,$

Now, define $A\subseteq\mathbb{R}^{n}\times\mathbb{R}^{n}$ as follows.

$\displaystyle A$
$\displaystyle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\\{(y,w):\ y\in
x+\left(1-\frac{1}{n}\right)uB^{n}\ \wedge\ w\in y+\frac{u}{n}B_{X}\right\\}$
$\displaystyle=\left\\{(y,w):\ w\in
x+\left(1-\frac{1}{n}\right)uB^{n}+\frac{u}{n}B_{X}\ \wedge\
y\in\left(x+\left(1-\frac{1}{n}\right)uB^{n}\right)\cap\left(w+\frac{u}{n}B_{X}\right)\right\\}.$

By (46), $f(w)$ is well-defined for every $(y,w)\in A$. It therefore follows
from Fubini’s theorem that

$\displaystyle\iint_{A}\|f(w)-\Lambda(w)\|_{Y}^{\kappa\beta}\mathrm{d}w\mathrm{d}y$
$\displaystyle=\int_{x+\left(1-\frac{1}{n}\right)uB^{n}}\mathrm{d}y\int_{y+\frac{u}{n}B_{X}}\|f(w)-\Lambda(w)\|_{Y}^{\kappa\beta}\mathrm{d}w$
$\displaystyle=\int_{x+\left(1-\frac{1}{n}\right)uB^{n}+\frac{u}{n}B_{X}}\mathrm{vol}\left(\left(x+\left(1-\frac{1}{n}\right)uB^{n}\right)\cap\left(w+\frac{u}{n}B_{X}\right)\right)\|f(w)-\Lambda(w)\|_{Y}^{\kappa\beta}\mathrm{d}w$
$\displaystyle\leqslant\mathrm{vol}\left(\frac{u}{n}B_{X}\right)\int_{x+uB^{n}}\|f(w)-\Lambda(w)\|_{Y}^{\kappa\beta}\mathrm{d}w.$

Hence,

$\frac{1}{\mathrm{vol}\left(x+\left(1-\frac{1}{n}\right)uB^{n}\right)}\int_{x+\left(1-\frac{1}{n}\right)uB^{n}}\frac{\mathrm{d}y}{\mathrm{vol}\left(y+\frac{u}{n}B_{X}\right)}\int_{y+\frac{u}{n}B_{X}}\|f(w)-\Lambda(w)\|_{Y}^{\kappa\beta}\mathrm{d}w\\\
\leqslant\frac{1}{\mathrm{vol}\left(x+\left(1-\frac{1}{n}\right)uB^{n}\right)}\int_{x+uB^{n}}\|f(w)-\Lambda(w)\|_{Y}^{\kappa\beta}\mathrm{d}w\leqslant\frac{\left(2eK\beta^{15}n^{4}u\right)^{\kappa\beta}}{\left(1-\frac{1}{n}\right)^{n}\log
M}\leqslant\frac{\left(4eK\beta^{15}n^{4}u\right)^{\kappa\beta}}{\log M}.$

This implies that there exists $y\in x+\left(1-\frac{1}{n}\right)uB^{n}$ such
that

$\left(\frac{1}{\mathrm{vol}\left(y+\frac{u}{n}B_{X}\right)}\int_{y+\frac{u}{n}B_{X}}\|f(w)-\Lambda(w)\|_{Y}^{\kappa\beta}\mathrm{d}w\right)^{\frac{1}{\kappa\beta}}\leqslant\frac{4eK\beta^{15}n^{4}u}{\left(\log
M\right)^{\frac{1}{\kappa\beta}}}=\frac{4eK\beta^{15}n^{5}}{\left(\log
M\right)^{\frac{1}{\kappa\beta}}}\cdot\frac{u}{n}.$

Recalling Definition (1), since by (43) we are ensured that $u\geqslant
1/(Mn^{3/2})$, it follows that

$\forall\,M\in(1,\infty),\qquad r_{\kappa\beta}^{X\to
Y}\left(\frac{4eK\beta^{15}n^{5}}{\left(\log
M\right)^{\frac{1}{\kappa\beta}}}\right)\geqslant\frac{1}{Mn^{\frac{3}{2}}}.$
(47)

For $\varepsilon\in(0,1)$ choose
$M=e^{(4eK\beta^{15}n^{5}/\varepsilon)^{\kappa\beta}}$ in (47), thus yielding
Theorem 5 as follows.

$\forall\,\varepsilon\in(0,1),\qquad r_{\kappa\beta}^{X\to
Y}(\varepsilon)\geqslant\frac{1}{n^{\frac{3}{2}}}\exp\left(-\frac{(4eK\beta^{15}n^{5})^{\kappa\beta}}{\varepsilon^{\kappa\beta}}\right).$
∎

##  4\. Preliminaries on UMD Banach spaces

This section is devoted to the presentation of several analytic properties of
UMD Banach spaces that will be used extensively in the proof of Theorem 5.

Let $(Y,\|\cdot\|_{Y})$ be a UMD Banach space and fix $p\in(1,\infty)$. Denote
(as usual) by $\beta_{p}(Y)$ the infimum over those $\beta\in(1,\infty]$ such
that if $\\{M_{j}\\}_{j=0}^{\infty}$ is a $Y$-valued $p$-integrable martingale
defined on some probability space $(\Omega,\mathbb{P})$ then for every
$n\in\mathbb{N}$ and every
$\varepsilon_{1},\ldots,\varepsilon_{n}\in\\{-1,1\\}$ we have

$\int_{\Omega}\Big{\|}M_{0}+\sum_{j=1}^{n}\varepsilon_{j}\left(M_{j}-M_{j-1}\right)\Big{\|}_{Y}^{p}\mathrm{d}\mathbb{P}\leqslant\beta^{p}\int_{\Omega}\left\|M_{n}\right\|_{Y}^{p}\mathrm{d}\mathbb{P}.$
(48)

Thus, using the notation of the Introduction, we have $\beta_{2}(Y)=\beta(Y)$.
The following inequality is well-known; see [17] for its proof.

$\beta_{p}(Y)\lesssim\frac{p^{2}}{p-1}\beta(Y).$ (49)

We also record for future use that (48) implies (see [53, Thm. 4.4]) that for
every $a_{0},a_{1},\ldots,a_{n}\in\mathbb{R}$,

$\int_{\Omega}\Big{\|}a_{0}M_{0}+\sum_{j=1}^{n}a_{j}\left(M_{j}-M_{j-1}\right)\Big{\|}_{Y}^{p}\mathrm{d}\mathbb{P}\leqslant\left(\max_{j\in\\{0,\ldots,n\\}}|a_{j}|^{p}\right)\beta_{p}(X)^{p}\int_{\Omega}\left\|M_{n}\right\|_{Y}^{p}\mathrm{d}\mathbb{P}.$
(50)

In [33] Garling introduced two parameters $\beta_{p}^{+}(Y),\beta_{p}^{-}(Y)$,
defined to be the best constants in the following inequalities, which are
required to hold true for every martingale $\\{M_{j}\\}_{j=0}^{\infty}$ as
above.

$\mathbb{E}_{\varepsilon}\bigg{[}\int_{\Omega}\Big{\|}M_{0}+\sum_{j=1}^{n}\varepsilon_{n}\left(M_{j}-M_{j-1}\right)\Big{\|}_{Y}^{p}\mathrm{d}\mathbb{P}\bigg{]}\leqslant\beta_{p}^{+}(Y)^{p}\int_{\Omega}\left\|M_{n}\right\|_{Y}^{p}\mathrm{d}\mathbb{P},$

and

$\int_{\Omega}\left\|M_{n}\right\|_{Y}^{p}\mathrm{d}\mathbb{P}\leqslant\beta_{p}^{-}(Y)^{p}\mathbb{E}_{\varepsilon}\bigg{[}\int_{\Omega}\Big{\|}M_{0}+\sum_{j=1}^{n}\varepsilon_{n}\left(M_{j}-M_{j-1}\right)\Big{\|}_{Y}^{p}\mathrm{d}\mathbb{P}\bigg{]},$

where $\mathbb{E}_{\varepsilon}[\cdot]$ denotes the expectation with respect
to $\varepsilon=(\varepsilon_{1},\ldots,\varepsilon_{n})$ chosen uniformly at
random from $\\{-1,1\\}^{n}$. Garling’s inequalities are weaker than (48),
which is required to hold for every $\varepsilon\in\\{-1,1\\}^{n}$ rather than
only in expectation with respect to $\varepsilon$. Hence,

$\max\left\\{\beta_{p}^{+}(Y),\beta_{p}^{-}(Y)\right\\}\leqslant\beta_{p}(Y),$
(51)

but there are examples of Banach spaces $Y$ for which $\beta_{p}^{+}(Y)$ or
$\beta_{p}^{-}(Y)$ is markedly smaller than $\beta_{p}(Y)$; see [33, 34]. Some
of the ensuing estimates can be stated in terms of the parameters
$\beta_{p}(Y),\beta_{p}^{+}(Y),\beta_{p}^{-}(Y)$, but in order to avoid
cumbersome expressions we will sometimes state our bounds in terms of the
quantity $\beta(Y):=\beta_{2}(Y)$, by invoking (49) and (51). In fact, in our
setting we will always choose $2\leqslant p\asymp\beta(Y)$, in which case by
using (49) and (51) we will sometimes bound from above the quantities
$\beta_{p}(Y),\beta_{p}^{+}(Y),\beta_{p}^{-}(Y)$ by a universal constant
multiple of $p\,\beta(Y)\asymp\beta(Y)^{2}$. This choice has the advantage of
simplifying some of the ensuing discussion, but it yields bounds that could be
improved for some (quite exotic) UMD Banach spaces $Y$ by a straightforward
inspection of our proofs.

###  4.1. $\mathscr{R}$-boundedness

Let $(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ be Banach spaces. The space of
bounded linear operators from $X$ to $Y$ is denoted $\mathscr{L}(X,Y)$.
Following [10, 21], a set of operators $\mathcal{T}\subseteq\mathscr{L}(X,Y)$
is said to be $\mathscr{R}$-bounded if

$\mathbb{E}_{\varepsilon}\bigg{[}\Big{\|}\sum_{j=1}^{N}\varepsilon_{j}T_{j}x_{j}\Big{\|}_{Y}^{p}\bigg{]}\leqslant
C^{p}\mathbb{E}_{\varepsilon}\bigg{[}\Big{\|}\sum_{j=1}^{N}\varepsilon_{j}x_{j}\Big{\|}_{X}^{p}\bigg{]},$
(52)

for every $N\in\mathbb{N}$, every $x_{1},\ldots,x_{N}\in Y$ and every
$T_{1},\ldots,T_{N}\in\mathcal{T}$, and some (equivalently, by Kahane’s
inequality [50], for all) $p\in[1,\infty)$. The infimum over those $C$ in (52)
is denoted $\mathscr{R}_{p}(\mathcal{T})$.

The following result is due to Bourgain [12]; see also [30] for a proof.

######  Proposition 20 (Bourgain’s vector-valued Stein inequality).

Suppose that $(Y,\|\cdot\|_{Y})$ is a UMD Banach space and fix
$p\in(1,\infty)$. Let $\\{\mathscr{F}_{j}\\}_{j\in{{\mathbb{Z}}}}$ be an
increasing sequence of sub-$\sigma$-algebras on some probability space
$(\Omega,\mathscr{F},\mathbb{P})$. For every $j\in{{\mathbb{Z}}}$ let
$\mathscr{E}_{j}\in\mathscr{L}(L_{p}(\mathbb{P},Y),L_{p}(\mathbb{P},Y))$ be
the conditional expectation operator corresponding to $\mathscr{F}_{j}$, i.e.,
$\mathscr{E}_{j}f=\mathbb{E}[f|\mathscr{F}_{j}]$ for every $f\in
L_{p}(\mathbb{P},Y)$. Then

$\mathscr{R}_{p}\left(\\{\mathscr{E}_{j}\\}_{j\in{{\mathbb{Z}}}}\right)\leqslant\beta_{p}^{+}(Y).$

By a classical representation theorem for positive self-adjoint semigroups of
contractions due to Rota [70] (see also [73, Sec. VI] or [59, Thm. 2.5]),
Proposition 20 implies the following dimension independent
$\mathscr{R}$-boundedness estimate for the heat semigroup. Here and in what
follows, $\Delta$ denotes the Laplacian on $\mathbb{R}^{n}$.

######  Corollary 21 ($\mathscr{R}$-boundedness of the heat semigroup).

Let $(Y,\|\cdot\|_{Y})$ be a UMD Banach space, $n\in\mathbb{N}$, and
$p\in(1,\infty)$. Then

$\mathscr{R}_{p}\left(\\{e^{t\Delta}\\}_{t\in(0,\infty)}\right)\leqslant\beta_{p}^{+}(Y).$

###### Proof.

For any $\delta>0$, the operator $Q=e^{\frac{1}{2}\delta\Delta}$ satisfies the
assumptions of Rota’s theorem as formulated in [73, Sec. VI] or [59, Thm.
2.5]. Thus there exists a measure space $(\Omega,\mathscr{F},\mu)$ with an
increasing sequence of sub-$\sigma$-algebras
$\\{\mathscr{F}_{j}\\}_{j\in{{\mathbb{Z}}}}$ and yet another
sub-$\sigma$-algebra ${\mathscr{F}}^{\prime}$ with the corresponding
conditional expectation operators $\mathscr{E}_{j}$ and
${\mathscr{E}}^{\prime}$ such that

$\forall\,j\in\mathbb{N}\cup\\{0\\},\qquad
e^{j\delta\Delta}=Q^{2j}=J^{-1}{\mathscr{E}}^{\prime}\mathscr{E}_{-j}J,$

where $J:L_{p}(\mathbb{R}^{n},Y)\to L_{p}(\Omega,{\mathscr{F}}^{\prime},Y)$ is
an isometric isomorphism. Thus

$\displaystyle\mathscr{R}_{p}\big{(}\\{e^{j\delta\Delta}\\}_{j\in\mathbb{N}\cup\\{0\\}}\big{)}$
$\displaystyle=\mathscr{R}_{p}\big{(}\\{J{\mathscr{E}}^{\prime}\mathscr{E}_{-j}J\\}_{j\in\mathbb{N}\cup\\{0\\}}\big{)}$
$\displaystyle\leqslant\|J^{-1}\|_{L_{p}(\Omega,{\mathscr{F}}^{\prime},Y)\to
L_{p}(\mathbb{R}^{n},Y)}\cdot\|{\mathscr{E}}^{\prime}\|_{L_{p}(\Omega,\mathscr{F},\mu)\to
L_{p}(\Omega,\mathscr{F^{\prime}},\mu)}\cdot\mathscr{R}_{p}\big{(}\\{\mathscr{E}_{-j}\\}_{j\in\mathbb{N}\cup\\{0\\}}\big{)}\cdot\|J\|_{L_{p}(\mathbb{R}^{n},Y)\to
L_{p}(\Omega,{\mathscr{F}}^{\prime},Y)}$ $\displaystyle\leqslant\beta_{+}(Y),$

where we used Proposition 20 together with easy properties of
$\mathscr{R}$-bounds and the contractivity of conditional expectations.

Using basic properties of $\mathscr{R}$-bounds (cf. [76, Proposition 9.5]), if
$\mathscr{T}=\bigcup_{k=1}^{\infty}\mathscr{T}_{k}$ is a union of an
increasing sequence $\mathscr{T}_{k}\subseteq\mathscr{T}_{k+1}$, and
$\overline{\mathscr{T}}$ is the closure of $\mathscr{T}$ in the strong
operator topology, then
$\mathscr{R}_{p}(\overline{\mathscr{T}})=\mathscr{R}_{p}(\mathscr{T})=\lim_{k\to\infty}\mathscr{R}_{p}(\mathscr{T}_{k})$.
With $\mathscr{T}_{k}=\\{e^{j2^{-k}\Delta}\\}_{j\in\mathbb{N}\cup\\{0\\}}$, we
have $\overline{\mathscr{T}}=\\{e^{t\Delta}\\}_{t\in(0,\infty)}$ by the strong
continuity of $t\mapsto e^{t\Delta}$, and this proves the claim. ∎

Our next goal is to prove Theorem 24 below, which is a useful bound on the
norm of operators on UMD Banach spaces that admit a certain integral
representation in terms of the heat semigroup. Results in this spirit have
been implicitly used for a long time; see for examples the probabilistic
treatment of the Riesz transforms by Gundy and Varopoulos [37], and of the
Beurling–Ahlfors transform by Bañuelos and Méndez-Hernández [4]. Formulations
in the UMD-valued setting appear in [41, 42, 43]. The version below is
essentially a combination of some of these earlier results but it does not
appear as stated in the literature, so its proof is included here.

In what follows, we will use some aspects of the theory of vector-valued
stochastic integration with respect to a Brownian motion
$\\{B(t)=(B_{1}(t),\ldots,B_{n}(t))\\}_{t\in(0,\infty)}$ in $\mathbb{R}^{n}$,
starting at $0$. Here and below, a Brownian motion in $\mathbb{R}^{n}$ is
always understood to be a standard Brownian motion.

It should be noted that for our purposes it is enough to consider finite-
dimensional-valued functions, in which case the stochastic integrals can be
defined coordinate-wise in the classical sense. We refer to [51, Chapter 3]
for the relevant background (and much more). It might be helpful to note that
the formulae in [51], which often involve the _quadratic variation_ $\langle
M,N\rangle_{t}$ of two stochastic processes $M$ and $N$, take a simpler form
in the Brownian case of our interest, by using the identities $\langle
B_{i},B_{j}\rangle_{t}=\delta_{ij}t$ (see [51, Theorem 3.3.16]).

In particular, Itô’s formula (see [51, Theorem 3.3.6]) is valid in our
setting, since it holds true for each scalar-valued coordinate function. For a
comprehensive theory of vector-valued stochastic integration, whose full
strength is not needed here, see [78, 77].

Before stating Theorem 24 we describe some preliminary background and simple
estimates that will be used in its proof. First, we recall the following
decoupling inequalities due to Garling [32].

######  Theorem 22 (Garling’s decoupling inequalities).

For $n\in\mathbb{N}$, let
$\\{B(t)=(B_{1}(t),\ldots,B_{n}(t))\\}_{t\in(0,\infty)}$ be a Brownian motion
in $\mathbb{R}^{n}$, starting at $0$. Also, let
$\\{C(t)=(C_{1}(t),\ldots,C_{n}(t))\\}_{t\in(0,\infty)}$ be an independent
copy of $\\{B(t)\\}_{t\in(0,\infty)}$. Suppose that $(Y,\|\cdot\|_{Y})$ is a
UMD Banach space and $p\in(1,\infty)$. Let
$V=(V_{1},\ldots,V_{n}):(0,\infty)\to Y^{n}$ be a stochastic process that is
adapted to the same filtration as $\\{B(t))\\}_{t\in(0,\infty)}$, takes values
in a finite-dimensional subspace of $Y^{n}$, and satisfies

$\mathbb{E}\bigg{[}\Big{(}\int_{0}^{\infty}\sum_{j=1}^{n}\|V_{j}(t)\|_{Y}^{2}\mathrm{d}t\Big{)}^{\frac{p}{2}}\bigg{]}<\infty$
(53)

The finite dimensionality assumption and the integrability assumption (53)
guarantee the existence of the stochastic integrals below by the scalar-valued
theory. Then

$\mathbb{E}\bigg{[}\Big{\|}\int_{0}^{\infty}\sum_{j=1}^{n}V_{j}(t)\mathrm{d}B_{j}(t)\Big{\|}_{Y}^{p}\bigg{]}\leqslant\beta_{p}^{-}(Y)^{p}\mathbb{E}\bigg{[}\Big{\|}\int_{0}^{\infty}\sum_{j=1}^{n}V_{j}(t)\mathrm{d}C_{j}(t)\Big{\|}_{Y}^{p}\bigg{]},$

and

$\mathbb{E}\bigg{[}\Big{\|}\int_{0}^{\infty}\sum_{j=1}^{n}V_{j}(t)\mathrm{d}C_{j}(t)\Big{\|}_{Y}^{p}\bigg{]}\leqslant\beta_{p}^{+}(Y)^{p}\mathbb{E}\bigg{[}\Big{\|}\int_{0}^{\infty}\sum_{j=1}^{n}V_{j}(t)\mathrm{d}B_{j}(t)\Big{\|}_{Y}^{p}\bigg{]}.$

Continuing with the notation of Theorem 22, for every (operator-valued)
measurable111Here, and in what follows, given two Banach spaces
$(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ and an open subset
$\Omega\subseteq\mathbb{R}^{n}$, when we say that an operator-valued mapping
$\Phi:\Omega\to\mathscr{L}(X,Y)$ is measurable we mean measurability in the
strong operator topology, i.e., we require that for every $x\in X$ the mapping
$w\mapsto\Phi(w)x$ from $\Omega$ to $Y$ has the property that the inverse
image of every Borel subset of $Y$ is Lebesgue-measurable. mapping
$\Phi:(0,\infty)\to\mathscr{L}(Y,Y)$ we have

$\mathbb{E}\bigg{[}\Big{\|}\int_{0}^{\infty}\sum_{j=1}^{n}\Phi(t)V_{j}(t)\mathrm{d}C_{j}(t)\Big{\|}_{Y}^{p}\bigg{]}\leqslant\mathscr{R}_{p}\left(\Phi\right)^{p}\mathbb{E}\bigg{[}\Big{\|}\int_{0}^{\infty}\sum_{j=1}^{n}V_{j}(t)\mathrm{d}C_{j}(t)\Big{\|}_{Y}^{p}\bigg{]},$
(54)

where we use the notation

$\mathscr{R}_{p}(\Phi)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathscr{R}_{p}\left(\left\\{\Phi(t)\right\\}_{t\in(0,\infty)}\right).$
(55)

The estimate (54) follows directly from the definition of
$\mathscr{R}$-boundedness by approximating the integrals by Riemann sums; see
Exercise 4 in Section 9.4 of [76]. Alternatively, inequality (54) follows by
combining Theorem 6.14 and Theorem 9.13 of [76]. By (54) and Theorem 22 we see
that

$\mathbb{E}\bigg{[}\Big{\|}\int_{0}^{\infty}\sum_{j=1}^{n}\Phi(t)V_{j}(t)\mathrm{d}B_{j}(t)\Big{\|}_{Y}^{p}\bigg{]}\leqslant\beta_{p}^{+}(Y)^{p}\beta_{p}^{-}(Y)^{p}\mathscr{R}_{p}(\Phi)^{p}\mathbb{E}\bigg{[}\Big{\|}\int_{0}^{\infty}\sum_{j=1}^{n}V_{j}(t)\mathrm{d}B_{j}(t)\Big{\|}_{Y}^{p}\bigg{]}.$
(56)

In the same vein as the above discussion, by approximating the integrals by
Riemann sums it follows from (50) that if $\phi:(0,\infty)\to\mathbb{R}$ is
measurable then

$\mathbb{E}\bigg{[}\Big{\|}\int_{0}^{\infty}\sum_{j=1}^{n}\phi(t)V_{j}(t)\mathrm{d}B_{j}(t)\Big{\|}_{Y}^{p}\bigg{]}\\\
\leqslant\beta_{p}(Y)^{p}\|\phi\|_{L_{\infty}(0,\infty)}^{p}\mathbb{E}\bigg{[}\Big{\|}\int_{0}^{\infty}\sum_{j=1}^{n}V_{j}(t)\mathrm{d}B_{j}(t)\Big{\|}_{Y}^{p}\bigg{]}.$
(57)

We record for future use the following simple estimate.

######  Lemma 23.

Fix $n\in\mathbb{N}$ and let
$\\{B(t)=(B_{1}(t),\ldots,B_{n}(t))\\}_{t\in[0,\infty)}$ be a Brownian motion
in $\mathbb{R}^{n}$, starting at $0$. For every Banach space
$(Y,\|\cdot\|_{Y})$, every $p\in(1,\infty)$ and every smooth and compactly
supported $h:\mathbb{R}^{n}\to Y$ with a finite-dimensional range, we have

$\limsup_{\tau\to\infty}\bigg{(}\int_{\mathbb{R}^{n}}\mathbb{E}\bigg{[}\Big{\|}\sum_{j=1}^{n}\int_{0}^{\tau}\frac{\partial}{\partial
x_{j}}e^{\frac{\tau-t}{2}\Delta}h(B(t)+x)\mathrm{d}B_{j}(t)\Big{\|}_{Y}^{p}\bigg{]}\mathrm{d}x\bigg{)}^{\frac{1}{p}}\leqslant\|h\|_{L_{p}(\mathbb{R}^{n},Y)}.$
(58)

###### Proof.

If $\varphi\in L_{1}(\mathbb{R}^{n})$ then
$\mathbb{E}[\varphi(B(t)+x)]=e^{\frac{t}{2}\Delta}\varphi(x)$ for every
$t\in[0,\infty)$ and $x\in\mathbb{R}^{n}$. Hence,

$\int_{\mathbb{R}^{n}}\mathbb{E}[\varphi(B(t)+x)]\mathrm{d}x=\int_{\mathbb{R}^{n}}e^{\frac{t}{2}\Delta}\varphi(x)\mathrm{d}x=\int_{\mathbb{R}^{n}}\varphi(x)\mathrm{d}x.$
(59)

For $(x,t)\in\mathbb{R}^{n}\times(0,\infty)$, let
$u(x,t)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}e^{\frac{t}{2}\Delta}h(x)$
denote the heat extension of $h$. By Itô’s formula (see [51, Theorem 3.3.6])
applied to the function $t\mapsto u(B(t)+x,\tau-t)$, for every
$x\in\mathbb{R}^{n}$ we have

$\displaystyle\sum_{j=1}^{n}\int_{0}^{\tau}\frac{\partial u}{\partial
x_{j}}(B(t)+x,\tau-t)\mathrm{d}B_{j}(t)$
$\displaystyle=u(B(\tau)+x,0)-u(x,\tau)-\int_{0}^{\tau}\bigg{(}-\frac{\partial
u}{\partial t}+\frac{1}{2}\Delta_{x}u\bigg{)}(B(t)+x,\tau-t)\mathrm{d}t$ (60)
$\displaystyle=h(B(\tau)+x)-e^{\frac{\tau}{2}\Delta}h(x).$ (61)

Consequently,

$\displaystyle\bigg{(}\int_{\mathbb{R}^{n}}\mathbb{E}\bigg{[}\Big{\|}\sum_{j=1}^{n}\int_{0}^{\tau}\frac{\partial}{\partial
x_{j}}e^{\frac{\tau-t}{2}\Delta}h(B(t)+x)\mathrm{d}B_{j}(t)\Big{\|}_{Y}^{p}\bigg{]}\mathrm{d}x\bigg{)}^{\frac{1}{p}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:use
ito}}}{{\leqslant}}\left(\int_{\mathbb{R}^{n}}\mathbb{E}\left[\left\|h(B(\tau)+x)\right\|_{Y}^{p}\right]\mathrm{d}x\right)^{\frac{1}{p}}+\left(\int_{\mathbb{R}^{n}}\mathbb{E}\left[\left\|e^{\frac{\tau}{2}\Delta}h(x)\right\|_{Y}^{p}\right]\mathrm{d}x\right)^{\frac{1}{p}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:markov
property}}}{{=}}\|h\|_{L_{p}(\mathbb{R}^{n},Y)}+\left\|e^{\frac{\tau}{2}\Delta}h\right\|_{L_{p}(\mathbb{R}^{n},Y)}.$
(62)

To deduce the desired bound (58) from (62), it remains to note that

$\forall\,p\in(1,\infty),\qquad\lim_{\tau\to\infty}\|e^{\frac{\tau}{2}\Delta}h\|_{L_{p}(\mathbb{R}^{n},Y)}=0.$
(63)

Indeed, by Young’s inequality we have the following point-wise estimate

$\forall\,x\in\mathbb{R}^{n},\qquad\left\|e^{\frac{\tau}{2}\Delta}h(x)\right\|_{Y}=\left\|k_{\frac{\tau}{2}}*h(x)\right\|_{Y}\leqslant\left\|k_{\frac{\tau}{2}}\right\|_{L_{q}(\mathbb{R}^{n})}\|h\|_{L_{p}(\mathbb{R}^{n},Y)},$
(64)

where $k:\mathbb{R}^{n}\to\mathbb{R}$ is the heat kernel and $q=p/(p-1)$.
Since $q\in(0,\infty)$, the $L_{q}$-norm of the heat kernel $k_{\tau/2}$
converges to $0$ as $\tau\to\infty$. Moreover,
$\|e^{\frac{\tau}{2}\Delta}h\|_{Y}$ is dominated point-wise by the
Hardy–Littlewood maximal function $M\|h\|_{Y}\in L_{p}(\mathbb{R}^{n})$. Hence
(63) follows from (64) by an application of the dominated convergence theorem.
∎

The following theorem is the main result of the present section: it
establishes an estimate that will be used several times in the ensuing
discussion.

######  Theorem 24.

Fix $n\in\mathbb{N}$ and $p\in(1,\infty)$. Suppose that for $t\in(0,\infty)$
we are given a bounded operator $A(t):Y\to Y$ such that the mapping
$A:(0,\infty)\to\mathscr{L}(Y,Y)$ is bounded and measurable. Also, suppose
that $T:L_{p}(\mathbb{R}^{n},Y)\to L_{p}(\mathbb{R}^{n},Y)$ is a linear
operator that has the following dual representation. For every sufficiently
nice $f$ in a dense subspace of $L_{p}(\mathbb{R}^{n},Y)$, and $g^{*}$ in a
dense subspace of $L_{q}(\mathbb{R}^{n},Y^{*})$, where $q=p/(p-1)$, we have

$\int_{\mathbb{R}^{n}}g^{*}(x)\left(Tf(x)\right)\mathrm{d}x=\sum_{j=1}^{n}\int_{0}^{\infty}\int_{\mathbb{R}^{n}}\bigg{(}\frac{\partial}{\partial
x_{j}}e^{\frac{t}{2}\Delta}g^{*}(x)\bigg{)}\bigg{(}A(t)\frac{\partial}{\partial
x_{j}}e^{\frac{t}{2}\Delta}f(x)\bigg{)}\mathrm{d}x\mathrm{d}t.$ (65)

Then, recalling the notation (55),

$\|T\|_{L_{p}(\mathbb{R}^{n},Y)\to
L_{p}(\mathbb{R}^{n},Y)}\leqslant\beta_{p}^{+}(Y)\beta_{p}^{-}(Y)\mathscr{R}_{p}(A).$
(66)

Moreover, in the special case $A:(0,\infty)\to\mathbb{C}$ we have

$\|T\|_{L_{p}(\mathbb{R}^{n},Y)\to
L_{p}(\mathbb{R}^{n},Y)}\leqslant\beta_{p}(Y)\left\|A\right\|_{L_{\infty}(0,\infty)}.$
(67)

###### Proof.

By duality and the identity (65), the desired estimate (66) would follow if we
show that for every sufficiently nice $f\in L_{p}(\mathbb{R}^{n},Y)$ and
$g^{*}\in L_{q}(\mathbb{R}^{n},Y^{*})$,

$\limsup_{\tau\to\infty}\sum_{j=1}^{n}\int_{0}^{\tau}\int_{\mathbb{R}^{n}}\left(\frac{\partial}{\partial
x_{j}}e^{\frac{s}{2}\Delta}g^{*}(x)\right)\left(A(s)\frac{\partial}{\partial
x_{j}}e^{\frac{s}{2}\Delta}f(x)\right)\mathrm{d}x\mathrm{d}s\\\
\leqslant\beta_{p}^{+}(Y)\beta_{p}^{-}(Y)\mathscr{R}_{p}(A)\|f\|_{L_{p}(\mathbb{R}^{n},Y)}\|g^{*}\|_{L_{q}(\mathbb{R}^{n},Y^{*})}.$
(68)

Let us consider $f,g$ smooth and compactly supported, and taking values in
finite-dimensional subspaces of $Y$ and $Y^{*}$, respectively.

For every $(x,s)\in\mathbb{R}^{n}\times(0,\infty)$ denote for the sake of
simplicity

$\gamma_{j}^{*}(x,s)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\partial}{\partial
x_{j}}e^{\frac{s}{2}\Delta}g^{*}(x)\qquad\mathrm{and}\qquad\phi_{j}(x,s)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}A(s)\frac{\partial}{\partial
x_{j}}e^{\frac{s}{2}\Delta}f(x).$

Let $\\{B(t)=(B_{1}(t),\ldots,B_{n}(t))\\}_{t\in[0,\infty)}$ be a Brownian
motion in $\mathbb{R}^{n}$, starting at $0$. It follows from the identity (59)
that for every $\tau\in(0,\infty)$,

$\sum_{j=1}^{n}\int_{0}^{\tau}\int_{\mathbb{R}^{n}}\left(\frac{\partial}{\partial
x_{j}}e^{\frac{s}{2}\Delta}g^{*}(x)\right)\left(A(s)\frac{\partial}{\partial
x_{j}}e^{\frac{s}{2}\Delta}f(x)\right)\mathrm{d}x\mathrm{d}s\\\
=\int_{\mathbb{R}^{n}}\mathbb{E}\bigg{[}\sum_{j=1}^{n}\int_{0}^{\tau}\gamma_{j}^{*}(B(t)+x,\tau-t)\left(\phi_{j}(B(t)+x,\tau-t)\right)dt\bigg{]}\mathrm{d}x=\int_{\mathbb{R}^{n}}\mathbb{E}\left[G^{*}_{\tau}(x)(F_{\tau}(x))\right]\mathrm{d}x,$
(69)

where we introduce the notations

$F_{\tau}(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{j=1}^{n}\int_{0}^{\tau}A(\tau-t)\frac{\partial}{\partial
x_{j}}e^{\frac{\tau-t}{2}\Delta}f(B(t)+x)\mathrm{d}B_{j}(t),$ (70)

and

$G^{*}_{\tau}(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{j=1}^{n}\int_{0}^{\tau}\frac{\partial}{\partial
x_{j}}e^{\frac{\tau-t}{2}\Delta}g^{*}(B(t)+x)\mathrm{d}B_{j}(t).$ (71)

(69) is a well-known identity (see [51, Proposition 2.17]) for scalar-valued
functions, and it follows from this for the vector-valued functions with a
finite-dimensional range that we consider here.

By Hölder’s inequality it follows from (69) that

$\sum_{j=1}^{n}\int_{0}^{\tau}\int_{\mathbb{R}^{n}}\left(\frac{\partial}{\partial
x_{j}}e^{\frac{s}{2}\Delta}g^{*}(x)\right)\left(A(s)\frac{\partial}{\partial
x_{j}}e^{\frac{s}{2}\Delta}f(x)\right)\mathrm{d}x\mathrm{d}s\\\
\leqslant\left(\int_{\mathbb{R}^{n}}\mathbb{E}\left[\|F_{\tau}(x)\|_{Y}^{p}\right]\mathrm{d}x\right)^{\frac{1}{p}}\left(\int_{\mathbb{R}^{n}}\mathbb{E}\left[\|G_{\tau}^{*}(x)\|_{Y^{*}}^{q}\right]\mathrm{d}x\right)^{\frac{1}{q}}.$
(72)

Recalling (71), by Lemma 23 we have

$\limsup_{\tau\to\infty}\left(\int_{\mathbb{R}^{n}}\mathbb{E}\left[\|G_{\tau}^{*}(x)\|_{Y^{*}}^{q}\right]\mathrm{d}x\right)^{\frac{1}{q}}\leqslant\|g^{*}\|_{L_{q}(\mathbb{R}^{n},Y^{*})}.$
(73)

Recalling (70), it follows from (56) and (57) that if we set
$K=\beta_{p}^{+}(Y)\beta_{p}^{-}(Y)\mathscr{R}_{p}(A)$ if $Y\neq\mathbb{R}$,
and $K=\beta_{p}(Y)\|A\|_{L_{\infty}(0,\infty)}$ if $A$ is scalar-valued, then

$\int_{\mathbb{R}^{n}}\mathbb{E}\left[\|F_{\tau}(x)\|_{Y}^{p}\right]\mathrm{d}x\leqslant
K^{p}\mathbb{E}\bigg{[}\Big{\|}\sum_{j=1}^{n}\int_{0}^{\tau}\frac{\partial}{\partial
x_{j}}e^{\frac{\tau-t}{2}\Delta}f(B(t)+x)\mathrm{d}B_{j}(t)\Big{\|}_{Y}^{p}\bigg{]}.$

Another application of Lemma 23 now implies that

$\limsup_{\tau\to\infty}\left(\int_{\mathbb{R}^{n}}\mathbb{E}\left[\|F_{\tau}(x)\|_{Y}^{p}\right]\mathrm{d}x\right)^{\frac{1}{p}}\leqslant\|f\|_{L_{q}(\mathbb{R}^{n},Y)}.$
(74)

The desired estimate (68) is a consequence of (72), (73) and (74). ∎

###  4.2. Vector-valued multipliers

Let $(Y,\|\cdot\|_{Y})$ be a Banach space. The Fourier transform of $f\in
L_{1}(\mathbb{R}^{n},Y)$ will be denoted below by
$\mathcal{F}f:\mathbb{R}^{n}\to Y$, where we use the normalization

$\mathcal{F}f(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^{n}}e^{-i\langle
x,y\rangle}f(x)\mathrm{d}y.$

A possible formulation of Parseval’s identity in this vector-valued setting is
to say that for functions $f:\mathbb{R}^{n}\to Y$ and $g^{*}:\mathbb{R}^{n}\to
Y^{*}$ that are either smooth and compactly supported, or Fourier transforms
of such functions, we have

$\int_{\mathbb{R}^{n}}\mathcal{F}g^{*}(x)(\mathcal{F}f(x))\mathrm{d}x=\int_{\mathbb{R}^{n}}g^{*}(x)(f(-x))\mathrm{d}x.$
(75)

If $(X,\|\cdot\|_{X})$ is an additional Banach space and
$\mathfrak{m}:\mathbb{R}^{n}\to\mathscr{L}(X,Y)$ is measurable then the
multiplier associated to $\mathfrak{m}$ is defined as usual by considering for
every smooth and compactly supported $f:\mathbb{R}^{n}\to X$ (or the Fourier
transform of such a function) the function
$T_{\mathfrak{m}}f:\mathbb{R}^{n}\to Y$ given by

$T_{\mathfrak{m}}f\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(\mathcal{F}^{-1}\mathfrak{m})*f=\mathcal{F}^{-1}\left(x\mapsto\mathfrak{m}(x)\mathcal{F}f(x)\right).$

If $\mathfrak{m}$ is smooth and locally bounded at least away from the
coordinate hyperplanes, and the Fourier transform of $f:\mathbb{R}^{n}\to X$
is smooth and compactly supported away from these hyperplanes, then also
$T_{\mathfrak{m}}f$ has a smooth and compactly supported Fourier transform.
Hence, for such $f$ and smooth and compactly supported
$g^{*}:\mathbb{R}^{n}\to Y^{*}$ (or the Fourier transform of such a function),
Parseval’s identity applies and gives

$\int_{\mathbb{R}^{n}}g^{*}(x)(T_{\mathfrak{m}}f(x))\mathrm{d}x=\int_{\mathbb{R}^{n}}\mathcal{F}g^{*}(x)\left(\mathfrak{m}(-x)\mathcal{F}f(-x)\right)\mathrm{d}x.$
(76)

Also, under our choice of normalization of the Fourier transform,
$-\Delta=T_{\mathfrak{m}}$ for $\mathfrak{m}(x)=\|x\|_{2}^{2}$.

Theorem 24 can be used to bound the following multipliers, which arise as
Laplace transforms of $-\Delta$. (Another approach to such multipliers appears
in the recent survey [25, Section 2.2.1]; while it is presented there for
scalar-valued functions, it is based on principles that are valid in any UMD
space.) Suppose that $A:(0,\infty)\to\mathscr{L}(Y,Y)$ is measurable and
define $\alpha:\mathbb{R}\to\mathscr{L}(Y,Y)$ by

$\forall\,y\in
Y,\qquad\alpha(s)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}s\int_{0}^{\infty}e^{-st}A(t)y\mathrm{d}t.$

Then Theorem 24 applies to $\alpha(-\Delta)$, i.e., to the operator
$T_{\mathfrak{m}}$ where $\mathfrak{m}:\mathbb{R}^{n}\to\mathscr{L}(Y,Y)$ is
given by

$\forall(x,y)\in\mathbb{R}^{n}\times
Y,\qquad\mathfrak{m}(x)y=\alpha(\|x\|_{2}^{2})y=\|x\|_{2}^{2}\int_{0}^{\infty}e^{-t\|x\|_{2}^{2}}A(t)y\mathrm{d}t.$
(77)

Indeed, by Parseval’s identity (76), the representation (65) is a direct
consequence of (77). We therefore have the following dimension independent
bound, which holds true for every $p\in(1,\infty)$.

$\left\|\Delta\int_{0}^{\infty}e^{t\Delta}A(t)\mathrm{d}t\right\|_{L_{p}(\mathbb{R}^{n},Y)\to
L_{p}(\mathbb{R}^{n},Y)}\leqslant\beta_{p}(Y)^{2}\mathscr{R}_{p}(A),$

where we recall the notation (55). Also, if $A$ takes values in $\mathbb{C}$
then

$\left\|\Delta\int_{0}^{\infty}e^{t\Delta}A(t)\mathrm{d}t\right\|_{L_{p}(\mathbb{R}^{n},Y)\to
L_{p}(\mathbb{R}^{n},Y)}\leqslant\beta_{p}(Y)\|A\|_{L_{\infty}(0,\infty)}.$
(78)

Later we shall use (78) as a source of dimension-independent bounds for
multipliers that correspond to imaginary powers of the Laplacian.
Specifically, for every $s\in(0,\infty)$ and $u\in\mathbb{R}$,

$s^{iu}=s^{-(1-iu)}s=\frac{s}{\Gamma(1-iu)}\int_{0}^{\infty}t^{-iu}e^{-st}\mathrm{d}t.$

It therefore follows from (78) that

$\left\|(-\Delta)^{iu}\right\|_{L_{p}(\mathbb{R}^{n},Y)\to
L_{p}(\mathbb{R}^{n},Y)}\leqslant\frac{\beta_{p}(Y)}{|\Gamma(1-iu)|}\asymp\beta_{p}(Y)\frac{e^{|u|\arctan|u|}}{\sqrt{1+|u|}}\asymp\beta_{p}(Y)\frac{e^{\frac{\pi|u|}{2}}}{\sqrt{1+|u|}},$
(79)

where the penultimate step in (79) is a consequence of Stirling’s formula. For
ease of later reference, we record the bound that we have just proved as
Corollary 25 below. The reverse implication, i.e., that the boundedness
$(-\Delta)^{iu}$ on $L_{p}(\mathbb{R}^{n},Y)$ implies that $Y$ is UMD, is also
true; in fact it was pointed out in [41] that [36] implicitly contains the
estimate $\beta_{p}(Y)\leqslant\liminf_{u\to
0}\|(-\Delta)^{iu}\|_{L_{p}(\mathbb{R}^{n},Y)\to L_{p}(\mathbb{R}^{n},Y)}$.

######  Corollary 25.

Suppose that $p\in(1,\infty)$ and $(Y,\|\cdot\|_{Y})$ is a UMD Banach space.
Then for every $u\in\mathbb{R}$ and $n\in\mathbb{N}$ we have

$\left\|(-\Delta)^{iu}\right\|_{L_{p}(\mathbb{R}^{n},Y)\to
L_{p}(\mathbb{R}^{n},Y)}\lesssim\beta_{p}(Y)\frac{e^{\frac{\pi|u|}{2}}}{\sqrt{1+|u|}}.$

Our next corollary of Theorem 24 is a dimension-independent bound for a
multiplier that will be used in the ensuing proof of Theorem 5. We shall use
below the following integral representation.

$\forall(\theta,\alpha)\in(0,1)\times[0,\infty),\qquad\frac{1}{(1+\alpha)^{\theta}}=\frac{\sin(\pi\theta)}{\pi}\int_{0}^{1}\frac{\mathrm{d}s}{s^{1-\theta}(1-s)^{\theta}(1+\alpha
s)}.$ (80)

To verify the validity of (80), simply apply the change of variable
$s=t/(1+\alpha-\alpha t)$.

######  Corollary 26.

Fix $a\in(0,2]$ and $n\in\mathbb{N}$. Define
$\mathfrak{m}_{a}:\mathbb{R}^{n}\to\mathbb{R}$ by setting

$\forall,x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}\smallsetminus\\{0\\},\qquad\mathfrak{m}_{a}(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{|x_{1}|^{a}}{\|x\|_{2}^{a}}.$
(81)

Suppose that $(Y,\|\cdot\|_{Y})$ is a UMD Banach space and that
$p\in(1,\infty)$. Then

$\|T_{\mathfrak{m}_{a}}\|_{L_{p}(\mathbb{R}^{n},Y)\to
L_{p}(\mathbb{R}^{n},Y)}\leqslant\beta_{p}^{+}(Y)^{2}\beta_{p}^{-}(Y)\leqslant\beta_{p}(Y)^{3}.$
(82)

######  Remark 27.

When $Y=\mathbb{C}$ in Corollary 26, Bañuelos and Bogdan [3] obtained the
bound

$\|T_{\mathfrak{m}_{a}}\|_{L_{p}(\mathbb{R}^{n},\mathbb{R})\to
L_{p}(\mathbb{R}^{n},\mathbb{R})}\leqslant\max\left\\{p,\frac{p}{p-1}\right\\}-1.$
(83)

Note that $\beta_{p}(\mathbb{C})=\max\\{p,p/(p-1)\\}-1$, by a theorem of
Burkholder [16]. We are unable to recover the better estimate (83) for the
scalar-valued case of Corollary 26 using our method.

###### Proof of Corollary 26.

Write each $x\in\mathbb{R}^{n}$ as $x=(x_{1},x^{\prime})$, where
$x^{\prime}=(x_{2},\ldots,x_{n})\in\mathbb{R}^{n-1}$. For every $s\in(0,1]$
define $\gamma_{s}:\mathbb{R}^{n}\to\mathbb{R}$ by

$\forall\,x\in\mathbb{R}^{n},\qquad\gamma_{s}(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{x_{1}^{2}}{x_{1}^{2}+s\|x^{\prime}\|_{2}^{2}}=\int_{0}^{\infty}x_{1}^{2}e^{-(x_{1}^{2}+s\|x^{\prime}\|_{2}^{2})t}\mathrm{d}t.$
(84)

Then $\gamma_{1}=\mathfrak{m}_{2}$, and if $a\in(0,2)$ then by (80) with
$\theta=a/2\in(0,1)$ and $\alpha=\|x^{\prime}\|_{2}^{2}/x_{1}^{2}$ we have

$\mathfrak{m}_{a}=\int_{0}^{1}\gamma_{s}d\mu_{a}(s),$

where $\mu_{a}$ is the probability measure on $(0,1)$ whose density is
proportional to $s^{\frac{a}{2}-1}(1-s)^{-\frac{a}{2}}$. Therefore, in order
to prove the desired estimate (82) it suffices to show that for every
$s\in(0,1)$,

$\|T_{\gamma_{s}}\|_{L_{p}(\mathbb{R}^{n},Y)\to
L_{p}(\mathbb{R}^{n},Y)}\leqslant\beta_{p}^{+}(Y)^{2}\beta_{p}^{-}(Y).$

Consider the UMD Banach space
$Z\stackrel{{\scriptstyle\mathrm{def}}}{{=}}L_{p}(\mathbb{R}^{n-1},Y)$. By the
identification $L_{p}(\mathbb{R}^{n},Y)\cong L_{p}(\mathbb{R},Z)$, the
multiplier $T_{\gamma_{s}}$ can be thought of as an operator from
$L_{p}(\mathbb{R},Z)$ to $L_{p}(\mathbb{R},Z)$; this is how Theorem 24 will be
applied next, i.e., with $Y$ replaced by $Z$, while noting that
$\beta_{p}^{\pm}(Z)=\beta_{p}^{\pm}(Y)$.

We consider a test function $f:\mathbb{R}^{n}\to Y$ that is finite linear
combination of functions of the form $x\mapsto f_{1}(x_{1})f_{2}(x_{2})\cdots
f_{n}(x_{n})y$, where $y\in Y$ and each $f_{i}$ has a smooth Fourier
transform, compactly supported away from $0$, and a similar function
$g^{*}:\mathbb{R}^{n}\to Y^{*}$. Note that such functions are dense in
$L_{p}(\mathbb{R},Z)\cong L_{p}(\mathbb{R}^{n},Y)$ and in
$L_{q}(\mathbb{R},Z^{*})=L_{q}(\mathbb{R},L_{q}(\mathbb{R}^{n-1},Y^{*}))\cong
L_{q}(\mathbb{R}^{n},Y^{*})$, respectively, where $q=p/(p-1)$. Then by the
Parseval identity (76) we have

$\int_{\mathbb{R}}g^{*}(x_{1})(T_{\gamma_{s}}f(x_{1}))\mathrm{d}x_{1}=\int_{\mathbb{R}^{n}}g^{*}(x)(T_{\gamma_{s}}f(x))\mathrm{d}x=\int_{\mathbb{R}^{n}}\mathcal{F}g^{*}(x)\left(\frac{x_{1}^{2}}{x_{1}^{2}+s\|x^{\prime}\|_{2}^{2}}\mathcal{F}f(-x)\right)\mathrm{d}x.$

Consequently, by the second equality in (84),

$\int_{\mathbb{R}}g^{*}(x_{1})(T_{\gamma_{s}}f(x_{1}))\mathrm{d}x_{1}=\int_{0}^{\infty}\int_{\mathbb{R}^{n}}x_{1}e^{-\frac{t}{2}x_{1}^{2}}\mathcal{F}g^{*}(x)\left(e^{-ts\|x^{\prime}\|_{2}^{2}}x_{1}e^{-\frac{t}{2}x_{1}^{2}}\mathcal{F}f(-x)\right)\mathrm{d}x\mathrm{d}t\\\
=\int_{0}^{\infty}\int_{\mathbb{R}^{n}}\mathcal{F}\left(\frac{\partial}{\partial
x_{1}}e^{\frac{t}{2}\left(\frac{\partial}{\partial
x_{1}}\right)^{2}}g^{*}\right)(x)\left(\mathcal{F}\left(-e^{ts\Delta^{\prime}}\frac{\partial}{\partial
x_{1}}e^{\frac{t}{2}\left(\frac{\partial}{\partial
x_{1}}\right)^{2}}f\right)\right)(-x)\mathrm{d}x\mathrm{d}t,$

where $\Delta^{\prime}$ denotes the Laplacian on $\mathbb{R}^{n-1}$, i.e.,
with respect to the variable $x^{\prime}$. By the vector-valued Parseval
identity (75), we therefore have

$\int_{\mathbb{R}}g^{*}(x_{1})(T_{\gamma_{s}}f(x_{1}))\mathrm{d}x_{1}=\int_{0}^{\infty}\int_{\mathbb{R}}\left(\frac{\partial}{\partial
x_{1}}e^{\frac{u}{2}\left(\frac{\partial}{\partial
x_{1}}\right)^{2}}g^{*}(x_{1})\right)\left(A(u)\frac{\partial}{\partial
x_{1}}e^{\frac{u}{2}\left(\frac{\partial}{\partial
x_{1}}\right)^{2}}f\right)(x_{1})\mathrm{d}x_{1}\mathrm{d}u,$

where
$A(u)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}-e^{us\Delta^{\prime}}:L_{p}(\mathbb{R},Z)\to
L_{p}(\mathbb{R},Z)$. Recalling Corollary 21, it therefore follows from
Theorem 24 that

$\|T_{\gamma_{s}}\|_{L_{p}(\mathbb{R}^{n},Y)\to
L_{p}(\mathbb{R}^{n},Y)}=\|T_{\gamma_{s}}\|_{L_{p}(\mathbb{R},Z)\to
L_{p}(\mathbb{R},Z)}\\\
\leqslant\beta_{p}^{+}(Z)\beta_{p}^{-}(Z)\mathscr{R}_{p}\left(\left\\{e^{t\Delta^{\prime}}:L_{p}(\mathbb{R}^{n-1},Y)\to
L_{p}(\mathbb{R}^{n-1},Y)\right\\}_{t\in(0,\infty)}\right)\leqslant\beta_{p}^{+}(Y)^{2}\beta_{p}^{-}(Y).$

###  4.3. Littlewood–Paley decomposition

We need to introduce notation for the usual multi-scale bump functions that
occur in Littlewood–Paley decompositions. Let $\phi:\mathbb{R}\to[0,\infty)$
be smooth and supported on $[-2,-1/2]\cup[1/2,2]$; for concreteness we can
take

$\forall\,x\in\mathbb{R},\qquad\phi(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\\{\begin{array}[]{ll}e^{-\frac{1}{(|x|-1/2)(2-|x|)}}&\mathrm{if}\
|x|\in(1/2,2),\\\ 0&\mathrm{if}\
|x|\in[0,1/2]\cup[2,\infty).\end{array}\right.$ (85)

For $k\in{{\mathbb{Z}}}$ define $\psi_{k}:\mathbb{R}\to\mathbb{R}$ by

$\forall\,x\in\mathbb{R},\qquad\psi_{k}(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\phi(2^{k}x)}{\sum_{j\in{{\mathbb{Z}}}}\phi(2^{j}x)}.$
(86)

We also define $\omega_{k}:\mathbb{R}\to\mathbb{R}$ by

$\forall\,x\in\mathbb{R},\qquad\omega_{k}(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\psi_{k-1}(x)+\psi_{k}(x)+\psi_{k+1}(x).$
(87)

Thus $\omega_{k}\psi_{k}=\psi_{k}$, and therefore the corresponding
multipliers satisfy the identity $T_{\omega_{k}}T_{\psi_{k}}=T_{\psi_{k}}$.

For every $k\in{{\mathbb{Z}}}$ define $\vartheta_{k}:\mathbb{R}\to[0,\infty)$
by

$\forall\,x\in\mathbb{R},\qquad\vartheta_{k}(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\sin^{4}(2^{k}x)}{(2^{k}x)^{2}}.$
(88)

Like $\psi_{k}$, the function $\vartheta_{k}$ is roughly localized around
$|x|\asymp 2^{-k}$, but unlike $\psi_{k}$, it has long tails that are
supported over all of $\mathbb{R}$. The importance of the special “pseudo-bump
functions” $\\{\vartheta_{k}\\}_{k\in{{\mathbb{Z}}}}$ stems from the fact that
they can be directly related to averages of dyadic martingales, which leads to
the following form of the Littlewood–Paley inequality with a good constant.

######  Proposition 28.

Suppose that $p\in(1,\infty)$ and let $(Y,\|\cdot\|_{Y})$ be a UMD Banach
space. Then

$\forall\,f\in
L_{p}(\mathbb{R},Y),\qquad\bigg{(}\mathbb{E}_{\varepsilon\in\\{-1,1\\}^{{{\mathbb{Z}}}}}\Big{\|}\sum_{j\in{{\mathbb{Z}}}}\varepsilon_{j}T_{\vartheta_{j}}f\Big{\|}_{L_{p}(\mathbb{R},Y)}^{p}\bigg{)}^{\frac{1}{p}}\leqslant\beta_{p}^{+}(Y)\|f\|_{L_{p}(\mathbb{R},Y)}\leqslant\beta_{p}(Y)\|f\|_{L_{p}(\mathbb{R},Y)}.$

Proposition 28 is due implicitly to Bourgain [12], where it is proved as an
intermediate step towards a more usual form of the Littlewood–Paley inequality
involving the localized bump functions $\\{\psi_{k}\\}_{k\in{{\mathbb{Z}}}}$
in place of $\\{\vartheta_{k}\\}_{k\in{{\mathbb{Z}}}}$, but with a more
complicated dependence on the UMD constant. The above formulation of
Proposition 28 appears explicitly as the special case
$h=k=\mathbf{1}_{[0,1/2]}-\mathbf{1}_{[1/2,1]}$ of Proposition 5.10 in [40]
(where we are using here the notation of [40, Proposition 5.10]).

In subsequent arguments we shall use Proposition 28 in addition to some
auxiliary estimates concerning the bump functions $\\{\psi_{k}\\}_{k\in Z}$,
which are valid for arbitrary Banach space targets. These estimates rely on
the fact that if $\mathfrak{m}\in L_{1}(\mathbb{R})$ then by Young’s
inequality we have

$\forall\,f\in
L_{p}(\mathbb{R},Y),\qquad\|T_{\mathfrak{m}}f\|_{L_{p}(\mathbb{R},Y)}=\left\|\left(\mathcal{F}^{-1}\mathfrak{m}\right)*f\right\|_{L_{p}(\mathbb{R},Y)}\leqslant\left\|\mathcal{F}^{-1}\mathfrak{m}\right\|_{L_{1}(\mathbb{R})}\|f\|_{L_{p}(\mathbb{R},Y)}.$
(89)

######  Lemma 29.

Suppose that $p\in[1,\infty]$ and that $(Y,\|\cdot\|_{Y})$ is a Banach space.
Then for every $k\in{{\mathbb{Z}}}$,

$\forall\,f\in
L_{p}(\mathbb{R},Y),\qquad\|T_{\psi_{k}}f\|_{L_{p}(\mathbb{R},Y)}\lesssim\|T_{\vartheta_{k}}f\|_{L_{p}(\mathbb{R},Y)}.$
(90)

###### Proof.

Since for every $x\in\mathbb{R}$ and $k\in{{\mathbb{Z}}}$ we have
$\vartheta_{k}(x)=\vartheta_{0}(2^{k}x)$ and $\psi_{k}(x)=\psi_{0}(2^{k}x)$,
we also have $T_{\vartheta_{k}}f(x)=T_{\vartheta_{0}}(y\mapsto
f(2^{k}y))(2^{-k}x)$ and $T_{\psi_{k}}f(x)=T_{\psi_{0}}(y\mapsto
f(2^{k}y))(2^{-k}x)$. Consequently, it suffices to prove (90) when $k=0$.
Since $\vartheta_{0}$ is nonzero on the support of $\psi_{0}$, we can write
$T_{\psi_{0}}f=T_{\psi_{0}/\vartheta_{0}}T_{\theta_{0}}f$. The function
$\psi_{0}/\vartheta_{0}$ is smooth and compactly supported, therefore its
inverse Fourier transform $g=\mathcal{F}^{-1}(\psi_{0}/\vartheta_{0})$ belongs
to the Schwartz class of test functions $\mathscr{S}(\mathbb{R})$, and in
particular $g\in L_{1}(\mathbb{R})$. Thus (90) follows from (89) with
$\mathfrak{m}=\psi_{0}/\vartheta_{0}$ and $T_{\vartheta_{0}}f$ in place of
$f$. ∎

In order to facilitate the next two applications of (89), we record the
following simple observation. If $\mathfrak{m}\in L_{1}(\mathbb{R})$ is smooth
then for every $a\in(0,\infty)$ we have

$\left\|\mathcal{F}^{-1}\mathfrak{m}\right\|_{L_{1}(\mathbb{R})}\leqslant\bigg{(}\int_{\mathbb{R}}\frac{\mathrm{d}x}{1+(ax)^{2}}\bigg{)}\sup_{x\in\mathbb{R}}\left|(1+a^{2}x^{2})\left(\mathcal{F}^{-1}\mathfrak{m}\right)(x)\right|\\\
=\frac{\pi}{a}\left\|\mathcal{F}^{-1}(\mathfrak{m}-a^{2}\mathfrak{m}^{\prime\prime})\right\|_{L_{\infty}(\mathbb{R})}\leqslant\frac{\pi}{a}\left\|\mathfrak{m}-a^{2}\mathfrak{m}^{\prime\prime}\right\|_{L_{1}(\mathbb{R})}\leqslant\frac{\pi}{a}\left\|\mathfrak{m}\right\|_{L_{1}(\mathbb{R})}+\pi
a\left\|\mathfrak{m}^{\prime\prime}\right\|_{L_{1}(\mathbb{R})}.$ (91)

Choosing
$a=\sqrt{\|\mathfrak{m}\|_{L_{1}(\mathbb{R})}/\|\mathfrak{m}^{\prime\prime}\|_{L_{1}(\mathbb{R})}}$
in (91) and substituting the resulting estimate into (89) yields

$\forall\,f\in
L_{p}(\mathbb{R},Y),\qquad\|T_{\mathfrak{m}}f\|_{L_{p}(\mathbb{R},Y)}\leqslant
2\pi\sqrt{\|\mathfrak{m}\|_{L_{1}(\mathbb{R})}\|\mathfrak{m}^{\prime\prime}\|_{L_{1}(\mathbb{R})}}\cdot\|f\|_{L_{p}(\mathbb{R},Y)}.$
(92)

######  Lemma 30.

Fix $k\in{{\mathbb{Z}}}$ and $p\in[1,\infty]$. Let $(Y,\|\cdot\|_{Y})$ be a
Banach space. Then for every $f\in L_{p}(\mathbb{R},Y)$ and $y\in\mathbb{R}$
we have

$\left(\int_{\mathbb{R}}\left\|T_{\psi_{k}}f(x+y)-T_{\psi_{k}(x)}f(x)\right\|_{Y}^{p}\mathrm{d}x\right)^{\frac{1}{p}}\lesssim\min\left\\{1,\frac{|y|}{2^{k}}\right\\}\left\|T_{\psi_{k}}f\right\|_{L_{p}(\mathbb{R},Y)}.$
(93)

###### Proof.

Write
$g(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}T_{\psi_{k}}f(x+y)-T_{\psi_{k}}f(x)$.
The fact that the norm of $g$ in $L_{p}(\mathbb{R},Y)$ is at most twice the
norm of $T_{\psi_{k}}f$ in $L_{p}(\mathbb{R},Y)$ follows from the triangle
inequality in $L_{p}(\mathbb{R},Y)$. For the second estimate in (93), note
that $g=T_{\rho_{y}\omega_{k}}T_{\psi_{k}}f$, where $\rho_{y}(x)=e^{-ixy}-1$.
Hence, it remains to show that for every $h\in L_{p}(\mathbb{R},Y)$ and
$y\in[-2^{k},2^{k}]$ we have
$\|T_{\rho_{y}\omega_{k}}h\|_{L_{p}(\mathbb{R},Y)}\lesssim|y|2^{-k}\|h\|_{L_{p}(\mathbb{R},Y)}$.
By (92), this will follow if we prove that
$\|\rho_{y}\omega_{k}\|_{L_{1}(\mathbb{R})}\cdot\|(\rho_{y}\omega_{k})^{\prime\prime}\|_{L_{1}(\mathbb{R})}\lesssim(2^{-k}y)^{2}$.
Since $\rho_{y}(x)\omega_{k}(x)=(\rho_{2^{-k}y}\omega_{0})(2^{k}y)$ and for
every smooth $\mathfrak{m}\in L_{1}(\mathbb{R})$ the product
$\|x\mapsto\mathfrak{m}(\lambda
x)\|_{L_{1}(\mathbb{R})}\cdot\|(x\mapsto\mathfrak{m}(\lambda
x))^{\prime\prime}\|_{L_{1}(\mathbb{R})}$ is independent of
$\lambda\in(0,\infty)$, it suffices to show that

$\forall\,z\in[-1,1],\qquad\|\rho_{z}\omega_{0}\|_{L_{1}(\mathbb{R})}\lesssim|z|\qquad\mathrm{and}\qquad\|(\rho_{z}\omega_{0})^{\prime\prime}\|_{L_{1}(\mathbb{R})}\lesssim|z|.$
(94)

The point-wise estimate $|\rho_{z}(x)|\leqslant|zx|$ combined with the fact
that the function $x\mapsto x\omega_{0}(x)$ is in $L_{1}(\mathbb{R})$ implies
the first assertion in (94). For the second assertion in (94), compute
directly that

$(\rho_{z}\omega_{0})^{\prime\prime}(x)=-z^{2}e^{-ixz}\omega_{0}(x)-i2ze^{-ixz}\omega_{0}^{\prime}(x)+(e^{-ixz}-1)\omega_{0}^{\prime\prime}(x).$

We therefore have the following point-wise estimate, which holds true whenever
$|z|\leqslant 1$.

$|(\rho_{z}\omega_{0})^{\prime\prime}(x)|\leqslant|z|^{2}|\omega_{0}(x)|+2|z\omega_{0}^{\prime}(x)|+|zx\omega_{0}^{\prime\prime}(x)|\leqslant|z|\big{(}|\omega_{0}(x)|+2|\omega_{0}^{\prime}(x)|+|x\omega_{0}^{\prime\prime}(x)|\big{)}.$

The second assertion in (94) is now a consequence of the fact that the three
functions $\omega_{0},\omega_{0}^{\prime}$ and $x\mapsto
x\omega_{0}^{\prime\prime}(x)$ all belong to $L_{1}(\mathbb{R})$. This
concludes the proof of Lemma 30∎

######  Lemma 31.

Fix $k\in{{\mathbb{Z}}}$, $p\in[1,\infty]$ and $\alpha\in(0,\infty)$. Let
$(Y,\|\cdot\|_{Y})$ be a Banach space. Then for every smooth and compactly
supported $f:\mathbb{R}\to Y$ we have

$\left\|T_{\psi_{k}}f\right\|_{L_{p}(\mathbb{R},Y)}\lesssim(1+\alpha)2^{2\alpha(k+2)}\left\|T_{\psi_{k}}(-\Delta)^{\alpha}f\right\|_{L_{p}(\mathbb{R},Y)}.$
(95)

###### Proof.

Recalling the definition of $\omega_{k}:\mathbb{R}\to[0,\infty)$ that is given
in (87), since $T_{\omega_{k}}T_{\psi_{k}}=T_{\psi_{k}}$ we have
$T_{\psi_{k}}f=(-\Delta)^{-\alpha}T_{\omega_{k}}T_{\psi_{k}}(-\Delta)^{\alpha}f=T_{\xi_{k}}T_{\psi_{k}}(-\Delta)^{\alpha}f$,
where

$\forall\,x\in\mathbb{R}\smallsetminus\\{0\\},\qquad\xi_{k}(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\omega_{k}(x)}{|x|^{2\alpha}}=2^{2k\alpha}\frac{\omega_{0}(2^{k}x)}{|2^{k}x|^{2\alpha}}=2^{2k\alpha}\xi_{0}(2^{k}x).$

It therefore suffices to show that $\xi_{0}$ is the Fourier transform of a
function in $L_{1}(\mathbb{R})$ of norm at most a constant multiple of
$(1+\alpha)2^{4\alpha}$. To this end, we will again apply the estimate (92).
First, since $|x|^{-1}\leqslant 4$ on the support of $\omega_{0}$ and
$\omega_{0}\in L_{1}(\mathbb{R})$, we have
$\|\xi_{0}\|_{L_{1}(\mathbb{R})}\lesssim 2^{4\alpha}$. Also,

$\forall\,x\in\mathbb{R}\smallsetminus\\{0\\},\qquad\xi_{0}^{\prime\prime}(x)=\frac{2\alpha(2\alpha+1)}{|x|^{2\alpha}}\cdot\frac{\omega_{0}(x)}{x^{2}}-\frac{4\alpha}{|x|^{2\alpha}}\cdot\frac{\omega_{0}^{\prime}(x)}{|x|}+\frac{\omega_{0}^{\prime\prime}(x)}{|x|^{2\alpha}}.$

Using again that $|x|^{-1}\leqslant 4$ on the support of $\omega_{0}$,
combined with the fact that the three functions $\omega_{0}^{\prime\prime}$,
$x\mapsto\omega_{0}(x)/x^{2}$ and $x\mapsto\omega_{0}^{\prime}(x)/x$ are all
in $L_{1}(\mathbb{R})$, it follows that
$\|\xi_{0}^{\prime\prime}\|_{L_{1}(\mathbb{R})}\lesssim(2\alpha+1)^{2}2^{4\alpha}$.
So,

$\|T_{\xi_{0}}\|_{L_{p}(\mathbb{R},Y)\to
L_{p}(\mathbb{R},Y)}\stackrel{{\scriptstyle\eqref{eq:easy
multiplier2}}}{{\lesssim}}\sqrt{\|\xi_{0}\|_{L_{1}(\mathbb{R})}\|\xi_{0}^{\prime\prime}\|_{L_{1}(\mathbb{R})}}\leqslant\sqrt{4^{2\alpha}\cdot(2\alpha+1)^{2}4^{2\alpha}}\lesssim(1+\alpha)4^{2\alpha}.\qed$

###  4.4. Type and Cotype

For $p\in[1,2)$ and $q\in[2,\infty)$, the type $p$ constant and the cotype $q$
constant of a Banach space $(Y,\|\cdot\|_{Y})$, denoted $T_{p}(X)$ and
$C_{q}(Y)$, respectively, are defined to be the infimum over those
$T,C\in[1,\infty]$ such that for every $n\in\mathbb{N}$ and every
$x_{1},\ldots,x_{n}\in Y$,

$\frac{1}{C}\Big{(}\sum_{j=1}^{n}\|x_{j}\|_{Y}^{q}\Big{)}^{\frac{1}{q}}\leqslant\mathbb{E}\bigg{[}\Big{\|}\sum_{j=1}^{n}\varepsilon_{j}x_{j}\Big{\|}_{Y}\bigg{]}\leqslant
T\Big{(}\sum_{j=1}^{n}\|x_{j}\|_{Y}^{p}\Big{)}^{\frac{1}{p}},$ (96)

where the expectation is with respect to
$\varepsilon=(\varepsilon_{1},\ldots,\varepsilon_{n})\in\\{-1,1\\}^{n}$ chosen
uniformly at random. The smallest $T,C\in(0,\infty]$ for which (96) holds true
are denoted $T_{p}(Y),C_{q}(Y)$, respectively. Any UMD Banach space
$(Y,\|\cdot\|_{Y})$ admits an equivalent uniformly convex norm [61, 1], and
hence it has finite cotype [29, 66]. The following lemma makes this
qualitative statement quantitative in the case of cotype. A similar (and
simpler) argument yields a quantitative bound in the case of type as well (see
Remark 33 below), but in what follows only the case of cotype will be used.

######  Lemma 32 (Cotype in terms of $\beta(Y)$).

There exists a universal constant $\kappa\in(1,\infty)$ such that for every
$\beta\in[1,\infty)$, if $(Y,\|\cdot\|_{Y})$ is a UMD Banach space with
$\beta(Y)\leqslant\beta$ then then $(Y,\|\cdot\|_{Y})$ it has cotype
$\kappa\beta$, and moreover $C_{\kappa\beta}(Y)\leqslant\kappa$.

###### Proof.

The proof below is a (somewhat tedious) combination of several results that
appear in the literature. The key step is an examination of the proof of
Pisier’s quantitative version [68] of the Maurey–Pisier theorem [63] for
stable type. For $p\in(1,2)$ let $ST_{p}(Y^{*})$ be the stable type $p$
constant of the dual space $Y^{*}$. Namely, $ST_{p}(Y^{*})$ is the infimum
over those $S\in(0,\infty]$ such that for every $n\in\mathbb{N}$, every
$x_{1}^{*},\ldots,x_{n}^{*}\in Y^{*}$ satisfy

$\mathbb{E}\bigg{[}\Big{\|}\sum_{j=1}^{n}\theta_{j}x_{j}^{*}\Big{\|}_{Y^{*}}\bigg{]}\leqslant
S\Big{(}\sum_{j=1}^{n}\left\|x_{j}^{*}\right\|_{Y^{*}}^{p}\Big{)}^{\frac{1}{p}},$
(97)

where in (97) the expectation is with respect to i.i.d. standard symmetric
$p$-stable random variables $\\{\theta_{j}\\}_{j=1}^{n}$, i.e., the
characteristic function of $\theta_{1}$ is

$\forall\,t\in\mathbb{R},\qquad\mathbb{E}\left[e^{it\theta_{1}}\right]=e^{-|t|^{p}}.$
(98)

It follows from (98) that $\mathbb{E}\left[|\theta_{1}|\right]\asymp 1/(p-1)$;
see e.g. [28, Sec. XVII]. By Jensen’s inequality and Kahane’s inequality, this
implies that $T_{p}(Y^{*})\leqslant(p-1)ST_{p}(Y^{*})$. Since
$C_{q}(Y)\leqslant T_{p}(Y^{*})$, where $q=p/(p-1)$ (see e.g. [62, Sec. 6]),
we deduce that

$ST_{p}(Y^{*})\gtrsim\frac{C_{q}(Y)}{p-1}\asymp qC_{q}(Y).$ (99)

Suppose from now on that $p\in(1,3/2]$ (in the argument below we only use that
$p$ is bounded away from $2$ by a universal constant). Equivalently,
$q\in[3,\infty)$. It follows from the proof of the main theorem of [68] that
there exists a universal constant $c\in(0,1)$ such that if $m\in\mathbb{N}$
satisfies

$m^{\frac{1}{q}}\leqslant\frac{c}{q}ST_{p}(Y^{*})\leqslant
c(p-1)ST_{p}(Y^{*}).$ (100)

then $Y^{*}$ contains a $2$-isomorphic copy of $\ell_{p}^{m}$. Unfortunately,
while the dependence of $m$ on $ST_{p}(Y^{*})$ that is stated in (100) is also
stated explicitly in [68], the dependence on $p$, which is crucial for us
here, is not computed in [68]. However, one can verify (100) by examining the
dependencies on $p$ of certain constants that appear in [68], and substituting
these dependencies into the proof of [68]. Specifically, the constant $C_{p}$
of Proposition 1.3 of [68] was computed in [58] to be

$C_{p}=\left(\frac{1}{\int_{0}^{\infty}\frac{\sin
v}{v^{p}}dv}\right)^{\frac{1}{p}}=2\left(\frac{\Gamma\left(\frac{p+1}{2}\right)}{\sqrt{\pi}\Gamma\left(1-\frac{p}{2}\right)}\right)^{\frac{1}{p}}.$

Thus, recalling that $p\in(1,3/2)$, we see that $C_{p}$ is bounded above and
below by positive universal constants. The parameter $\Phi$ of Lemma 1.4 of
[68] can be estimated via a direct computation (e.g., using the last line of
page 975 of [31]) to give $\Phi\lesssim 1/(p-1)$. The proof of [68] uses only
two additional unspecified parameters, denoted $K$ and $\eta$, that appear in
Lemma 1.5 of [68]. In Proposition 2 of [47] it is shown that one can take
$K=2$ and $\eta=(2-p)/(8p(q+1)^{q})$. A direct substitution of these estimates
into the proof of Theorem 2.1 of [68] now yields (100).

Note that

$\beta(\ell_{p}^{m})\gtrsim\min\left\\{q,\left(\log
m\right)^{\frac{1}{p}}\right\\}=\min\left\\{q,\left(\log
m\right)^{1-\frac{1}{q}}\right\\}.$ (101)

While (101) is folklore, we did not find it in the literature so we briefly
sketch the relevant computation. Let $k\in\mathbb{N}$ be the largest integer
such that $2^{k}\leqslant m$. Let $\mu$ be the uniform probability measure on
the discrete hypercube $\\{-1,1\\}^{k}$, and think of $\ell_{p}^{m}$ as
containing an isometric copy of $L_{p}(\mu)$. Define
$M_{0},\ldots,M_{k}:\\{-1,1\\}^{k}\to L_{p}(\mu)$ by setting $M_{0}\equiv 1$
and for $j\in\\{1,\ldots,k\\}$ defining

$\forall\,\varepsilon,\delta\in\\{-1,1\\}^{k},\qquad
M_{j}(\varepsilon)(\delta)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\prod_{\ell=1}^{k}(1+\varepsilon_{\ell}\delta_{\ell})=2^{k}\mathbf{1}_{\\{(\varepsilon_{1},\ldots,\varepsilon_{j})=(\delta_{1},\ldots,\delta_{j})\\}}.$

Then $\\{M_{j}\\}_{j=0}^{k}$ is a martingale with respect to the natural
coordinate filtration of $\\{-1,1\\}^{k}$.

Observe that

$\bigg{(}\int_{\\{-1,1\\}^{k}}\left\|M_{k}(\varepsilon)\right\|_{L_{p}(\mu)}^{2}\mathrm{d}\mu(\varepsilon)\bigg{)}^{\frac{1}{2}}=2^{\frac{k(p-1)}{p}}=2^{\frac{k}{q}}.$
(102)

For every $\varepsilon,\delta\in\\{-1,1\\}^{k}$ write
$j(\varepsilon,\delta)=0$ if $\varepsilon_{1}\neq\delta_{1}$ and otherwise let
$j(\varepsilon,\delta)$ be the largest $j\in\\{1,\ldots,k\\}$ such that
$\varepsilon_{i}=\delta_{i}$ for all $i\in\\{1,\ldots,j\\}$. With this
notation,

$S(\varepsilon,\delta)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Big{(}\sum_{j=1}^{k}\left(M_{j}(\varepsilon)(\delta)-M_{j-1}(\varepsilon)(\delta)\right)^{2}\Big{)}^{\frac{1}{2}}\asymp
2^{j(\varepsilon,\delta)}-1.$ (103)

Note that for every $j\in\\{0,\ldots,k\\}$ we have

$\mu\times\mu\left(\left\\{(\varepsilon,\delta)\in\\{-1,1\\}^{k}\times\\{-1,1\\}^{k}:\
j(\varepsilon,\delta)=j\right\\}\right)\asymp\frac{1}{2^{j}}.$ (104)

Hence, using the triangle inequality in $L_{p}(\mu)$ and Khinchine’s
inequality, we have

$\bigg{(}\int_{\\{-1,1\\}^{k}}\int_{\\{-1,1\\}^{k}}\Big{\|}\sum_{j=1}^{k}\eta_{j}\left(M_{j}(\varepsilon)-M_{j-1}(\varepsilon)\right)\Big{\|}_{L_{p}(\mu)}^{2}\mathrm{d}\mu(\varepsilon)\mathrm{d}\mu(\eta)\bigg{)}^{\frac{1}{2}}\\\
\gtrsim\bigg{(}\int_{\\{-1,1\\}^{k}}\int_{\\{-1,1\\}^{k}}S(\varepsilon,\delta)^{p}\mathrm{d}\mu(\varepsilon)\mathrm{d}\mu(\delta)\bigg{)}^{\frac{1}{p}}\stackrel{{\scriptstyle\eqref{eq:S
lower}\wedge\eqref{eq:product measure estimate
j}}}{{\gtrsim}}\Big{(}\sum_{j=1}^{k}2^{j(p-1)}\Big{)}^{\frac{1}{p}}\asymp
2^{\frac{k}{q}}\cdot\min\left\\{q,k^{\frac{1}{p}}\right\\},$

which, when contrasted with (102), implies (101).

A combination of (99), (100) and (101) implies that there exist universal
$a,b\in(0,1/2)$ such that

$\forall\,q\in[3,\infty),\qquad\beta(X)\geqslant
a\min\left\\{q,\big{(}q\log\left(1+bC_{q}(X)\right)\big{)}^{1-\frac{1}{q}}\right\\}.$

For every $q\geqslant\beta(X)/a$ this gives
$a(q\log\left(1+bC_{q}(X)\right))^{2/3}\leqslant
a(q\log\left(1+bC_{q}(X)\right))^{1-1/q}\leqslant\beta(X)$. Hence,
$C_{q}(X)\leqslant e^{3a/(2e)}/(ba^{3/2})$, since $q\geqslant\beta(X)/a$. By
choosing $\kappa=\max\\{1/a,e^{3a/(2e)}/(ba^{3/2})\\}$, the proof of Lemma 32
is complete. ∎

######  Remark 33.

The same argument as in the proof of Lemma 32, without the need to use
duality, shows that $T_{\kappa\beta/(\kappa\beta-1)}(Y)\leqslant\kappa$. Since
we shall not need this fact below, the details are omitted.

###  4.5. UMD-valued Riesz potentials, Sobolev spaces and interpolation

Fix $n\in\mathbb{N}$ and $s,p\in(0,\infty)$. Suppose that $(Y,\|\cdot\|_{Y})$
is a Banach space. If $f:\mathbb{R}^{n}\to Y$ is smooth and compactly
supported then its homogeneous $(s,p)$-Riesz potential (semi)norm is defined
as usual by

$\|f\|_{H_{s,p}(\mathbb{R}^{n},Y)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\|(-\Delta)^{\frac{s}{2}}f\right\|_{L_{p}(\mathbb{R}^{n},Y)}=\left\|T_{\xi\mapsto\|\xi\|_{2}^{s}}f\right\|_{L_{p}(\mathbb{R}^{n},Y)}.$
(105)

($\|\cdot\|_{H_{s,p}(\mathbb{R}^{n},Y)}$ is sometimes denoted
$\|\cdot\|_{\dot{H}_{s,p}(\mathbb{R}^{n},Y)}$, but we use a simpler notation
since nonhomogeneous Riesz potentials do not occur in what follows.) The
Banach space $H_{s,p}(\mathbb{R}^{n},Y)$ is the completion of the smooth and
compactly support functions $f:\mathbb{R}^{n}\to Y$ under the norm
$\|\cdot\|_{H_{s,p}(\mathbb{R}^{n},Y)}$.

Throughout the ensuing discussion we shall use standard notation and basic
facts from complex interpolation theory, as appearing in [9]. The following
lemma provides quantitative control on the behavior of the spaces
$H_{s,p}(\mathbb{R}^{n},Y)$ under complex interpolation when $Y$ is a UMD
Banach space.

######  Lemma 34.

Let $(Y,\|\cdot\|_{Y})$ be a UMD Banach space. Fix $p\in[1,\infty)$ and
$s,\sigma\in(0,\infty)$ with $s<\sigma$. Suppose also that $\theta\in(0,1)$
and define $t=(1-\theta)s+\theta\sigma$. Then every $f\in
H_{t,p}(\mathbb{R}^{n},Y)$ satisfies

$\left\|f\right\|_{[H_{s,p}(\mathbb{R}^{n},Y),H_{\sigma,p}(\mathbb{R}^{n},Y)]_{\theta}}\lesssim\beta_{p}(Y)e^{\frac{\pi(\sigma-s)}{4}\sqrt{\theta(1-\theta)}}\left\|f\right\|_{H_{t,p}(\mathbb{R}^{n},Y)}.$

###### Proof.

Consider the strip
$S\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{z\in\mathbb{C}:\ \Re
z\in(0,1)\\}$. For every $M\in(0,\infty)$ define an auxiliary mapping
$\Phi_{M}:\overline{S}\to
H_{s,p}(\mathbb{R}^{n},Y)+H_{\sigma,p}(\mathbb{R}^{n},Y)$ by

$\forall\,z\in\overline{S},\qquad\Phi_{M}(z)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}e^{M(z(z-1)-\theta(\theta-1))}(-\Delta)^{\frac{t-s-z(\sigma-s)}{2}}f.$

Then $\Phi_{M}$ is holomorphic on $S$ and satisfies $\Phi_{M}(\theta)=f$. By
the definition of the complex interpolation space
$[H_{s,p}(\mathbb{R}^{n},Y),H_{\sigma,p}(\mathbb{R}^{n},Y)]_{\theta}$, we
therefore have

$\left\|f\right\|_{[H_{s,p}(\mathbb{R}^{n},Y),H_{\sigma,p}(\mathbb{R}^{n},Y)]_{\theta}}\leqslant\inf_{M>0}\sup_{b\in\mathbb{R}}\max\left\\{\|\Phi_{M}(ib)\|_{H_{s,p}(\mathbb{R}^{n},Y)},\|\Phi_{M}(1+ib)\|_{H_{\sigma,p}(\mathbb{R}^{n},Y)}\right\\}.$
(106)

For every $(a,b)\in[0,1]\times\mathbb{R}$ we have

$(-\Delta)^{\frac{s+a(\sigma-s)}{2}}\Phi_{M}(a-bi)=e^{Ma(a-1)+M\theta(1-\theta)-Mb^{2}-M(2a-1)bi}(-\Delta)^{\frac{ib(\sigma-s)}{2}}(-\Delta)^{\frac{t}{2}}f.$

Recalling (105), we therefore obtain the estimate

$\displaystyle\|\Phi_{M}(a$
$\displaystyle+bi)\|_{H_{s+a(\sigma-s),p}(\mathbb{R}^{n},Y)}$
$\displaystyle\leqslant
e^{Ma(a-1)+M\theta(1-\theta)-Mb^{2}}\left\|(-\Delta)^{\frac{ib(\sigma-s)}{2}}\right\|_{L_{p}(\mathbb{R}^{n},Y)\to
L_{p}(\mathbb{R}^{n},Y)}\|f\|_{H_{t,p}(\mathbb{R}^{n},Y)}$
$\displaystyle\lesssim\beta_{p}(Y)e^{Ma(a-1)+M\theta(1-\theta)}\cdot
e^{-Mb^{2}+\frac{\pi|b|(\sigma-s)}{4}}\|f\|_{H_{t,p}(\mathbb{R}^{n},Y)},$
(107)

where in (4.5) we used Corollary 25. The function $b\mapsto-
Mb^{2}+\pi|b|(\sigma-s)/4$ attains its maximum on $\mathbb{R}$ at
$b=\pi(\sigma-s)/(8M)$. It therefore follows from (4.5) that

$\sup_{b\in\mathbb{R}}\max\left\\{\|\Phi_{M}(ib)\|_{H_{s,p}(\mathbb{R}^{n},Y)},\|\Phi_{M}(1+ib)\|_{H_{\sigma,p}(\mathbb{R}^{n},Y)}\right\\}\lesssim\beta_{p}(Y)e^{M\theta(1-\theta)+\frac{\pi^{2}(\sigma-s)^{2}}{64M}}\|f\|_{H_{t,p}(\mathbb{R}^{n},Y)}.$

In combination with (106) we therefore have

$\displaystyle\left\|f\right\|_{[H_{s,p}(\mathbb{R}^{n},Y),H_{\sigma,p}(\mathbb{R}^{n},Y)]_{\theta}}$
$\displaystyle\lesssim\beta_{p}(Y)\left(\inf_{M\in(0,\infty)}e^{M\theta(1-\theta)+\frac{\pi^{2}(\sigma-s)^{2}}{64M}}\right)\|f\|_{H_{t,p}(\mathbb{R}^{n},Y)}$
$\displaystyle=\beta_{p}(Y)e^{\frac{\pi(\sigma-s)}{4}\sqrt{\theta(1-\theta)}}\left\|f\right\|_{H_{t,p}(\mathbb{R}^{n},Y)},$
(108)

where for (4.5) the optimal choice of $M\in(0,\infty)$ is
$M=\pi(\sigma-s)/(8\sqrt{\theta(1-\theta)})$. ∎

Suppose that $\Omega\subseteq\mathbb{R}^{n}$ is open (for our purposes
$\Omega$ will always be either a multiple of $B^{n}$ or all of
$\mathbb{R}^{n}$). If $(Y,\|\cdot\|_{Y})$ is a Banach space and
$p\in[1,\infty]$ then for every smooth $f:\Omega\to Y$ denote

$\|f\|_{W_{1,p}(\Omega,Y)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{j=1}^{n}\left\|\frac{\partial
f}{\partial x_{j}}\right\|_{L_{p}(\Omega,Y)}.$ (109)

Thus $\|\cdot\|_{W_{p,1}(\Omega,Y)}$ is the (homogeneous) first order Sobolev
(semi)norm of $f$. The corresponding Sobolev space $W_{p,1}(\Omega,Y)$ is the
completion of the space of all smooth and compactly supported functions
$f:\mathbb{R}^{n}\to Y$ under the norm $\|\cdot\|_{W_{p,1}(\Omega,Y)}$. For
$(s,p)\in(0,1)\times[1,\infty)$, the order $s$ fractional (homogeneous)
Sobolev (semi)norm of $f:\Omega\to Y$ is defined by

$\|f\|_{W_{s,p}(\Omega,Y)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigg{(}\iint_{\Omega\times\Omega}\frac{\|f(x)-f(y)\|_{Y}^{p}}{\|x-y\|_{2}^{n+ps}}\mathrm{d}x\mathrm{d}y\bigg{)}^{\frac{1}{p}}.$
(110)

While the notation $\|\cdot\|_{\dot{W}_{s,p}(\Omega,Y)}$ is sometimes used in
the literature, we shall use the above simpler notation because nonhomogeneous
Sobolev norms do not occur in what follows.

Our next goal is to relate UMD-valued Riesz potentials to Sobolev norms. The
following lemma treats the case of first order Sobolev norms, and also
contains a comparison between Riesz potentials that will be needed later; it
is a simple consequence of the boundedness of the Hilbert transform on UMD
Banach spaces, combined with the method of rotations.

######  Lemma 35.

Suppose that $(Y,\|\cdot\|_{Y})$ is a UMD Banach space, $n\in\mathbb{N}$ and
$p\in(1,\infty)$. Then every $f\in W_{1,p}(\mathbb{R}^{n},Y)$ satisfies

$\|f\|_{H_{1,p}(\mathbb{R}^{n},Y)}\leqslant\beta_{p}(Y)^{2}\|f\|_{W_{1,p}(\mathbb{R}^{n},Y)}.$

Moreover, if $s\in(1,\infty)$ and $j\in\\{1,\ldots,n\\}$ then for every smooth
$f:\mathbb{R}^{n}\to Y$ we have

$\left\|\frac{\partial f}{\partial
x_{j}}\right\|_{H_{s-1,p}(\mathbb{R}^{n},Y)}\leqslant\beta_{p}(Y)^{2}\|f\|_{H_{s,p}(\mathbb{R}^{n},Y)}.$
(111)

###### Proof.

Suppose that $K:\mathbb{R}^{n}\to Y$ is odd, continuous on
$\mathbb{R}^{n}\smallsetminus\\{0\\}$, and positively homogeneous of order
$-n$, i.e., $K(tx)=t^{-n}K(x)$ for every $t\in(0,\infty)$ and
$x\in\mathbb{R}^{n}\smallsetminus\\{0\\}$. For $f\in L_{p}(\mathbb{R}^{n},X)$
consider the corresponding Calderòn–Zygmund singular integral

$\forall\,x\in\mathbb{R}^{n},\qquad
T_{K}f(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\int_{\mathbb{R}^{n}}K(x-y)f(y)\mathrm{d}y.$
(112)

It follows from the method of rotations, as presented by Iwaniec and Martin in
[44], that

$\|T_{K}\|_{L_{p}(\mathbb{R}^{n},Y)\to
L_{p}(\mathbb{R}^{n},Y)}\leqslant\frac{\pi}{2}\left(\int_{S^{n-1}}|K(z)|\mathrm{d}\sigma(z)\right)\|H\|_{L_{p}(\mathbb{R},Y)\to
L_{p}(\mathbb{R},Y)},$ (113)

where $\sigma$ is the surface area measure on the Euclidean sphere $S^{n-1}$
and $H:L_{p}(\mathbb{R},Y)\to L_{p}(\mathbb{R},Y)$ is the Hilbert transform,
i.e.,

$\forall\,\varphi\in L_{p}(\mathbb{R},Y),\qquad
H\varphi(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{\pi}\int_{\mathbb{R}}\frac{\varphi(y)}{x-y}\mathrm{d}y.$
(114)

The integrals in (112) and (114) exist in the sense of principal values. The
estimate (113) is presented in [44, Proposition 5.1] in the case
$Y=\mathbb{C}$, but the same proof applies to the Banach space-valued setting
without any change (the proof is based on an integral identity that is
estimated using convexity of the norm. As such, the vector-valued and scalar-
valued cases are identical; here the Banach space $Y$ can be general and the
UMD property isn’t used).

For $j\in\\{1,\ldots,n\\}$ consider the Riesz transform given by

$\forall\,x\in\mathbb{R}^{n},\qquad
R_{j}f(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}\int_{\mathbb{R}^{n}}\frac{x_{j}-y_{j}}{\|x-y\|_{2}^{n+1}}f(y)\mathrm{d}y.$

By (113) we have,

$\displaystyle\|R_{j}\|_{L_{p}(\mathbb{R}^{n},Y)\to L_{p}(\mathbb{R}^{n},Y)}$
$\displaystyle\leqslant\frac{\pi\Gamma(\frac{n+1}{2})\int_{S^{n-1}}|z_{1}|\mathrm{d}\sigma(z)}{2\pi^{\frac{n+1}{2}}}\|H\|_{L_{p}(\mathbb{R},Y)\to
L_{p}(\mathbb{R},Y)}$ $\displaystyle=\|H\|_{L_{p}(\mathbb{R},Y)\to
L_{p}(\mathbb{R},Y)}\leqslant\beta_{p}(Y)^{2},$ (115)

where the bound $\|H\|_{L_{p}(\mathbb{R},Y)\to
L_{p}(\mathbb{R},Y)}\leqslant\beta_{p}(Y)^{2}$ that was used in (4.5) is
implicit in the important work of Burkholder [15] and explicit in [32, Theorem
3].

Note that for every $j\in\\{1,\ldots,n\\}$ we have
$R_{j}=(-\Delta)^{-1/2}\frac{\partial}{\partial x_{j}}$, as follows directly
by computing the Fourier transform (see e.g. [72, Chapter III])).
Consequently,

$\|f\|_{H_{1,p}(\mathbb{R}^{n},Y)}\stackrel{{\scriptstyle\eqref{eq:def
bessel}}}{{=}}\left\|(-\Delta)^{-\frac{1}{2}}\Delta
f\right\|_{L_{p}(\mathbb{R}^{n},Y)}=\Big{\|}\sum_{j=1}^{n}R_{j}\frac{\partial
f}{\partial
x_{j}}\Big{\|}_{L_{p}(\mathbb{R}^{n},Y)}\stackrel{{\scriptstyle\eqref{eq:Wp1
def}\wedge\eqref{eq:use
IM}}}{{\leqslant}}\beta_{p}(Y)^{2}\|f\|_{W_{1,p}(\mathbb{R}^{n},Y)}.$

Finally, to deduce (111), proceed as follows.

$\left\|\frac{\partial f}{\partial
x_{j}}\right\|_{H_{s-1,p}(\mathbb{R}^{n},Y)}\stackrel{{\scriptstyle\eqref{eq:def
bessel}}}{{=}}\left\|(-\Delta)^{\frac{s-1}{2}}\frac{\partial f}{\partial
x_{j}}\right\|_{L_{p}(\mathbb{R}^{n},Y)}=\left\|(-\Delta)^{-\frac{1}{2}}\frac{\partial}{\partial
x_{j}}(-\Delta)^{\frac{s}{2}}f\right\|_{L_{p}(\mathbb{R}^{n},Y)}\\\
=\left\|R_{j}(-\Delta)^{\frac{s}{2}}f\right\|_{L_{p}(\mathbb{R}^{n},Y)}\stackrel{{\scriptstyle\eqref{eq:use
IM}}}{{\leqslant}}\beta_{p}(Y)^{2}\left\|(-\Delta)^{\frac{s}{2}}f\right\|_{L_{p}(\mathbb{R}^{n},Y)}\stackrel{{\scriptstyle\eqref{eq:def
bessel}}}{{=}}\beta_{p}(Y)^{2}\|f\|_{H_{s,p}(\mathbb{R}^{n},Y)}.$

The following theorem asserts a useful comparison between UMD-valued
fractional Sobolev norms and the corresponding Riesz potentials.

######  Theorem 36.

Fix $n\in\mathbb{N}$, $s\in(0,1)$ and $p\in[2,\infty)$. Suppose that
$(Y,\|\cdot\|_{Y})$ is a UMD Banach space of cotype $p$. Then every $f\in
H_{s,p}(\mathbb{R}^{n},Y)$ satisfies

$\|f\|_{W_{s,p}(\mathbb{R}^{n},Y)}\lesssim\frac{C_{p}(Y)\beta_{p}(Y)^{4}(nV_{n})^{\frac{1}{p}}}{s(1-s)}\|f\|_{H_{s,p}(\mathbb{R}^{n},Y)},$
(116)

###### Proof of Theorem 36.

The proof below proceeds via a reduction to a slightly stronger statement in
the one dimensional case $n=1$. So, assume for the moment that we already
proved that

$\forall\,g\in
H_{s,p}(\mathbb{R},Y),\qquad\|g\|_{W_{s,p}(\mathbb{R},Y)}\lesssim\frac{C_{p}(Y)\beta_{p}(Y)}{s(1-s)}\|g\|_{H_{s,p}(\mathbb{R},Y)}.$
(117)

We shall now deduce the desired estimate (116) from (117), and then proceed to
prove (117).

For every $z\in S^{n-1}$ and $w\in z^{\perp}\subseteq\mathbb{R}^{n}$ define
$g_{z,w}:\mathbb{R}\to Y$ by setting

$\forall\,t\in\mathbb{R},\qquad
g_{z,w}(t)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f(w+tz).$ (118)

By changing to polar coordinates we see that

$\displaystyle\|f\|_{W_{s,p}(\mathbb{R}^{n},Y)}^{p}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:s sobolev}}}{{=}}$
$\displaystyle\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{\|f(x+y)-f(x)\|_{Y}^{p}}{\|y\|_{2}^{n+ps}}\mathrm{d}x\mathrm{d}y$
$\displaystyle=$
$\displaystyle\frac{1}{2}\iiint_{S^{n-1}\times\mathbb{R}^{n}\times\mathbb{R}}\frac{\|f(x+rz)-f(x)\|_{Y}^{p}}{|r|^{1+ps}}\mathrm{d}\sigma(z)\mathrm{d}x\mathrm{d}r$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:s sobolev}\wedge\eqref{eq:def
gzw}}}{{=}}$
$\displaystyle\frac{1}{2}\int_{S^{n-1}}\mathrm{d}\sigma(z)\int_{z^{\perp}}\|g_{z,w}\|_{W_{s,p}(\mathbb{R},Y)}^{p}\mathrm{d}w.$

Hence, using (117) we deduce that

$\|f\|_{W_{s,p}(\mathbb{R}^{n},Y)}\lesssim\frac{C_{p}(Y)\beta_{p}(Y)}{s(1-s)}\left(\int_{S^{n-1}}\mathrm{d}\sigma(z)\int_{z^{\perp}}\|g_{z,w}\|_{H_{s,p}(\mathbb{R},Y)}^{p}\mathrm{d}w\right)^{\frac{1}{p}}.$
(119)

For every $z\in S^{n-1}$, $w\in z^{\perp}$ and $t\in\mathbb{R}$ denote

$h_{z,w}(t)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left(-\langle
z,\nabla\rangle^{2}\right)^{\frac{s}{2}}f(w+tz)\stackrel{{\scriptstyle\eqref{eq:def
gzw}}}{{=}}\left(-\frac{\partial^{2}}{\partial
t^{2}}\right)^{\frac{s}{2}}g_{z,w}(t).$

With this notation, for every $z\in S^{n-1}$ we have

$\int_{z^{\perp}}\|g_{z,w}\|_{H_{s,p}(\mathbb{R},Y)}^{p}\mathrm{d}w\stackrel{{\scriptstyle\eqref{eq:def
bessel}}}{{=}}\int_{z^{\perp}}\left\|h_{z,w}\right\|_{L_{p}(\mathbb{R},Y)}^{p}\mathrm{d}w=\left\|\left(-\langle
z,\nabla\rangle^{2}\right)^{\frac{s}{2}}f\right\|_{L_{p}(\mathbb{R}^{n},Y)}^{p}\\\
=\left\|\left(-\langle
z,\nabla\rangle^{2}\right)^{\frac{s}{2}}(-\Delta)^{-\frac{s}{2}}(-\Delta)^{\frac{s}{2}}f\right\|_{L_{p}(\mathbb{R}^{n},Y)}^{p}\stackrel{{\scriptstyle\eqref{eq:def
bessel}}}{{\leqslant}}\left\|T_{m_{z}}\right\|_{L_{p}(\mathbb{R}^{n},Y)\to
L_{p}(\mathbb{R}^{n},Y)}^{p}\left\|f\right\|_{H_{s,p}(\mathbb{R}^{n},Y)}^{p},$
(120)

where we set $m_{z}(x)=|\langle z,x\rangle|^{s}/\|x\|_{2}^{s}$ for
$x\in\mathbb{R}^{n}$ and $z\in S^{n-1}$. By rotation invariance, it follows
from Corollary 26 that the norm of the multiplier $T_{m_{z}}$ as an operator
from $L_{p}(\mathbb{R}^{n},Y)$ to itself is at most a constant multiple of
$\beta_{p}(Y)^{3}$. By combining this bound with (119) and (120), we get

$\|f\|_{W_{s,p}(\mathbb{R}^{n},Y)}\lesssim\frac{C_{p}(Y)\beta_{p}(Y)^{4}\sigma(S^{n-1})^{\frac{1}{p}}}{s(1-s)}\left\|f\right\|_{H_{s,p}(\mathbb{R}^{n},Y)}.$

Since $\sigma(S^{n-1})=nV_{n}$, this concludes the deduction of (116) from
(117).

It remains to prove (117). The case $Y=\mathbb{R}$ of (117) is given in [9,
Theorem 6.2.5], without explicit dependence on the relevant parameters. The
proof of (117) below consists of an adaptation of the argument of [9, Theorem
6.2.5] to the UMD-valued setting, while tracking the bounds.

Recalling the Littlewood–Paley partition of unity
$\\{\psi_{j}\\}_{j\in{{\mathbb{Z}}}}$ given in (86), for every
$y\in\mathbb{R}$ we have

$\displaystyle\left(\int_{\mathbb{R}}\left\|g(x+y)-g(x)\right\|_{Y}^{p}\mathrm{d}x\right)^{\frac{1}{p}}$
$\displaystyle\leqslant\sum_{j\in{{\mathbb{Z}}}}\left(\int_{\mathbb{R}}\left\|T_{\psi_{j}}g(x+y)-T_{\psi_{j}(x)}g(x)\right\|_{Y}^{p}\mathrm{d}x\right)^{\frac{1}{p}}$
(121)
$\displaystyle\lesssim\sum_{j\in{{\mathbb{Z}}}}\min\Big{\\{}1,\frac{|y|}{2^{j}}\Big{\\}}\left\|T_{\psi_{j}}g\right\|_{L_{p}(\mathbb{R},Y)},$
(122)

where in (121) we used the fact that $\sum_{j\in{{\mathbb{Z}}}}\psi_{j}\equiv
1$ and the triangle inequality in $L_{p}(\mathbb{R},Y)$, and in (122) we used
Lemma 30. Now,

$\displaystyle\|g\|_{W_{s,p}(\mathbb{R},Y)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:use
localization}}}{{\lesssim}}$
$\displaystyle\Big{(}\sum_{r\in{{\mathbb{Z}}}}\int_{2^{r}}^{2^{r+1}}\Big{(}\sum_{j\in{{\mathbb{Z}}}}\min\Big{\\{}1,\frac{|y|}{2^{j}}\Big{\\}}\left\|T_{\psi_{j}}g\right\|_{L_{p}(\mathbb{R},Y)}\Big{)}^{p}\frac{\mathrm{d}y}{|y|^{1+ps}}\Big{)}^{\frac{1}{p}}$
(123) $\displaystyle\asymp$
$\displaystyle\Big{(}\sum_{r\in{{\mathbb{Z}}}}\Big{(}\sum_{u\in{{\mathbb{Z}}}}\frac{\min\left\\{1,2^{u}\right\\}}{2^{su}}\cdot
2^{-s(r-u)}\left\|T_{\psi_{r-u}}g\right\|_{L_{p}(\mathbb{R},Y)}\Big{)}^{p}\Big{)}^{\frac{1}{p}}$
(124) $\displaystyle\leqslant$
$\displaystyle\sum_{u\in{{\mathbb{Z}}}}\Big{(}\sum_{r\in{{\mathbb{Z}}}}\frac{\min\left\\{1,2^{pu}\right\\}}{2^{psu}}\cdot
2^{-ps(r-u)}\left\|T_{\psi_{r-u}}g\right\|_{L_{p}(\mathbb{R},Y)}^{p}\Big{)}^{\frac{1}{p}}$
$\displaystyle=$
$\displaystyle\Big{(}\sum_{j\in{{\mathbb{Z}}}}2^{-psj}\left\|T_{\psi_{j}}g\right\|_{L_{p}(\mathbb{R},Y)}^{p}\Big{)}^{\frac{1}{p}}\sum_{u\in{{\mathbb{Z}}}}\frac{\min\left\\{1,2^{u}\right\\}}{2^{su}}$
$\displaystyle\asymp$
$\displaystyle\frac{1}{s(1-s)}\Big{(}\sum_{j\in{{\mathbb{Z}}}}2^{-psj}\left\|T_{\psi_{j}}g\right\|_{L_{p}(\mathbb{R},Y)}^{p}\Big{)}^{\frac{1}{p}},$
(126)

where for (124) use the fact that for each $r\in{{\mathbb{Z}}}$ in the
integrand of the corresponding summand that appears in the right hand side of
(123) we have $|y|\asymp 2^{r}$, and make the change of variable $u=r-j$, and
for (4.5) use the triangle inequality in $\ell_{p}({{\mathbb{Z}}})$.

Recalling the functions $\\{\vartheta_{j}\\}_{j\in{{\mathbb{Z}}}}$ given in
(88), an application of Proposition 28 shows that

$\mathbb{E}_{\varepsilon\in\\{-1,1\\}^{{{\mathbb{Z}}}}}\bigg{[}\Big{\|}\sum_{j\in{{\mathbb{Z}}}}\varepsilon_{j}T_{\vartheta_{j}}(-\Delta)^{\frac{s}{2}}g\Big{\|}_{L_{p}(\mathbb{R},Y)}\bigg{]}\leqslant\beta_{p}(Y)\left\|(-\Delta)^{\frac{s}{2}}g\right\|_{L_{p}(\mathbb{R},Y)}=\beta_{p}(Y)\|g\|_{H_{s,p}(\mathbb{R},Y)}.$
(127)

At the same time, since $L_{p}(\mathbb{R},Y)$ has cotype $p$ with constant
$C_{p}(Y)$,

$\mathbb{E}_{\varepsilon\in\\{-1,1\\}^{{{\mathbb{Z}}}}}\bigg{[}\Big{\|}\sum_{j\in{{\mathbb{Z}}}}\varepsilon_{j}T_{\vartheta_{j}}(-\Delta)^{\frac{s}{2}}g\Big{\|}_{L_{p}(\mathbb{R},Y)}\bigg{]}\gtrsim\frac{1}{C_{p}(Y)}\Big{(}\sum_{j\in{{\mathbb{Z}}}}\left\|T_{\vartheta_{j}}(-\Delta)^{\frac{s}{2}}g\right\|_{L_{p}(\mathbb{R},Y)}^{p}\Big{)}^{\frac{1}{p}}\\\
\gtrsim\frac{1}{C_{p}(Y)}\Big{(}\sum_{j\in{{\mathbb{Z}}}}\left\|T_{\psi_{j}}(-\Delta)^{\frac{s}{2}}g\right\|_{L_{p}(\mathbb{R},Y)}^{p}\Big{)}^{\frac{1}{p}}\gtrsim\frac{1}{C_{p}(Y)}\Big{(}\sum_{j\in{{\mathbb{Z}}}}2^{-psj}\left\|T_{\psi_{j}}g\right\|_{L_{p}(\mathbb{R},Y)}^{p}\Big{)}^{\frac{1}{p}},$
(128)

where in the penultimate step of (128) we used Lemma 90 and in the final step
of (128) we used Lemma 31. By combining (127) and (128), we see that

$\Big{(}\sum_{j\in{{\mathbb{Z}}}}2^{-psj}\left\|T_{\psi_{j}}g\right\|_{L_{p}(\mathbb{R},Y)}^{p}\Big{)}^{\frac{1}{p}}\lesssim
C_{p}(Y)\beta_{p}(Y)\|g\|_{H_{s,p}(\mathbb{R},Y)}.$ (129)

A substitution of (129) into (126) now yields the desired estimate (117). ∎

######  Remark 37.

The left side of (129) is the Besov norm $\|g\|_{B^{s}_{p,p}(\mathbb{R},Y)}$.
Hence (129) asserts a quantitative embedding of $H_{s,p}(\mathbb{R},Y)$ into
$B^{s}_{p,p}(\mathbb{R},Y)$ when $Y$ is a UMD space of cotype $p$. A