1 Introduction
Let
$(X,\|\cdot \|_X)$
be a Banach space. For
$p\in [1,\infty )$
, the vector-valued
$L_p$
norm of a function
$f:\Omega \to X$
defined on a measure space
$(\Omega ,\mathscr {F},\unicode{x3bc} )$
is given by
$\|f\|_{L_p(\Omega ,\unicode{x3bc} ;X)}^p = \int _\Omega \|f(\unicode{x3c9} )\|_X^p\mathop {}\!\mathrm {d}\unicode{x3bc} (\unicode{x3c9} )$
. When
$\Omega $
is a finite set and
$\unicode{x3bc} $
is the normalized counting measure, we will simply write
$\|f\|_{L_p(\Omega ;X)}$
.
Let
$\mathscr {C}_n=\{-1,1\}^n$
be the discrete hypercube. For
$i\in \{1,\dots ,n\}$
, the ith partial derivative of a function
$f:\mathscr {C}_n\to X$
is defined by
$$ \begin{align} \forall \ \varepsilon\in\mathscr{C}_n, \ \ \ \partial_if(\varepsilon) \stackrel{\mathrm{def}}{=} \frac{f(\varepsilon)-f(\varepsilon_1,\dots ,\varepsilon_{i-1},-\varepsilon_i,\varepsilon_{i+1},\dots ,\varepsilon_n)}{2}. \end{align} $$
In [Reference PisierPis86], Pisier showed that for every
$n\in \mathbb N$
and
$p\in [1,\infty )$
, every
$f:\mathscr {C}_n\to X$
satisfies
with
$\mathfrak {P}_p^n(X) \leqslant 2e\log n$
. Showing that
$\mathfrak {P}_p^n(X)$
is bounded by a constant depending only on p and the geometry of the given Banach space X, is of fundamental importance in the theory of nonlinear type (see [Reference PisierPis86, Reference Naor and SchechtmanNS02]). The first positive and negative results in this direction were obtained by Talagrand in [Reference TalagrandTal93], who showed that
$\mathfrak {P}_p^n(\mathbb R) \asymp _p1$
and
$\mathfrak {P}_p^n(\ell _\infty ) \asymp _p\log n$
for every
$p\in [1,\infty )$
.
Talagrand’s dimension-independent scalar-valued inequality (2) was greatly generalized in the range
$p\in (1,\infty )$
by Naor and Schechtman [Reference Naor and SchechtmanNS02]. Recall that a Banach space
$(X,\|\cdot \|_X)$
is called a UMD space if for every
$p\in (1,\infty )$
, there exists a constant
$\unicode{x3b2} _p\in (0,\infty )$
such that for every
$n\in \mathbb N$
, every probability space
$(\Omega , \mathscr {F},\unicode{x3bc} )$
and every filtration
$\{\mathscr {F}_i\}_{i=0}^n$
of sub-
$\unicode{x3c3} $
-algebras of
$\mathscr {F}$
, every martingale
$\{\mathscr {M}_i:\Omega \to X\}_{i=0}^n$
adapted to
$\{\mathscr {F}_i\}_{i=0}^n$
satisfies
$$ \begin{align} \max_{\unicode{x3b4}=(\delta_1,\ldots,\delta_n)\in\mathscr{C}_n}\Big\| \sum_{i=1}^n \delta_i (\mathscr{M}_i-\mathscr{M}_{i-1}) \Big\|_{L_p(\Omega,\unicode{x3bc};X)} \leqslant \beta_p \|\mathscr{M}_n-\mathscr{M}_0\|_{L_p(\Omega,\unicode{x3bc};X)}. \end{align} $$
The least constant
$\unicode{x3b2} _p\in (0,\infty )$
for which (3) holds is called the
$\text{UMD}_p$
constant of X and is denoted by
$\unicode{x3b2} _p(X)$
. In [Reference Naor and SchechtmanNS02], Naor and Schechtman proved that for every UMD Banach space X and
$p\in (1,\infty )$
,
$$ \begin{align} \mathop{\mathrm{sup}}\limits_{n\in\mathbb N}\mathfrak{P}_p^n(X) \leqslant \beta_p(X). \end{align} $$
Their result was later strengthened by Hytönen and Naor [Reference Hytönen and NaorHN13] in terms of the random martingale transform inequalities of Garling; see [Reference GarlingGar90]. Recall that a Banach space
$(X,\|\cdot \|_X)$
is a
$\text{UMD}^+$
space if for every
$p\in (1,\infty )$
there exists a constant
$\unicode{x3b2} _p^+\in (0,\infty )$
such that for every martingale
$\{\mathscr {M}_i:\Omega \to X\}_{i=0}^n$
as before, we have
$$ \begin{align} \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i (\mathscr{M}_i-\mathscr{M}_{i-1}) \Big\|^p_{L_p(\Omega,\unicode{x3bc};X)}\Big)^{1/p} \leqslant \beta_p^+ \|\mathscr{M}_n-\mathscr{M}_0\|_{L_p(\Omega,\unicode{x3bc};X)}. \end{align} $$
Similarly, X is a
$\text{UMD}^-$
Banach space if for every
$p\in (1,\infty )$
there exists a constant
$\unicode{x3b2} _p^-\in (0,\infty )$
such that for every martingale
$\{\mathscr {M}_i:\Omega \to X\}_{i=0}^n$
as before, we have
$$ \begin{align} \|\mathscr{M}_n-\mathscr{M}_0\|_{L_p(\Omega,\unicode{x3bc};X)} \leqslant \beta_p^- \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i (\mathscr{M}_i-\mathscr{M}_{i-1}) \Big\|^p_{L_p(\Omega,\unicode{x3bc};X)}\Big)^{1/p}. \end{align} $$
The least positive constants
$\unicode{x3b2} _p^+, \unicode{x3b2} _p^-$
for which (5) and (6) hold are respectively called the
$\text{UMD}_p^+$
and
$\text{UMD}_p^-$
constants of X and are denoted by
$\unicode{x3b2} _p^+(X)$
and
$\unicode{x3b2} _p^-(X)$
. In [Reference Hytönen and NaorHN13], Hytönen and Naor showed that for every Banach space X whose dual
$X^\ast $
is a
$\text{UMD}^+$
space and
$p\in (1,\infty )$
,
$$ \begin{align} \mathop{\mathrm{sup}}\limits_{n\in\mathbb N} \mathfrak{P}_p^n(X) \leqslant \beta_{p/(p-1)}^+(X^\ast). \end{align} $$
In fact, in [Reference Hytönen and NaorHN13, Theorem 1.4], the authors proved a generalization (see (28)) of inequality (2) for a family of n functions
$\{f_i:\mathscr {C}_n\to X\}_{i=1}^n$
under the assumption that the dual of X is
$\text{UMD}^+$
.
The main result of the present note is a different inequality of this nature with respect to a Fourier-analytic parameter of X. For a Banach space
$(X,\|\cdot \|_X)$
and
$p\in (1,\infty )$
, let
$\mathfrak {s}_p(X)\in (0,\infty ]$
be the least constant
$\mathfrak {s} \in (0,\infty ]$
such that the following holds. For every probability space
$(\Omega , \mathscr {F},\unicode{x3bc} )$
,
$n\in \mathbb N$
and filtration
$\{\mathscr {F}_i\}_{i=1}^n$
of sub-
$\unicode{x3c3} $
-algebras of
$\mathscr {F}$
with corresponding vector-valued conditional expectations
$\{\mathscr {E}_i\}_{i=1}^n$
, every sequence of functions
$\{f_i:\Omega \to X\}_{i=1}^n$
satisfies
$$ \begin{align} \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i\mathscr{E}_i f_i\Big\|_{L_p(\Omega,\unicode{x3bc};X)}^p\Big)^{1/p} \leqslant \mathfrak{s}\Big( \frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_if_i\Big\|_{L_p(\Omega,\unicode{x3bc};X)}^p\Big)^{1/p}. \end{align} $$
The square function inequality (8) originates in Stein’s classical work [Reference SteinSte70], where he showed that
$\mathfrak {s}_p(\mathbb R) \asymp _p 1$
for every
$p\in (1,\infty )$
. In the vector-valued setting which is of interest here, it has been proved by Bourgain in [Reference BourgainBou86] that for every
$\text{UMD}^+$
Banach space and
$p\in (1,\infty )$
,
$$ \begin{align} \mathfrak{s}_p(X) \leqslant\beta_p^+(X). \end{align} $$
For a function
$f:\mathscr {C}_n\to X$
and
$i\in \{0,1,\dots ,n\}$
denote by
$$ \begin{align} \forall \ \varepsilon\in\mathscr{C}_n, \ \ \ \mathscr{E}_i f (\varepsilon) \stackrel{\mathrm{def}}{=} \frac{1}{2^{n-i}} \sum_{\delta_{i+1},\ldots,\delta_n\in\{-1,1\}} f(\varepsilon_1,\dots ,\varepsilon_i,\delta_{i+1},\dots ,\delta_n), \end{align} $$
so that
$\mathscr {E}_n f = f$
and
$\mathscr {E}_0 f = \frac {1}{2^n} \sum _{\unicode{x3b4} \in \mathscr {C}_n} f(\unicode{x3b4} )$
. The main result of this note is the following theorem.
Theorem 1 Fix
$p\in (1,\infty )$
and let
$(X,\|\cdot \|_X)$
be a Banach space with
$\mathfrak {s}_p(X)<\infty $
. If, additionally, X is a
$\text{UMD}^-$
space, then for every
$n\in \mathbb N$
and functions
$f_1,\dots ,f_n:\mathscr {C}_n\to X$
, we have
$$ \begin{align} \Big\| \sum_{i=1}^n (\mathscr{E}_i f_i - \mathscr{E}_{i-1}f_i)\Big\|_{L_p(\mathscr{C}_n;X)} \leqslant \mathfrak{s}_p(X) \beta_p^-(X) \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i \partial_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p}. \end{align} $$
Choosing
$f_1=\cdots =f_n=f$
, we deduce that the constants in Pisier’s inequality (2) satisfy
$$ \begin{align} \mathop{\mathrm{sup}}\limits_{n\in\mathbb N} \mathfrak{P}_p^n(X) \leqslant \mathfrak{s}_p(X) \beta_p^-(X). \end{align} $$
Combining (12) with Bourgain’s inequality (9), we deduce that
$\mathrm {sup}_{n\in \mathbb N} \mathfrak {P}_p^n(X) \leqslant \unicode{x3b2} _p^+(X)\unicode{x3b2} _p^-(X)$
, which is weaker than Naor and Schechtman’s bound (4). Nevertheless, it appears to be unknown (see [Reference PisierPis16, p. 197]) whether every Banach space X with
$\mathfrak {s}_p(X)<\infty $
is necessarily a
$\text{UMD}^+$
space. Therefore, it is conceivable that there exist Banach spaces X for which inequality (12) does not follow from the previously known results of [Reference Naor and SchechtmanNS02, Reference Hytönen and NaorHN13]. We will see in Proposition 5 below that if the dual
$X^\ast $
of a Banach space X is
$\text{UMD}^+$
, then X satisfies the assumptions of Theorem 1. Therefore, Theorem 1 also contains the aforementioned result of [Reference Hytönen and NaorHN13].
Moreover, Theorem 1 implies an inequality similar to [Reference Hytönen and NaorHN13, Theorem 1.4] (see also Remark 3 below for comparison), under different assumptions. We will need some standard terminology from discrete Fourier analysis. Recall that every function
$f:\mathscr {C}_n\to X$
can be expanded in a Walsh series as
$$ \begin{align} f = \sum_{A\subseteq\{1,\ldots,n\}} \widehat{f}(A) w_A, \end{align} $$
where
$\widehat {f}(A)\in X$
and the Walsh function
$w_A:\mathscr {C}_n\to \{-1,1\}$
is given by
$w_A(\varepsilon ) = \prod _{i\in A}\varepsilon _i$
for
$\varepsilon =(\varepsilon _1,\dots ,\varepsilon _n)\in \mathscr {C}_n$
and
$A\neq \varnothing $
. As usual, we agree that
$w_\varnothing \equiv 1$
. Moreover, the fractional hypercube Laplacian of a function
$f:\mathscr {C}_n\to X$
is given by
$$ \begin{align} \forall \ \unicode{x3b1}\in\mathbb R, \ \ \ \ \Delta^\unicode{x3b1}\Big( \sum_{A\subseteq\{1,\ldots,n\}} \widehat{f}(A) w_A \Big) \stackrel{\mathrm{def}}{=} \sum_{\substack{A\subseteq\{1,\ldots,n\} \\ A\neq\varnothing}} |A|^\unicode{x3b1} \widehat{f}(A) w_A. \end{align} $$
Corollary 2 Fix
$p\in (1,\infty )$
and let
$(X,\|\cdot \|_X)$
be a Banach space with
$\mathfrak {s}_p(X)<\infty $
. If, additionally, X is a
$\text{UMD}^-$
space, then for every
$n\in \mathbb N$
and functions
$f_1,\dots ,f_n:\mathscr {C}_n\to X$
, we have
$$ \begin{align} \Big\| \sum_{i=1}^n \Delta^{-1}\partial_i f_i \Big\|_{L_p(\mathscr{C}_n;X)} \leqslant \mathfrak{s}_p(X) \beta_p^-(X) \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i \partial_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p}. \end{align} $$
Asymptotic notation In what follows we use the convention that for
$a,b\in [0,\infty ]$
the notation
$a\gtrsim b$
(respectively
$a\lesssim b$
) means that there exists a universal constant
$c\in (0,\infty )$
such that
$a\geqslant cb$
(respectively
$a\leqslant cb$
). Moreover,
$a\asymp b$
stands for
$(a\lesssim b)\wedge (a\gtrsim b)$
. The notations
$\lesssim _\unicode{x3be} , \gtrsim _\chi $
and
$\asymp _\unicode{x3c8} $
mean that the implicit constant c depends on
$\unicode{x3be} , \chi $
and
$\unicode{x3c8} $
respectively.
2 Proofs
We first present the proof of Theorem 1.
Proof Proof of Theorem 1
For a function
$h:\mathscr {C}_n\to X$
and
$i\in \{1,\dots ,n\}$
consider the averaging operator
$$ \begin{align} \forall \ \varepsilon\in\mathscr{C}_n, \ \ \ \ \mathsf{E}_i h(\varepsilon) \stackrel{\mathrm{def}}{=} \frac{h(\varepsilon)+h(\varepsilon_1,\dots ,\varepsilon_{i-1},-\varepsilon_i,\varepsilon_{i+1},\dots ,\varepsilon_n)}{2} = (\mathsf{id}-\partial_i)h(\varepsilon), \end{align} $$
where
$\mathsf{id}$
is the identity operator. Then, for every
$i\in \{0,1,\dots ,n\}$
we have the identities
$$ \begin{align} \mathscr{E}_i h= \mathsf{E}_{i+1} \circ\cdots\circ \mathsf{E}_n h = \mathbb{E}[h|\mathscr{F}_i], \end{align} $$
where
$\mathscr {F}_i = \unicode{x3c3} (\varepsilon _1,\dots ,\varepsilon _i)$
. Since for every
$i\in \{1,\dots ,n\}$
,
$$ \begin{align} \mathbb{E}\big[ \mathscr{E}_i f_i - \mathscr{E}_{i-1} f_i \big| \mathscr{F}_{i-1}\big] = 0, \end{align} $$
the sequence
$\{\mathscr {E}_i f_i - \mathscr {E}_{i-1} f_i \}_{i=1}^n$
is a martingale difference sequence and thus the
$\text{UMD}^-$
condition and (8) imply that
$$ \begin{align} \Big\| \sum_{i=1}^n (\mathscr{E}_i f_i - \mathscr{E}_{i-1}f_i)\Big\|_{L_p(\mathscr{C}_n;X)} & \stackrel{(6)}{\leqslant} \beta_p^-(X) \Big( \frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i (\mathscr{E}_i f_i - \mathscr{E}_{i-1}f_i)\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p} \nonumber\\ & \stackrel{(16)}{=} \beta_p^-(X) \Big( \frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i \mathscr{E}_i \partial_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p} \nonumber\\ & \stackrel{(8)}{\leqslant} \mathfrak{s}_p(X) \beta_p^-(X) \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i \partial_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p} \end{align} $$
which completes the proof. ▪
We will now derive Corollary 2 from Theorem 1. The proof follows a symmetrization argument of [Reference Hytönen and NaorHN13].
Proof Proof of Corollary 2
As noticed in (19) above, (11) can be equivalently written as
$$ \begin{align} \Big\| \sum_{i=1}^n \mathscr{E}_i \partial_i f_i \Big\|_{L_p(\mathscr{C}_n;X)} \leqslant \mathfrak{s}_p(X) \beta_p^-(X) \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i \partial_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p}. \end{align} $$
Fix a permutation
$\unicode{x3c0} \in S_n$
and consider the filtration
$\{\mathscr {F}_i^\unicode{x3c0} \}_{i=0}^n$
given by
$\mathscr {F}_i^\unicode{x3c0} = \unicode{x3c3} (\varepsilon _{\unicode{x3c0} (1)},\dots ,\varepsilon _{\unicode{x3c0} (i)})$
with corresponding conditional expectations
$\{\mathscr {E}_i^\unicode{x3c0} \}_{i=0}^n$
. Repeating the argument of the proof of Theorem 1 for this filtration and the martingale difference sequence
$\{\mathscr {E}_i^\unicode{x3c0} f_{\unicode{x3c0} (i)} - \mathscr {E}_{i-1}^\unicode{x3c0} f_{\unicode{x3c0} (i)}\}_{i=1}^n$
, we see that for every
$\unicode{x3c0} \in S_n$
,
$$ \begin{align} \Big\| \sum_{i=1}^n \mathscr{E}_i^\unicode{x3c0} \partial_{\unicode{x3c0}(i)} f_{\unicode{x3c0}(i)} \Big\|_{L_p(\mathscr{C}_n;X)} & \leqslant \mathfrak{s}_p(X) \beta_p^-(X) \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i \partial_{\unicode{x3c0}(i)} f_{\unicode{x3c0}(i)}\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p} \nonumber\\ & =\mathfrak{s}_p(X) \beta_p^-(X) \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i \partial_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p}, \end{align} $$
since
$(\unicode{x3b4} _1,\dots ,\unicode{x3b4} _n)$
has the same distribution as
$(\unicode{x3b4} _{\unicode{x3c0} (1)},\dots ,\unicode{x3b4} _{\unicode{x3c0} (n)})$
. An obvious adaptation of (10) along with (13) shows that for every
$h:\mathscr {C}_n\to X$
,
$$ \begin{align} \mathscr{E}_i^\unicode{x3c0} h = \sum_{A\subseteq\{\unicode{x3c0}(1),\ldots,\unicode{x3c0}(i)\}} \widehat{h}(A) w_A \end{align} $$
where
$\widehat {h}(A)$
are the Walsh coefficients of h. Therefore, expanding each
$f_{\unicode{x3c0} (i)}$
as a Walsh series (13) we have
$$ \begin{align} \forall \ i\in\{1,\dots ,n\}, \ \ \ \ \mathscr{E}^\pi_i\partial_{\unicode{x3c0}(i)} f_{\unicode{x3c0}(i)} = \sum_{\substack{A\subseteq\{1\ldots,n\} \\ \max \unicode{x3c0}^{-1}(A)=i}} \widehat{f_{\unicode{x3c0}(i)}}(A) w_A \end{align} $$
and therefore
$$ \begin{align} \sum_{i=1}^n \mathscr{E}^\pi_i\partial_{\unicode{x3c0}(i)} f_{\unicode{x3c0}(i)} = \sum_{A\subseteq\{1,\ldots,n\}} \widehat{f_{\unicode{x3c0}(\max \unicode{x3c0}^{-1}(A))}}(A) w_A. \end{align} $$
Averaging (24) over all permutations
$\unicode{x3c0} \in S_n$
and using the fact that
$\unicode{x3c0} (\max \unicode{x3c0} ^{-1}(A))$
is uniformly distributed in A, we get
\begin{align*}\frac{1}{n!}\sum_{\unicode{x3c0}\in S_n} \sum_{i=1}^n \mathscr{E}^\pi_i\partial_{\unicode{x3c0}(i)} f_{\unicode{x3c0}(i)} &=\!\!\! \sum_{\substack{A\subseteq\{1,\ldots,n\}\\ A\neq\varnothing}} \frac{1}{|A|} \sum_{i\in A} \widehat{f_i}(A) w_A\\& = \sum_{i=1}^n \sum_{\substack{A\subseteq\{1,\ldots,n\} \\ i\in A}} \frac{1}{|A|} \widehat{f_i}(A) w_A = \sum_{i=1}^n \Delta^{-1}\partial_i f_i. \end{align*}
Hence, by convexity we finally deduce that
$$ \begin{align} \Big\| \sum_{i=1}^n \Delta^{-1}\partial_i f_i\Big\|_{L_p(\mathscr{C}_n;X)} & \leqslant \frac{1}{n!}\sum_{\unicode{x3c0}\in S_n} \Big\| \sum_{i=1}^n \mathscr{E}_i^\unicode{x3c0} \partial_{\unicode{x3c0}(i)} f_{\unicode{x3c0}(i)} \Big\|_{L_p(\mathscr{C}_n;X)} \nonumber\\ & \stackrel{(21)}{\leqslant} \mathfrak{s}_p(X) \beta_p^-(X) \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i \partial_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p}, \end{align} $$
which completes the proof. ▪
Remark 3 In [Reference Hytönen and NaorHN13], Hytönen and Naor obtained a different extension of Pisier’s inequality (2) for Banach spaces whose dual is
$\text{UMD}^+$
. For a function
$F:\mathscr {C}_n\times \mathscr {C}_n\to X$
and
$i\in \{1,\dots ,n\}$
, let
$F_i:\mathscr {C}_n\to X$
be given by
$$ \begin{align} \forall \ \varepsilon\in\mathscr{C}_n, \ \ \ \ F_i(\varepsilon) \stackrel{\mathrm{def}}{=} \frac{1}{2^n}\sum_{\unicode{x3b4}\in\mathscr{C}_n} \delta_i F(\varepsilon,\unicode{x3b4}). \end{align} $$
In [Reference Hytönen and NaorHN13, Theorem 1.4], it was shown that for every
$p\in (1,\infty )$
and every function
$F:\mathscr {C}_n\times \mathscr {C}_n\to X$
,
$$ \begin{align} \Big\| \sum_{i=1}^n \Delta^{-1}\partial_i F_i\Big\|_{L_p(\mathscr{C}_n;X)} \leqslant \beta_{p/(p-1)}^+(X^\ast) \|F\|_{L_p(\mathscr{C}_n\times\mathscr{C}_n;X)}. \end{align} $$
In fact, since every Banach space whose dual is
$\text{UMD}^+$
is K-convex (see [Reference PisierPis16] and Section 3 below) the validity of inequality (27) is equivalent to its validity for functions of the form
$F(\varepsilon ,\unicode{x3b4} ) = \sum _{i=1}^n\unicode{x3b4} _i F_i(\varepsilon )$
, where
$F_1,\dots ,F_n:\mathscr {C}_n\to X$
. In other words, [Reference Hytönen and NaorHN13, Theorem 1.4] is equivalent to the fact that if
$X^\ast $
is
$\text{UMD}^+$
, then for every
$F_1,\dots ,F_n:\mathscr {C}_n\to X$
and
$p\in (1,\infty )$
,
$$ \begin{align} \Big\| \sum_{i=1}^n \Delta^{-1}\partial_i F_i\Big\|_{L_p(\mathscr{C}_n;X)} \leqslant A_p(X) \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i F_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p}, \end{align} $$
up to the value of the constant
$A_p(X)$
. In particular, applying (28) to
$F_i=\partial _i f_i$
, one recovers Corollary 2, so inequality (28) of [Reference Hytönen and NaorHN13] is formally stronger than (15) in the class of spaces whose dual is
$\text{UMD}^+$
.
3 Concluding Remarks
In this section we will compare our result with existing theorems in the literature. Recall that a Banach X space is K-convex if X does not contain the family
$\{\ell _1^n\}_{n=1}^\infty $
with uniformly bounded distortion. We will need the following lemma.
Lemma 4 If a space
$(X,\|\cdot \|_X)$
satisfies
$\mathfrak {s}_p(X)<\infty $
for some
$p\in (1,\infty )$
, then X is K-convex.
Proof It is well known since Stein’s work [Reference SteinSte70] that inequality (8) does not hold for
$p\in \{1,\infty \}$
even for scalar valued functions. In fact, an inspection of the argument in [Reference SteinSte70, p. 105] shows that for every
$n\in \mathbb N$
there existn functions
$g_1,\dots ,g_n:\mathscr {C}_n\to \{0,1\}$
such that for every
$q\in (2,\infty )$
,
$$ \begin{align} \Big\| \Big(\sum_{i=1}^n \big(\mathscr{E}_i g_i\big)^2\Big)^{1/2}\Big\|_{L_q(\mathscr{C}_n;\mathbb R)} \gtrsim \Big( \int_0^n y^{q/2} e^{-y}\mathop{}\!\mathrm{d} y \Big)^{1/q} \Big\| \Big(\sum_{i=1}^n g_i^2\Big)^{1/2}\Big\|_{L_q(\mathscr{C}_n;\mathbb R)}, \end{align} $$
where
$\{\mathscr {E}_i\}_{i=0}^n$
are the conditional expectations (10). Using the fact that
$L_\infty (\mathscr {C}_n;\mathbb R)$
is 2-isomorphic to
$L_{n}(\mathscr {C}_n;\mathbb R)$
, we thus deduce that
$$ \begin{align} \Big\| \Big(\sum_{i=1}^n \big(\mathscr{E}_i g_i\big)^2\Big)^{1/2}\Big\|_{L_\infty(\mathscr{C}_n;\mathbb R)} & \gtrsim \Big( \int_0^n y^{n/2} e^{-y}\mathop{}\!\mathrm{d} y \Big)^{1/n} \Big\| \Big(\sum_{i=1}^n g_i^2\Big)^{1/2}\Big\|_{L_\infty(\mathscr{C}_n;\mathbb R)} \\ & \asymp \sqrt{n}\Big\| \Big(\sum_{i=1}^n g_i^2\Big)^{1/2}\Big\|_{L_\infty(\mathscr{C}_n;\mathbb R)} \nonumber \end{align} $$
Therefore, by duality in
$L_\infty (\mathscr {C}_n;\ell _2^n)$
and Khintchine’s inequality [Reference KhintchineKhi23], we deduce that there exist n functions
$h_1,\ldots ,h_n:\mathscr {C}_n\to \mathbb R$
such that
$$ \begin{align} \frac{1}{2^n}\sum_{\unicode{x3b4}\in\mathscr{C}_n}\Big\| \sum_{i=1}^n \delta_i \mathscr{E}_i h_i\Big\|_{L_1(\mathscr{C}_n;\mathbb R)} \gtrsim \frac{\sqrt{n}}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i h_i\Big\|_{L_1(\mathscr{C}_n;\mathbb R)}. \end{align} $$
Suppose that a Banach space X with
$\mathfrak {s}_p(X)<\infty $
is not K-convex, so that there exists a constant
$K\in [1,\infty )$
such that for every
$n\in \mathbb N$
, there exists a linear operator
$\mathsf{J}_n:L_1(\mathscr {C}_n;\mathbb R)\to X$
satisfying
$$ \begin{align} \forall \ h\in L_1(\mathscr{C}_n;\mathbb R), \ \ \ \ \|h\|_{L_1(\mathscr{C}_n;\mathbb R)} \leqslant \|\mathsf{J}_nh\|_X \leqslant K\|h\|_{L_1(\mathscr{C}_n;\mathbb R)}. \end{align} $$
Consider the functions
$H_1,\dots ,H_n:\mathscr {C}_n\to L_1(\mathscr {C}_n;\mathbb R)$
given by
$$ \begin{align} \forall \ \varepsilon,\varepsilon'\in\mathscr{C}_n, \ \ \ \big[H_i(\varepsilon)\big](\varepsilon') = h_i(\varepsilon_1\varepsilon_1',\dots ,\varepsilon_n\varepsilon_n'), \end{align} $$
where
$h_i\in L_1(\mathscr {C}_n;\mathbb R)$
are the functions satisfying (31). Then, for every
$i\in \{1,\dots ,n\}$
, we have
$[\mathscr {E}_iH_i(\varepsilon )](\varepsilon ') = \mathscr {E}_ih_i(\varepsilon _1\varepsilon _1',\dots ,\varepsilon _n\varepsilon _n')$
and, by translation invariance, for every
$\varepsilon ,\unicode{x3b4} \in \mathscr {C}_n$
we have
\begin{align*} &\Big\| \sum_{i=1}^n \delta_i \mathscr{E}_i H_i(\varepsilon)\Big\|_{L_1(\mathscr{C}_n;\mathbb R)} = \Big\| \sum_{i=1}^n \delta_i \mathscr{E}_i h_i\Big\|_{L_1(\mathscr{C}_n;\mathbb R)}\quad\mbox{and}\\ & \Big\| \sum_{i=1}^n \delta_i H_i(\varepsilon)\Big\|_{L_1(\mathscr{C}_n;\mathbb R)} = \Big\| \sum_{i=1}^n \delta_i h_i\Big\|_{L_1(\mathscr{C}_n;\mathbb R)} \end{align*}
Therefore, considering the mappings
$f_1,\dots ,f_n:\mathscr {C}_n\to X$
given by
$f_i = \mathsf{J}_n \circ H_i$
, we see that
$$ \begin{align} \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i\mathscr{E}_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p} \gtrsim K^{-1} \sqrt{n} \Big( \frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_if_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p}, \end{align} $$
thus showing that
$\mathfrak {s}_p(X)\gtrsim K^{-1} \sqrt {n}$
, which is a contradiction.▪
Recall that the X-valued Rademacher projection is defined to be
$$ \begin{align} \mathsf{Rad}\Big( \sum_{A\subseteq\{1,\ldots,n\}} \widehat{f}(A) w_A\Big) \stackrel{\mathrm{def}}{=} \sum_{i=1}^n \widehat{f}(\{i\}) w_{\{i\}}. \end{align} $$
A deep theorem of Pisier [Reference PisierPis82] asserts that a Banach space is K-convex if and only if
$$ \begin{align} \forall \ r\in(1,\infty), \ \ \ \ \mathsf{K}_r(X) \stackrel{\mathrm{def}}{=} \mathop{\mathrm{sup}}\limits_{n\in\mathbb N}\big\|\mathsf{Rad}\big\|_{L_r(\mathscr{C}_n;X)\to L_r(\mathscr{C}_n;X)} <\infty. \end{align} $$
In particular, it follows from Lemma 4 that
$\mathfrak {s}_p(X)<\infty $
for some
$p\in (1,\infty )$
implies that
$\mathsf{K}_r(X)<\infty $
for every
$r\in (1,\infty )$
. We proceed by showing that Banach spaces belonging to the class considered in [Reference Hytönen and NaorHN13, Theorem 1.4] satisfy the assumptions of Theorem 1.
Proposition 5 Let
$(X,\|\cdot \|_X)$
be a Banach space. If
$X^\ast $
is a
$\text{UMD}^+$
space, then X is a
$\text{UMD}^-$
space and
$\mathfrak {s}_p(X)<\infty $
for every
$p\in (1,\infty )$
.
Proof The fact that if
$X^\ast $
is
$\text{UMD}^+$
, then X is
$\text{UMD}^-$
has been proved by Garling in [Reference GarlingGar90, Theorem 1], so we only have to prove that
$\mathfrak {s}_p(X)<\infty $
. Let
$f_1,\dots ,f_n:\mathscr {C}_n\to X$
and
$G^\ast :\mathscr {C}_n\times \mathscr {C}_n\to X^\ast $
be such that
$$ \begin{align} \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i \mathscr{E}_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p} = \frac{1}{4^n} \sum_{\varepsilon,\unicode{x3b4}\in\mathscr{C}_n} \big\langle G^\ast(\varepsilon,\unicode{x3b4}),\sum_{i=1}^n\delta_i \mathscr{E}_if_i(\varepsilon)\big\rangle \end{align} $$
and
$\|G^\ast \|_{L_q(\mathscr {C}_n\times \mathscr {C}_n;X^\ast )} = 1$
, where
$\tfrac {1}{p}+\tfrac {1}{q}=1$
. Let
$G_i^\ast :\mathscr {C}_n\to X^\ast $
be given by
$$ \begin{align} \forall \ \varepsilon\in\mathscr{C}_n, \ \ \ \ G_i^\ast(\varepsilon) = \frac{1}{2^n}\sum_{\unicode{x3b4}\in\mathscr{C}_n} \delta_i G^\ast(\varepsilon,\unicode{x3b4}). \end{align} $$
Then, since
$X^\ast $
is
$\text{UMD}^+$
, we deduce that
$X^\ast $
is also K-convex (this is proved in [Reference GarlingGar90] but it also follows by combining Bourgain’s inequality (9) with Lemma 4) and thus
$$ \begin{align} \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i G^\ast_i\Big\|_{L_q(\mathscr{C}_n;X^\ast)}^q\Big)^{1/q} \stackrel{(38)}{=} \Big(\frac{1}{4^n}\sum_{\varepsilon,\unicode{x3b4}\in\mathscr{C}_n} \big\| \mathsf{Rad}_\unicode{x3b4} G^\ast(\varepsilon,\unicode{x3b4})\big\|_X^q\Big)^{1/q} \leqslant \mathsf{K}_q(X^\ast). \end{align} $$
Hence, we have
$$ \begin{align} &\Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i \mathscr{E}_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p} \stackrel{(37)\wedge(38)}{=} \frac{1}{4^n} \sum_{\varepsilon,\unicode{x3b4}\in\mathscr{C}_n} \big\langle \sum_{i=1}^n\delta_i G_i^\ast(\varepsilon),\sum_{i=1}^n\delta_i \mathscr{E}_if_i(\varepsilon)\big\rangle \\ & = \frac{1}{2^n}\sum_{\varepsilon\in\mathscr{C}_n} \langle G_i^\ast(\varepsilon), \mathscr{E}_if_i(\varepsilon)\rangle = \frac{1}{2^n}\sum_{\varepsilon\in\mathscr{C}_n} \langle \mathscr{E}_iG_i^\ast(\varepsilon), f_i(\varepsilon)\rangle \nonumber\\ &= \frac{1}{4^n} \sum_{\varepsilon,\unicode{x3b4}\in\mathscr{C}_n} \Big\langle \sum_{i=1}^n\delta_i \mathscr{E}_iG_i^\ast(\varepsilon),\sum_{i=1}^n\delta_i f_i(\varepsilon)\Big\rangle \nonumber\\ & \leqslant \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i \mathscr{E}_i G^\ast_i\Big\|_{L_q(\mathscr{C}_n;X^\ast)}^q\Big)^{1/q} \cdot \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p}. \nonumber \end{align} $$
Therefore, combining (40) with (8) and (39), we deduce that
$$ \begin{align} \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} & \Big\| \sum_{i=1}^n \delta_i \mathscr{E}_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p} \\ & \stackrel{(8)}{\leqslant} \mathfrak{s}_q(X^\ast) \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i G^\ast_i\Big\|_{L_q(\mathscr{C}_n;X^\ast)}^q\Big)^{1/q} \cdot \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p} \nonumber\\ & \stackrel{(39)}{\leqslant} \mathfrak{s}_q(X^\ast) \mathsf{K}_q(X^\ast) \cdot \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i f_i\Big\|_{L_p(\mathscr{C}_n;X)}^p\Big)^{1/p}, \nonumber \end{align} $$
which shows that
$\mathfrak {s}_p(X) \leqslant \mathsf{K}_q(X^\ast ) \mathfrak {s}_q(X^\ast )$
.▪
We conclude by observing that spaces satisfying the assumptions of Theorem 1 are necessarily superreflexive (see [Reference PisierPis16, Chapter 11] for the relevant terminology).
Lemma 6 If a
$\text{UMD}^-$
Banach space
$(X,\|\cdot \|_X)$
satisfies
$\mathfrak {s}_p(X)<\infty $
, then X is superreflexive.
Proof A theorem of Pisier [Reference PisierPis73] asserts that a Banach space X is K-convex if and only if X has nontrivial Rademacher type. Therefore, we deduce from Lemma 4 that if
$\mathfrak {s}_p(X)<\infty $
for some
$p\in (1,\infty )$
, then there exist
$s\in (1,2]$
and
$T_s(X)\in (0,\infty )$
such that
$$ \begin{align} \forall \ x_1,\dots ,x_n\in X, \ \ \ \ \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n}\Big\|\sum_{i=1}^n \delta_i x_i\Big\|_X^s\Big)^{1/s} \leqslant T_s(X) \Big(\sum_{i=1}^n \|x_i\|_X^s\Big)^{1/s}. \end{align} $$
Therefore, if X also satisfies the
$\text{UMD}^-$
property, we deduce that for every X-valued martingale
$\{\mathscr {M}_i:\Omega \to X\}_{i=0}^n$
,
$$ \begin{align} \|\mathscr{M}_n - \mathscr{M}_0\|_{L_s(\Omega,\unicode{x3bc};X)} & \leqslant \beta_s^-(X) \Big(\frac{1}{2^n} \sum_{\unicode{x3b4}\in\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i (\mathscr{M}_i-\mathscr{M}_{i-1}) \Big\|^s_{L_s(\Omega,\unicode{x3bc};X)}\Big)^{1/s} \nonumber\\ & \stackrel{(42)}{\leqslant} \beta_s^-(X)T_s(X) \Big( \sum_{i=1}^n \|\mathscr{M}_i-\mathscr{M}_{i-1}\|^s_{L_s(\Omega,\unicode{x3bc};X)}\Big)^{1/s}, \end{align} $$
which means that X has martingale type s. Combining this with well-known results linking martingale type and superreflexivity (see [Reference PisierPis16, Chapters 10-11]), we reach the desired conclusion.▪
Therefore, Theorem 1 establishes that
$\mathfrak {P}_p^n(X)\asymp _p 1$
for X in a (strict, see [Reference GarlingGar90, Reference QiuQiu12]) subclass of all superreflexive spaces. According to a result of the author and A. Naor (see [Reference EskenazisEsk19, Chapter 4]), the bound
$\mathfrak {P}_p^n(X) = o(\log n)$
holds for every superreflexive Banach space X and
$p\in (1,\infty )$
.
Remark added in proofs. After the submission of this paper, Ivanisvili, van Handel and Volberg circulated a preprint [Reference Ivanisvili, van Handel and VolbergIvHV20] showing that a Banach space satisfies
$\mathop {\mathrm {sup}} _n\mathfrak {P}_p^n(X)<\infty $
for every (equivalently, for some)
$p\in [1,\infty )$
if and only if X has finite cotype.
Acknowledgment
I would like to thank Assaf Naor for helpful discussions.
