Various sharp estimates for semi-discrete Riesz transforms of the second order

TL;DR

This paper establishes sharp $L^p$ and related estimates for second order Riesz transforms on certain Lie groups, combining stochastic integral representations with martingale inequalities to improve understanding of these operators.

Contribution

It provides new sharp $L^p$ estimates and extends the analysis of second order Riesz transforms to semi-discrete Lie groups, integrating stochastic methods with harmonic analysis.

Findings

Derived sharp $L^p$ estimates using Choi type constants.

Established weak-type, logarithmic, and exponential bounds.

Proved optimal $L^q o L^p$ estimates for the transforms.

Abstract

We give several sharp estimates for a class of combinations of second order Riesz transforms on Lie groups ${G}={G}_{x} \times {G}_{y}$ that are multiply connected, composed of a discrete abelian component ${G}_{x}$ and a connected component ${G}_{y}$ endowed with a biinvariant measure. These estimates include new sharp $L^p$ estimates via Choi type constants, depending upon the multipliers of the operator. They also include weak-type, logarithmic and exponential estimates. We give an optimal $L^q \to L^p$ estimate as well. It was shown recently by Arcozzi, Domelevo and Petermichl that such second order Riesz transforms applied to a function may be written as conditional expectation of a simple transformation of a stochastic integral associated with the function. The proofs of our theorems combine this stochastic integral representation with a number of deep estimates for pairs of martingales under strong differential subordination by Choi, Banuelos and Osekowski. When two continuous directions are available, sharpness is shown via the laminates technique. We show that sharpness is preserved in the discrete case using Lax-Richtmyer theorem.

Equations

$$\text{mathbbm}E(M_0^{\alpha ,f}|{\mathcal{Z}}_0=z).$$

$$({\partial }^{+}f/\partial x_i)(x,y):=f(x+g_i,y)-f(x,y):=({\partial }_i^{+}f)(x,y)$$

$$({\partial }^{-}f/\partial x_i)(x,y):=f(x,y)-f(x-g_i,y):=({\partial }_i^{-}f)(x,y).$$

$${\hat{\nabla }}_{x}f(x):=(X_1^{+}f,X_2^{+}f,\dots ,X_1^{-}f,X_2^{-}f,\dots )(x)=\sum _{i=1}^m\sum _{\tau =\pm }{X}_i^{\tau }f(x)$$

$$X_i^{+}=(0,\dots ,1,\dots ,0)\times 0_m,X_i^{-}=0_m\times (0,\dots ,1,\dots ,0).$$

$$(U,V{)}_{{\hat{T}}_x\text{mathbbm}{G}_x}:=\frac{1}{2}\sum _{i=1}^m\sum _{\tau =\pm }{U}_i^{\tau }{V}_i^{\tau }.$$

$${\hat{\nabla }}_{z}f(z)$$

$$(\phi ,\psi {)}_{L^2(\text{mathbbm}G)}$$

$${\mathbf{R}}_{\text{mathbbm}M}:={\nabla }_{\text{mathbbm}M}\circ (-{\Delta }_{\text{mathbbm}M}{)}^{-1/2}$$

$$(U,V{)}_{L^2(\hat{T}\text{mathbbm}G)}$$

$$U(z)=\sum _{i=1}^m\sum _{\tau =\pm }{U}_i^{\tau }(z){\hat{X}}_i^{\tau }+\sum _{j=1}^n{U}_j(z){\hat{Y}}_j,$$

$${\hat{\nabla }}_z^{\ast }U(z)$$

$${\Delta }_{z}f(z)$$

$${\hat{\mathbf{R}}}_{z}f:=\left({\hat{\nabla }}_{z}f\right)\circ (-{\Delta }_{z}f{)}^{-1/2}$$

$$R_i^{\pm }=X_i^{\pm }\circ (-{\Delta }_z{)}^{-1/2}\mathrm{and}{R}_j=Y_j\circ (-{\Delta }_z{)}^{-1/2}.$$

$$R_i^2:=R_i^{+}{R}_i^{-}=R_i^{-}{R}_i^{+}.$$

$$R_{\alpha }^2:=\sum _{i=1}^m{\alpha }_i^x{R}_i^2+\sum _{j,k=1}^n{\alpha }_{jk}^y{R}_j{R}_k,$$

$$R_{\alpha }^2=\left({\hat{\mathbf{R}}}_z,{\mathbf{A}}_{\alpha }{\hat{\mathbf{R}}}_z\right)$$

$$\boldsymbol{A}_{\alpha}:=\left(\begin{array}[]{cc}\boldsymbol{A}_{\alpha}^{x}&\boldsymbol{0}\\ \boldsymbol{0}&\boldsymbol{A}_{\alpha}^{y}\end{array}\right)$$

$${\mathbf{A}}_{\alpha }^x=\mathrm{diag}({\alpha }_1^x,\dots ,{\alpha }_m^x,{\alpha }_1^x,\dots ,{\alpha }_m^x)\in \text{mathbbm}{C}^{2m\times 2m},{\mathbf{A}}_{\alpha }^y=({\alpha }_{jk}^y{)}_{j,k=1\dots n}\in \text{mathbbm}{C}^{n\times n}.$$

$$\Vert R_{\alpha }^2{\Vert }_p⩽{\Vert {\mathbf{A}}_{\alpha }\Vert }_2(p^{\ast }-1),$$

$${\Vert {\mathbf{A}}_{\alpha }\Vert }_2=\max \left({\Vert {\mathbf{A}}_{\alpha }^x\Vert }_2,{\Vert {\mathbf{A}}_{\alpha }^y\Vert }_2\right)=\max \left(|{\alpha }_1^x|,\dots ,|{\alpha }_m^x|,{\Vert {\mathbf{A}}_{\alpha }^y\Vert }_2\right).$$

$${\mathfrak{C}}_{0,1,p}=\frac{p}{2}+\frac{1}{2}\log \left(\frac{1+e^{-2}}{2}\right)+\frac{{\beta }_2}{p}+\dots .,$$

$$||f||{}_{L^{p,\infty }(\text{mathbbm}G,\text{mathbbm}C)}=\sup \left\{{\mu }_z(E{)}^{1/p-1}{\int }_{E}fd{\mu }_z\right\},$$

$$\Vert R_{\alpha }^2{\Vert }_{L^p(\text{mathbbm}G,\text{mathbbm}C)\to L^{p,\infty }(\text{mathbbm}G,\text{mathbbm}C)}⩽{\Vert {\mathbf{A}}_{\alpha }\Vert }_2\cdot \{\begin{array}{ll}{\left(\frac{1}{2}\Gamma \left(\frac{2p-1}{p-1}\right)\right)}^{1-1/p}& \text{mbox}if1<p\le 2,\\ {\left(\frac{p^{p-1}}{2}\right)}^{1/p}& \text{mbox}ifp\ge 2.\end{array}$$

$${\int }_E|R_{\alpha }^{2}f|d{\mu }_z⩽{\Vert {\mathbf{A}}_{\alpha }\Vert }_2\cdot \left(K{\int }_{\text{mathbbm}G}\Psi (|f|)d{\mu }_z+\frac{{\mu }_z(E)}{2(K-1)}\right).$$

$${\int }_{\text{mathbbm}G}\Phi \left(\frac{|R_{\alpha }^{2}f|}{K{\Vert {\mathbf{A}}_{\alpha }\Vert }_2}\right)d{\mu }_z⩽\frac{||f||{}_{L^1(\text{mathbbm}G,\text{mathbbm}C)}}{2K(K-1)}.$$

$$\Vert R_{\alpha }^{2}f{\Vert }_{L^p(E,\text{mathbbm}C)}⩽C_{p,q}{\Vert {\mathbf{A}}_{\alpha }\Vert }_2\Vert f{\Vert }_{L^q(\text{mathbbm}G,\text{mathbbm}C)}{\mu }_z(E{)}^{1/p-1/q}.$$

$$(\mathcal{I}f,g)$$

$$Y_j\circ {\Delta }_z$$

Full text

It was shown recently by Arcozzi-Domelevo-Petermichl that such second order Riesz transforms applied to a function may be written as conditional expectation of a simple transformation of a stochastic integral associated with the function.

The proofs of our theorems combine this stochastic integral representation with a number of deep estimates for pairs of martingales under strong differential subordination by Choi, Banuelos and Osȩkowski.

When two continuous directions are available, sharpness is shown via the laminates technique. We show that sharpness is preserved in the discrete case using Lax-Richtmyer theorem.

Adam Osekowski is supported by Narodowe Centrum Nauki Poland (NCN), grant DEC-2014/14/E/ST1/00532

Stefanie Petermichl is supported by ERC project CHRiSHarMa DLV-862402

1. Introduction

Sharp, classical $L^{p}$ norm inequalities for pairs of differentially subordinate martingales date back to the celebrated work of Burkholder [15] in 1984 where the optimal constant is exhibited. See also from the same author [17][18]. The relation between differentially subordinate martingales and CZ (i.e. Caldéron–Zygmund) operators is known at least since Gundy–Varopoulos [32]. Banuelos–Wang [12] were the first to exploit this connection to prove new sharp inequalities for singular intergrals. This intersection of probability theory with classical questions in harmonic analysis has lead to much interest and a vast literature has been accumulating on this line of research.

In this article we state a number of sharp estimates that hold in the very recent, new direction concerning the semi–discrete setting, applying it to a family of second order Riesz transforms on multiply–connected Lie groups. We recall their representation through stochastic integrals using jump processes on multiply–connected Lie groups from [3]. In this representation formula jump processes play a role, but the strong differential subordination holds between the martingales representing the test function and the operator applied to the test function.

The usual procedure for obtaining (sharp) inequalities for operators of Calderón–Zygmund type from inequalities for martingales is the following. Starting with a test function $f$, martingales are built using Brownian motion or background noise and harmonic functions in the upper half space $\mathbbm{R}^{+}\times\mathbbm{R}^{n}$. Through the use of Itøformula, it is shown that the martingale arising in this way from $Rf$, where $R$ is a Riesz transform in $\mathbbm{R}^{n}$, is a martingale transform of the martingale arising from $f$. The two form a pair of martingales that have differential subordination and (in case of Hilbert or Riesz transforms) orthogonality. One then derives sharp martingale inequalities under hypotheses of strong differential subordination (and orthogonality) relations.

In the case of Riesz transforms of the second order, the use of heat extensions in the upper half space instead of Poisson extensions originated in the context of a weighted estimate in Petermichl–Volberg [44] and was used to prove $L^{p}$ estimates for the second order Riesz transforms based on the results of Burkholder in Nazarov–Volberg [50] as part of their best-at-time estimate for the Beurling–Ahlfors operator, whose real and imaginary parts themselves are second order Riesz transforms. We mention the recent version on discrete abelian groups Domelevo–Petermichl [24] also using a type of heat flow. These proofs are deterministic. The technique of Bellman functions was used. This deterministic strategy does well when no orthogonality is present and when strong subordination is the only important property. Stochastic proofs (aside from giving better estimates in some situations) also have the advantage that once the integral representation is known, the proofs are a very concise consequence of the respective statements on martingales.

In [3] the authors proved sharp $L^{p}$ estimates for semi–discrete second order Riesz transforms $R_{\alpha}^{2}$ using stochastic integrals. There is an array of Riesz transforms of the second order that are treated, indexed my a matrix index $\alpha$ (see below for precisions on acceptable $\alpha$). The following representation formula of semi-discrete second order Riesz transforms $R^{2}_{\alpha}$ à la Gundy–Varopoulos (see [32]) is instrumental:

Theorem**.**

**(Arcozzi–Domelevo–Petermichl, 2016)

The second order Riesz transform $R_{\alpha}^{2}f$ of a function $f\in L^{2}(\mathbbm{G})$ as defined in (1.1) can be written as the conditional expectation**

[TABLE]

Here $M_{t}^{\alpha,f}$ is a suitable martingale transform of a martingale $M^{f}_{t}$ associated to $f$, and $\mathcal{Z}_{t}$ is a suitable random walk on $\mathbbm{G}$

We remark that the $L^{p}$ estimates of the discrete Hilbert transform on the integers are still open. It is a famous conjecture that this operator has the same norm as its continuous counterpart.

These known $L^{p}$ norm inequalities use special functions found in the results of Pichorides [45], Verbitsky[18], Essén [28], Banuelos–Wang [12] when orthogonality is present in addition to differential subordination or Burkholder [15][16][17], Wang [51] when differential subordination is the only hypothesis.

The aim of the present paper is to establish new estimates for semi–discrete Riesz transforms by using the martingale representation above together with recent martingale inequalities found in the literature.

Here is a brief description of the new results in this paper.

• •

In the case where the function $f$ is real valued, we can obtain better estimates for $R_{\alpha}^{2}$ than in the general case. These estimates depend upon the make of the matrix index $\alpha$. The precise statement is found in Theorem 1.2.

• •

We prove a refined sharp weak type estimate using a weak type norm defined just before the statement of Theorem 1.3 .

• •

We prove logarithmic and exponential estimates, in a sense limiting (in $p$) cases of the classical sharp $L^{p}$ estimate. See Theorem 1.4.

• •

We consider the norm estimates of the $R^{2}_{\alpha}:L^{q}\to L^{p}$, spaces of different exponent. The statement is found in Theorem 1.5.

1.1. Differential operators and Riesz transforms

First order derivatives and tangent planes

We will consider Lie groups $\mathbbm{G}:=\mathbbm{G}_{x}\times\mathbbm{G}_{y},$ where $\mathbbm{G}_{x}$ is a discrete abelian group with a fixed set $G$ of $m$ generators, and their reciprocals, and $\mathbbm{G}_{y}$ is a connected, Lie group of dimension $n$ endowed with a biinvariant metric. The choice of the set $G$ of generators in $\mathbbm{G}_{x}$ corresponds to the choice of a bi-invariant metric structure on $\mathbbm{G}_{x}$. We will use on $\mathbbm{G}_{x}$ the multiplicative notation for the group operation. We will define a product metric structure on $\mathbbm{G}$, which agrees with the Riemannian structure on the first factor, and with the discrete “word distance” on the second. We will at the same time define a “tangent space”  $T_{z}\mathbbm{G}$ for $\mathbbm{G}$ at a point $z=(x,y)\in(\mathbbm{G}_{x}\times\mathbbm{G}_{y})=\mathbbm{G}$. We will do this in three steps.

First, since $\mathbbm{G}_{y}$ is an $n$-dimensional connected Lie group with Lie algebra $\mathfrak{G}_{y}$. We can identify each left-invariant vector field $Y$ in $\mathfrak{G}_{y}$ with its value at the identity $e$, $\mathfrak{G}_{y}\equiv T_{e}\mathbbm{G}_{y}$. Since $\mathbbm{G}$ is compact, it admits a bi-invariant Riemannian metric, which is unique up to a multiplicative factor. We normalize it so that the measure $\mu_{y}$ associated with the metric satisfies $\mu_{y}(\mathbbm{G}_{y})=1$. The measure $\mu_{y}$ is also the normalized Haar measure of the group. We denote by $\char 60\relax\cdot,\cdot\char 62\relax_{y}$ be the corresponding inner product on $T_{y}\mathbbm{G}_{y}$ and by $\boldsymbol{\nabla_{y}}f(y)$ the gradient at $y\in\mathbbm{G}_{y}$ of a smooth function $f:\mathbbm{G}_{y}\rightarrow\mathbbm{R}$. Let $Y_{1},\ldots,Y_{n}$ be a orthonormal basis for $\mathfrak{G}_{y}$. The gradient of $f$ can be written $\boldsymbol{\nabla_{y}}f=Y_{1}(f)Y_{1}+\ldots+Y_{n}(f)Y_{n}$.

Second, in the discrete component $\mathbbm{G}_{x}$, let $\mathfrak{G}_{x}=(g_{i})_{i=1,\ldots,m}$ be a set of generators for $\mathbbm{G}_{x}$, such that for $i\neq j$ and $\sigma=\pm 1$ we have $g_{i}\neq g_{j}^{\sigma}$. The choice of a particular set of generators induces a word metric, hence, a geometry, on $\mathbbm{G}_{x}$. Any two sets of generators induce bi-Lipschitz equivalent metrics.

At any point $x\in\mathbbm{G}_{x}$, and given a direction $i\in\{1,\ldots,m\}$, we can define the right and the left derivative at $x$ in the direction $i$:

[TABLE]

[TABLE]

Comparing with the continuous component, this suggests that the tangent plane $\hat{T}_{x}\mathbbm{G}_{x}$ at a point $x$ of the discrete group $\mathbbm{G}_{x}$ might actually be split into a “right” tangent plane $T_{x}^{+}\mathbbm{G}_{x}$ and a “left” tangent plane $T_{x}^{-}\mathbbm{G}_{x}$, according to the direction with respect to which discrete differences are computed. We consequently define the augmented discrete gradient $\widehat{\boldsymbol{\nabla}}_{x}f(x)$, with a hat, as the $2m$–vector of $\hat{T}_{x}\mathbbm{G}_{x}:=T_{x}^{+}\mathbbm{G}_{x}\oplus T_{x}^{-}\mathbbm{G}_{x}$ accounting for all the local variations of the function $f$ in the direct vicinity of $x$; that is, the $2m$–column–vector

[TABLE]

with $X_{i}^{\tau}\in\hat{T}_{x}\mathbbm{G}_{x},$ where we noted the discrete derivatives $X_{i}^{\pm}f:=\partial_{i}^{\pm}f$ and introduced the discrete $2m$–vectors $X_{i}^{\pm}$ as the column vectors of $\mathbbm{Z}^{2m}$

[TABLE]

Here the $1$’s in $X_{i}^{\pm}$ are located at the $i$–th position of respectively the first or the second $m$–tuple. Notice that those vectors are independent of the point $x$. The scalar product on $\hat{T}_{x}\mathbbm{G}_{x}:=T_{x}^{+}\mathbbm{G}_{x}\oplus T_{x}^{-}\mathbbm{G}_{x}$ is defined as

[TABLE]

We chose to put a factor $\frac{1}{2}$ in front of the scalar product to compensate for the fact that we consider both left and right differences.

Finally, for a function $f$ defined on the cartesian product $\mathbbm{G}:=\mathbbm{G}_{x}\times\mathbbm{G}_{y}$, the (augmented) gradient $\widehat{\boldsymbol{\nabla}}_{z}f(z)$ at the point $z=(x,y)$ is an element of the tangent plane $\hat{T}_{z}\mathbbm{G}:=\hat{T}_{x}\mathbbm{G}_{x}\oplus T_{y}\mathbbm{G}_{y}$, that is a $(2m+n)$–column–vector

[TABLE]

where $\hat{X}_{i}^{\tau}$ and $\hat{Y}_{j}(z)$ can be identified with column vectors of size $(2m+n)$ with obvious definitions and scalar product $(\cdot,\cdot)_{\hat{T}_{z}\mathbbm{G}_{z}}$.

Let $\mathrm{d}\mu_{z}:=\mathrm{d}\mu_{x}\mathrm{d}\mu_{y}$, $\mathrm{d}\mu_{x}$ being the counting measure on $\mathbbm{G}_{x}$ and $\mathrm{d}\mu_{y}$ being  the Haar measure on $\mathbbm{G}_{y}$. The inner product of $\varphi,\psi$ in $L^{2}(\mathbbm{G})$ is

[TABLE]

Finally, we make the following hypotheses

Hypothesis

We assume everywhere in the sequel:

The discrete component $\mathbbm{G}_{x}$ of the Lie group $\mathbbm{G}$ is an abelian group 2. 2.

The connected component $\mathbbm{G}_{y}$ of the Lie group $\mathbbm{G}$ is a Lie group that can be endowed with a biinvariant Riemannian metric, so that the family $(Y_{j})_{j=1,\ldots,n}$ commutes with $\Delta_{y}$.

Notice that this includes compact Lie groups $\mathbbm{G}_{y}$ since those can be endowed with a biinvariant metric. It also includes the usual Euclidian spaces since those are commutative.

Riesz transforms

Following [1][2], recall first that for a compact Riemannian manifold $\mathbbm{M}$ without boundary, one denotes by $\boldsymbol{\nabla}_{\mathbbm{M}}$, $\operatorname{div}_{\mathbbm{M}}$ and $\Delta_{\mathbbm{M}}:=\operatorname{div}_{\mathbbm{M}}\boldsymbol{\nabla}_{\mathbbm{M}}$ respectively the gradient, the divergence and the Laplacian associated with $\mathbbm{M}$. Then $-\Delta_{\mathbbm{M}}$ is a positive operator and the vector Riesz transform is defined as the linear operator

[TABLE]

acting on $L^{2}_{0}(\mathbbm{M})$ ($L^{2}$ functions with vanishing mean). It follows that if $f$ is a function defined on $\mathbbm{M}$ and $y\in\mathbbm{M}$ then $\boldsymbol{R}_{\mathbbm{M}}f(y)$ is a vector of the tangent plane $T_{y}\mathbbm{M}$.

Similarly on $\mathbbm{M}=\mathbbm{G}$, we define $\boldsymbol{\nabla}_{\mathbbm{G}}:=\widehat{\boldsymbol{\nabla}}_{z}$ as before, and then we define the divergence operator as its formal adjoint, that is $-\operatorname{div}_{\mathbbm{G}}=-\widehat{\operatorname{div}}_{z}:=\widehat{\boldsymbol{\nabla}}_{z}^{\ast}$, with respect to the natural $L^{2}$ inner product of vector fields:

[TABLE]

We have the $L^{2}$-adjoints $(X_{i}^{\pm})^{\ast}=-X_{i}^{\mp}$ and $Y_{j}^{\ast}=-Y_{j}$. If $U\in\hat{T}\mathbbm{G}$ is defined by

[TABLE]

we define its divergence $\widehat{\boldsymbol{\nabla}}_{z}^{\ast}U$ as

[TABLE]

The Laplacian $\Delta_{\mathbbm{G}}$ is as one might expect:

[TABLE]

where we denoted $X_{i}^{2}:=X_{i}^{+}X_{i}^{-}=X_{i}^{-}X_{i}^{+}$. We have chosen signs so that $-\Delta_{\mathbbm{G}}\geqslant 0$ as an operator. The Riesz vector $(\hat{\boldsymbol{R}}_{z}f)(z)$ is the $(2m+n)$–column–vector of the tangent plane $\hat{T}_{z}\mathbbm{G}$ defined as the linear operator

[TABLE]

We also define transforms along the coordinate directions:

[TABLE]

Plan of the paper

In the next two sections, we present successively the main results of the paper and recall the weak formulations involving second order Riesz transforms and semi-discrete heat extensions. Section 2 introduces the stochastic setting for our problems. This includes in Subsection 2.1 semi-discrete random walks, martingale transforms and quadratic covariations. Subsection 2.2 presents a set of martingale inequalities already known in the literature. Finally, in Section 3 we give the proof of the main results.

1.2. Main results

In this text, we are concerned with second order Riesz transforms and combinations thereof. We first define the square Riesz transform in the (discrete) direction $i$ to be

[TABLE]

Then, given $\alpha:=((\alpha^{x}_{i})_{i=1\ldots m},(\alpha^{y}_{jk})_{j,k=1\ldots n})\in\mathbbm{C}^{m}\times\mathbbm{C}^{n\times n}$, we define $R_{\alpha}^{2}$ to be the following combination of second order Riesz transforms:

[TABLE]

where the first sum involves squares of discrete Riesz transforms as defined above, and the second sum involves products of continuous Riesz transforms. This combination is written in a condensed manner as the quadratic form

[TABLE]

where $\boldsymbol{A}_{\alpha}$ is the $(2m+n)\times(2m+n)$ block matrix

[TABLE]

with

[TABLE]

In the theorems below, we assume that $\mathbbm{G}$ is a Lie group and $R^{2}_{\alpha}$ is a combination of second order Riesz transforms as defined above. The first application of the stochastic integral formula, Theorem 1.1 was done in [3], while the other applications, Theorems 1.2 1.3 1.4 and 1.5 are new.

Theorem 1.1**.**

(Arcozzi–Domelevo–Petermichl, 2016) For any $1\char 60\relax p\char 60\relax\infty$ we have

[TABLE]

where, as previously, $p^{*}=\max\{p,p/(p-1)\}$.

Above, we have set:

[TABLE]

In the case where $\mathbbm{G}=\mathbbm{G}_{x}$ only consists of the discrete component, this was proved in [25][24] using the deterministic Bellman function technique. In the case where $\mathbbm{G}=\mathbbm{G}_{y}$ is a connected compact Lie group, this was proved in [8] using Brownian motions defined on manifolds and projections of martingale transforms.

In the case where the function $f$ is real valued, we can obtain better estimates. For any real numbers $a\char 60\relax b$ and any $1\char 60\relax p\char 60\relax\infty$, let $\mathfrak{C}_{a,b,p}$ be the constants introduced in Bañuelos and Osȩkowski [10].

Theorem 1.2**.**

Assume that $a\boldsymbol{I}\leqslant\boldsymbol{A}_{\alpha}\leqslant b\boldsymbol{I}$ in the sense of quadratic forms. Then $R^{2}_{\alpha}:L^{p}(\mathbbm{G},\mathbbm{R})\rightarrow L^{p}(\mathbbm{G},\mathbbm{R})$ enjoys the norm estimate $\|R_{\alpha}^{2}\|_{p}\leqslant\mathfrak{C}_{a,b,p}$.

We should point out here that the constants $\mathfrak{C}_{a,b,p}$ appear in earlier works of Burkholder [15] (for $a=-b$: then $\mathfrak{C}_{a,b,p}=b(p^{\ast}-1)$), and in the paper [19] by Choi (in the case when one of $a$, $b$ is zero). The Choi constants are not explicit; an approximation of $\mathfrak{C}_{0,1,p}$ is known and writes as

[TABLE]

with $\beta_{2}=\log^{2}\left(\tfrac{1+e^{-2}}{2}\right)+\tfrac{1}{2}\log\left(\tfrac{1+e^{-2}}{2}\right)-2\left(\tfrac{e^{-2}}{1+e^{-2}}\right)^{2}.$

Coming back to complex-valued functions, we will also establish the following weak-type bounds. We consider the norms

[TABLE]

where the supremum is taken over the class of all measurable subsets $E$ of $\mathbbm{G}$ of positive measure.

Theorem 1.3**.**

For any $1\char 60\relax p\char 60\relax\infty$ we have

[TABLE]

We will also prove the following logarithmic and exponential estimates, which can be regarded as versions of Theorem 1.1 for $p=1$ and $p=\infty$. Consider the Young functions $\Phi,\,\Psi:[0,\infty)\to[0,\infty)$, given by $\Phi(t)=e^{t}-1-t$ and $\Psi(t)=(t+1)\log(t+1)-t.$

Theorem 1.4**.**

Let $K\char 62\relax 1$ be fixed.

(i) For any measurable subset $E$ of $\mathbbm{G}$ and any $f$ on $\mathbbm{G}$ we have

[TABLE]

(ii) For any $f:\mathbbm{G}\to\mathbbm{C}$ bounded by $1$,

[TABLE]

Our final result concerns another extension of Theorem 1.1, which studies the action of $R_{\alpha}^{2}$ between two different $L^{p}$ spaces. For $1\leq p\char 60\relax q\char 60\relax\infty$, let $C_{p,q}$ be the constant defined by Osȩkowski in [40].

Theorem 1.5**.**

For any $1\leq p\char 60\relax q\char 60\relax\infty$, any measurable subset $E$ of $\mathbbm{G}$ and any $f\in L^{q}(\mathbbm{G})$ we have

[TABLE]

An interesting feature is that all the estimates in the five theorems above are sharp when the group $\mathbbm{G}=\mathbbm{G}_{x}\times\mathbbm{G}_{y}$ and $\dim(\mathbbm{G}_{y})+\dim^{\infty}(\mathbbm{G}_{x})\geqslant 2$, where $\dim^{\infty}(\mathbbm{G}_{x})$ denotes the number of infinite components of $\mathbbm{G}_{x}$.

1.3. Weak formulations

Let $f:\mathbbm{G}\rightarrow\mathbbm{C}$ be given. The heat extension $\tilde{f}(t)$ of $f$ is defined as $\tilde{f}(t):=e^{t\Delta_{z}}f=:P_{t}f$. We have therefore $\tilde{f}(0)=f$. The aim of this section is to derive weak formulations for second order Riesz transforms. We start with the weak formulation of the identity operator $\mathcal{I}$, that is obtained by using semi-discrete heat extensions (see [3] for details).

Assume $f$ in $L^{2}(\mathbbm{G})$ and $g$ in $L^{2}(\mathbbm{G})$. Let $\bar{f}$ be the average of $f$ on $\mathbbm{G}$ if $\mathbbm{G}$ has finite measure and zero otherwise. Then

[TABLE]

and the sums and integrals that arise converge absolutely.

In order to pass to the weak formulation for the squares of Riesz transforms, we first observe that the following commutation relations hold

[TABLE]

This is an easy consequence of the hypothesis made on the Lie group. Following [3], the following weak formulation for second order Riesz transforms holds

Assume $f$ in $L^{2}(\mathbbm{G})$ and $g$ in $L^{2}(\mathbbm{G})$, then

[TABLE]

and the sums and integrals that arise converge absolutely.

2. Stochastic integrals and martingale transforms

In what follows, we assume that we have a complete probability space $(\Omega,\mathcal{F},\mathbbm{P})$ with a càdlàg (i.e. right continuous left limit) filtration $(\mathcal{F}_{t})_{t\geqslant 0}$ of sub-$\sigma$–algebras of $\mathcal{F}$. We assume as usual that $\mathcal{F}_{0}$ contains all events of probability zero. All random walks and martingales are adapted to this filtration.

We define below a continuous-time random process $\mathcal{Z}$ with values in $\mathbbm{G}$, $\mathcal{Z}_{t}:=(\mathcal{X}_{t},\mathcal{Y}_{t})\in\mathbbm{G}_{x}\times\mathbbm{G}_{y}$, having infinitesimal generator $L=\Delta_{z}$. The pure-jump component $\mathcal{X}_{t}$ is a compound Poisson jump process on the discrete set $\mathbbm{G}_{x}$, wheras the continuous component $\mathcal{Y}_{t}$ is a standard brownian motion on the manifold $\mathbbm{G}_{y}$. Then, Itô’s formula ensures that semi-discrete “harmonic” functions $f:\mathbbm{R}^{+}\times\mathbbm{G}\rightarrow\mathbbm{C}$ solving the backward heat equation $(\partial_{t}+\Delta_{z})f=0$ give rise to martingales $M_{t}^{f}:=f(t,\mathcal{Z}_{t})$ for which we define a class of martingale transforms.

2.1. Stochastic integrals, Martingale transforms and quadratic covariations

Stochastic integrals on Riemannian manifolds and Itô

integral

Following Emery [26][27], see also Arcozzi [1][2], we define the Brownian motion $\mathcal{Y}_{t}$ on $\mathbbm{G}_{y}$, a compact Riemannian manifold, as the process $\mathcal{Y}_{t}\ :\ \Omega\rightarrow(0,T)\times\mathbbm{G}_{y}$ such that for all smooth functions $f:\mathbbm{G}_{y}\rightarrow\mathbbm{R}$, the quantity

[TABLE]

is an $\mathbbm{R}$–valued continuous martingale. For any adapted continuous process $\Psi$ with values in the cotangent space $T^{\ast}\mathbbm{G}_{y}$ of $\mathbbm{G}_{y}$, if $\Psi_{t}(\omega)\in T^{\ast}_{Y_{t}(\omega)}\mathbbm{G}_{y}$ for all $t\geqslant 0$ and $\omega\in\Omega$, then one can define the continuous Itô integral $I_{\Psi}$ of $\Psi$ as

[TABLE]

so that in particular

[TABLE]

The integrand above involves the $1$–form of $T_{y}^{\ast}\mathbbm{G}_{y}$

[TABLE]

A pure jump process on $\mathbbm{G}_{x}$

We will now define the discrete $m$–dimensional process $\mathcal{X}_{t}$ on the discrete abelian group $\mathbbm{G}_{x}$ as a generalized compound Poisson process. In order to do this we need a number of independent variables and processes:

First, for any given $1\leqslant i\leqslant m$, let $\mathcal{N}^{i}_{t}$ be a càdlàg Poisson process of parameter $\lambda$, that is

[TABLE]

The sequence of instants where the jumps of the $\mathcal{N}^{i}_{t}$ occur is noted $(T^{i}_{k})_{k\in\mathbbm{N}}$, with the convention $T^{i}_{0}=0$.

Second, we set

[TABLE]

Almost surely, for any two distinct $i$ and $j$, we have $\{T_{k}^{i}\}_{k\in\mathbbm{N}}\cap\{T_{k}^{j}\}_{k\in\mathbbm{N}}=\emptyset$. Let therefore $\{T_{k}\}_{k\in\mathbbm{N}}=\cup_{i=1}^{m}\{T_{k}^{i}\}_{k\in\mathbbm{N}}$ be the ordered sequence of instants of jumps of $\mathcal{N}_{t}$ and let $i_{t}\equiv i_{t}(\omega)$ be the index of the coordinate where the jump occurs at time $t$. We set $i_{t}=0$ if no jump occurs. The random variables $i_{t}$ are measurable: $i_{t}=(\mathcal{N}^{1}_{t}-\mathcal{N}_{t^{-}}^{1},\mathcal{N}^{2}_{t}-\mathcal{N}_{t^{-}}^{2},\ldots,\mathcal{N}^{m}_{t}-\mathcal{N}_{t^{-}}^{m})\cdot(1,2,\ldots,m)$. In differential form,

[TABLE]

Third, we denote by $(\tau_{k})_{k\in\mathbbm{N}}$ a sequence of independent Bernoulli variables

[TABLE]

Finally, the random walk $\mathcal{X}_{t}$ started at $\mathcal{X}_{0}\in\mathbbm{G}_{x}$ is the càdlàg compound Poisson process (see e.g. Protter [48], Privault [46, 47]) defined as

[TABLE]

where $G_{i}^{\tau}=(0,\ldots,0,\tau g_{i},0,\ldots,0)$ when $i\neq 0$  and $(0,\ldots,0)$ when $i=0$.

Stochastic integrals on discrete groups

We recall for the convenience of the reader the derivation of stochastic integrals for jump processes. We will emphasize the fact that the corresponding Itô’s formula involves the action of a discrete $1$–form written in a well-chosen local coordinate system of the discrete augmented cotangent plane (see details below). Let $1\leqslant k\leqslant\mathcal{N}_{t}$ and let $(T_{k},i_{k},\tau_{k})$ be respectively the instant, the axis and the direction of the $k$–th jump. We set $T_{0}=0$. Let $f:=f(t,x)$, $t\in\mathbbm{R}^{+}$, $x\in\mathbbm{G}_{x}$ a function defined on $\mathbbm{R}^{+}\times\mathbbm{G}_{x}$. Then

[TABLE]

At an instant $s$, the integrand in the last term writes as

[TABLE]

where we introduced, for all $1\leqslant i\leqslant m$,

[TABLE]

Notice that, for any given $1\leqslant i\leqslant m$, up to a normalisation factor, the system of coordinate $(X_{i}^{2},X_{i}^{0})$ is obtained thanks to a rotation of $\pi/4$ of the canonical system of coordinate $(X_{i}^{+},X_{i}^{-})$. Finally,

[TABLE]

where we set $\mathrm{d}\mathcal{X}^{i}_{s}=\tau_{\mathcal{N}_{s}}\mathrm{d}\mathcal{N}^{i}_{s}$. It is easy to see that $\mathrm{d}\mathcal{X}^{i}_{s}$ is the stochastic differential of a martingale. Here and in the sequel, we take $\lambda=2$.

Discrete Itô integral

The stochastic integral above shows that Itô formula (2.1) for continuous processes has a discrete counterpart involving stochastic integrals for jump processes, namely we have the discrete Itô integral

[TABLE]

This has a more intrinsic expression similar to the continuous Itô integral (2.1). If we regard the discrete component $\mathbbm{G}_{x}$ as a “discrete Riemannian” manifold, then this discrete Itô integral involves discrete vectors (resp. $1$–forms) defined on the augmented discrete tangent (resp. cotangent) space $\hat{T}_{x}\mathbbm{G}_{x}$ (resp. $\hat{T}_{x}^{\ast}\mathbbm{G}_{x}$) of dimension $2m$ defined as

[TABLE]

Let $\mathrm{d}\widehat{\mathcal{W}}_{s}\in\hat{T}_{\mathcal{X}_{s}}\mathbbm{G}_{x}$ be the vector and $\widehat{\boldsymbol{\mathrm{d}}}f\in\hat{T}_{\mathcal{X}_{s}}^{\ast}\mathbbm{G}_{x}$ be the $1$–form respectively defined as:

[TABLE]

[TABLE]

We have with these notations

[TABLE]

where the factor $1/2$ is included in the pairing $\langle\cdot,\cdot\rangle_{\hat{T}_{x}^{\ast}\mathbbm{G}_{x}\times\hat{T}_{x}\mathbbm{G}_{x}}$.

Semi–discrete stochastic integrals

Let finally $\mathcal{Z}_{t}=(\mathcal{X}_{t},\mathcal{Y}_{t})$ be a semi-discrete random walk on the cartesian product $\mathbbm{G}=\mathbbm{G}_{x}\times\mathbbm{G}_{y}$, where $\mathcal{X}_{t}$ is the random walk above defined on $\mathbbm{G}_{x}$ with generator $\Delta_{x}$ and where $\mathcal{Y}_{t}$ is the Brownian motion defined on $\mathbbm{G}_{y}$ with generator $\Delta_{y}$. For $f:=f(t,z)=f(t,x,y)$ defined from $\mathbbm{R}^{+}\times\mathbbm{G}$ onto $\mathbbm{C}$, we have easily the stochastic integral involving both discrete and continuous parts:

[TABLE]

where the semi-discrete Itô integral writes as

[TABLE]

Martingale transforms

We are interested in martingale transforms allowing us to represent second order Riesz transforms. Let $f(t,z)$ be a solution to the heat equation $\partial_{t}-\Delta_{z}=0$. Fix $T\char 62\relax 0$ and $\mathcal{Z}_{0}\in\mathbbm{G}$. Then define

[TABLE]

This is a martingale since $f(T-t)$ solves the backward heat equation $\partial_{t}+\Delta_{z}=0$, and we have in terms of stochastic integrals

[TABLE]

Given $\boldsymbol{A}_{\alpha}$ the $\mathbbm{C}^{(2m+n)\times(2m+n)}$ matrix defined earlier, we note $M_{t}^{\alpha,f,T,\mathcal{Z}_{0}}$ the martingale transform $\boldsymbol{A}_{\alpha}\ast M_{t}^{f,T,\mathcal{Z}_{0}}$ defined as

[TABLE]

where the first integral involves the $L^{2}$ scalar product on $\hat{T}_{z}\mathbbm{G}\times\hat{T}_{z}\mathbbm{G}$ and the second integral involves the duality $\hat{T}_{z}^{\ast}\mathbbm{G}\times\hat{T}_{z}\mathbbm{G}$. In differential form:

[TABLE]

Quadratic covariation and subordination

We have the quadratic covariations (see Protter [48], Dellacherie–Meyer [22], or Privault [46, 47]). Since

[TABLE]

it follows that

[TABLE]

Differential subordination

Following Wang [51], given two adapted càdlàg Hilbert space valued martingales $X_{t}$ and $Y_{t}$, we say that $Y_{t}$ is differentially subordinate by quadratic variation to $X_{t}$ if $\char 124\relax Y_{0}\char 124\relax_{\mathbbm{H}}\leqslant\char 124\relax X_{0}\char 124\relax_{\mathbbm{H}}$ and $[Y,Y]_{t}-[X,X]_{t}$ is nondecreasing nonnegative for all $t$. In our case, we have

[TABLE]

Hence

[TABLE]

This means that $M^{\alpha,f}_{t}$ is differentially subordinate to $\|\boldsymbol{A}_{\alpha}\|_{2}M_{t}^{f}$.

2.2. Martingale inequalities under differential subordination

In the final part of the section we discuss a number of sharp martingale inequalities which hold under the assumption of the differential subordination imposed on the processes. Our starting point is the following celebrated $L^{p}$ bound.

Theorem 2.1**.**

(Wang, 1995) Suppose that $X$ and $Y$ are martingales taking values in a Hilbert space $\mathbbm{H}$ such that $Y$ is differentially subordinate to $X$. Then for any $1\char 60\relax p\char 60\relax\infty$ we have

[TABLE]

and the constant $p^{\ast}-1$ is the best possible, even if $\mathbbm{H}=\mathbbm{R}$.

This result was first proved by Burkholder in [15] in the following discrete-time setting. Suppose that $(X_{n})_{n\geq 0}$ is an $\mathbbm{H}$-valued martingale and $(\alpha_{n})_{n\geq 0}$ is a predictable sequence with values in $[-1,1]$. Let $Y:=\alpha*X$ be the martingale transform of $X$ defined for almost all $\omega\in\Omega$ by

[TABLE]

Then the above $L^{p}$ bound holds true and the constant $p^{\ast}-1$ is optimal. The general continuous-time version formulated above is due to Wang [51]. To see that the preceding discrete-time version is indeed a special case, treat a discrete-time martingale $(X_{n})_{n\geq 0}$ and its transform $(Y_{n})_{n\geq 0}$ as continuous-time processes via $X_{t}=X_{\lfloor t\rfloor}$, $Y_{t}=Y_{\lfloor t\rfloor}$ for $t\geq 0$; then $Y$ is differentially subordinate to $X$.

In 1992, Choi [19] established the following non-symmetric, discrete-time version of the $L^{p}$ estimate.

Theorem 2.2**.**

(Choi, 1992) Suppose that $(X_{n})_{n\geq 0}$ is a real-valued discrete time martingale and let $(Y_{n})_{n\geq 0}$ be its transform by a predictable sequence $(\alpha_{n})_{n\geq 0}$ taking values in $[0,1]$. Then there exists a constant $\mathfrak{C}_{p}$ depending only on $p$ such that $\|Y\|_{p}\leqslant\mathfrak{C}_{p}\|X\|_{p}$ and the estimate is best possible.

This result can be regarded as a non-symmetric version of the previous theorem, since the transforming sequence $(\alpha_{n})_{n\geq 0}$ takes values in a non-symmetric interval $[0,1]$. There is a natural question whether the estimate can be extended to the continuous-time setting; in particular, this gives rise to the problem of defining an appropriate notion of non-symmetric differential subordination. The following statement obtained by Bañuelos and Osȩkowski addresses both these questions. For any real numbers $a\char 60\relax b$ and any $1\char 60\relax p\char 60\relax\infty$, let $\mathfrak{C}_{a,b,p}$ be the constant introduced in [10].

Theorem 2.3**.**

(Banuelos–Osȩkowski, 2012) Let $(X_{t})_{t\geq 0}$ and $(Y_{t})_{t\geq 0}$ be two real-valued martingales satisfying

[TABLE]

for all $t\geqslant 0$. Then for all $1\char 60\relax p\char 60\relax\infty$, we have $\|Y\|_{p}\leqslant\mathfrak{C}_{a,b,p}\|X\|_{p}$.

The condition (2.4) is the continuous counterpart of the condition that the transforming sequence $(\alpha_{n})_{n\geq 0}$ takes values in the interval $[a,b]$. Thus, in particular, Choi’s constant $\mathfrak{C}_{p}$ is, in the terminology of the above theorem, equal to $\mathfrak{C}_{p,0,1}$.

We return to the context of the “classical” differential subordination introduced in the preceding subsection and study other types of martingale inequalities. The following statements, obtained by Bañuelos–Osȩkowski, [11] will allow us to deduce sharp weak-type and logarithmic estimates for Riesz transforms, respectively.

Theorem 2.4**.**

(Banuelos–Osȩkowski, 2015) Suppose that $X$ and $Y$ are martingales taking values in a Hilbert space $\mathbbm{H}$ such that $Y$ is differentially subordinate to $X$.

(i) Let $1\char 60\relax p\char 60\relax 2$. Then for any $t\geq 0$,

[TABLE]

(ii) Suppose that $2\char 60\relax p\char 60\relax\infty$. Then for any $t\geq 0$,

[TABLE]

Both estimates are sharp: for each $p$, the numbers $\frac{p^{-1/(p-1)}}{2}\Gamma\left(\frac{p}{p-1}\right)$ and $1-p^{-1}$ cannot be decreased.

Recall that $\Phi,\,\Psi:[0,\infty)\to[0,\infty)$ are conjugate Young functions given by $\Phi(t)=e^{t}-1-t$ and $\Psi(t)=(t+1)\log(t+1)-t$.

Theorem 2.5**.**

(Banuelos–Osȩkowski, 2015) Suppose that $X$ and $Y$ are martingales taking values in a Hilbert space $\mathbbm{H}$ such that $Y$ is differentially subordinate to $X$. Then for any $K\char 62\relax 1$ and any $t\geq 0$ we have

[TABLE]

For each $K$, the constant $(2(K-1))^{-1}$ appearing on the left, is the best possible (it cannot be replaced by any smaller number).

The following exponential estimate, established by Osȩkowski in [41], can be regarded as a dual statement to the above logarithmic bound.

Theorem 2.6**.**

(Osȩkowski, 2013) Assume that $X$, $Y$ are $\mathbbm{H}$-valued martingales such that $\char 124\relax\char 124\relax X\char 124\relax\char 124\relax_{\infty}\leq 1$ and $Y$ is differentially subordinate to $X$. Then for any $K\char 62\relax 1$ and any $t\geq 0$ we have

[TABLE]

Finally, we will need the following sharp $L^{q}\to L^{p}$ estimate, established by Osȩkowski in [43], which will allow us to deduce the corresponding estimate for Riesz transforms.

Theorem 2.7**.**

(Osȩkowski, 2014) Assume that $X$, $Y$ are $\mathbbm{H}$-valued martingales such that $Y$ is differentially subordinate to $X$. Then for any $1\leq p\char 60\relax q\char 60\relax\infty$ there is a constant $L_{p,q}$ such that

[TABLE]

Actually, the paper [43] identifies, for any $p$ and $q$ as above, the optimal (i.e., the least) value of the constant $L_{p,q}$ in the estimate above. As the description of this constant is a little complicated (and will not be needed in our considerations below), we refer the reader to that paper for the formal definition of $L_{p,q}$.

Let us conclude with the observation which will be crucial in the proofs of our main results. Namely, all the martingale inequalities presented above are of the form $\mathbb{E}\zeta(\char 124\relax Y_{t}\char 124\relax)\leq\mathbb{E}\xi(\char 124\relax X_{t}\char 124\relax)$, $t\geq 0$, where $\zeta$, $\xi$ are certain convex functions. This will allow us to successfully apply a conditional version of Jensen’s inequality.

3. Proofs of the main results

We turn our attention to the proofs of the estimates for $R_{\alpha}^{2}$ formulated in the introductory section. We will focus on Theorems 1.1, 1.2 and 1.3 only; the remaining statements are established by similar arguments. Also, we postpone the proof of the sharpness of these estimates to the next section.

3.1. Proof of Theorem 1.1

Recall that the subordination estimate (2.3) shows that the martingale transform $Y_{t}:=M_{t}^{\alpha}$ is differentially subordinate to the martingale $X_{t}:=\|\boldsymbol{A}_{\alpha}\|_{2}\hskip 1.00006ptM_{t}^{f}$. Therefore, by Theorem 2.1, we immediately obtain that

[TABLE]

for all $t\geq 0$. Since the operator $\mathcal{T}^{\alpha}$ is a conditional expectation of $M_{t}^{\alpha,f}$, an application of Jensen’s inequality proves the estimate $\|\mathcal{T}^{\alpha}\|_{p}\leqslant\|\boldsymbol{A}_{\alpha}\|_{2}\hskip 1.00006pt(p^{\ast}-1)$, which is the desired bound.

3.2. Proof of Theorem 1.2

The argument is the same as above and exploits the fine-tuned $L^{p}$ estimate of Theorem 2.3 applied to $X_{t}=M_{t}^{f}$ and $Y_{t}=M_{t}^{\alpha,f}$. It is not difficult to prove that the difference of quadratic variations above writes in terms of a jump part and a continuous part as

[TABLE]

which is nonpositive since we assumed precisely $a\boldsymbol{I}\leqslant\boldsymbol{A}_{\alpha}\leqslant b\boldsymbol{I}$. Thus, the estimate of Theorem 1.2 follows. The sharpness is established in a similar manner. $\Box$

3.3. Proof of Theorem 1.3

We will focus on the case $1\char 60\relax p\char 60\relax 2$; for remaining values of $p$ the argument is similar. An application of Theorem 2.4 to the processes $X_{t}=\|\boldsymbol{A}_{\alpha}\|_{2}\hskip 1.00006ptM_{t}^{f}$ and $Y_{t}=M_{t}^{\alpha,f}$ yields

[TABLE]

and hence, by Jensen’s inequality, we obtain

[TABLE]

Therefore, if $E$ is an arbitrary measurable subset of $\mathbbm{G}$, we get

[TABLE]

Apply this bound to $\lambda f$, where $\lambda$ is a nonnegative parameter, then divide both sides by $\lambda$ and optimize the right-hand side over $\lambda$ to get the desired assertion.

4. Sharpness

The proof of the sharpness of the different results is made in several steps. In some cases the sharpness for certain second order Riesz transform estimates in the continuous setting (such as in Theorem 1.1) is already known. In these cases we prove below the sharpness for the discrete (or semidiscrete) case by using sequences of finite difference approximates of continuous functions and their finite difference second order Riesz transforms. In other cases, we need to prove first sharpness for certain continuous second order Riesz transforms. The key point here is to transfer the sharp result for zigzag martingales into a sharp result for certain continuous second order Riesz transforms by the laminate technique. We will illustrate this for the weak-type estimate of Theorem 1.3 and establish the following statement.

Theorem 4.1**.**

Let $\Theta:[0,\infty)\to[0,\infty)$ be a given function and let $\lambda\char 62\relax 0$ be a fixed number. Assume further that there is a pair $(F,G)$ of finite martingales starting from $(0,0)$ such that $G$ is a $\pm 1$-transform of $F$ and

[TABLE]

Then there is a function $f:\mathbb{R}^{2}\to\mathbb{R}$ supported on the unit disc $\mathbb{D}$ of $\mathbb{R}^{2}$ such that

[TABLE]

We will prove this statement with the use of laminates, important family of probability measures on matrices. It is convenient to split this section into several separate parts. For the sake of convenience, and to make this section as self contained as possible, we recall the preliminaries on laminates and their connections to martingales from [14] and [39], Section 4.2.

4.1. Laminates

Assume that $\mathbb{R}^{m\times n}$ stands for the space of all real matrices of dimension $m\times n$ and $\mathbb{R}^{n\times n}_{sym}$ denote the subclass of $\mathbb{R}^{n\times n}$ which consists of all symmetric matrices of dimension $n\times n$.

Definition 4.2**.**

A function $f:\mathbb{R}^{m\times n}\to\mathbb{R}$ is said to be rank-one convex, if for all $A,B\in\mathbb{R}^{m\times n}$ with $\textrm{rank }B=1$, the function $t\mapsto f(A+tB)$ is convex

For other equivalent definitions of rank-one convexity, see [21, p. 100]. Suppose that $\mathcal{P}=\mathcal{P}(\mathbb{R}^{m\times n})$ is the class of all compactly supported probability measures on $\mathbb{R}^{m\times n}$. For a measure $\nu\in\mathcal{P}$, we define

[TABLE]

the associated center of mass or barycenter of $\nu.$

Definition 4.3**.**

We say that a measure $\nu\in\mathcal{P}$ is a laminate (and write $\nu\in\mathcal{L}$), if

[TABLE]

for all rank-one convex functions $f$. The set of laminates with barycenter [math] is denoted by $\mathcal{L}_{0}(\mathbb{R}^{m\times n})$.

Laminates can be used to obtain lower bounds for solutions of certain PDEs, as observed by Faraco in [30]. In addition, laminates appear naturally in the context of convex integration, where they lead to interesting counterexamples, see e.g. [5], [20], [34], [37] and [49]. For our results here we will be interested in the case of $2\times 2$ symmetric matrices. The key observation is that laminates can be regarded as probability measures that record the distribution of the gradients of smooth maps: see Corollary 4.7 below. We briefly explain this and refer the reader to the works [33], [37] and [49] for full details.

Definition 4.4**.**

Let $U$ be a subset of $\mathbb{R}^{2\times 2}$ and let $\mathcal{PL}(U)$ denote the smallest class of probability measures on $U$ which

• (i)

contains all measures of the form $\lambda\delta_{A}+(1-\lambda)\delta_{B}$ with $\lambda\in[0,1]$ and satisfying $\textrm{rank}(A-B)=1$;

• (ii)

is closed under splitting in the following sense: if $\lambda\delta_{A}+(1-\lambda)\nu$ belongs to $\mathcal{PL}(U)$ for some $\nu\in\mathcal{P}(\mathbb{R}^{2\times 2})$ and $\mu$ also belongs to $\mathcal{PL}(U)$ with $\overline{\mu}=A$, then also $\lambda\mu+(1-\lambda)\nu$ belongs to $\mathcal{PL}(U)$.

The class $\mathcal{PL}(U)$ is called the prelaminates in $U$.

It follows immediately from the definition that the class $\mathcal{PL}(U)$ only contains atomic measures. Also, by a successive application of Jensen’s inequality, we have the inclusion $\mathcal{PL}\subset\mathcal{L}$. The following are two well known lemmas in the theory of laminates; see [5], [33], [37], [49].

Lemma 4.5**.**

Let $\nu=\sum_{i=1}^{N}\lambda_{i}\delta_{A_{i}}\in\mathcal{PL}(\mathbb{R}^{2\times 2}_{sym})$ with $\overline{\nu}=0$. Moreover, let $0\char 60\relax r\char 60\relax\tfrac{1}{2}\min\char 124\relax A_{i}-A_{j}\char 124\relax$ and $\delta\char 62\relax 0$. For any bounded domain $\mathcal{B}\subset\mathbb{R}^{2}$ there exists $u\in W^{2,\infty}_{0}(\mathcal{B})$ such that $\|u\|_{C^{1}}\char 60\relax\delta$ and for all $i=1\dots N$

[TABLE]

Lemma 4.6**.**

Let $K\subset\mathbb{R}^{2\times 2}_{sym}$ be a compact convex set and suppose that $\nu\in\mathcal{L}(\mathbb{R}^{2\times 2}_{sym})$ satisfies $\operatorname*{supp}\nu\subset K$. For any relatively open set $U\subset\mathbb{R}^{2\times 2}_{sym}$ with $K\subset U$, there exists a sequence $\nu_{j}\in\mathcal{PL}(U)$ of prelaminates with $\overline{\nu}_{j}=\overline{\nu}$ and $\nu_{j}\overset{*}{\rightharpoonup}\nu$, where $\overset{*}{\rightharpoonup}$ denotes weak convergence of measures.

Combining these two lemmas and using a simple mollification, we obtain the following statement, proved by Boros, Shékelyhidi Jr. and Volberg [14]. It exhibits the connection between laminates supported on symmetric matrices and second derivatives of functions. It will be our main tool in the proof of the sharpness. Recall that $\mathbb{D}$ denotes the unit disc of $\mathbb{C}$.

Corollary 4.7**.**

Let $\nu\in\mathcal{L}_{0}(\mathbb{R}^{2\times 2}_{sym})$. Then there exists a sequence $u_{j}\in C_{0}^{\infty}(\mathbb{D})$ with uniformly bounded second derivatives, such that

[TABLE]

for all continuous $\phi:\mathbb{R}^{2\times 2}_{sym}\to\mathbb{R}$.

4.2. Biconvex functions and a special laminate

The next step in our analysis is devoted to the introduction of a certain special laminate. We need some additional notation. A function $\zeta:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is said to be biconvex if for any fixed $z\in\mathbb{R}$, the functions $x\mapsto\zeta(x,z)$ and $y\mapsto\zeta(z,y)$ are convex. Now, take the martingales $F$ and $G$ appearing in the statement of Theorem 4.1. Then the martingale pair

[TABLE]

is finite, starts from $(0,0)$ and has the following zigzag property: for any $n\geq 0$ we have $\mathtt{F}_{n}=\mathtt{F}_{n+1}$ with probability $1$ or $\mathtt{G}_{n}=\mathtt{G}_{n+1}$ almost surely; that is, in each step $(\mathtt{F},\mathtt{G})$ moves either vertically, or horizontally. Indeed, this follows directly from the assumption that $G$ is a $\pm 1$-transform of $F$. This property combines nicely with biconvex functions: if $\zeta$ is such a function, then a successive application of Jensen’s inequality gives

[TABLE]

The distribution of the terminal variable $(\mathtt{F}_{\infty},\mathtt{G}_{\infty})$ gives rise to a probability measure $\nu$ on $\mathbb{R}^{2\times 2}_{sym}$: put

[TABLE]

where $\operatorname*{diag}(x,y)$ stands for the diagonal matrix \left(\begin{array}[]{cc}x&0\\ 0&y\end{array}\right). Observe that $\nu$ is a laminate of barycenter [math]. Indeed, if $\psi:\mathbb{R}^{2\times 2}\to\mathbb{R}$ is a rank-one convex, then $(x,y)\mapsto\psi(\operatorname*{diag}(x,y))$ is biconvex and thus, by (4.1),

[TABLE]

Here we used the fact that $(\mathtt{F},\mathtt{G})$ is finite, so $(\mathtt{F}_{\infty},\mathtt{G}_{\infty})=(\mathtt{F}_{n},\mathtt{G}_{n})$ for some $n$.

4.3. A proof of Theorem 4.1

Consider a continuous function $\phi:\mathbb{R}^{2\times 2}_{sym}\to\mathbb{R}$ given by

[TABLE]

By Corollary 4.7, there is a functional sequence $(u_{j})_{j\geq 1}\subset C_{0}^{\infty}(\mathbb{D})$ such that

[TABLE]

Therefore, for sufficiently large $j$, we have

[TABLE]

Setting $f=\Delta u_{j}$, we obtain the desired assertion.

In the remaining part of this subsection, let us briefly explain how Theorem 4.1 yields the sharpness of weak-type and logarithmic estimates for second-order Riesz transforms (in the classical setting). We will focus on the weak-type bounds for $1\char 60\relax p\char 60\relax 2$ - the remaining estimates can be treated analogously. Suppose that $\lambda_{p}$ is the best constant in the estimate

[TABLE]

valid for all pairs $(F,G)$ of finite martingales starting from [math] such that $G$ is a $\pm 1$-transform of $F$. The value of $\lambda_{p}$ appears in the statement of Theorem 9 above, the fact that it is already the best for martingale transforms follows from the examples exhibited in [38]. For any $\varepsilon\char 62\relax 0$, Theorem 4.1 yields the existence of $f:\mathbb{R}^{2}\to\mathbb{R}$, supported on the unit disc, such that

[TABLE]

That is, if we set $A=\{\char 124\relax(R_{1}^{2}-R_{2}^{2})f\char 124\relax\geq\lambda_{p}-\varepsilon\}$, we get

[TABLE]

However, if the weak-type estimate holds with a constant $c_{p}$, Young’s inequality implies

[TABLE]

Therefore, the inequality (4.2) enforces that

[TABLE]

(since $\varepsilon$ was arbitrary). This estimate is equivalent to

[TABLE]

which is the desired sharpness.

4.4. From continuous to discrete sharp estimates

We claim that the sharp bounds found for the continuous second order Riesz transforms also hold in the case of purely discrete groups. Groups of mixed type would be treated in the same manner. We illustrate those results only for the sharpness in Theorem 1.1 and in Theorem 1.3 since other results follow the same lines. Precisely, we show that the sharpness in the discrete case is inherited from the sharpness of the continuous case through the use of the so–called fundamental theorem of finite difference methods from Lax and Richtmyer [35] (see also [36]). This result states that stability and consistency of the finite difference scheme implies convergence of the approximate finite difference solution towards the continuous solution, in a sense that we detail below.

Finite difference Riesz transforms

Let $u=R^{2}_{i}f$ be the $i$-th second order Riesz transform in $\Omega:=\mathbbm{R}^{N}$ of a function $f\in L^{p}$. The function $u$ is the unique solution to the Poisson problem in $\mathbbm{R}^{N}$, $\Delta u=\partial^{2}_{i}f$ in $\mathbbm{R}^{N}$ (see [29]). This is a problem of the form $Au=Bf$, where $A=\Delta$ and $B=\partial^{2}_{ij}$. Introduce now a finite difference grid of step–size $h\char 62\relax 0$, that is the grid $\Omega_{h}:=h\mathbbm{Z}^{N}$. The functions $v_{h}$ defined on $\Omega_{h}$ are equipped with the $L^{p}_{h}$ norm defined as

[TABLE]

It is common to identify a finite difference function $v_{h}$ defined on the grid $\Omega_{h}$ with the piecewise constant function (also denoted) $v_{h}:\mathbbm{R}^{N}\rightarrow\mathbbm{C}$ such that $v_{h}(x)=v_{h}(y)$ for all $x$’s in the open cube $\Omega(y)$ of volume $h^{N}$ centered around the grid point $y\in\Omega_{h}$. With this notation, we might write finite difference integrals in the form

[TABLE]

The finite difference second order Riesz transform $u_{h}=R^{2}_{i}f_{h}$ of $f_{h}$ is the solution to the problem $A_{h}u_{h}=B_{h}^{2}f_{h}$, where $A_{h}:=\Delta_{h}$ is the finite difference Laplacian and $B_{h}:=\partial_{i,h}^{2}$ the $3$–point finite difference second order derivative. Precisely, for any $x\in\Omega_{h}$, any $v_{h}:\Omega_{h}\rightarrow\mathbbm{R}$,

[TABLE]

[TABLE]

It is classical that we have the consistency of the discrete problem with respect to the continuous problem, that is for given smooth functions $u$ and $f$ we have $\Delta_{h}u=\Delta u+\mathcal{O}(h)$ and $\partial^{2}_{i,h}f=\partial_{i}^{2}f+\mathcal{O}(h)$, where the coefficients in $\mathcal{O}(h)$ include as a factor up to fourth–order derivatives of $u$ or $f$. This implies in particular that $B_{h}f=Bf+\mathcal{O}(h)$ in $L_{h}^{p}$ for any given smooth function $f$ with compact support. It is also classical that $(-\Delta_{h})^{-1}$ is bounded in $L_{h}^{p}$ uniformly w.r.t. $h$. This is the $L^{p}$ stability of the finite difference scheme. The fundamental theorem of finite difference methods implies the $L^{p}$ convergence of the sequence of discrete second order Riesz transforms $u_{h}$ towards the continuous second order Riesz transform $u$.

Discrete Riesz tranforms on Lie-Group

Observe that the finite difference Riesz transform $u_{h}=R^{2}_{i,h}f_{h}$ defined on the grid $\Omega_{h}$, also gives rise to a Riesz transform on the Lie group $\Omega_{1}=\mathbbm{Z}^{N}$. This is a consequence of the homogeneity of order zero of the Riesz transforms. Indeed, the equation $\Delta_{h}u_{h}=\partial^{2}_{i,h}f_{h}$ rewrites as $\Delta_{1}u_{1}=\partial^{2}_{i,1}f_{1}$, where $u_{1}(y):=u_{h}(y/h)$, $f_{1}(y):=f_{h}(y/h)$ for all $y\in\mathbbm{Z}^{N}$, and where $\Delta_{1}$ and $\partial^{2}_{i,1}$ are the discrete differential operators defined on $\mathbbm{Z}^{N}$. We have also $\|u_{h}\|_{L^{p}_{h}}=h^{N/p}\|u_{1}\|_{L^{p}_{1}}$ and $\|f_{h}\|_{L^{p}_{h}}=h^{N/p}\|f_{1}\|_{L^{p}_{1}}$. Notice that for all $h$, this ensures that $\|u_{h}\|_{L^{p}_{h}}/\|f_{h}\|_{L^{p}_{h}}=\|u_{1}\|_{L^{p}_{1}}/\|f_{1}\|_{L^{p}_{1}}$.

Sharpness for Theorem 1.1 in the discrete setting

In the continuous setting, the sharpness was proved in [31] based on the combintation $R^{2}_{\alpha}=R_{1}^{2}-R_{2}^{2}$ of second order Riesz tranforms. Let $u^{(k)}=R_{\alpha}^{2}f^{(k)}$ a sequence of second order Riesz transforms yielding the sharp constant $C_{p}$ in the estimate, that is $\|u^{(k)}\|_{p}/\|f^{(k)}\|_{p}\rightarrow C_{p}$ as $k$ goes to infinity. For each $k\in\mathbbm{N}$ and $h\char 62\relax 0$, introduce the finite difference approximation $f_{h}^{(k)}$ of $f^{(k)}$ and the corresponding finite difference Riesz transform. Thanks to the convergence of the finite difference scheme, we can extract a subsequence $f_{h_{k}}^{(k)}$ such that $\|u_{1}^{(k)}\|_{p}/\|f_{1}^{(k)}\|_{p}=\|u_{h_{k}}^{(k)}\|_{p}/\|f_{h_{k}}^{(k)}\|_{p}\rightarrow C_{p}$. Therefore $C_{p}$ is also the sharp constant for the second order Riesz transforms in $\mathbbm{Z}^{N}$.

Sharpness for Theorem 1.3 in the discrete setting

Recall that we have a bound of the form

[TABLE]

for a certain constant $C_{p}$ that is known to be sharp in the case of continuous second order Riesz transforms. In order to prove sharpness when the Lie group $\mathbbm{G}$ does not have enough continuous components, it suffices again to approximate a sequence of continuous extremizers by a sequence of finite difference approximations. Take $\mathbbm{G}=\mathbbm{R}^{N}$. For any $\varepsilon\char 62\relax 0$, let $f$, $u:=R^{2}_{\alpha}f$, and $E$ with finite measure chosen so that

[TABLE]

We can assume without loss of generality that $f$ is a smooth function with compact support. Let $f_{h}$ a finite difference approximation of $f$ defined as its $L^{2}$ projection on the grid, and $u_{h}$ its discrete second order Riesz transform both defined on $\Omega_{h}:=h\mathbbm{Z}^{N}$. Since $\mu_{z}(E)$ is the finite $N$–dimensional Lebesgue measure of $E$, we use outer measure approximations of $E$ followed by approximations from below by a finite number of small enough cubes of size $h$ centered around the grid points of $\Omega_{h}$, to define a “finite difference” approximation $E_{h}$ of $E$ such that

[TABLE]

when $h$ goes to zero. Since the discrete Riesz transforms are stable in $L^{2}$, the Lax-Richtmyer theorem ensures that $\|u_{h}\|_{L^{2}_{h}}\rightarrow\|u\|_{L^{2}}$ which implies $\|u_{h}\|_{L^{1}_{h}(E)}\rightarrow\|u\|_{L^{1}(E)}$ and also $\|u_{h}\|_{L^{1}_{h}(E_{h})}\rightarrow\|u\|_{L^{1}(E)}$. Therefore for $h$ small enough,

[TABLE]

Let as before $u_{1}(y):=u_{h}(y/h)$, $f_{1}(y):=f_{h}(y/h)$ for all $y\in\Omega_{1}:=\mathbbm{Z}^{N}$, and $E_{1}:=E/h$. We have sucessively $\mu_{h}(E)=h^{N}\mu_{1}(E_{1})$, $\|u_{h}\|_{L^{1}_{h}(E_{h})}=h^{N}\|u_{1}\|_{L^{1}_{h}(E_{1})}$ and $\|f_{h}\|_{L^{p}}=h^{N/p}\|f_{1}\|_{L^{p}}$. This yields immediately

[TABLE]

allowing us to prove sharpness for the class of discrete groups we are interested in.