Lower bound of Riesz transform kernels revisited and commutators on stratified Lie groups

TL;DR

This paper revisits lower bounds of Riesz transform kernels on stratified Lie groups, establishes new two-weight estimates and endpoint characterizations for commutators with BMO functions, and analyzes the Heisenberg group case.

Contribution

It provides a new lower bound for Riesz transform kernels and extends endpoint estimates and characterizations for their commutators on stratified Lie groups.

Findings

New lower bound for Riesz transform kernels.

Bloom-type two weight estimates for commutators.

Weak type (1,1) characterizations for Riesz commutators.

Abstract

Let $\mathcal G$ be a stratified Lie group and $\{\X_j\}_{1 \leq j \leq n}$ a basis for the left-invariant vector fields of degree one on $\mathcal G$. Let $\Delta = \sum_{j = 1}^n \X_j^2 $ be the sub-Laplacian on $\mathcal G$ and the $j^{\mathrm{th}}$ Riesz transform on $\mathcal G$ is defined by $R_j:= \X_j (-\Delta)^{-\frac{1}{2}}$, $1 \leq j \leq n$. In this paper we give a new version of the lower bound of the kernels of Riesz transform $R_j$ and then establish the Bloom-type two weight estimates as well as a number of endpoint characterisations for the commutators of the Riesz transforms and BMO functions, including the $L\log^+L(\mathcal G)$ to weak $L^1(\mathcal G)$, $H^1(\mathcal G)$ to $L^1(\mathcal G)$ and $L^\infty(\mathcal G)$ to BMO$(\mathcal G)$. Moreover, we also study the behaviour of the Riesz transform kernel on a special case of stratified Lie group: the Heisenberg group, and then we obtain the weak type $(1,1)$ characterisations for the Riesz commutators.

Equations

$$[b,H]f(x)=b(x)Hf(x)-H(bf)(x)$$

$$\mathrm{BMO}(\mathcal{G}):=\{b\in L_{loc}^1(\mathcal{G}):\Vert b{\Vert }_{\mathrm{BMO}(\mathcal{G})}<\infty \},$$

$$\Vert b{\Vert }_{\mathrm{BMO}(\mathcal{G})}:=\underset{B}{\sup }\frac{1}{|B|}{\int }_B|b(g)-b_B|dg,$$

$$\rho (g,g_{\ast })=r,|K_j(g_1,g_2)|\ge C{r}^{-Q},\forall g_1\in B(g,\eta r),g_2\in B(g_{\ast },\eta r).$$

$$|K_j(g_1,g_2)|\ge C\rho (g_1,g_2{)}^{-Q},|K_j(g_2,g_1)|\ge C\rho (g_1,g_2{)}^{-Q},$$

$$|B(g,R)\cap G|\approx R^Q,$$

$$\Vert [b,R_j](f){\Vert }_{L_{\lambda }^p(\mathcal{G})}\lesssim \Vert b{\Vert }_{{\mathrm{BMO}}_{\nu }(\mathcal{G})}([\lambda {]}_{A_p(\mathcal{G})}\cdot [\mu {]}_{A_p(\mathcal{G})}{)}^{\max \{1,\frac{1}{p-1}\}}\Vert f{\Vert }_{L_{\mu }^p(\mathcal{G})}.$$

$$\Vert b{\Vert }_{{\mathrm{BMO}}_{\nu }(\mathcal{G})}\lesssim \Vert [b,R_j]{\Vert }_{L_{\mu }^p(\mathcal{G})\to L_{\lambda }^p(\mathcal{G})}.$$

$$\displaystyle\big{|}\{g\in\mathcal{G}:\ |[b,R_{j}](f)(g)|>\lambda\}\big{|}\leq\theta_{b}\int_{\mathcal{G}}{|f(g)|\over\lambda}\bigg{(}1+\log^{+}\Big{(}{|f(g)|\over\lambda}\Big{)}\bigg{)}dg.$$

$$K_j(g)\ne 0,a.e.g\in {\mathbb{H}}^n.$$

$$\displaystyle\mathfrak{g}=\bigoplus_{i=1}^{k}V_{i},\quad[V_{1},V_{i}]=V_{i+1},\ \mbox{ for $i\leq k-1$ and $[V_{1},V_{k}]=0$;}$$

$$\exp :\mathfrak{g}⟶\mathcal{G},$$

$$\displaystyle\delta_{r}\bigg{(}\sum_{i=1}^{k}v_{i}\bigg{)}=\sum_{i=1}^{k}r^{i}v_{i},\quad\mbox{with $v_{i}\in V_{i}$}.$$

$${\mathrm{X}}_j\left(f\circ {\delta }_r\right)=r\left({\mathrm{X}}_{j}f\right)\circ {\delta }_r,1\le j\le n,r>0,f\in C^1,$$

$${\delta }_{\frac{1}{r}}\circ \Delta \circ {\delta }_r=r^2\Delta ,\forall r>0.$$

$${\tau }_g(g^{\prime })=g\circ g^{\prime }.$$

$${\delta }_{\lambda }(g)={\delta }_{\lambda }\left(x_1,x_2,\cdots ,x_N\right)=\left({\lambda }^{{\alpha }_1}{x}_1,{\lambda }^{{\alpha }_2}{x}_2,\cdots ,{\lambda }^{{\alpha }_N}{x}_N\right),$$

$$Q=\sum _{i=1}^{k}i\dim V_i.$$

$$p_h(g)=h^{-\frac{Q}{2}}p({\delta }_{\frac{1}{\sqrt{h}}}(g)),\forall h>0,g\in \mathcal{G}.$$

$$K_j\in C^{\infty }(\mathcal{G}\setminus \{0\}),K_j({\delta }_r(g))=r^{-Q}{K}_j(g),\forall g\ne 0,r>0,1\le j\le n,$$

$$K_j(g)=\frac{1}{\sqrt{\pi }}{\int }_0^{+\infty }{h}^{-\frac{1}{2}}{\mathrm{X}}_j{p}_h(g)dh=\frac{1}{\sqrt{\pi }}{\int }_0^{+\infty }{h}^{-\frac{Q}{2}-1}\left({\mathrm{X}}_{j}p\right)({\delta }_{\frac{1}{\sqrt{h}}}(g))dh.$$

$$\rho (g,g^{\prime })=\rho (g^{\prime -1}\circ g)=\rho (g^{-1}\circ g^{\prime }),\forall g,g^{\prime }\in \mathcal{G}.$$

$$|\rho (g_1,g_2)-\rho (g_1,g_3)|\le C_{\rho }\rho (g_2,g_3)\forall g_1,g_2,g_3\in \mathcal{G}.$$

$$C_1\rho (g_1,g_2)\le d(g_1,g_2)\le C_2\rho (g_1,g_2).$$

$$\Vert b{\Vert }_{\mathrm{BMO}(G)}^{\prime }=\underset{B}{\sup }\underset{c}{\inf }\frac{1}{|B|}{\int }_B|b(x)-c|dx.$$

$$\left|\{x\in B:b(x)<m_b(B)\}\right|\ge \frac{1}{2}|B|$$

$$\left|\{x\in B:b(x)>m_b(B)\}\right|\ge \frac{1}{2}|B|.$$

$$\Vert b{\Vert }_{{\mathrm{BMO}}_{\nu }(\mathcal{G})}:=\underset{B}{\sup }\frac{1}{\nu (B)}{\int }_B|b(g)-b_B|dg.$$

$$[w{]}_{A_p}:=\underset{B}{\sup }\left(\frac{1}{|B|}{\int }_{B}w(g)dg\right){\left(\frac{1}{|B|}{\int }_B{\left(\frac{1}{w(g)}\right)}^{1/(p-1)}dg\right)}^{p-1}<\infty .$$

$$[z,t]\circ [z^{\prime },t^{\prime }]$$

$$\displaystyle:=\Big{[}z_{1}+z_{1}^{\prime},\cdots,z_{n}+z_{n}^{\prime},t+t^{\prime}+2\mbox{Im}\big{(}\sum_{j=1}^{n}z_{j}\bar{z}^{\prime}_{j}\big{)}\Big{]}.$$

Full text

1. Introduction and statement of main results

The Calderón-Zygmund theory of singular integrals has a central role in modern harmonic analysis with extensive applications to other fields such as partial differential equations and complex analysis. A prototype of singular integral on the real line is the Hilbert transform which is bounded on $L^{p}(\mathbb{R})$ for $1<p<\infty$. At the end-point $p=1$, the Hilbert transform is bounded from $L^{1}(\mathbb{R})$ to $L^{1,\infty}(\mathbb{R})$ and is bounded from the Hardy space $H^{1}(\mathbb{R})$ into $L^{1}(\mathbb{R})$ while at the end-point $p=\infty$, the Hilbert transform is bounded from $L^{\infty}(\mathbb{R})$ to the BMO space BMO$(\mathbb{R})$. Further study on singular integrals and the related partial differential equations leads to the commutator $[b,H]$ of the Hilbert transform $H$ and a BMO function $b$ defined by

[TABLE]

for suitable functions $f$ (introduced by A.P. Calderón [4]). It is well-known that the commutator of the Hilbert transform has the following properties:

a) $\|[b,H]\|_{L^{p}(\mathbb{R})\to L^{p}(\mathbb{R})}\approx\|b\|_{{\rm BMO}(\mathbb{R})}$ ([7]);

b) $\|[b,H]\|_{L_{\mu}^{p}(\mathbb{R})\to L^{p}_{\lambda}(\mathbb{R})}\approx\|b\|_{{\rm BMO}_{\nu}(\mathbb{R})}$ with $\mu,\lambda\in A_{p}$ for $1<p<\infty$ and \nu=\big{(}{\mu\over\lambda}\big{)}^{1\over p}, where $A_{p}$ denotes the Muckenhoupt weights ([2],[15]);

c) $[b,H]$ is bounded from $L\log^{+}L(\mathbb{R})$ to $L^{1,\infty}(\mathbb{R})$ if and only if $b\in{\rm BMO}(\mathbb{R})$ ([24],[13],[1]);

d) $[b,H]$ is bounded from $H^{1}(\mathbb{R})$ to $L^{1}(\mathbb{R})$ if and only if $b$ equals a constant almost everywhere ([14]);

e) $[b,H]$ is bounded from $L_{c}^{\infty}(\mathbb{R})$ to ${\rm BMO}(\mathbb{R})$ if and only if $b$ equals a constant almost everywhere ([14]);

f) $[b,H]$ is of weak type $(1,1)$ if and only if $b$ is in $L^{\infty}(\mathbb{R})$ ([1]).

For more details, we refer to the references listed above. We also point out that there are quite a number of recent results on the characterisations of commutators in the above forms for singular integrals in different settings, see for example [10, 22, 9, 21, 25, 24, 15, 19, 13, 8, 1].

Inspired by these classical results above, it is natural to ask whether these results hold on the Heisenberg group $\mathbb{H}^{n}$. Note that in several complex variables, the Heisenberg group $\mathbb{H}^{n}$ is the boundary of the Siegel upper half space, whose roles are holomorphically equivalent to the unit sphere and the unit ball in $\mathbb{C}^{n}$. And hence, the role of the Riesz transform on $\mathbb{H}^{n}$ is similar to the role of Hilbert transform on the real line.

The first four authors in [9] established an analogous results of a) above in the more general setting of stratified Lie groups by studying the behaviour of the Riesz transform kernels and obtaining a lower bound on the kernel. Note that the Riesz transform kernel does not have an explicit representation, and hence the key difficulty is to obtain a suitable version of kernel lower bounds. To overcome this, in [9] they studied and made good use of the group structures and the dilations related to the stratified condition, and then established a first version of the kernel lower bound for Riesz transforms.

However, it is not clear whether the kernel lower bound for Riesz transforms introduced in [9] is enough to further study analogous results of b), c), d) and e) on stratified Lie groups.

The aim of this paper is three fold. First, we provide a better understanding of the Riesz transform kernel behaviour on stratified Lie groups, which is stronger than the lower bound obtained in [9] and is new in the literature. Second, we apply these kernel lower bounds to study the endpoint characterisations of boundedness of commutators, i.e., we establish similar versions of b), c), d) and e) for the Riesz commutators (but with some improvement) on stratified Lie groups. Third, we also study the characterisation of boundedness from $L^{1}$ to $L^{1,\infty}$. However, we establish this result only on Heisenberg groups since its proof requires the result that the kernel of the Riesz transform on Heisenberg group is non-zero almost everywhere, which we will prove in this paper. It is still an open question for this result on general stratified Lie groups.

To be more specific, suppose $\mathcal{G}$ is a stratified Lie group. Let $\{{\mathrm{X}}_{j}\}_{1\leq j\leq n}$ be a basis for the left-invariant vector fields of degree one on $\mathcal{G}$. Let $\Delta=\sum_{j=1}^{n}{\mathrm{X}}_{j}^{2}$ be the sub-Laplacian on $\mathcal{G}$. Consider the $j^{\mathrm{th}}$ Riesz transform on $\mathcal{G}$ which is defined as $R_{j}:={\mathrm{X}}_{j}(-\Delta)^{-\frac{1}{2}}$.

It is well-known that the Riesz transform $R_{j}$ is a Calderón–Zygmund operator on $\mathcal{G}$, i.e., it is bounded on $L^{2}(\mathcal{G})$ and the kernel satisfies the corresponding size and smoothness conditions, see for example [6, 11]. Moreover, it is also well-known that from [6], the set comprising the identity operator together with Riesz transforms, $\{I,R_{1},R_{2},\ldots,R_{n}\}$ characterises the Hardy space $H^{1}(\mathcal{G})$ introduced and studied by Folland and Stein [11].

We now recall the BMO space on $\mathcal{G}$, which is the dual space of $H^{1}(\mathcal{G})$ [11, Chapter 5], defined as

[TABLE]

where

[TABLE]

and $b_{B}:={1\over|B|}\int_{B}b(g)\,dg$, where $B$ denotes the ball on $\mathcal{G}$ defined via a homogeneous norm $\rho$. We also mention that the weighted BMO space ${\rm BMO}_{\nu}(\mathcal{G})$ is defined via replacing the measure $|B|$ in (4.1) by $\nu(B)$. See Section 2 for details.

The first part of main results of this paper is to establish an enhanced version of the lower bound of the Riesz transform kernel on stratified Lie groups and then study the properties of commutators, including the following:

${\rm b})^{\prime}$ $\|[b,R_{j}]\|_{L_{\mu}^{p}(\mathcal{G})\to L^{p}_{\lambda}(\mathcal{G})}\approx\|b\|_{{\rm BMO}_{\nu}(\mathcal{G})}$ with $\mu,\lambda\in A_{p}(\mathcal{G})$ for $1<p<\infty$ and \nu=\big{(}{\mu\over\lambda}\big{)}^{1\over p};

${\rm c})^{\prime}$ $[b,R_{j}]$ is bounded from $L\log^{+}L(\mathcal{G})$ to $L^{1,\infty}(\mathcal{G})$ if and only if $b\in{\rm BMO}(\mathcal{G})$;

${\rm d})^{\prime}$ $[b,R_{j}]$ is bounded from $H^{1}(\mathcal{G})$ to $L^{1}(\mathcal{G})$ if and only if $b$ equals a constant almost everywhere;

${\rm e})^{\prime}$ $[b,R_{j}]$ is bounded from $L_{c}^{\infty}(\mathcal{G})$ to ${\rm BMO}(\mathcal{G})$ if and only if $b$ equals a constant almost everywhere.

The second part of the main results of this paper is to consider the characterisation of the endpoint boundedness of the commutator with respect to the weak type $(1,1)$, i.e., we aim to study:

$f)^{\prime}$ $[b,R_{j}]$ is of weak type $(1,1)$ if and only if $b$ is in $L^{\infty}(\mathcal{G})$.

However, it is not clear whether one can establish this result on general stratified Lie groups $\mathcal{G}$. Here we study the characterisation of boundedness of commutator from $L^{1}$ to $L^{1,\infty}$ by focusing on the special case of stratified Lie group: the Heisenberg group $\mathbb{H}^{n}$, which can be identified with $\mathbb{C}^{n}\times\mathbb{R}$ with the group structure (for all the notation on $\mathbb{H}^{n}$