You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We study multivariable spectral multipliers $F(L_1,L_2)$ acting on Cartesian product of ambient spaces of two self-adjoint operators $L_1$ and $L_2$. We prove that if $F$ satisfies Hörmander type differentiability condition then the operator $F(L_1,L_2)$ is of Calderón-Zygmund type. We apply obtained results to the analysis of quasielliptic operators acting on product of some fractal spaces. The existence and surprising properties of quasielliptic operators have been recently observed in works of Bockelman, Drenning and Strichartz. This paper demonstrates that Riesz type operators corresponding to quasielliptic operators are continuous on $L^p$ spaces. This solves the problem posed in [@BS (1.3) p. 1363].'
address: 'Adam Sikora, Mathematical Sciences Institute, Bldg 27, Australian National University, ACT 0200 Australia '
author:
- Adam Sikora
date: '28/07/2008'
title: Multivariable spectral multipliers and analysis of quasielliptic operators on fractals
---

I dedicate this paper to the memory of my teachers Andrzej Hulanicki and Tadeusz Pytlik.

Introduction
============

Suppose that $L$ is a self-adjoint operator acting on $L^2(X,\mu)$, where $X$ is a measure space with measure $\mu$. Such an operator admits a spectral resolution $E_L(\lambda)$ and for any bounded Borel function $F\colon {\mathbf{R}}\to {\mathbf{C}}$, we define the operator $F(L)$ by the formula $$F(L)=\int_{\mathbf{R}}F(\lambda) d E_L(\lambda).$$ By the spectral theorem the operator $F(L)$ is continuous on $L^2(X,\mu)$. Spectral multiplier theorems investigate sufficient conditions on function $F$ which ensure that the operator $F(L)$ extends to a bounded operator on $L^q$ for some $q$ in the range $1\le q \le \infty$. The theory of spectral multipliers constitutes an important field of Harmonic analysis and there exists a vast literature devoted to the topic, see for example [@Ale1; @Ale2; @ChS; @CoS; @DM; @DOS; @He1; @Her1; @Mi; @MM; @MRS] and references within. The main aim of this paper is to develop a theory of multivariable spectral multipliers of two self-adjoint independent operators acting on the Cartesian product of their ambient spaces. One could also consider here three or more operators but, for simplicity, we limit the discussion to the two dimensional version.

Multivariable spectral multiplier
---------------------------------

We begin our discussion by explaining the definition of multivariable spectral multipliers. We consider two self-adjoint operators $L_j$, $j=1,2$ acting on spaces $L^2(X_j)$. The tensor product operators $L_{1} \otimes 1$ and $ 1 \otimes L_{2}$ act on $L^2({X}_1 \times
{X}_2)$, where ${X}_1 \times
{X}_2$ is the Cartesian product of $X_1$ and $X_2$ with the product measure $\mu=\mu_1 \times \mu_2$. To simplify notation we will write $L_1$ and $L_2$ instead of $L_{1}
\otimes 1$ and $ 1 \otimes L_{2}$. Note that there is a unique spectral decomposition $E$ such that for all Borel subsets $A\subset {\mathbf{R}}^2$, $E(A)$ is a projection on $L^2({X}_1
\times{X}_2)$ and such that for any Borel subsets $A_j \subset {\mathbf{R}}$, $j=1,2$ one has $$E(A_1\times{A}_2)=E_{L_1}(A_1)\otimes E_{L_2}(A_2).$$ Hence for any function $F\colon {\mathbf{R}}^2   \to {\mathbf{C}}$ one can define the operator $F(L_1,L_2)$ acting as operators on space $L^2({X}_1 \times {X}_2,)$ by the formula $$\label{equwd}
  F(L_1,L_2)=\int_{{\mathbf{R}}^2}F(\lambda_1,\lambda_2) d E(\lambda_1,\lambda_2).$$ A straightforward variation of classical spectral theory arguments shows that for any bounded Borel function $F\colon {\mathbf{R}}^2   \to {\mathbf{C}}$ the operator $F(L_1,L_2)$ is continuous on $L^2({X}_1 \times {X}_2)$ and its norm is bounded by $\|F\|_{L^\infty}$. In this paper we are looking for necessary smoothness conditions on function $F$ so that the operator $F(L_1,L_2)$ is bounded on a range of other $L^p({X}_1 \times
{X}_2)$ spaces. The condition on function $F$ which we use is a variant of the differentiability condition in Hörmdander-Mikhlin Fourier multiplier result, see [@Her1; @Mi] and [@Her2 Theorem 7.9.6].

One motivation for multivariable spectral multiplier results comes from Riesz transform like operators, for example $$\frac{aL_1+bL_2}{cL_1+dL_2} \quad \mbox{or} \quad \frac{L_1L_2}{(cL_1+dL_2)^2},$$ where $a,b\in {\mathbf{R}}$ are arbitrary real numbers and $c,d>0$. Here we assume that the operators $L_i$ are positive for $i=1,2$. It is easy to note that then such operators are bounded on $L^2({X}_1 \times
{X}_2)$. We prove that under standard assumptions such operators are of Calderón-Zygmund type, that is, they are bounded on all $L^p({X}_1 \times
{X}_2)$ spaces for $1<p<\infty$ and of weak type $(1,1)$. Of course one can consider a much larger family of operators of this type. For the Laplace operators acting on some fractal spaces and for some real numbers $c,d$ the above operators are still bounded on $L^2({X}_1 \times
{X}_2)$ (and as we prove it in this paper on other $L^p({X}_1 \times
{X}_2)$ spaces $1<p< \infty$) [*even though $cd <0$*]{}. We describe this intriguing phenomenon in the following section.

Product of fractals and quasielliptic operators
-----------------------------------------------

In [@BS] and [@DS] it is shown that for the Laplace operators defined on some fractal spaces, the set of ratios of eigenvalues have gaps. This means that there are intervals $(\alpha,\beta)$, $0<\alpha<\beta$, such that for any two eigenvalues $\lambda_i$, $\lambda_j$ of the Laplacian acting on the same fractal spaces one has $\frac{\lambda_j}{\lambda_i}\notin (\alpha,\beta)$. For example if one considers the Dirichlet or Neumann Laplacian on Sierpiński Gasket $SG$ then $\frac{\lambda_j}{\lambda_i}\notin (\alpha,\beta)$, where $$\label{gap}
\alpha=\lim_{n\to \infty} \frac{\psi_n(5)}{\psi_n(3)}\approx 2.0611 \quad \mbox{and} \quad 
 \beta=\lim_{n\to \infty} \frac{\psi_n(3)}{\psi_{n+1}(3)}\approx 2.4288$$ and $\psi (x) = (5 -\sqrt{25 - 4x})/2$, see [@BS]. The existence of gaps in the set of ratios of eigenvalues has a surprising consequence. Namely one can consider the product of two copies of such fractal spaces and operators $L_1$ and $L_2$ which are copies of the same Laplace operator acting on first and second variable respectively. Now it is easy to notice that if $(\alpha,\beta)$ is a gap in the set of ratios of eigenvalues $\alpha<\gamma < \beta$, $a,b\in {\mathbf{R}}$, $c,d>0$ and $\frac{d}{c}=\gamma$ then the operators $$\label{qell}
\frac{aL_1+bL_2}{cL_1-dL_2}  \quad \mbox{or} \quad \frac{L_1L_2}{(cL_1-dL_2)^2},$$ are bounded on $L^2$. Following [@BS] we call operators of the form ${cL_1-dL_2} $ [*quasielliptic*]{}. It was asked in [@BS] whether the above operators are bounded on other $L^p$ spaces. This question is the initial motivation of this paper. We prove that these operators are indeed bounded on all $L^p$ spaces for $1<p<\infty$ and of weak type $(1,1)$.

The idea of multivariable spectral multipliers is of independent interest. It seems to be possible to obtain more general versions of multivariable spectral multipliers and some Marcinkiewicz type variations of these results. However, here we concentrate on obtaining a possibly simple proof of weak type $(1,1)$ for Riesz transform type operators corresponding to quasielliptic operators.

Doubling condition
------------------

Before we state our main result we have to describe our basic assumptions. As it is usually the case in theory of spectral multipliers we require the doubling condition and some version of Gaussian estimates for semigroups generated by operators $L_j$, $j=1,2$, see [@Ale2; @DOS]. We assume that the considered ambient spaces $X_j$ $j=1,2$ are equipped with a Borel measure $\mu_j$ and distances $\rho_j$. Let $B(x,r)= \{ y \in X \colon \, {\rho}(x,y) <  r \}$ be the open ball centred at $x$ and radius $r$. We suppose throughout that ${X_1},X_2$ satisfy the doubling condition, that is there exists a constant $C$ such that $$\label{doubling1.1}
    \mu_j(B(x_j, 2r)) \le C  \mu_j(B(x_j, r)) \quad \forall x_j \in X_j,
\forall r > 0, j=1,2.$$ Note that (\[doubling1.1\]) implies that there exist positive constants $C$ and $d_1,d_2$ such that $$\label{d}
    \mu_j(B(x_j, t r)) \le C (1+t)^{d_j} \mu_j(B(x_j, r))  \quad \forall
t > 0,
    x_j \in X_j, r > 0.$$ In the sequel we always assume that (\[d\]) holds. Note that all fractal spaces which we discuss here satisfy condition (\[d\]). In fact for these spaces $\mu(B(x, r)) \sim r^d$ for all $r\le 1$, see [@Ki; @Str2].

Kernels of operators
--------------------

Suppose that $T$ is a bounded operator on $L^2(X).$ We say that a measurable function $K_T\colon X^2 \to {\mathbf{C}}$ is the (singular) kernel of $T$ if $$\label{abcde}
 \langle T f_1,f_2\rangle  =
 \int_X T f_1\overline{f_2} d\mu  =
 \int_X K_{T}(x,y) f_1(y)\overline{f_2(x)} d\mu(x) d\mu(y).$$ for all $f_1,f_2\in C_c(X)$ (for all $f_1,f_2\in C_c(X)$ such that ${\operatorname{supp}}f_1 \cap {\operatorname{supp}}f_2 = \emptyset$ respectively). It is well known that if $T$ is bounded from $L^1(X)$ to $L^q(X)$, where $1<q$, then $T$ is a kernel operator, and $$\|T\|_{L^1 \to L^q}
= \sup_{y\in X} \| K_{T}(\cdot , y)\|_{L^{q}}\, .$$ In addition, if $S$ is continuous on $L^q(X)$ then for almost all $y\in X$ $$\label{ker1}
K_{ST,y}=SK_{T,y}$$ where $K_{T,y}(x)=K_T(x,y)$. Next we denote the weak type $(1,1)$ norm of an operator $T$ on a measure space $(X,\mu)$ by $\|T\|_{L^1 \to L^{1,\infty}}
=\sup \lambda  \ \mu  ( \{x\in  X : \ |Tf(x)| >  \lambda \})$, where the supremum is taken over $\lambda > 0$ and functions $f$ with $L^1$ norm less than one; this is often called the “operator norm”, though in fact it is not a norm.

In the sequel we will always require the following Gaussian estimates for the heat kernel corresponding to the operators $L_1$ and $L_2$.

\[1b\] Let $L_j$, $j=1,2$ be self-adjoint positive definite operators. We assume that the semigroups generated by $L_j$ on $L^2(X_j)$ have the kernels $p^{[j]}_{t}(x_j,y_j)=K_{\exp(-tL_j)}(x_j,y_j)$ defined by [(\[abcde\])]{} which for some constants $C_j,b_j>0$ and ${m}>1$ satisfy the following Gaussian upper bounds $$\label{1.3kernel}
    \vert  p^{[j]}_{t}(x_j,y_j)  \vert \le C_j\mu(B(y_j, t^{1/m}))^{-1} \exp
    \Big(-b_j\frac{{\rho}_j(x_j,y_j)^{m/(m-1)}}{t^{1/(m-1)}} \Big)$$ for all $t>0$ and $j\in \{1,2\}$.

We will call $p^{[j]}_{t}(x,y)$ the heat kernels associated with $L_j$. Such estimates are typical for elliptic or sub-elliptic differential operators of order $m$ (see e.g. [@Da2]). But such estimates hold also for most of the Laplace type operators acting on fractals, see [@Ki; @Str2].

Note that Assumption \[1b\] implies that the operator $K_{\exp(sL_1+tL_2)}$ has a $L^\infty$ kernel given by the following formula $$K_{\exp(sL_1+tL_2)}((x_1,x_2),(y_1,y_2))=p^{[1]}_{s}(x_1,y_1)p^{[2]}_t(x_2,y_2)$$ for all $s,t>0$.

Main result
===========

We define a family of dilations $\{\delta_t\}_{t>0}$ acting on functions $F\colon {\mathbf{R}}^2 \to C$ by the formula $$\delta_{t}F(\lambda_1,\lambda_2)  =   F(t\lambda_1,t\lambda_2).$$ Next let us recall that the norms in Sobolev spaces $W_{s}^{p}({\mathbf{R}}^n)$ are defined by the formula $$\|F\|_{W_{s}^{p}({\mathbf{R}}^n)}=\|(\Delta+I)^{s/2}F\|_{L^p({\mathbf{R}}^n)},$$ where $\Delta$ is the standard Laplace operator. Now we can formulate our main spectral multiplier result.

\[main\] Suppose that spaces $X_j$ satisfy doubling condition [(\[d\])]{} with constants $d_1$ and $d_2$. Further assume that operators $L_j$ satisfy Assumption [\[1b\]]{} with the same order $m$. Let $F\colon {\mathbf{R}}^2 \to {\mathbf{C}}$ be a continuous function and let $\eta \in
C_c^\infty(0,\infty)$ be an auxiliary nonzero cut-off function. Suppose that for some $s>\frac{d_1+d_2}{2}$ $$\label{hc}
\sup_{t> 0} \Vert \widetilde{\eta} \, \delta_{t }F
\Vert_{W^\infty_s}<\infty,$$ where $\widetilde{\eta}(x,y)=\eta(x+y)$. Then the operator $F(L_1,L_2)$ is of weak type $(1,1)$ and is bounded on $L^q(X)$, $X=X_1\times X_2$ for all $1<q<\infty$. In addition $$\|F({L_1,L_2})\|_{L^1 \to  L^{1,\infty}}
    \le  C_s \sup_{t >   0} \Vert \widetilde{\eta}
    \, \delta_t F \Vert_{W^\infty_s({\mathbf{R}}^2)}.$$

[*Remarks*]{} 1. We assume that $L_1$ and $L_2$ are positive so $F(L_1,L_2)$ depends only on the restriction of $F$ to $[0,\infty)^2$. However, it is easier to state Theorem \[main\] if one considers functions $F\colon {\mathbf{R}}^2 \to {\mathbf{C}}$.

2\. The condition on function $F$ in Theorem \[main\] is similar to the condition in Hörmdander-Mikhlin Fourier multiplier result, see [@Her1; @Mi] and [@Her2 Theorem 7.9.6]. The only difference is that we use the space $W^\infty_s$ instead of $W^2_s$. The significance of this difference is discussed in [@DOS].

3\. It is not difficult to check that if for all $|I| \le \left[\frac{d_1+d_2}{2}\right]+1$, where $[a]$ is an integer part of $a$, $$\sup_{\lambda\in {\mathbf{R}}^2}|\lambda|^{|I|}|\partial^I F(\lambda)| < \infty,$$ then $F$ satisfies condition (\[hc\]), see [@Ale1; @Ale2; @LM]. In a sense condition (\[hc\]) is a fractional exponent version of the above condition in which $|I|$ must be an integer. The above condition is an illuminating illustration of condition (\[hc\]). In fact condition (\[hc\]) can be stated equivalently using interpolation between integer cases of the above definition.

Note that $\eta$ is only an auxiliary function and that condition (\[hc\]) does not depend of $\eta$.

4\. It would be interesting to obtain a version of Theorem \[main\] with condition (\[hc\]) replaced by $$\sup_{\lambda\in {\mathbf{R}}^2}|\lambda^I\partial^I F(\lambda)| < \infty,$$ for all $|I|\le l$ for some sufficiently large $l$, see [@MRS Theorem 1.3].

5\. It could be also interesting to try to obtain some multivariable spectral multipliers results similar to Theorem \[main\] using the techniques developed in [@DR; @LLM].

Notation
--------

In subsequent sections we use the following notation. We put $X_P=X_1\times X_2$, $x_P=(x_1,x_2)\in X_P$ and $y_P=(y_1,y_2)$. Next $\mu_P=\mu_1\times \mu_2$ and we set $\rho_P(x_P,y_P)
=\max\{(\rho_1(x_1,y_1),\rho_2(x_2,y_2)\}$. If the discussed results hold separately for both $X_1$ and $X_2$ spaces we just skip index $i,j$ and use $X,x,y,\mu$ etc.

Proof of Theorem \[main\]
=========================

We split the proof of Theorem \[main\] into a few lemmas. First we show the following straightforward consequences of Assumption \[1b\]

\[ltu\] Suppose that [(\[1.3kernel\])]{} and [(\[d\])]{} hold. Then for all $r,t>0$ $$\int_{X-B(y,r)}|p_{t}(x,y)|^2 d\mu(x)
    \le   C    \mu(B(y,t^{1/m}))^{-1}\exp
    \left(-b\frac{r^{{m}/{(m-1)}}}{ {t^{1/(m-1)}}}\right).$$ In particular $$\|p_{t}(x,{\, \cdot \,})\|^2_{L^2(X)}=
    \|p_{t}({\, \cdot \,},x)\|^2_{L^2(X)} \le C \mu_j(B(x,t^{1/m}))^{-1}.$$

By (\[1.3kernel\]) and (\[d\]) (see also [@CD Lemma 2.1]) $$\begin{aligned}
\int\limits_{X-B(y,r)}\!\!|p_{t}(x,y)|^2d\mu(x) \le
\frac{C}{\mu(B(y,t^{1/m}))^2}
\!\!\int\limits_{X-B(y,r)}\!\!\exp\left(-2b\sqrt[m-1]{\rho(x,y)^m/t}\right) d\mu(x)\\
\le \frac{C\exp(-b\sqrt[m-1]{r^m/t})}{\mu(B(y,t^{1/m}))^2}
\int_{X}\exp\big(-b\sqrt[m-1]{\rho(x,y)^{m}/t}\big) d\mu(x)\\
\le \frac{C\mu(B(y,t^{1/m}))}{\exp(-b\sqrt[m-1]{r^{m}/t})}.\end{aligned}$$

Second we prove the following lemma

\[P\] Suppose that $\|p^{[j]}_{t}(x_j, {\, \cdot \,})\|^2_{L^2(X_j,\mu_j)} \le C
\mu_j(B(x_j,t^{{1}/{m}}))^{-1}$. Then $$\begin{aligned}
    \Vert  K_{{F}(L_1,L_2)}  (x_P,{\, \cdot \,})  \Vert^2_{L^2(X_P)}=
    \Vert  K_{\overline{F}(L_1,L_2)}  ({\, \cdot \,},x_P)  \Vert^2_{L^2(X_P)}
    \nonumber \\ \le C  \Vert F \Vert_{L^\infty}^2 \prod_{j=1,2}\mu_j(B(x_j,R^{-1}))^{-1}
    \end{aligned}$$ for any Borel function $F \colon {\mathbf{R}}^2 \to {\mathbf{C}}$ such that ${\operatorname{supp}}F\subset [0,{R}^{{m}}]\times [0,{R}^{{m}}]$.

Set $$G_1=\frac{F}{G_2}, \quad \mbox{\rm where}  \quad
G_2(\lambda_1,\lambda_2)=\exp(-{R}^{-{m}}(\lambda_1+\lambda_2)).$$ Then $$\Vert G_1(L_1,L_2) \Vert_{L^2(X_P) \to L^2(X_P)} \le
\|G_1\|_{L^\infty}\le  e\|F\|_{L^\infty}.$$ By (\[ker1\]) the operator $F(L_1,L_2)$ has the kernel given by the formula $$K_{F(L_1,L_2)}(x_P,y_P)
= \left[{G}_1(L_1,L_2)K_{G_2(L_1,L_2)}({\, \cdot \,},y_P)\right](x_P).$$ Now $$\begin{aligned}
&&\int_{X_P} \vert   K_{F(L_1,L_2)}(x_P,y_P) \vert^{2}  d \mu_P(y_P)\label{GF}
\\ &&\le \Vert
{G}_1(L_1,L_2) \Vert_{L^2 \to L^2}^2  \prod_{j=1,2}\Vert
p^{[j]}_{{R}^{-m}}({\, \cdot \,}, x_j ) \Vert_{L^2(X_j)}^2\nonumber
\\ &&\le C   \Vert G_1\Vert^2_{L^\infty}
\prod_{j=1,2}\mu_j(B(x_j,{R}^{-1}))^{-1} \le C  \Vert F
\Vert^2_{L^\infty}\prod_{j=1,2}\mu_j(B(x_j,{R}^{-1}))^{-1}  .\nonumber\end{aligned}$$

Third we show that (compare [@DOS; @Ou])

\[jeden\] For any $s_1,s_2 \ge 0$ there exists a constant $C$ such that $$\begin{aligned}
\label{sss}
    \int_{X_1\times X_2} \prod_{j=1,2}\Big(|p^{[j]}_{
(1+i\tau_j){R}^{-m}}(x_j,y_j)|^2\rho_j(x_j,y_j)^{s_j}\Big) d
\mu_1\times\mu_2(x_1,x_2) \nonumber
   \\ \le C  \prod_{j=1,2}\Big(\mu_j(B(y_j,{R}^{-1}))^{-1}
{R}^{-s_j}(1+|\tau_j|)^{s_j}\Big)
    \end{aligned}$$ for all $\tau_j\in {\mathbf{R}}$ and ${R}>0$.

Note that the integral (\[sss\]) is a product of integrals and it is enough to show that the above estimates hold for $j=1$ and $j=2$ separately, that is $$\begin{aligned}
    \int_{X_j}| p^{[j]}_{
(1+i\tau_j){R}^{-{m}}}(x_j,y_j)|^2\rho_j(x_j,y_j)^{s_j} d
\mu_j(x_j)
 \\   \le C   \mu_j(B(y_j,{R}^{-1}))^{-1} {R}^{-s_j}(1+|\tau_j|)^{s_j}\end{aligned}$$ holds for $j=1$ and $j=2$. The proof for $j=1$ and $j=2$ is the same so to simplify notation we skip the index $j$. The rest of the proof follows closely the proof in [@DOS] and we describe it here for the sake of completeness. First, we assume that $\|f\|_{L^2(X)}=1$ and that ${\operatorname{supp}}f \subset X-B(y,r)$. Next, we define the holomorphic function $F_y\colon \{z\in {\mathbf{C}}\colon\ \Re
e\,z > 0\} \to {\mathbf{C}}$ by the formula $$F_y(z)=e^{-zR^m}\mu(B(y,1/R))
  \bigg(\int_X p_z(x,y)f(x) d\mu(x)\bigg)^2.$$ By the same argument as in (\[GF\]) if we put $z=|z|e^{i\theta}$, then $\Re e \, z =|z|\cos\theta$ and $\|p_{z}({\, \cdot \,},y)\|^2_{L^2}=\|p_{|z|\cos\theta}({\, \cdot \,},y)\|^2_{L^2}$. Hence by Lemma \[ltu\] $$\begin{aligned}
|F_y(z)| & \le &e^{-R^m|z|\cos\theta}\mu(B(y,1/R))\|p_{|z|\cos\theta}({\, \cdot \,},y )\|^2_{L^2}
\\ & \le & Ce^{-R^m|z|\cos\theta}
\frac{\mu(B(y,1/R))}{\mu(B(y,\sqrt[m]{|z|\cos\theta}))}
\le
Ce^{-R^m|z|\cos\theta}\Big(1+\frac{1}{R^{m}|z|\cos\theta}\Big)^{d/m}
\\      & \le & CR^{-d}(|z|\cos\theta)^{-d/m}.\end{aligned}$$ Similarly for $\theta=0$ by Lemma \[ltu\] $$|F_{y}(|z|)|\le C R^{-d} |z|^{-d/m}
  \exp\bigg(-\frac{br^{m/(m-1)}}{|z|^{1/(m-1)}}\bigg).$$ Now let us recall the following version of Phragmen-Lindelöf Theorem

\[dav\] Suppose that function $F$ is analytic on the half-plane ${\mathbf{C}}_+=\{ z\in {\mathbf{C}}\colon \Re e\, z >
0 \}$ and that $$\vert F (|z|e^{i\theta}) \vert \le a_1 (|z|\cos\theta)^{-{\beta_1}}$$ $$\vert F(|z|) \vert \le a_1 |z|^{-{\beta_1}}\exp(-a_2|z|^{-{\beta_2}})$$ for some $a_1, a_2 > 0$, ${\beta_1} \ge 0$, ${\beta_2}\in(0,1]$, all $z \in {\mathbf{C}}_+$. Then $$\vert F(|z|e^{i\theta})  \vert   \le  a_1 2^{\beta_1}
    (|z|\cos\theta)^{-{\beta_1}}
    \exp\Big(-\frac{a_2{\beta_2}}{2}|z|^{-{\beta_2}}\cos\theta\Big)$$ for all $z \in {\mathbf{C}}_+$.

Now if $|z|e^{i\theta}=(1+i\tau)R^{-m}$, then $|z|=R^{-m}(1+|\tau|^2)^{1/2}$, $\cos\theta=(1+|\tau|^2)^{-1/2}$ and $|z|\cos\theta  =   R^{-m}$. Putting $a_1=CR^{-d}$, $a_2=br^{m/(m-1)}$, ${\beta_1}=d/m$ and ${\beta_2}=1/(m-1)$ in Lemma \[dav\] we conclude that $$\begin{aligned}
|F_y((1+i\tau)R^{-m})| \le C' \exp{\Big(-b'(rR/(1+|\tau|))^{m/(m-1)}\Big)}.\end{aligned}$$ Hence $$\begin{aligned}
\mu(B(y,1/R))\int_{X-B(y,r)} |p_{(1+i\tau)R^{-m}}(x,y)|^2d \mu(x) \\ \le
 C \exp{\Big(-b'(rR/(1+|\tau|))^{m/(m-1)}\Big)}.\end{aligned}$$ Finally, we have $$\begin{aligned}
&&\hspace{-2cm}\int_X |p_{(1+i\tau)R^{-m}}(x,y)|^2{\rho}(x,y)^s  d \mu(x)\\
&=& \sum_{k \ge 0} \int_{ k (1+|\tau|)R^{-1} \le {\rho}(x,y) \le (k+1)
(1+|\tau|)R^{-1}}
|p_{(1+i\tau)R^{-m}}(x,y)|^2{\rho}(x,y)^s  d \mu(x)\\
&\le& (1+|\tau|)^s R^{-s}\sum_{k \ge 0} (k+1)^s \int_{ X - B(y,k
(1+|\tau|)R^{-1}) }
|p_{(1+i\tau)R^{-m}}(x,y)|^2  d \mu(x)\\
&\le& C\mu(B(y,1/R))^{-1}R^{-s}(1+|\tau|)^{s}.
 \end{aligned}$$

Next we show that

\[waga\] Suppose that $L_1$ and $L_2$ satisfy Assumption [(\[1b\])]{}, $R>0$ and $s=s_1+s_2$. Then for any ${\varepsilon}> 0$ there exists a constant $C=C(s,{\varepsilon})$ such that $$\begin{aligned}
\int_{X} \vert K_{F(L_1,L_2)}  ((x_1,x_2),(y_1,y_2))\vert^{2}
\prod_{j=1,2}(1 + {R}\rho_j(x_j,y_j) )^{s_j}
d\mu_1\times \mu_2(x_1,x_2) \nonumber\\
\le C    \Vert \delta_{R^m} F
\Vert_{W^\infty_{\frac{s}{2} + {\varepsilon}}}^{2} \prod_{j=1,2}\mu_j(B(y_j,R^{-1}))^{-1} \end{aligned}$$ for all Borel functions $F$ such that ${\operatorname{supp}}F \subseteq [0, {R}^{{m}}]^2$.

Set $$G(\lambda_1,\lambda_2)= e^{\lambda_1+\lambda_2}\delta_{{R}^{m}}
F(\lambda_1,\lambda_2).$$ In virtue of the Fourier inversion formula $$\begin{aligned}
&&F(L_1,L_2)=G(R^{-m}L_1, R^{-m}L_2)\exp{(-R^{-m}(L_1+L_2))
}\\&&\hspace{0.5cm}=\frac{1}{4\pi^2}\int_{{\mathbf{R}}^2}
\exp{\left( \sum_{j=1,2}(i\tau_j-1)R^{-m}L_j \right)}
\widehat{G}(\tau_1,\tau_2)
d \tau_1 d \tau_2,\end{aligned}$$ where $\widehat{G}$ is the Fourier transform of the function $G$. Hence $$\begin{aligned}
 &&K_{F(L_1,L_2)}((x_1,x_2),(y_1,y_2))  \\&&\hspace{0.5cm}= \frac{1}{4\pi^2}
\int_{{\mathbf{R}}^2}\widehat{G}(\tau_1,\tau_2)
\prod\limits_{j=1,2} p^{[j]}_{(1-i\tau_j){R}^{-m}}(x_j,y_j)   d \tau_1 d
\tau_2.\end{aligned}$$ Thus by Lemma \[jeden\] and Lemma \[ltu\] $$\begin{aligned}
    &&\Bigg( \int_{X_1\times X_2} \vert
K_{F(L_1,L_2)}((x_1,x_2),(y_1,y_2))
  \vert^{2}  \nonumber \\ &&\hspace{2cm} \times(1 + R{\rho}_1(x_1,y_1)
)^{s_1}(1 + R{\rho}_2(x_2,y_2) )^{s_2}
d\mu_1\times \mu_2(x_1,x_2) \Bigg)^{\frac{1}{2}}   \nonumber\\
  &&\le\int\limits_{{\mathbf{R}}^2} \! \vert  \widehat{G}(\tau_1,\tau_2)\vert
  \Big(\prod_{j=1,2}\int\limits_{X_j} \! |p^{[j]}_{(1-i  \tau_j)R^{-m}}(x_j,y_j)|^2
(1+R{\rho}_j(x_j,y_j))^{s_j}
  d\mu_j(x_j)\Big)^{\frac{1}{2}} d  \tau_1 d \tau_2  \nonumber\\
  &&\le C\Big( \prod_{j=1,2}\mu_j(B(y_j,R^{-m})^{-\frac{1}{2}}\Big)
    \int_{{\mathbf{R}}^2}
  \vert \widehat{G}(\tau_1,\tau_2) \vert
(1+|\tau_1|)^{s_1/2}(1+|\tau_2|)^{s_2/2}
  d \tau_1 d \tau_2  \nonumber\\
  &&\hspace{1cm} \le  C  \Big(\prod_{j=1,2}\mu_j(B(y_j,R^{-m})^{-\frac{1}{2}}\Big)\left(\int_{{\mathbf{R}}^2} \vert \widehat{G}(\tau_1,\tau_2) \vert^{2}
  ({1+\tau_1^2+\tau_2^{2}})^{\frac{s_1+s_2+{\varepsilon}+2}{2}}\right)^{\frac{1}{2}}
\nonumber \\ &&\hspace{2cm}
 \times 
  \left(\int_{{\mathbf{R}}^2}({1+\tau_1^2+\tau_2^{2}})^{\frac{-2-{\varepsilon}}{2}}\right)^{\frac{1}{2}}
\label{dwa}\\
  &&  \hspace{4cm} \le C   
  \|G\|_{W^2_{\frac{s_1+s_2+2+{\varepsilon}}{2}}}\prod_{j=1,2}\mu_j(B(y_j,R^{-{m}})^{-\frac{1}{2}}.
\nonumber\end{aligned}$$ However, ${\operatorname{supp}}F  \subseteq  [0, {R}^{{m}}]^2$ and ${\operatorname{supp}}\delta_{{R}^m}F
\subseteq [0,1]^2$ so $$\label{dwa2}
  \|G\|_{W^2_{{(s+2+{\varepsilon})}/{2}}}
\le C \|\delta_{R^{m}} F\|_{W^2_{(s+2+{\varepsilon})/{2}}}
\le C \|\delta_{R^{m}} F\|_{W^\infty_{(s+2+{\varepsilon})/{2}}}.$$ From the last two estimates one can obtain a multiplier result in which the required order of differentiability of the function $ F$ is of $1$ greater than that of Lemma \[waga\]. To get rid of this additional $1$ we use Mauceri-Meda interpolation argument, see [@MM] and [@DOS]. First we note that the estimates from Lemma \[waga\] are equivalent to the following inequality $$\begin{aligned}
\int_{X} \vert K_{\delta_{R^{-m}}F(L_1,L_2)}
((x_1,x_2),(y_1,y_2))\vert^{2} \prod_{j=1,2}(1 +
R\rho_j(x_j,y_j) )^{s_j}
d\mu_1\times \mu_2(x_1,x_2)
\nonumber
\\ \le C  
\Vert F\Vert_{W^\infty_{ {\frac{s_1+s_2}{2}}+ {\varepsilon}}}^{2} \prod_{j=1,2}  \mu_j(B(y_j,R^{-1}))^{-1} \label{dww}\end{aligned}$$ for all bounded Borel functions $F$ such that ${\operatorname{supp}}F\subset [0,1]^2$. Now we define the linear operator $K_{y_1,y_2,{R}} \colon
L^\infty([0,1]^2) \to
L^2{(X_1\times X_2,\mu_1\times \mu_2)}$ by the formula $$K_{y_1,y_2,{R}}(F)=K_{\delta_{R^{-m}} F(L_1,L_2)}({\, \cdot \,},(y_1,y_2)).$$ By Lemma \[P\] $$\Big\Vert K_{y_1,y_2,{R}}\Big\Vert_{L^\infty([0,1]^2) \to L^2(X_1 \times
X_2 ,\mu_1 \times \mu_2)}^2\le C  \prod_{j=1,2} 
\mu_j(B(y_j,R^{-1}))^{-1}.$$ Next we put $L^2_{y_1,y_2,s_1,s_2,{R}}=L^2(X_1\times
X_2,\mu_{y_1,y_2,s_1,s_2,{R}})$, where $$d\mu_{y_1,y_2,s_1,s_2,R}(x_1,x_2)
=\prod_{j=1,2}(1+R\rho_j(x_j,y_j))^{s_j}d\mu_1\times \mu_2(x_1,x_2)$$ By (\[dwa\]) and (\[dwa2\]) $$\Big\Vert K_{y,R}\Big\Vert_{W^\infty_{(s_1+s_2+2+{\varepsilon})/2}([0,1]^2)
\to L^2_{y_1,y_2,s_1,s_2,R}}^2
\le C    \prod_{j=1,2}  \mu_j(B(y_j,R^{-1}))^{-1}.$$ By interpolation, for every $\theta \in (0,1)$ there exists a constant $C$ such that $$\begin{aligned}
&& \Vert \delta_{R}F(L_1,L_2)({\, \cdot \,},(y_1,y_2) )
\Vert_{L^{2}_{y_1,y_2,\theta s_1, \theta s_2,R}} ^2\\&& \le  {C}\Vert F \Vert_{{[L^\infty, W^\infty_{(s_1+s_2+2+{\varepsilon})/2}]_{[\theta]}}}^2 \prod_{j=1,2}
\mu_j(B(y_j,R^{-1}))^{-1}.\end{aligned}$$ In particular, for all $s>0$, $ \theta \in (0, 1)$ and ${\varepsilon}' >
{\varepsilon}$ $$\begin{aligned}
&&\Vert \delta_{R^{-m}}F(L_1,L_2)({\, \cdot \,},(y_1,y_2) )
\Vert_{L^{2}_{y_1,y_2,\theta s_1, \theta s_2,R}}^2 \\&& \le  {C}\Vert F \Vert_{{ W^\infty_{(\theta s_1+\theta s_2+2\theta +\theta
{\varepsilon}')/2}}}^2 \prod_{j=1,2}
\mu_j(B(y_j,R^{-1}))^{-1}.\end{aligned}$$ Hence putting $s'_j={s_j}/{\theta}$ in the above inequality and taking $\theta$ small enough we obtain $$\begin{aligned}
&&\Vert \delta_{R^{-m}}F(L_1,L_2)({\, \cdot \,},(y_1,y_2) )
\Vert_{L^{2}_{y_1,y_2,s'_1, s'_2,R}}^2 \\ &&\le  {C} \Vert F \Vert_{{ W^\infty_{(s'_1+s'_2)/2+{\varepsilon}"}}}^2\prod_{j=1,2}
\mu_j(B(y_j,R^{-1}))^{-1}.\end{aligned}$$ This proves (\[dww\]) and Lemma \[P\]

Next we show the following simple consequence of doubling condition (\[d\]).

\[waga2\] Suppose that [(\[d\])]{} holds. Then for any $s_1 > d_1$ and $s_2 >
d_2$ there exists ${\varepsilon}>0$ such that $$\int_{{X_P-B(y_P,r)}}\prod_{j=1,2}(1 + {{R}}\rho_j(x_j,y_j) )^{-s_j}
 d\mu(x) \le C  {(1+rR)^{-{\varepsilon}}}\prod_{j=1,2}
\mu(B(y_j,{R}^{-1}))$$ where $B(y_P,r)=\{x_P \in X_P \colon \, \rho_P(x_P,y_P) < r\}$.

Choose ${\varepsilon}>0$ such that $s'_j= s_j-{\varepsilon}>d_j$ and note that $$\inf_{{X_P-B(y_P,r)}}\prod_{j=1,2}(1 + {R}\rho_j(x_j,y_j) )^{-{\varepsilon}}\le
{(1+rR)^{-{\varepsilon}}}.$$ Hence $$\begin{aligned}
\int_{{X_P-B(y_P,r)}}\prod_{j=1,2}(1 + {R}\rho_j(x_j,y_j) )^{-s_j}
 d\mu_P(x_P)
\\ \le {(1+rR)^{-{\varepsilon}}} \int_{{X_P}}     \prod_{j=1,2}(1 +
{R}\rho_j(x_j,y_j) )^{-s'_j}
d\mu_P(x_P).\end{aligned}$$ Next $$\int_{{X_P}}     \prod_{j=1,2}(1 + {R}\rho_j(x_j,y_j) )^{-s'_j}
d\mu_P(x_P) =\prod_{j=1,2} \int_{{X_j}}    (1 + {R}\rho_j(x_j,y_j) )^{-s'_j}
d\mu_j(x_j)$$ so it is enough to show that for $j=1$ and $j=2$ one has $$\int_{{X_j}}  (1 + {R}\rho_j(x_j,y_j) )^{-s'_j}
d\mu_j(x_j) \le \mu_j(B(y_j,{R}^{-1})).$$ As before we skip the index $j$ in the proof. $$\begin{aligned}
\label{rr1}
\int\limits_{X} (1+ R{\rho}(x,y)  )^{-s}d\mu(x) \le  \mu(B(y, R^{-1}))  +  \sum_{k \ge 0}
\int\limits_{2^k \le R{\rho}(x,y)  \le 2^{k+1}} (R  {\rho}(x,y))^{-s}d\mu(x)
 \\
\le  \mu(B(y, R^{-1})) +\sum_{k \ge 0} (2^k)^{-s}  \mu(B(y, 2^{k+1}/R)) \le
C\sum_{k \ge
0} (2^k)^{d-s} \mu(B(y, R^{-1}))\\
\le  \mu(B(y, R^{-1})).\end{aligned}$$

To prove that operator is of weak type (1,1) we usually use estimates for the gradient of the kernel. The following theorem replaces the gradient estimates in the proof of Theorem \[main\].

\[CZ\] Suppose that $\| F \|_{L^\infty} \le C_1$ and that $$\label{lap1}
\sup_{r\in{\mathbf{R}}^+}\sup_{y_P\in X_P}\int_{ X_P-B(y_P,r)}
|K_{F(1-\Phi_{r})(L_1,L_2)}(x_P,y_P)| d \mu_P(x_P) \leq C_1 ,$$ where $\Phi_{r}(L_1,L_2)=\exp(-r^m(L_1+L_2) )$. Then $$\| F(L_1,L_2)\|_{L^1\to L^{1,\infty}} \le C C_1.$$

Theorem \[CZ\] follows from [@DM Theorem 2]. Indeed one can check that the family of operators $\exp(-tL_P)$, where $L_P=L_1+L_2$ satisfies all assumptions from [@DM]. Hence any operator $T$ which is bounded on $L^2$ and whose kernel $K_{T(1-\Phi_{r})}(L_1,L_2)$ satisfies condition (\[lap1\]) is of weak type $(1,1)$. See also [@CD; @He1] and [@CoS] for similar results.

[*Remark*]{}. In this note we define Calderón-Zygmund type operators in the sense of Theorem \[CZ\] and [@DM Theorem 2]. Note that this does not mean that the singular kernel of the operator $T$ is continuous outside the diagonal as it is the case for classical Calderón-Zygmund operators. For quasielliptic operators which we discuss below continuity of the kernel is a question of independent interest.

Choose a function $\eta$ in $C_c^\infty(0,1)$ such that $$\sum_{n\in {\mathbf{Z}}} \eta(2^{nm}\lambda) = 1 \qquad\forall \lambda \in {\mathbf{R}}_+.$$ Then $$F(1-\Phi_{r})(L_1,L_2) =\sum_{n\in {\mathbf{Z}}} {\widetilde{\eta}}_n F(1-\Phi_{r})(L_1,L_2),$$ where ${\widetilde{\eta}}_n=\delta_{2^{-nm}}{\widetilde{\eta}}$. Recall that ${\widetilde{\eta}}(\lambda_1,\lambda_2)=\eta(\lambda_1+\lambda_2)$. By Lemma \[waga\] and Lemma \[waga2\] for any $s_j>{d_j}$, $j=1,2$ there exists ${\varepsilon}, {\varepsilon}' >0$ such that $$\begin{aligned}
&&
\int_{X_P-B(y_P,r)} | K_{{\widetilde{\eta}}_nF(1-\Phi_{r})(L_1,L_2)}(x_P,y_P)|{d}
\mu_P(x_P)\\  &\le&  
\Big(\int_{X_P}
\vert K_{{\widetilde{\eta}}_n F(1-\Phi_{r})(L_1,L_2)}
(x_P,y_P)\vert^{2}\prod_{j=1,2} (1 +   2^n{\rho}(x_j,y_j) )^{s_j}
d\mu_P(x_P)\Big)^{1/2}\nonumber  \\
&&\times\Big(\int_{{X_P-B(y_P,r)}}\prod_{j=1,2} (1 +   2^n{\rho}(x_j,y_j)
)^{-s_j}
d\mu_P(x_P)\Big)^{1/2}    \nonumber  \\
 &\leq&  C
(1+2^{n}r)^{-{\varepsilon}/2} \Big( \prod_{j=1,2}\mu_j(B(y_j,{2}^{-n}))^{1/2}\Big) \nonumber  \\
&&\times \Big( \prod_{j=1,2}\mu_j(B(y_j,{2}^{-n}))^{-1/2}  \Big)  \|\delta_{2^{nm}}   [{\widetilde{\eta}}_n
F(1-\Phi_{r})]\|_{W^\infty_{\frac{s_1+s_2+{\varepsilon}'}{2}}}\nonumber
\\
 &\leq&  C
 (1+2^{n}r)^{-{\varepsilon}/2}\|\delta_{2^{nm}}[{\widetilde{\eta}}_n
F(1-\Phi_{r})]\|_{W^\infty_{\frac{s_1+s_2+{\varepsilon}'}{2}}}.\nonumber\end{aligned}$$ Now for any Sobolev space $W_s^{p}({\mathbf{R}})$ and any integer $k>s$ $$\begin{aligned}
\|\delta_{2^{nm}}[{\widetilde{\eta}}_n   F(1-\Phi_{r})]\|_{W^{\infty}_{s}}   \leq   C
\|\delta_{2^{nm}}[{\widetilde{\eta}}_n F]\|_{W_s^{p}}
\|\delta_{2^{nm}}[1-\Phi_{r}]\|_{C_k([1/4,1])}\\ \le
\frac{C 2^{nm}r^m}{1+2^{nm}r^m} \|\delta_{2^{nm}} [{\widetilde{\eta}}_n F]\|_{W_{s}^{\infty}}
=\frac{C (r2^n)^{m}}{1+(r2^n)^{m}} \|{\widetilde{\eta}}\delta_{2^{nm}}  F\|_{W_s^{\infty}}.\end{aligned}$$ Hence for $s=\frac{s_1+s_2+{\varepsilon}'}{2}$ $$\begin{aligned}
\label{koniec}
&&\hspace{-1cm}\sup_{y_P\in X_P}  \int_{X_P-B(y_P,r)}
|K_{F(1-\Phi_{r})(L_1,L_2)}(x_P,y_P)| d \mu_P(x_P)   \\
&\leq& C \sum_{n\in {\mathbf{Z}}} \frac{(r2^n)^{m}}{1+(r2^n)^{m}} \, (1+2^{n}r)^{-{\varepsilon}/2}
 \|{\widetilde{\eta}}\delta_{2^{nm}}  F\|_{W_s^{p}}
\leq C\sup_{t>0}  \|{\widetilde{\eta}}\delta_{t}  F\|_{W_s^{p}} \nonumber\end{aligned}$$ as required to prove Theorem \[main\].

Analysis on fractals and quasielliptic operators {#chfra}
================================================

Many interesting examples of spaces and operators satisfying Assumption \[1b\] are described in the theory of Brownian Motion and Laplace like operators on fractals, see for example [@Ki; @Str2]. A compelling instance of such an operator is the Laplace operator on the Sierpiński gasket $SG$ (Neumann or Dirichlet). Assumption \[1b\] and condition (\[d\]) hold with $d =\log 3/ (\log 5
-\log 3)=$ and $m=d+1=\log5/ (\log5-\log3) $, see [@BP; @Str1; @Str2]. We are especially interested in this example because the set of ratios of eigenvalues for the Laplacian on $SG$ has gaps so it can be used to construct quasielliptic operators. Indeed the main results of [@BS Theorem 2.1] says that

For any two eigenvalues $\lambda_i, \lambda_j$ of the Laplacian on SG $$\frac{\lambda_i}{\lambda_j}\notin (\alpha,\beta)$$ where $\alpha$ and $\beta$ are defined by [(\[gap\])]{}.

There are more examples of Laplacian on fractals such that the sets of ratios of their eigenvalues have gaps. For instance this is the case for hierarchical fractals introduced by Humbly $HH(b)$. For definitions and a precise formulation of results, see [@DS Theorems 5.2 and 5.3]. It also can be noticed that the discussed result that holds on $SG$ also holds in the setting of [@Te]. It is easy to note that if $(\alpha,\beta)$ is a gap in the set of ratios of eigenvalues $\alpha<\gamma < \beta$, $a,b\in {\mathbf{R}}$, $c,d>0$ and $\frac{d}{c}=\gamma$ then the operators defined by (\[qell\]) are bounded on $L^2$. It seems likely that this short description of results concerning quasielliptic operators and gaps in the sets of ratios of eigenvalues will be soon outdated. Therefore, we state our result in an abstract way which could be applied to all quasielliptic operators.

\[one\] Suppose that $L$ is a positive self-adjoint operator acting on $L^2(X)$. We say that the interval $(\alpha, \beta)$ is a gap in the set of ratios of eigenvalues of $L$ if $$\frac{\lambda_i}{\lambda_j}\notin (\alpha,\beta)$$ for all $\lambda_i,\lambda_j$ in the $L^2$ spectrum of the operator $L$.

Next we consider the Cartesian product of two copies of metric measure space $X$ and operators $L_{1}=L \otimes 1$ and $ L_2=1 \otimes L$ acting on $L^2(X^2)$.

\[frac\] Suppose that metric measure space $(X,\mu, \rho)$ satisfies condition [(\[d\])]{} with doubling constant $d$ and that operator $L$ satisfy Assumption [\[1b\]]{}. Next assume that $(\alpha, \beta)$ is a gap in the set of ratios of eigenvalues of $L$ and that $$\mbox{\rm supp}\, \omega \subset \{(\lambda_1,\lambda_2)\colon
\, \left|\frac{\lambda_1}{\lambda_2}-\gamma \right| \le \sigma\}$$ for some $\gamma, \sigma$ such that $\alpha < \gamma-\sigma<\gamma+\sigma < \beta$. Assume in addition that for some $s>{d}$ $$\sup_{t> 0} \Vert \widetilde{\eta} \, \delta_{t }\left[(1-\omega) F\right]
\Vert_{W^\infty_s}<\infty,$$ where $ \widetilde{\eta}$ is defined in the same way as in Theorem [\[main\]]{}. Then the operator $F(L_1,L_2)$ is of weak type $(1,1)$ and is bounded on $L^q(X^2)$ for all $1<q<\infty$.

It is not difficult to note that if $(\alpha, \beta)$ is a gap in the set of ratios of eigenvalues of $L$ then $\omega F(L_1,L_2)=0$ so $F(L_1,L_2)=(1-\omega) F(L_1,L_2)$ and Theorem \[frac\] follows from Theorem \[main\].

Below we describe a straightforward consequence of Theorem \[frac\].

\[ko\] Suppose that for some $c,d>0$, $cL_1-dL_2$ is a quasielliptic operator, that is $\alpha< \frac{d}{c}< \beta$, where $(\alpha, \beta)$ is a gap in the set of ratios of eigenvalues of $L$ described in Definition [\[one\]]{}. Then for all constants $a,b \in {\mathbf{C}}$ the operators $$\frac{aL_1+bL_2}{cL_1-dL_2} \quad \mbox{and} \quad
\frac{L_1L_2}{(cL_1-dL_2)^2}$$ are bounded on all $L^p$ spaces for $1<p<\infty$ and of a weak type $(1,1)$.

Corollary \[ko\] follows from Theorem \[frac\] if one chooses function $\omega$ such that $\omega(\lambda_1,\lambda_2)=1$ for all $\lambda_1,\lambda_2$ such that $\left|\frac{\lambda_1}{\lambda_2}-\frac{d}{c}\right| \le {\varepsilon}$ for some small ${\varepsilon}>0$.

If $\alpha$ and $\beta$ are defined by [(\[gap\])]{} then the Neumann and Dirichlet Laplacian acting on Sierpiński gasket $GS$ satisfy all assumption of Theorem [\[frac\]]{} and Corollary [\[ko\]]{}.

[**Acknowledgement:** ]{} The author is indebted to Robert Strichartz for alerting him to the main research problem studied in the paper and hosting him at Cornell University.

[10]{}

G. Alexopoulos. Spectral multipliers on [L]{}ie groups of polynomial growth. , 120(3):973–979, 1994.

G. Alexopoulos. Spectral multipliers for Markov chains. , 56(3):833–852, 2004.

M. T. Barlow and E. A. Perkins. Brownian motion on the [S]{}ierpiński gasket. , 79(4):543–623, 1988.

. , [56]{}(3):1361–1375, [2007.]{}

Th. Coulhon and X. T. Duong. Riesz transforms for $1\leq p\leq 2$. , 351(3):1151–1169, 1999.

M. Christ and C. D. Sogge. The weak type $L\sp 1$ convergence of eigenfunction expansions for pseudodifferential operators. [*Invent. Math.*]{} [94]{} (1988), no. 2, 421-453.

M. Cowling and A. Sikora. A spectral multiplier theorem on ${\rm SU}(2)$. 238 (2001), no. 1, 1–36.

E. B. Davies. Uniformly elliptic operators with measurable coefficients. , 132(1):141–169, 1995.

S. Drenning, R. S. Strichartz. Spectral Decimation on Hambly’s Homogeneous Hierarchical Gaskets. Preprint, 2007.

X. T. Duong and A. McIntosh. Singular integral operators with non-smooth kernels on irregular domains. , 15(2):233–265, 1999.

X. T. Duong, El-Maati Ouhabaz and A. Sikora. Plancherel-type estimates and sharp spectral multipliers. 196(2):443–485, 2002.

X. T. Duong and D.W Robinson. Semigroup kernels, Poisson bounds, and holomorphic functional calculus. 142(1):443–485, 1996.

W. Hebisch. A multiplier theorem for [S]{}chrödinger operators. , 60/61(2):659–664, 1990.

L. H[ö]{}rmander. Estimates for translation invariant operators in ${L}\sp{p}$spaces. , 104:93–140, 1960.

L. H[ö]{}rmander. . Springer-Verlag, Berlin, 1983.

J. Kigami. Cambridge University Press, 2001.

F. Lancien, G. Lancien and C. Le Merdy. A joint functional calculus for sectorial operators with commuting resolvents. , 77(2):387–414, 1998.

N. Lohoué and M. Marias. Invariants géométriques des espaces localement symétriques et théorème de multiplicateurs. Preprint, 2007.

G. Mauceri and S. Meda. Vector-valued multipliers on stratified groups. , 6(3-4):141–154, 1990.

D. Muller, F. Ricci, Fulvio and E.M. Stein. , 2: 99–233, 1995.

S. G. Mikhlin. Multidimensional singular integrals and integral equations. Translated from the Russian by W. J. A. Whyte. Pergamon Press, Oxford-New York-Paris 1965.

El-Maati Ouhabaz. Gaussian estimates and holomorphy of semigroups. , 123(5):1465–1474, 1995.

R. S. Strichartz. Function spaces on fractals. , 198(1): 43–83, 2003.

R. S. Strichartz. Princeton University Press, Princeton, 2006

A. Teplyaev. Spectral analysis on infinite Sierpiński gaskets. , 159(2):537–567, 1998.