# Weak and Strong Type \(A_1\)–\(A_\infty \) Estimates for Sparsely Dominated Operators

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## Abstract

*T*satisfying a sparse domination property

*T*. Moreover, we also establish a dual weak type \((q_0',q_0')\) estimate. In a last part, we give a result on the optimality of the weighted strong type bounds including those previously obtained by Bernicot, Frey, and Petermichl.

## Keywords

Sparse domination Muckenhoupt weights Sharp weighted bounds## Mathematics Subject Classification

42B20 42B25## 1 Introduction

*L*is the Laplace operator with respect to Neumann boundary conditions. Generally, the Riesz transform in such a setting does not satisfy any pointwise regularity estimates and therefore falls outside of the class of Calderón–Zygmund operators. However, it satisfies a sparse domination property which does in fact yield the quantitative weighted bounds from the \(A_2\) conjecture. In \(\mathbf {R}^n\), foregoing the full range of \(1<p<\infty \), one can consider the Riesz transform for elliptic operators \(L=-{{\mathrm{div}}}(A\nabla )\) for

*A*with bounded, complex coefficients. Such operators are only bounded in \(L^p\) for a certain range \(p_0<p<q_0\), and it was established in [6] that they satisfy a sparse domination property

For Calderón–Zygmund operators, weighted weak type (1, 1) estimates were established by Lerner et al. [33] and later improved upon by Hytönen and Pérez [26]. In this article, we establish the corresponding \((p_0,p_0)\) estimate in the more general setting described above. The arguments used in [33] rely on introducing weights in the classical arguments involving Calderón–Zygmund decompositions \(f=g+b\) and the vanishing mean value property of the ‘bad’ part *b* in combination with the Hörmander condition of the kernel of the operator. In general, the operators we are considering here need not be integral operators at all and for the more general operators such as the Riesz transform associated to an elliptic operator, an argument by Blunck and Kunstmann [8] (see also [23]) gave a weak type \((p_0,p_0)\) boundedness using an adapted \(L^{p_0}\) Calderón–Zygmund decomposition, where a certain cancellation of the operator with respect to the semigroup generated by the elliptic operator replaces the regularity estimates of the kernel. Weights were then introduced into this argument by Auscher and Martell [2], but it seems like these techniques do not yield optimal bounds in terms of the constants of the weights. Therefore, we give a new argument to establish the corresponding bounds while still recovering the old bounds found in [26].

Here, in order to combine the previous approaches and to tie the theory together, we deduce quantitative weighted bounds directly from sparse domination assumptions. We introduce weights into a weak boundedness argument for sparse operators where there exists a Calderón–Zygmund decomposition with the property that the ‘bad’ part *b* cancels completely. We then combine this with generalizations of the main lemmata used in [33]. Moreover, we leave the Euclidean setting and extend the results to more general doubling metric measure spaces including certain bounded domains and Riemannian manifolds as was also studied in [8] and [2, 3].

In a last part we show that the strong type weighted estimates are optimal, given a precise control of the asymptotic behaviour of the unweighted \(L^p\) operator norm of *T* at the endpoints \(p=p_0\) and \(p=q_0\). We give an example of such an operator in the case \(p_0=1\), \(q_0=n\).

### 1.1 The Setting

*B*and which satisfies the doubling property, i.e. there is a \(C>0\) such that

*B*, where 2

*B*denotes the ball with the same centre as

*B*and whose radius is twice that of the radius of

*B*. Taking the smallest such

*C*we define \(\nu :=\log _2 C\), which we refer to as the

*doubling dimension*. We write \(|E|:=\mu (E)\) and for each measurable set

*E*of finite non-zero measure and each \(0<p\le \infty \) we will write

*E*. We write \(\langle f,g\rangle :=\int f g\,\mathrm {d}\mu \), and define \(p'=p/(p-1)\in [1,\infty ]\) for \(1\le p\le \infty \).

The collection \(\mathscr {D}\) is used as a replacement for the collection of balls or the collection of all cubes in \(\mathbf {R}^n\), which is justified by the fact that for any ball \(B(x;r)\subseteq \mathbf {R}^n\) there is a cube \(Q\in \mathscr {D}\) so that \(B(x;r)\subseteq Q\) and \({{\mathrm{diam}}}(Q)\le \rho r\) for a constant \(\rho =\rho (n)>0\), and for any cube \(P\subseteq \mathbf {R}^n\) there is a cube \(Q\in \mathscr {D}\) such that \(P\subseteq Q\) and \(\ell (Q)\le 6\ell (P)\), where \(\ell (R)\) denotes the side length of a cube *R*.

We say that a collection \(\mathscr {S}\subseteq \mathscr {D}\) is called \(\eta \)-*sparse* for \(0<\eta \le 1\) if for each \(\alpha \in \big \{0,\frac{1}{3},\frac{2}{3}\big \}^n\) there is a pairwise disjoint collection \((E_Q)_{Q\in \mathscr {S}\cap \mathscr {D}^\alpha }\) of measurable sets so that \(E_Q\subseteq Q\) and \(|Q|\le \eta ^{-1}|E_Q|\).

### Remark 1.1

Since \(\mathbf {R}^n\) is connected and unbounded, the doubling property implies that \(\mu (\mathbf {R}^n)=\infty \) [21]. We are working in \(\mathbf {R}^n\) for notational reasons only; since our applications lie in a more general framework, our arguments are written so that they work with minimal adaptations in general doubling metric measure spaces *X*. Our main results remain true even when \(\mu (X)<\infty \), for example when *X* is a bounded Lipschitz domain in \(\mathbf {R}^n\). We will detail how this can be seen in Sect. 4.

We let \(\mathcal {D}\) be a space of test functions on \(\mathbf {R}^n\) with the property that it is dense in \(L^p(w)\) for all \(1\le p<\infty \) and all weights \(w\in A_\infty \), for example, \(\mathcal {D}=C_c^\infty (\mathbf {R}^n)\).

### Definition 1.2

*T*be a (sub)linear operators, initially defined on \(\mathcal {D}\), with the following property: There are \(1\le p_0<q<q_0\le \infty \) and constants \(c>0\) and \(0<\eta \le 1\) so that for each pair of functions \(f,g\in \mathcal {D}\) there is an \(\eta \)-sparse collection \(\mathscr {S}\subseteq \mathscr {D}\) so that

*sparsely dominated operators*.

If \(T\in S(p_0,q_0)\), then it extends to a bounded operator on \(L^p\) for all \(p_0<p<q_0\), see Proposition 2.2. For examples of operators in this class we refer the reader to Sect. 1.3

When writing that a constant \(C=C(T)>0\) depends on *T*, we mean that it depends on the constants \(c,\eta \) in the domination property (1.2). We remark that the sum on the right-hand side of (1.2) can be split into \(3^n\) sums by considering the different dyadic grids, simplifying the proofs by only having to consider a single dyadic grid at a time. Finally, we remark that if *T* is linear, then \(T\in S(p_0,q_0)\) if and only if \(T^*\in S(q_0',p_0')\), where \(T^*\) denotes the dual operator of *T*.

We will write \(A\lesssim B\) when there is a constant \(C>0\), independent of the important parameters, so that \(A\le C B\). Moreover we write \(A\simeq B\) if \(A\lesssim B\) and \(B\lesssim A\).

### 1.2 Main Results

*T*will be of strong type (

*p*,

*p*) for any \(p_0<p<q_0\) and of weak type \((p_0,p_0)\), see Proposition 2.2. As a matter of fact,

*T*will satisfy weighted boundedness for various classes of weights. It has been shown in [6] that for \(p_0<p<q_0\) and any \(w\in A_{p/p_0}\cap {{\mathrm{RH}}}_{(q_0/p)'}\) we have

*T*one has

*T*one has

To continue on along this line of results, we establish the following:

### Theorem 1.3

Our result (1.7) recovers (1.6) when setting \(p_0=1\), \(q_0=\infty \). One shows that (1.8) follows from (1.7) by applying (2.1) and Proposition 2.1(ii). This result recovers the exponent in (1.5) when \(q_0=\infty \).

*T*and all weights \(w\in A_1\) one has

Both the proofs of (1.9) and (1.10) rely on taking a Calderón–Zygmund decomposition \(f=g+b\). Here, the Hörmander condition of the kernel of *T* is used to deal with the ‘bad’ part *b*, using an argument that can already be found in [37] (namely, they use [18, Lemma 3.3, p. 413]). Since we are making no such assumptions on our operators, which may not even be integral operators, we rely on new methods to deal with this term, using only sparse domination. We establish the following result:

### Theorem 1.4

We note that in particular we recover the bound (1.10). It is of interested to point out that we get this bound even for operators outside of the class of Calderón–Zygmund operators that are in \(S(1,\infty )\), see Example 1.8.

We also establish a dual result of the type first studied in [33], generalizing the result [26, Theorem 1.23]. Here we denote by \(T^*\) the dual operator of *T* for linear *T*.

### Theorem 1.5

Using the ideas of [35], we then establish optimality of the weighted estimates in terms of the asymptotic behaviour of the unweighted \(L^p\) operator norm of *T* at the endpoints \(p=p_0\) and \(p=q_0\). We refer to Definition 5.1 for the definition of the exponents \(\alpha _T(p_0)\) and \(\gamma _T(q_0)\).

### Theorem 1.6

In the example of the Riesz transform on two copies of \(\mathbf {R}^n\) glued smoothly along their unit circles [9], it is known that \(q_0=n\) and \(\gamma _T(q_0)=(n-1)/n\), and thus the weighted estimate is optimal. See Example 5.5.

### 1.3 Examples

There is a wealth of examples of sparsely dominated operators. Other than the class of Calderón–Zygmund operators, our main examples can be found in [6, Sect. 3]. See also the earlier work [2]. We point out several examples of particular interest here.

### Example 1.7

*A*be a complex, bounded, measurable matrix-valued function in \(\mathbf {R}^n\) satisfying the ellipticity condition \({{\mathrm{Re}}}(A(x)\xi \cdot \overline{\xi })\ge \lambda |\xi |^2\) for all \(\xi \in \mathbf {C}^n\) and a.e. \(x\in \mathbf {R}^n\). Then one can define a maximal accretive operator

### Example 1.8

(Riesz transform associated to Neumann Laplacian) Suppose \(\Delta \) is the Laplace operator associated with Neumann boundary conditions in a bounded convex doubling domain in \(\mathbf {R}^n\). As studied in [40], the Riesz transform \(\nabla \Delta ^{-1/2}\) will not in general have a kernel satisfying pointwise regularity estimates and is thus not in the class of Calderón–Zygmund operators. However, this operator does belong to the class \(S(1,\infty )\) and will therefore satisfy the bound (1.6). Note that for this example we need to apply our results to a metric measure space other than \(\mathbf {R}^n\). We refer the reader to Sect. 4 for an overview of the theory in bounded domains.

### Example 1.9

(Fourier multipliers) Let *m* be the function in \(\mathbf {R}^n\) defined by \(m(\xi )=1-|\xi |^2\) for \(|\xi |\le 1\) and \(m(\xi )=0\) elsewhere. For \(\delta \ge 0\), the Bochner–Riesz operator \(\mathcal {B}^\delta \) is defined as the Fourier multiplier \(\mathcal {B}^\delta f:=(m^\delta \hat{f})^{\vee }\). Then, for any \(\delta >0\) there exists a \(1<p_0<2\) so that for any \(0<\varepsilon <2-p_0\) we have \(\mathcal {B}^\delta \in S(p_0+\varepsilon , 2)\). For details we refer the reader to [5].

## 2 Preliminaries

### 2.1 Notation

*w*with a Borel measure by setting

*w*.

We provide some facts about the classes \(A_1\) and \(A_\infty \) that we will use.

### Proposition 2.1

- (i)
\(A_q=\bigcup _{1\le p<q}A_p\) for \(1<q\le \infty \) and \({{\mathrm{RH}}}_s=\bigcup _{s<r\le \infty }{{\mathrm{RH}}}_r\) for \(1\le s<\infty \). In particular we have \(w\in A_\infty \) if and only if \(w\in A_p\) for some \(1\le p<\infty \).

- (ii)For \(1\le p<\infty \), \(1\le s<\infty \) we have \(w\in A_p\cap {{\mathrm{RH}}}_s\) if and only if \(w^s\in A_{s(p-1)+1}\). Moreover, we have$$\begin{aligned}{}[w^s]_{A_{s(p-1)+1}}\le \left( [w]_{A_p}[w]_{{{\mathrm{RH}}}_s}\right) ^s. \end{aligned}$$
- (iii)There are constants \(c,\kappa >0\) depending only on the doubling dimension \(\nu \), so that for every \(w\in A_1\) we have$$\begin{aligned} \mathscr {M}^{\mathscr {B}}_q w\le c[w]_{A_1}w\quad \text {for }1\le q\le 1+\frac{1}{\kappa [w]_{A_\infty }}. \end{aligned}$$

### Proof

For (i) we refer the reader to [20, 41]. Property (ii) can be found in [28].

Property (iii) is a consequence of [27, Theorem 1.1]. Indeed, this result states that there are constants \(c,\kappa >0\) depending only on \(\nu \) such that for any ball *B* we have \(\langle w\rangle _{q(w),B}\le c\langle w\rangle _{1,2B}\), where \(q(w):=1+1/(\kappa [w]_{A_\infty })\). Thus, (iii) follows from Hölder’s inequality and the definition of \(A_1\). \(\square \)

### 2.2 Weighted Boundedness of Sparsely Dominated Operators

We wish to give some heuristic arguments as to why we can expect certain weighted boundedness of sparsely dominated operators. We start with the following observation:

### Proposition 2.2

Let \(1\le p_0<q_0\le \infty \) and \(T\in S(p_0,q_0)\). Then *T* is of strong type (*p*, *p*) for all \(p_0<p<q_0\) and of weak type \((p_0,p_0)\).

The verification of the strong boundedness is by now standard, see also [13]. While the weak type boundedness should be well known, we could not find a precise reference for the cases where \(p_0>1\). For the case \(p_0=1\) we refer the reader to [12, Theorem E], see also [4, Proposition 6]. For completeness we give a proof of the general case here, which we defer to the end of this section.

We will show that if an operator *T* lies in \(S(p_0,q_0;\mu )\), then *T* must also lie in \(S(q_-,q_+;w)\) for appropriate weights *w*, and for certain \(q_-<q_+\) depending on *w*. Then Proposition 2.2 implies that *T* satisfies weighted boundedness.

*w*with the same collections \((E_Q)_{Q\in \mathscr {S}\cap \mathscr {D}^\alpha }\).

*f*,

*g*, and by applying the sparse domination property to the pair

*f*,

*gw*, we find a sparse collection \(\mathscr {S}\subseteq \mathscr {D}\) so that by (2.2) we have

*T*is of weak type \((p_0,p_0)\) with respect to such weights.

*T*in terms of the characteristic constants of the weight in the situations

### Proof of Proposition 2.2

By splitting into \(3^n\) terms, we may assume without loss of generality that our sparse domination occurs in a single dyadic grid \(\mathscr {D}^\alpha \) throughout our arguments.

*T*extends to a bounded operator in \(L^p\).

*f*with \(\Vert f\Vert _{p_0}=1\) and \(E\subseteq \mathbf {R}^n\) of finite positive measure we define

*K*is chosen large enough to ensure that \(|E|\le 2|E'|\). Let \(h\in \mathcal {D}\) with \(|h|\le \chi _{E'}\). Then we can find a sparse collection \(\mathscr {S}\subseteq \mathscr {D}^\alpha \) such that

*g*, \((b_P)_{P\in \mathscr {P}}\) so that \(|f|^{p_0}=g+\sum _{P\in \mathscr {P}}b_P\) and

### Remark 2.3

The cancellation of the ‘bad‘ part *b* in our proofs occurs because we are able to perform our Calderón–Zygmund decomposition in the same dyadic grid as where the sparse domination occurs, see Lemma 4.6. The usual Whitney decomposition argument that is used for Calderón–Zygmund decompositions in general doubling metric measure spaces, as can be found for example in [11, 39], is not precise enough for this particular argument and we need to adapt the results so that they work with our dyadic grids.

## 3 Proofs of the Main Results

Throughout these proofs we fix \(\alpha \in \big \{0,\frac{1}{3},\frac{2}{3}\big \}^n\) and only consider cubes taken from the grid \(\mathscr {D}^\alpha \). We also only consider the dyadic maximal operators \(\mathscr {M}_p\) to be taken with respect to this grid to facilitate some of the arguments and for simpler constants in our estimates. Recall that \(\mathcal {D}\) denotes a space of functions in \(\mathbf {R}^n\) which has the property that it is dense in \(L^p(w)\) for all \(1\le p<\infty \) and all weights \(w\in A_\infty \).

As an analogue to [32, Lemma 3.2] and [26, Lemma 6.1], our main lemma is the following:

### Lemma 3.1

We point out that a similar type of result is established in [15, Theorem B].

### Remark 3.2

*p*if and only if \(p_0>1\). As a matter of fact, we shall see in the proof of Lemma 3.4 that this constant appears in an application of Kolmogorov’s Lemma to the maximal operator. This extra term is what causes the additional terms in the quantitative bounds for \(p_0>1\) in Theorem 1.4 and at this moment we are unsure whether it can be removed or not.

We break up the proof of the main lemma into a sequence of lemmata.

### Lemma 3.3

### Proof

### Lemma 3.4

*f*such that \(\mathscr {M}f<\infty \). For this result we refer the reader to [10, Proposition 2] or [20, Theorem 9.2.7].

### Proof

By combining the two cases, the assertion follows. \(\square \)

### Proof of Lemma 3.1

### Proof of Theorem 1.3

### Proof of Theorem 1.4

*E*, we set \(\Omega :=\{\mathscr {M}^{\mathscr {B}}(|f|^{p_0})> 2c[w]_{A_1} w(E)^{-1}\}\), where \(\mathscr {M}^{\mathscr {B}}\) denotes the uncentred maximal operator with respect to all balls \(B\subseteq \mathbf {R}^n\) and where \(c=c(n,\nu )>0\) is the constant appearing in the inequality \(\Vert \mathscr {M}^{\mathscr {B}}\phi \Vert _{L^{1,\infty }(w)}\le c[w]_{A_1}\Vert \phi \Vert _{L^1(w)}\), which is a consequence of (3.1). We have

*g*,

*b*so that \(|f|^{p_0}=g+b\), where

*h*satisfying \(|h|\le \chi _{E'}\) and \(hw\in \mathcal {D}\), we apply the sparse domination property to the pair

*f*,

*hw*to find a sparse collection \(\mathscr {S}\subseteq \mathscr {D}^\alpha \) so that, by using Lemma 3.1 with the weight \(w\chi _{E'}\), for all \(p_0<p<q_0\) and \(1<q<\infty \) we have

*b*cancel in the exact same way as they do in the proof of Proposition 2.2.

*g*as follows: We remark that for a cube \(P\in \mathscr {D}^\alpha \) we have

### Proof of Theorem 1.5

*B*(

*P*) containing

*P*so that \(B(P)\cap \Omega ^c\ne \emptyset \) and \(|B(P)|\lesssim |P|\), where the implicit constant depends only on

*n*and \(\nu \), see also the proof of Lemma 4.6. Moreover, we obtain functions

*g*,

*b*so that \(|f|^{q_0'}=g+b\), where

*h*satisfying \(|h|\le \chi _{E'}\) and \(hw^{1/q_0}\in \mathcal {D}\), and fix a \(p_0<p<q_0\) to be chosen later. We apply the sparse domination property to the pair \(hw^{1/q_0}\),

*f*to find a sparse collection \(\mathscr {S}\subseteq \mathscr {D}^\alpha \) so that, by applying Lemma 3.1 with the weight \(w^{1/(q_0/p)'}\), we find that for all \(1<q<\infty \) we have

*b*cancel in the same way as before.

## 4 Extensions of the Results to Spaces of Homogeneous Type

*X*is a set equipped with a quasimetric

*d*, i.e. a mapping satisfying the usual properties of a metric except for the triangle inequality, which is replaced by the estimate

*X*satisfying the doubling property, i.e. there is a \(C>0\) such that

*C*we set \(\nu :=\log _2 C\). Furthermore, we write \(|E|:=\mu (E)\) for all Borel sets \(E\subseteq X\). The doubling property implies that for \(x\in X\) and \(R\ge r>0\) we have

*X*is separable [7, Proposition 1.6].

Finally, we make the assumption that Lebesgue’s Differentiation Theorem holds. This holds, for example, when *X* is a domain in \(\mathbf {R}^n\). Indeed, more generally, if \(A=1\) (that is, (*X*, *d*) is a metric space) and \(\mu \) is an inner regular Borel outer measure, then Lebesgue’s Differentiation Theorem holds, see [22, Sect. 14]. This assumption is used for the \(L^\infty \) bound on the good part in our Calderón–Zygmund decompositions.

*X*is unbounded and where

*X*is bounded separately, the latter situation being simpler. To facilitate this, we impose that the underlying quasimetric space (

*X*,

*d*) has exactly one of the following properties:

- (I)There is a constant \(\gamma >0\) so thatfor all \(x\in X\), \(r>0\);$$\begin{aligned} {{\mathrm{diam}}}(B(x;r))\ge \gamma r \end{aligned}$$(4.3)
- (II)
\({{\mathrm{diam}}}X<\infty \).

*X*is unbounded. The extra assumption for the unbounded case is not too restrictive in the sense that the unbounded spaces in our applications usually do satisfy property (I). We point out that when (

*X*,

*d*) is a connected metric space, then it satisfies either (I) or (II):

### Proposition 4.1

Suppose *X* is metric, connected, and unbounded. Then (I) holds with \(\gamma =1\).

### Proof

Let \(r>\varepsilon >0\). The assumptions on *X* imply that \(X\ne \overline{B(x;r-\varepsilon )}\cup B(x;r)^c\) and thus we can pick \(y\in B(x;r)\backslash \overline{B(x;r-\varepsilon )}\) so that \({{\mathrm{diam}}}(B(x;r))\ge d(x,y)\ge r-\varepsilon \), proving the result. \(\square \)

A non-connected example where (I) holds with \(\gamma =1/2\) is the subset \((-\infty ,0)\cup (1,2)\) of the real line. An example where (I) fails is any metric space that has an isolated point.

We will use the following definition of a dyadic system in *X*.

### Definition 4.2

*X*and a collection of points \((z^k_j)_{j\in J_k}\), then we call \((\mathscr {D}_k)_{k\in \mathbf {Z}}\) a

*dyadic system*in

*X*with parameters \(c_0\), \(C_0\), \(\delta \), if it satisfies the following properties:

- (i)for all \(k\in \mathbf {Z}\) we have$$\begin{aligned} X=\bigcup _{j\in J_k}Q_j^k; \end{aligned}$$
- (ii)
for \(l\ge k\), if \(Q\in \mathscr {D}_l\) and \(Q'\in \mathscr {D}_k\), we have that either \(Q\cap Q'=\emptyset \) or \(Q\subseteq Q'\);

- (iii)for each \(k\in \mathbf {Z}\) and \(j\in J_k\) we have$$\begin{aligned} B(z^k_j;c_0\delta ^k)\subseteq Q^k_j\subseteq B(z^k_j;C_0\delta ^k); \end{aligned}$$
- (iv)
for \(l\ge k\), if \(Q^l_{j'}\subseteq Q^k_j\), then \(B(z^l_{j'};C_0\delta ^k)\subseteq B(z^k_j;C_0\delta ^k)\).

The elements of a dyadic system are called cubes. We call \(z^k_j\) the *centre* of \(Q^k_j\). If \(Q\in \mathscr {D}_k\), then we call the unique cube \(Q'\in \mathscr {D}_{k-1}\) so that \(Q\subseteq Q'\), the *parent* of *Q*. Furthermore, we say that *Q* is a *child* of \(Q'\). Note that it is possible that for a cube *Q* there exists more than one \(k\in \mathbf {Z}\) so that \(Q\in \mathscr {D}_k\). Hence, when speaking of a child or the parent of *Q*, this should be with respect to a specific \(k\in \mathbf {Z}\) where \(Q\in \mathscr {D}_k\) to avoid ambiguity.

For a detailed discussion on the construction of dyadic systems and for the following theorem we refer the reader to [25] and references therein.

### Theorem 4.3

*K*, so that there are dyadic system \(\mathscr {D}^1,\ldots ,\mathscr {D}^K\) in

*X*with parameters \(c_0\), \(C_0\), \(\delta \) so that for each \(x\in X\) and \(r>0\) there exists an \(\alpha \in \{1,\ldots , K\}\) and \(Q\in \mathscr {D}^\alpha \) so that

Writing \(\mathscr {D}:=\cup _{\alpha =1}^K\mathscr {D}^\alpha \), one defines the respective notions for weight classes accordingly. Likewise, we say that a collection \(\mathscr {S}\subseteq \mathscr {D}\) is called \(\eta \)-*sparse* for \(0<\eta \le 1\) if for each \(\alpha \in \big \{1,\ldots , K\big \}\) there is a pairwise disjoint collection \((E_Q)_{Q\in \mathscr {S}\cap \mathscr {D}^\alpha }\) of measurable sets so that \(E_Q\subseteq Q\) and \(|Q|\le \eta ^{-1}|E_Q|\).

For our main results we require that the Calderón–Zygmund decompositions we take are adapted to the dyadic grids obtained from this theorem. The standard Calderón–Zygmund decomposition as found in [11] is not precise enough for these purposes, see also Remark 2.3.

*T*that satisfy the property that there is a constant \(c>0\) and an \(0<\eta \le 1\) so that for each pair of functions

*f*,

*g*in an appropriately large class of functions on

*X*there is an \(\eta \)-sparse collection \(\mathscr {S}\subseteq \mathscr {D}\) so that

### Theorem 4.4

Let \(1\le p_0<q_0\le \infty \) and suppose that (*X*, *d*) satisfies either property (I) or property (II). Then for \(T\in S(p_0,q_0)\), the results of Theorems 1.3 and 1.4 remain true, where the dependence on *n* of the constants changes to dependence on the parameters of the dyadic system (and also \(\gamma \) in the case (I)). Similarly, the results of Theorem 1.5 remain true in the case that property (I) is satisfied.

The main difficulty arises when one wants to take Calderón–Zygmund decompositions. We remark that in the cases (I) and (II) one can use the standard maximal cube arguments and localization arguments, respectively, to conclude that our dyadic maximal operators satisfy the usual weak and strong boundedness results. The Lemmata in Sect. 3 all follow in the more general setting in the same way as they have been presented, where we replace the set of test functions \(\mathcal {D}\) by another appropriate class of functions that is dense in \(L^p(w)\) for all \(1\le p<\infty \), \(w\in A_\infty \) such as the linear span of the indicator functions functions over the balls in *X*.

From now on we consider a fixed dyadic system \(\mathscr {D}^*=\cup _{k\in \mathbf {Z}}\mathscr {D}_k\) in *X* with parameters \(c_0\), \(C_0\), \(\delta \).

We first assume that we are in the easier case (II). We define the maximal operator \(\mathscr {M}\) with respect to the cubes \(Q\in \mathscr {D}^*\) by \(\mathscr {M}f:=\sup _{Q\in \mathscr {D}^*}\langle f\rangle _{1,Q}\chi _Q\).

### Lemma 4.5

*A*, so that

### Proof

Fix \(k_0\in \mathbf {Z}\) small enough so that \(c_0\delta ^{k_0}>{{\mathrm{diam}}}X\). Then for any \(x\in X\) we have \(B(x;c_0\delta ^{k_0})=X\). Hence, it follows from property (iii) of dyadic systems that \(\mathscr {D}_{k_0}=\{X\}\).

*x*so that \(\langle f\rangle _{1,P_x}>\lambda \). By minimality of \(k_x\), it follows that \(\langle f\rangle _{1,p(P_x)}\le \lambda \), where \(p(P_x)\in \mathscr {D}_{k_x-1}\) denotes the parent of \(P_x\). By (4.2) and property (iii) of dyadic systems this implies that

It remains to show that the hereby obtained collection \(\mathcal {P}=(P_x)_{x\in X}\) is pairwise disjoint. Indeed, assume that \(P_1,P_2\in \mathcal {P}\) so that \(P_1\cap P_2\ne \emptyset \). We have either \(P_1\subseteq P_2\) or \(P_2\subseteq P_1\) by property (ii) of dyadic systems. Without loss of generality we assume the first. Pick \(x\in X\) so that \(P_1=P_x\). Since \(x\in P_2\) and \(\langle f\rangle _{1,P_2}>\lambda \), minimality of \(k_x\) implies that \(P_2\in \mathscr {D}_l\) for some \(l\ge k_x\). Again by property (ii) of dyadic systems, this implies that \(P_2\subseteq P_1\), proving that \(P_1=P_2\). The assertion follows. \(\square \)

Next, we consider the case (I). We define the maximal operator \(\mathscr {M}^{\mathscr {B}}\) with respect to the balls \(B\subseteq X\) by \(\mathscr {M}^{\mathscr {B}} f:=\sup _{B}\langle f\rangle _{1,B}\chi _B\).

### Lemma 4.6

*A*, and \(\gamma \), so that

### Theorem 4.7

### Proof

We set \(k_x:=\min K_x\in \mathbf {Z}\). Then \(Q^{k_x}_x\in \mathscr {E}\) while \(p(Q^{k_x}_x)=Q^{k_x-1}_x\notin \mathscr {E}\). Hence, \(Q^{k_x}_x\in \mathscr {P}\), proving that \(x\in \cup _{P\in \mathscr {P}}P\), as desired.

### Proof of Lemma 4.6

We apply the Whitney Decomposition Theorem to write \(\Omega =\cup _{P\in \mathscr {P}}P\).

### Proof of Theorem 4.4

In both cases (I) and (II), the proof of Theorem 1.3 holds mutatis mutandis. Moreover, in the case (I), the same is true for Theorem 1.4, where one uses Lemma 4.6, and for Theorem 1.5, where one uses Theorem 4.7.

*X*is bounded, we have \(w(X)<\infty \). Thus, by (3.1), we have

## 5 Optimality of Weighted Strong Type Estimates

In this section we are going to show that the weighted strong type estimates in (1.3) and (1.8) are optimal, given a certain asymptotic behaviour of the unweighted \(L^p\) operator norm of *T*. Such asymptotic behaviour is directly linked to lower bounds on the (generalized) kernel of the operator, see Example 5.5. We improve upon the result in [6], where it was shown that the estimate (1.3) is optimal for sparse forms. Indeed, here we are directly using properties of the operator *T* itself rather than only its sparse bounds.

Our method is an adaptation of the results of Fefferman and Pipher [16] and Luque et al. [35]. We deduce sharpness of weighted bounds from the asymptotic behaviour of the unweighted \(L^p\) norm of *T* as *p* tends to \(p_0\) and \(q_0\), respectively. The proof exploits the known sharp behaviour of the Hardy–Littlewood maximal function via the iteration algorithm of Rubio de Francia.

*X*obtained from Theorem 4.3. Then we define

Let us first define the critical exponents that determine the asymptotic behaviour of the unweighted \(L^p\) operator norm of *T*.

### Definition 5.1

*T*be a bounded operator on \(L^p\) for all \(p_0<p<q_0\). We define

*w*we have \(w\in A_{s/p_0}\cap {{\mathrm{RH}}}_{(q_0/s)'}\) if and only if \(w^{(q_0/s)'}\in A_{\phi (s)}\).

We establish the following connection between the weighted strong type estimates for *T* and the asymptotic behaviour of the unweighted \(L^p\) operator norm at the endpoints \(p=p_0\) and \(p=q_0\).

### Theorem 5.2

*T*be a bounded operator on \(L^p\) for all \(p_0<p<q_0\). Suppose that for some \(p_0<s<q_0\) and for all \(w\in A_{s/p_0}\cap {{\mathrm{RH}}}_{(q_0/s)'}\),

We also establish a version involving the \(A_1\) characteristics. Its proof follows the same lines as the one for Theorem 5.2 and will therefore be omitted.

### Theorem 5.3

*T*be a bounded operator on \(L^p\) for all \(p_0<p<q_0\). Suppose that for some \(p_0<s<q_0\) and for all \(w\in A_1\cap {{\mathrm{RH}}}_{(q_0/s)'}\),

### Proof of Theorem 5.2

- (A)
\(h\le \mathcal {R}h\),

- (B)
\(\Vert \mathcal {R}h\Vert _p \le 2 \Vert h\Vert _p\),

- (C)
\([(\mathcal {R}h)^{p_0}]_{A_1} \le 2^{p_0} \Vert \mathscr {M}\Vert _{p/p_0}\).

- (A)
\( h \le \mathcal {R}h\),

- (B)
\(\Vert \mathcal {R}h\Vert _{(p/s)'} \le 2 \Vert h\Vert _{(p/s)'}\),

- (C)
\([(\mathcal {R}h)^q]_{A_1} \le 2^q \Vert M\Vert _{(p/s)'/q}\).

For the application of these results to sparsely dominated operators, we make the following observation.

### Proposition 5.4

### Proof

From the above, we can deduce optimality of the weighted estimates as stated in Theorem 1.6.

### Proof of Theorem 1.6

Let us give an example of an operator *T* for which the exponent \(\gamma _T(q_0)\) is known.

### Example 5.5

Let *M* be a complete \(C^\infty \) Riemannian manifold *M* of dimension \(n \ge 3\). Assume that *M* is the union of a compact part and a finite number of Euclidean ends, e.g. two copies of \(\mathbf {R}^n\) glued smoothly along their unit circles. Then it was shown in [9] that in the case that the number of ends is at least two, the corresponding Riesz transform *T* is bounded from \(L^p(M)\) to \(L^p(M;T^*M)\) if and only if \(1<p<n\). More precisely, it was shown in [9, Lemma 5.1] that the kernel of *T* decays only to order \(n-1\). A straightforward calculation, analogous to the classical results (see e.g. [39, p.42]), shows that this implies \(\gamma _T(q_0)=\gamma _T(n)=\frac{n-1}{n}\).

## Notes

### Acknowledgements

Dorothee Frey would like to thank Carlos Pérez for pointing out the results of [35]. Bas Nieraeth is grateful to Frédéric Bernicot and José Manuel Conde-Alonso for pointing out weak type results for sparse operators.

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