Vertical maximal functions on manifolds with ends

Abstract

We consider the setting of manifolds with ends which are obtained by compact perturbation (gluing) of ends of the form ${\mathbb {R}}^{n_i}\times {\mathcal {M}}_i$. We investigate the family of vertical resolvent $\{\sqrt{t}\nabla (1+t\Delta )^{-m}\}_{t>0}$, where $m\ge 1$. We show that the family is uniformly continuous on all $L^p$ for $1\le ~p~\le ~\min _{i}n_i$. Interestingly, this is a closed-end condition in the considered setting. We prove that the corresponding maximal function is bounded in the same range except that it is only weak-type (1, 1) for $p=1$. The Fefferman-Stein vector-valued maximal function is again of weak-type (1, 1) but bounded if and only if $1<p<\min _{i}n_i$, and not at $p=\min _{i}n_i$.

1 Introduction

Maximal functions play a significant role in the history and development of Harmonic Analysis, see [21, Chapters II and V]. This research area is closely related to the notion of square functions, Riesz transform, and other singular integrals. Here our starting point is the theory of vertical maximal functions and we study the same setting for which the square functions were investigated in [2]. This is a situation which involves manifolds with ends and includes examples of spaces of non-doubling class. The description of Riesz transform continuity in this setting was provided in [5] and [16], see also [4].

This point motivates and initiates our study here in understanding the behaviour of certain maximal functions on $L^p$ spaces.

There are two main themes for which we believe readers could find our manuscript interesting:

• Maximal functions and Fefferman-Stein vector-valued maximal functions.

• Uniform continuity of gradient of heat kernels or resolvents and its relation with Riesz transform.

Let us recall $L^{p_0}$-boundedness condition of the gradient of the heat semigroup generated by the negative Laplace-Beltrami operator $\Delta$

We do not study vertical maximal functions and related ideas in their full generality but rather in illuminating setting of manifolds with ends which provides several surprising and inspiring examples illustrating the above two points. It became clear from the beginning of our study that we should investigate these points in the context of two other central topics of harmonic analysis:

• Square functions: We are interested in the following vertical square functions

  $$\begin{aligned} \Bigg (\int _0^\infty |\sqrt{t}\nabla \psi (t\Delta ) x|^2\frac{dt}{t}\Bigg )^{1/2}, \end{aligned}$$

  (1)

  where $\psi (t\Delta )$ can either be the semigroup generated by $\Delta$ for $\psi (z)=e^z$ or a resolvent operator, that is, $\psi (z):=(1+z)^{-m}$ for some $m\ge 1$.

• R-boundedness: We refer to Sect. 4 for the description.

Our investigation sheds light on the significance of results described in [17] as well as suggests several interesting future directions in Analysis in Banach spaces.

We consider here the dichotomy of

• Horizontal maximal functions, see (3).

• Vertical maximal functions, see (4).

The horizontal variant involves only a generator of the semigroup and resolvent. The more unexpected behaviour is related to vertical type estimates which involve gradient action, which we discuss in detail below. The vertical estimates are the most interesting part of our discussion.

The classical semigroup theory is based on the equivalence between the resolvent and semigroup approach, see Pazy [19]. We believe that in the setting of manifolds with ends the resolvent approach is more natural and effective. It is likely to be a better approach in studying compact perturbation of the standard Laplace operator and many similar settings. Thus, building harmonic analysis theory based on resolvent rather than on heat kernel is one of the directions which we would like to investigate here. A more comprehensive approach to the resolvent based harmonic analysis requires more extensive work going beyond our discussion here, and which we are going to study somewhere else.

Our study here is devoted to the setting of manifolds with ends. In general, these spaces usually do not satisfy the doubling condition, and this is a very significant aspect of our investigation. In a metric measure space $(X,d,\mu )$, the doubling condition means that there exists a constant $C>0$ such that

for all $x\in X$ and $t>0$, where the notation B(x, t) is used to denote the ball of radius t centered at the point x.

We start with the classical definition of maximal functions introduced by Hardy and Littlewood in 1930. The following Hardy-Littlewood maximal operator given in terms of the averages over balls B(x, t) has been an essential part of harmonic analysis:

$$\begin{aligned} Mf(x):=\sup _{t>0}\frac{1}{\mu (B(x,t))}\int _{B(x,t)}|f(y)|dy. \end{aligned}$$

The classical result shows the boundedness of M on $L^p({\mathbb {R}}^d)$ spaces for $1<p\le \infty$ along with weak-type (1,1) bounds. These maximal functions are studied in various forms till date and Stein’s book [21] is a great source to know some classical results on them.

The following Stein’s maximal function

$$\begin{aligned} M^{exp}f(x):=\sup _{t>0}|e^{t\Delta }f(x)| \end{aligned}$$

defined using the semigroup generated by the negative Laplace-Beltrami operator $\Delta$ is of significant importance in the characterisation of Hardy spaces, see for instance [3]. On Euclidean spaces, this maximal function is dominated by the Hardy-Littlewood maximal operator and hence is bounded on $L^p$ spaces with weak-type (1,1) estimates on $L^1$. In [20, Page 73], Stein showed the boundedness of $M^{exp}$ on $L^p$ spaces for $1<p\le \infty$ implicitly, where he proved a more general result for the semigroup generated by a self-adjoint operator that satisfies contraction property $\Vert e^{t\Delta }f\Vert _{L^p}\le \Vert f\Vert _{L^p}$. Similar result was obtained by Le Merdy and Xu in [18, Corollary 4.2] for the derivative of the heat semigroup. The weak-type estimates were used to be obtained with the help of the standard Calderón-Zygmund theory of 70’s and 80’s, specifically on the Euclidean spaces or more generally on doubling metric measure spaces. This theory no longer works in non-doubling situations.

In certain non-doubling situations, more precisely on manifolds with ends, Duong-Li-Sikora [10] showed the weak-type (1,1) estimates for both Hardy-Littlewood maximal operator and Stein’s maximal operator. These non-doubling complete Riemannian manifolds are formed as connected sums and are of great interest to us in this article.

We start with a definition of a manifold with finitely many ends. We refer readers to [13,14,15] for a more detailed description of the setting. Here we use notations similar to [2].

Definition 1.1

We say that a manifold ${\mathcal {V}}$ is a connected sum of a finite number of complete and connected manifolds ${\mathcal {V}}_1,\ldots ,{\mathcal {V}}_l$ of the same dimension, denoted by

$$\begin{aligned} {\mathcal {V}}={\mathcal {V}}_1\#{\mathcal {V}}_2\#\cdots \#{\mathcal {V}}_l \end{aligned}$$

if there exists some compact subset $K\subset {\mathcal {V}}$, with non-empty interior, for which ${\mathcal {V}}\setminus K$ can be expressed as the disjoint union of open subsets $E_i\subset {\mathcal {V}}$ for $i=1,\ldots ,l$, where $E_i\simeq {\mathcal {V}}_i{\setminus } K_i$ for some compact subset $K_i\subset {\mathcal {V}}_i$.

Fix $N\in {\mathbb {N}}^*={\mathbb {N}}\setminus \{0\}$. We consider an N-dimensional complete Riemannian manifold ${\mathcal {M}}$ (unless otherwise stated) that is obtained by taking the connected sum of $l \ge 2$ copies of manifolds which are Cartesian products of Euclidean spaces ${\mathbb {R}}^{n_i}$ with compact Riemannian manifolds ${\mathcal {M}}_i$ and $n_i+\text {dim }{\mathcal {M}}_i=N$. That is, we are interested in smooth Riemannian manifolds of the form

$$\begin{aligned} {\mathcal {M}}:=({\mathbb {R}}^{n_1}\times {\mathcal {M}}_1)\#\cdots \#({\mathbb {R}}^{n_l}\times {\mathcal {M}}_l). \end{aligned}$$

(2)

As mentioned in the previous definition, it is possible to choose compact subset $K\subset {\mathcal {M}}$ with non-empty interior, and open subsets $E_i\subset {\mathcal {M}}$ such that ${\mathcal {M}}\setminus K$ can be expressed as a disjoint union of $E_i$. This makes ${\mathcal {M}}$ a manifold with ends where $E_i$ are referred to as the ends and K is considered to be the center of ${\mathcal {M}}$. The Riemannian structure on ${\mathcal {M}}$ restricted to each end coincides with the Cartesian product of Euclidean spaces ${\mathbb {R}}^{n_i}$ and some metric on ${\mathcal {M}}_i$. For more geometrical interpretation of these manifolds, see [13,14,15].

Note that, an interesting case occurs where the dimensions $n_i$ of the Euclidean spaces are not all the same. That is, the ends have different ‘asymptotic dimension’. In that situation the manifold is not a doubling space. We can think of this intuitively as a “connected sum of Euclidean spaces of different dimensions”. Such a class of manifolds was studied in detail by Grigor’yan and Saloff-Coste, who obtained upper and lower bounds on the heat kernel on such manifolds, see [14, 15]. Duong-Li-Sikora used this to obtain weak-type (1,1) estimates for the Hardy-Littlewood maximal function and Stein’s maximal function. At this point, no bounds similar to [14, 15] are known for the gradient as well as the derivative of the heat kernel in the setting of manifolds with ends.

Therefore, in this article, we investigate the boundedness of the following resolvent based horizontal and vertical maximal functions, respectively, in the class of manifolds with ends. Here $\Delta$ is the negative Laplace-Beltrami operator on ${\mathcal {M}}$ and $\nabla$ is the gradient corresponding to the Riemannian structure

$$\begin{aligned} M_m^{res,\Delta }f(x)=\sup _{t>0}|t\Delta (1+t\Delta )^{-m}f(x)|,\;\;\; m\in {\mathbb {N}}, \end{aligned}$$

(3)

$$\begin{aligned} M_m^{res,\nabla }f(x)=\sup _{t>0}|\sqrt{t}\nabla (1+t\Delta )^{-m}f(x)|,\;\;\;m\in {\mathbb {N}}. \end{aligned}$$

(4)

For (3) and (4) we are able to use pointwise estimates from [16], see also Propositions 2.1 and 2.2 below. Our main aim is to prove the following theorems in the setting of Riemannian manifolds with ends. Thus, we assume ${\mathcal {M}}$ to be a manifold with ends defined in (2) with $n_i\ge 3$ and $n^* =\min n_i$. We start our discussion here with a version of $(G_p)$-condition which should be connected with the results from [1].

Theorem A

Let $m\in {\mathbb {N}}$. The following estimates on the gradient of the resolvent operator holds:

$$\begin{aligned}{} & {} \Vert \;|{\sqrt{t}} \nabla (1+t\Delta )^{-m}|\;\Vert _{n^*\rightarrow n^*}\le {C} \quad \forall {t>0}, \end{aligned}$$

(5)

$$\begin{aligned}{} & {} \quad \Vert \;|{\sqrt{t}} \nabla (1+t\Delta )^{-m}|\;\Vert _{1\rightarrow 1}\le {C} \quad \forall {t>0}, \end{aligned}$$

(6)

and

$$\begin{aligned} \Vert \;|{\sqrt{t}} \nabla (1+t\Delta )^{-m}|\;\Vert _{p \rightarrow p} \le {C} (\sqrt{t})^{1-n^*/p} \quad \forall {t\ge 1} \end{aligned}$$

(7)

for all $p\in [n^*, \infty ]$. Moreover for some $c>0$

$$\begin{aligned} \Vert \;|{\sqrt{t}} \nabla (1+t\Delta )^{-1}|\;\Vert _{p \rightarrow p} \ge {c} (\sqrt{t})^{1-n^*/p} \quad \forall {t\ge 1} \end{aligned}$$

(8)

for all $p\in [n^*, \infty ]$.

Remark 1.1

Note that (5) and (6) are similar to the estimates valid for the standard Laplace operator on Euclidean space ${\mathbb {R}}^n$. However, from (7) and (8) it follows that for $t \ge 1$ the norm $\Vert \;|{\sqrt{t}} \nabla (1+t\Delta )^{-1}|\;\Vert _{p \rightarrow p}$ is comparable to $(\sqrt{t})^{1-n^*/p}$ for $p\in [n^*, \infty ]$. For $p>n^*$ this asymptotic essentially differs from the standard Laplace operator behaviour.

The next statement is a maximal function version of the condition $(G_p)$. It is valid in the same range of p except of weak-type (1, 1) for $L^1$.

Theorem B

Assume that ${\mathcal {M}}$ is a manifold with ends defined in (2). Then on $L^p({\mathcal {M}})$

• (a) The vertical maximal operator $M_m^{res,\nabla }$ is bounded if and only if $1< p \le n^*$, that is,

  $$\begin{aligned} \Vert M_m^{res,\nabla }f(x)\Vert _{p \rightarrow p } \le C < \infty \end{aligned}$$

  if and only if $1<p \le n^*$ (for some constant $C>0$).

• (b) The horizontal maximal operator $M^{res,\Delta }_m$ is bounded for all $1<p \le \infty$, that is,

  $$\begin{aligned} \Vert M_m^{res,\Delta }f(x)\Vert _{p \rightarrow p } \le c < \infty \end{aligned}$$

  for all $1<p \le \infty$ (for some constant $c>0$).

Both operators $M_m^{res,\nabla }$ and $M_m^{res,\Delta }$ are also of weak-type (1, 1).

Remark 1.2

The negative result in the case of the vertical maximal operator comes as a consequence of the resolvent based $(G_p)$ condition proved in Theorem A. Interestingly, $(G_p)$ condition is not open ended unlike the one with the gradient of the heat kernel given in [1]. Note that in the setting considered in [8] condition $(G_p)$ is proven to be open ended.

Remark 1.3

It is known that $L^p$ continuity for square function (1) implies $(G_p)$ type version of conditions (5) and (6) for $L^p$ spaces, see [6, Proposition 5.3] and [7]. Such square function estimates for all $1<p<n^*$ were obtained in [2]. It was also verified that the square function estimates are not valid for $n^* \le p$. It means that this approach cannot be used to verify (5) for $p=n^*$. The results described in [2] cannot lead to (7) and (8). In [6], the following problem is also stated: “One might ask whether boundedness of Square function in (1) is in turn equivalent to $(G_p)$. To the best of our knowledge, this question is also open in general.” One can see that in the setting which we consider here the equivalence is not valid for $p=n^*$.

The next theorem gives us the vector-valued version of Theorem B. Similar estimate for the Hardy-Littlewood maximal function was obtained by Fefferman and Stein in [11] on Euclidean spaces. The equation (9) below is a variant of the Fefferman-Stein maximal inequality for the vertical maximal operator. There are many such results for the Hardy-Littlewood maximal operator on different spaces. See, for instance, [12, Theorem 1.2] on spaces of homogeneous type and [9, Theorem 3.1] again on spaces of homogeneous type but for the lattice valued maximal function. The inequality in (9) for the vertical maximal operator comes naturally on a doubling manifold as a result of the Hardy-Littlewood maximal inequality and is mentioned in Proposition 2.3. In Theorem C below we show this result on manifolds with ends.

Theorem C

Consider the vertical maximal operator $M^{res,\nabla }_m$ on $L^p({\mathcal {M}})$. There exists a constant $A_{p,{\mathcal {M}}}>0$ (depending on p and ${\mathcal {M}}$) such that

$$\begin{aligned} \bigg |\bigg |\bigg (\sum _{i=1}^\infty (M_m^{res,\nabla }f_i)^2\bigg )^{1/2}\bigg |\bigg |_{L^p} \le A_{p,{\mathcal {M}}}\bigg |\bigg |\bigg (\sum _{i=1}^\infty |f_i|^2\bigg )^{1/2}\bigg |\bigg |_{L^p} \end{aligned}$$

(9)

for all $1<p<n^*$ and of weak-type (1, 1). The estimates are false for $n^* \le p$.

Another crucial idea for Harmonic analysis which is closely related to our discussion of Maximal functions is continuity of the Riesz transform acting on $L^p$ spaces:

and some constant $C>0$. This is a basic issue which was already initiated in the setting of complete non-compact manifolds in [22] by Strichartz. It was proved in [16] that the Riesz transform $T=\nabla \Delta ^{-1/2}$ acts as a bounded operator from $L^p({\mathcal {M}}) \rightarrow L^p({\mathcal {M}}; T{\mathcal {M}})$ if and only if $1<p<n^*$ and that T is also of weak-type (1, 1) in this class of manifolds. The results described in [16] which includes non-doubling settings were continuation of studies developed in [4] and [5].

An interesting application of the Riesz transform was given by Cometx and Ouhabaz in [6]. They showed that the boundedness of the Riesz transform implies R-boundedness of the set $\{\sqrt{t}\nabla e^{t\Delta }:t>0\}$ which is equivalent to the boundedness of the vertical square function in (1) for the semigroup. We know that, in our setting of manifolds with ends, the range of p for the boundedness of $(R_p)$ and R-boundedness of the set $\{\sqrt{t}\nabla e^{t\Delta }:t>0\}$ coincides, see [2, 16]. This implies that these two conditions are equivalent in our setting. However, we do not know any direct proof for the inverse implication, that is, from R-boundedness to $(R_p)$ condition.

2 Preliminaries

Throughout this article, we fix ${\mathcal {M}}=({\mathbb {R}}^{n_1}\times {\mathcal {M}}_1)\#\cdots \#({\mathbb {R}}^{n_l}\times {\mathcal {M}}_l)$ as a complete Riemannian manifold with $l\ge 2$ ends, unless otherwise stated. We note again that $\Delta$ is the negative Laplace-Beltrami operator on ${\mathcal {M}}$ and $\nabla$ is the gradient corresponding to the Riemannian structure. The notation K will be used to denote the compact subset (also the center) of ${\mathcal {M}}$ and the sets $K_i$ and $E_i$ will be as given in Definition 1.1. That is, ${\mathcal {M}}\setminus K$ can be expressed as the disjoint union of the ends $E_i$, where $E_i\simeq {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i{\setminus } K_i$.

In what follows, we always assume that each $n_i$ is at least 3. Recall that we set $n^*=\min _{i} n_i$.

Notation. For $x\in {\mathbb {R}}^n$, we define $\left<x\right>:=(1+|x|^2)^{1/2}$. We use the notation d(x, y) to denote the distance between two points x and y in some ambient Riemannian manifold. We also employ the notation $``\lesssim ''$ and $``\gtrsim ''$ to denote inequalities “up to a constant”. That is, for any two quantities $a, b\in {\mathbb {R}}$, by $a\lesssim b$ we mean that there exists a constant $C>0$ such that $a\le C\cdot b$. The same holds for $``\gtrsim ''$. We denote equivalence “up to a constant” by $a\simeq b$ which means that there exist constants c and $c'$ such that $\frac{1}{c}a\le b\le c'a$. Lastly, for a function g(x, y) of two variables, we use the notation $\nabla _xg(x,y)$ to denote the gradient with respect to the first variable.

We write the Stein’s maximal function in the resolvent form as

$$\begin{aligned} M_m^{res}f(x)=\sup _{t>0}|(1+t\Delta )^{-m}f(x)|, \end{aligned}$$

and keep the notation $M^{res,\Delta }_m$ and $M^{res,\nabla }_m$, defined in (3) and (4), for the resolvent based horizontal and vertical maximal operators, respectively. With the help of the standard integral representation we get the following relation between the resolvent and the semigroup generated by $\Delta$.

$$\begin{aligned} (1+t\Delta )^{-m}= \frac{1}{\Gamma (m)}\int _0^\infty e^{-s} s^{m-1}e^{st\Delta }ds, \end{aligned}$$

where $\Gamma (m)$ is the Gamma function of m. This allows us to control the function $M^{res}_m f(x)$ by the Stein’s maximal function $M^{exp} f(x)$. Indeed,

$$\begin{aligned} \sup _{t>0}|(1+t\Delta )^{-m}f(x)| \le&\ \frac{1}{\Gamma (m)}\int _0^\infty e^{-s} s^{m-1}\sup _{t>0}|e^{st\Delta }f(x)|ds\nonumber \\ =&\ \sup _{t>0}|e^{t\Delta }f(x)| \frac{1}{\Gamma (m)}\int _0^\infty e^{-s} s^{m-1}ds \lesssim \sup _{t>0}|e^{t\Delta }f(x)|. \end{aligned}$$

(10)

Thus, $M^{res}_m$ is bounded on $L^p$ whenever $M^{exp}$ is.

2.1 Higher-order Resolvents of the Laplacian on ${\mathcal {M}}$

In this section we recall, from [2], several properties of the resolvent $(\Delta +k^2)^{-m}$ for $0<k\le 1$. For the origin of these resolvent parametrix construction ideas see also [5] and [16].

We first recall the following key lemma from [16] which was crucial for the parametrix construction of the resolvent $(\Delta +k^2)^{-1}$ on ${\mathcal {M}}$. For each end ${\mathbb {R}}^{n_i}\times {\mathcal {M}}_i$ we choose a point $x_i^\circ$ such that $x_i^\circ \in K_i$, where $K_i$ are compact subsets of ${\mathbb {R}}^{n_i}\times {\mathcal {M}}_i$.

Lemma 2.1

(Key Lemma) Assume that each $n_i$ is at least 3. Let $v\in C_c^\infty ({\mathcal {M}},{\mathbb {R}})$. Then there exists a function $u:{\mathcal {M}}\times {\mathbb {R}}_+\rightarrow {\mathbb {R}}$ such that $(\Delta +k^2)^{-1}u=v$ and such that on ith end we have:

$$\begin{aligned} |u(x,k)|\le C ||v||_\infty \langle d(x_i^\circ ,x)\rangle ^{-(n_i-2)}\exp (-kd(x_i^\circ ,x))\;\;\forall x\in {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i,\\ |\nabla u(x,k)|\le C||v||_\infty \langle d(x_i^\circ ,x)\rangle ^{-(n_i-1)}\exp (-kd(x_i^\circ ,x))\;\;\forall x\in {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i \end{aligned}$$

for some $C>0$.

With the help of the above Lemma, it was shown in [16] that the resolvent operator $(\Delta +k^2)^{-1}$ can be decomposed into four components. That is, for $0<k\le 1$, the resolvent can be written as

$$\begin{aligned} (\Delta +k^2)^{-1}=\sum _{j=1}^4 G_j(k). \end{aligned}$$

We recall below the definition of each of these components. Let $\phi _i\in C^\infty ({\mathcal {M}})$ be a function supported entirely in ${\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i$ that is identically equal to 1 everywhere outside of a compact set. Define $v_i:=-\Delta \phi _i$ and note that $v_i$ is compactly supported. Let $u_i$ be the function for $v_i$ whose existence is asserted by Lemma 2.1. The $G_1(k)$ term is entirely supported on the diagonal ends and has the following kernel

$$\begin{aligned} G_{1}(k)(x,y):= \sum _{i = 1}^{l} {(\Delta _{{\mathbb {R}}^{n_{i}}\times {\mathcal {M}}_{i}} + k^{2})}^{-1}(x,y) \phi _{i}(x) \phi _{i}(y). \end{aligned}$$

Let $G_{int}(k)$ be an interior parametrix for the resolvent that is supported close to the compact subset

$$\begin{aligned} K_{\Delta }:=\{(x,x): x\in K\}\subset {\mathcal {M}}^2 \end{aligned}$$

and agrees with the resolvent of $\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}$ in a small neighbourhood of $K_\Delta$ intersected with the support of $\nabla \phi _i(x)\phi _i(y)$. This gives us the $G_2(k)$ term whose kernel has the following representation

$$\begin{aligned} G_{2}(k)(x,y):= \bigg (1 - \sum _{i = 1}^{l} \phi _{i}(x) \phi _{i}(y)\bigg )G_{int}(k)(x,y). \end{aligned}$$

The $G_{3}(k)$ term has the nice property that its kernel is multiplicatively separable into functions of x and y. That is,

$$\begin{aligned} G_{3}(k)(x,y):= \sum _{i = 1}^{l} (\Delta _{{\mathbb {R}}^{n_{i}}\times {\mathcal {M}}_{i}} + k^{2})^{-1}(x_{i}^{\circ },y) u_{i}(x,k) \phi _{i}(y), \end{aligned}$$

where $(\Delta + k^{2})u_i=-\Delta \phi _i =\nu _i$. The final term $G_4(k)$ is a correction term, and for this we first define the error term E(k) as

$$\begin{aligned} (\Delta + k^{2})(G_{1}(k) + G_{2}(k) + G_{3}(k)) = I + E(k). \end{aligned}$$

The operator $G_4(k)$ is given by

$$\begin{aligned} G_{4}(k)(x,y):= - (\Delta + k^{2})^{-1} v_{y}(x), \end{aligned}$$

where $v_{y}(x):= E(k)(x,y)$ and is compactly supported. The error term was computed in [16] and has the following representation

$$\begin{aligned} E(k)=\sum _{i=1}^l(E_1^i(k)+E_2^i(k))+E_3(k). \end{aligned}$$

Here

$$\begin{aligned}{} & {} E_1^i(k)(x,y):=-2\nabla \phi _i(x)\phi _i(y)\\{} & {} \quad \left[\nabla _x(\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-1}(x,y)-\nabla _xG_{int}(k)(x,y)\right], \\{} & {} \quad E_2^i(k)(x,y)\\{} & {} \quad :=\phi _i(y)v_i(x)\left(-(\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-1}(x,y) +G_{int}(k)(x,y)+(\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-1}(x_i^{\circ },y)\right) \end{aligned}$$

for $i=1,\ldots ,l$, and

$$\begin{aligned} E_3(k)(x,y):=\left((\Delta +k^2)G_{int}(k)(x,y)-\delta _y(x)\right)\left(1-\sum _{i=1}^l\phi _i(x)\phi _i(y)\right), \end{aligned}$$

where $\delta _y$ is the Dirac-delta function centered at y.

Since we are interested in the higher-order resolvents on ${\mathcal {M}}$, by the above decomposition we can write $(\Delta +k^2)^{-m}$ as

$$\begin{aligned} (\Delta +k^2)^{-m}=\frac{(-1)^{m-1}}{(m-1)!}\partial _{k^2}^{(m-1)}(\Delta +k^2)^{-1}=\sum _{j=1}^4 H_j^{(m)}(k), \end{aligned}$$

where $H^{(m)}_j(k):=\frac{(-1)^{m-1}}{(m-1)!}\partial _{k^2}^{(m-1)}G_j(k)$. For simplicity, we use the shorthand notation $H_j(k)$.

Now, following [16], for $a=1,2$ we define weight functions $\omega _a: {\mathcal {M}} \times [0, 1] \rightarrow (0,\infty )$ by

$$\begin{aligned} \omega _a(x,k) = {\left\{ \begin{array}{ll} \hspace{2cm} 1, &{}\quad x \in K, \\ \langle d(x_i^\circ ,x)\rangle ^{-(n_i -a)} \exp (-{ckd(x_i^\circ ,x)}), &{}\quad x\in {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i,\;1\le i\le l. \end{array}\right. }\nonumber \\ \end{aligned}$$

(11)

We recall below the propositions from [2] that estimates the terms $H_3(k)$ and $H_4(k)$ using these weight functions.

Proposition 2.1

[2, Proposition 3.4] The kernel of the operator $H_3(k)$ satisfies

$$\begin{aligned} |H_3(k)(x,y)| \lesssim k^{-2(m-1)}\omega _2(x,k)\omega _2(y,k) \end{aligned}$$

and

$$\begin{aligned} |\nabla _x H_3(k)(x,y)| \lesssim k^{-2(m-1)}\omega _1(x,k)\omega _2(y,k) \end{aligned}$$

for all $x,y\in {\mathcal {M}}$, and $0<k\le 1$.

Proposition 2.2

[2, Proposition 3.5] The kernel of the operator $H_4(k)$ satisfies

$$\begin{aligned} |H_4(k)(x,y)| \lesssim k^{-2(m-1)}\omega _2(x,k)\omega _1(y,k) \end{aligned}$$

and

$$\begin{aligned} |\nabla _x H_4(k)(x,y)| \lesssim k^{-2(m-1)}\omega _1(x,k)\omega _1(y,k) \end{aligned}$$

for all $x,y\in {\mathcal {M}}$, and $0<k\le 1$.

Let ${\mathcal {N}}$ be a compact Riemannian manifold. Then ${\mathbb {R}}^n\times {\mathcal {N}}$ is a manifold that satisfies the doubling property. On this manifold, we show in the following lemma that the vertical maximal operator is bounded on $L^p$ for $1<p\le \infty$ and satisfies weak type (1,1) estimates.

Lemma 2.2

The vertical maximal operator $M^{exp,\nabla }f(x)=\sup _{t>0}|\sqrt{t}\nabla e^{-t\Delta _{{\mathbb {R}}^n\times {\mathcal {N}}}}f(x)|$ is bounded on $L^p({\mathbb {R}}^n\times {\mathcal {N}})$ for $1<p\le \infty$ and is weak-type (1, 1). The same statement holds for $M^{res,\nabla }_m=\sup _{t>0}|\sqrt{t}\nabla ({1+t\Delta _{{\mathbb {R}}^n\times {\mathcal {N}}}})^{-m}f(x)|$.

Proof

The heat kernel on ${\mathbb {R}}^n\times {\mathcal {N}}$ has the form

$$\begin{aligned} e^{-t\Delta _{{\mathbb {R}}^n\times {\mathcal {N}}}}(x,y) \lesssim \left( \frac{1}{ t^{n/2}}+\frac{1}{ t^{N/2}}\right) \exp \left( -\frac{d(x,y)^2}{4t}\right) , \end{aligned}$$

where $N=\text {dim } {\mathcal {N}}+n$, and satisfies the following spatial derivative estimate

$$\begin{aligned} |\nabla e^{-t\Delta _{{\mathbb {R}}^n\times {\mathcal {N}}}}(x,y)|\lesssim t^{-1/2} \left( \frac{1}{ t^{n/2}}+\frac{1}{ t^{N/2}}\right) \exp \left( -\frac{d(x,y)^2}{4t}\right) . \end{aligned}$$

On controlling the right hand side by an infinite sum we obtain

$$\begin{aligned} |\sqrt{t}\nabla e^{-t\Delta _{{\mathbb {R}}^n\times {\mathcal {N}}}}(x,y)|\lesssim \sum _{k=1}^\infty \frac{k^{-N/2}e^{-k}}{B(x,\sqrt{kt})}\chi _{B(x,\sqrt{kt})}. \end{aligned}$$

Thus,

$$\begin{aligned} M^{exp,\nabla }f(x)&=\sup _{t>0}|\sqrt{t}\nabla e^{-t\Delta _{{\mathbb {R}}^n\times {\mathcal {N}}}}f(x)| =\sup _{t>0}\left|\sqrt{t}\nabla \int _{{\mathbb {R}}^n\times {\mathcal {N}}}e^{-t\Delta _{{\mathbb {R}}^n\times {\mathcal {N}}}}(x,y)f(y)dy\right|\\ {}&\lesssim \sup _{t>0}\left|\int _{{\mathbb {R}}^n\times {\mathcal {M}}}\sum _{k=1}^\infty \frac{k^{-N/2}e^{-k}}{B(x,\sqrt{kt})}\chi _{B(x,\sqrt{kt})}\right|\\&\lesssim \sup _{t>0}\left|\sum _{k=1}^\infty k^{-N/2}e^{-k}\frac{1}{B(x,\sqrt{kt})}\int _{B(x,\sqrt{kt})}f(y)dy\right|\\&\lesssim Mf(x),\;\;\;\; \text {since} \sum _{k=1}^\infty k^{-N/2}e^{-k}<\infty . \end{aligned}$$

Here Mf is the Hardy-Littlewood maximal function which is bounded on $L^p({\mathbb {R}}^n\times {\mathcal {N}})$ for $1<p\le \infty$ and is weak-type (1,1). This implies that $M^{exp,\nabla }f$, and hence $M^{res,\nabla }_mf$, are weak-type (1,1) and bounded on $L^p({\mathbb {R}}^n\times {\mathcal {N}})$ for all $1<p\le \infty$. $\square$

Proposition 2.3

Consider again the vertical maximal operator $M^{exp,\nabla }$ on $L^p({\mathbb {R}}^n\times {\mathcal {N}})$. Then, for any sequence $(f_j)_{j=1}^\infty \in L^p({\mathbb {R}}^n\times {\mathcal {N}})$

$$\begin{aligned} \left|\left|\Bigg (\sum _{j=1}^\infty |M^{exp,\nabla }f_j|^2\Bigg )^{1/2}\right|\right|_{1,\infty } \le C\left|\left|\Bigg (\sum _{j=1}^\infty |f_j|^2\Bigg )^{1/2}\right|\right|_1 \end{aligned}$$

for some constant $C>0$ and

$$\begin{aligned} \left|\left|\Bigg (\sum _{j=1}^\infty |M^{exp,\nabla }f_j|^2\Bigg )^{1/2}\right|\right|_p \le A_{n,p,{\mathcal {N}}}\left|\left|\Bigg (\sum _{j=1}^\infty |f_j|^2\Bigg )^{1/2}\right|\right|_p \end{aligned}$$

for all $1<p<\infty$ and some constant $A_{n,p,{\mathcal {N}}}$ (depending on p and the manifold ${\mathbb {R}}^n\times {\mathcal {N}}$).

Proof

Using the similar idea as in the proof of Lemma 2.2 we obtain that $M^{exp,\nabla }$ is controlled by the Hardy-Littlewood maximal operator M. This implies that for a sequence of functions $(f_j)_{j=1}^\infty$ in $L^p({\mathbb {R}}^n\times {\mathcal {N}})$ we have

$$\begin{aligned} \bigg |\bigg |\bigg (\sum _{j=1}^\infty |M^{exp,\nabla }f_j|^2\bigg )^{1/2}\bigg |\bigg |_p&\lesssim \bigg |\bigg |\bigg (\sum _{j=1}^\infty |Mf_j|^2\bigg )^{1/2}\bigg |\bigg |_p \le A_{n,p,{\mathcal {N}}}\bigg |\bigg |\bigg (\sum _{j=1}^\infty |f_j|^2\bigg )^{1/2}\bigg |\bigg |_p, \end{aligned}$$

where the last inequality follows from [12, Theorem 1.2], see also [21, Theorem 1, p. 51], and holds for all $1<p<\infty$ and is also weak-type (1,1). $\square$

The following lemma for bounded linear operators on a Banach space X is crucial to prove the vector-valued version of the vertical maximal function. The statement is quoted from [21, Section 2.5, Chapter X, p. 450] and we include its proof here for completeness.

Lemma 2.3

Suppose S is a bounded linear operator on $L^p(X,d\mu )$, where X is a Banach space. Let $f=(f_j)_{j=1}^\infty$ be a sequence in $L^p(X,d\mu )$. Then we have the following relation

$$\begin{aligned} \left|\left|\Big (\sum _{j=1}^\infty |Sf_j|^2\Big )^{1/2}\right|\right|_p\le ||S||_{p\rightarrow p}\left|\left|\Big (\sum _{j=1}^\infty |f_j|^2\Big )^{1/2}\right|\right|_p. \end{aligned}$$

(12)

Proof

To prove the above relation it is enough to show that for any $J\in {\mathbb {N}}$ we have

$$\begin{aligned} \left|\left|\Big (\sum _{j=1}^{J} |Sf_j|^2\Big )^{1/2}\right|\right|_p\le ||S||_{p\rightarrow p}\left|\left|\Big (\sum _{j=1}^{J} |f_j|^2\Big )^{1/2}\right|\right|_p. \end{aligned}$$

(13)

For a unit vector $\omega =(\omega _{J})\in {\mathbb {C}}^{J}$ define $f_\omega :=\left<f,\omega \right>$ such that $Sf_\omega =\left<Sf,\omega \right>$, where $Sf=(Sf_j)_{j=1}^{J}$. Let $\sigma _{J}$ be the boundary of the sphere ${\mathbb {S}}^{J}$ in ${\mathbb {C}}^{J}$. Consider

$$\begin{aligned}&\int _{\sigma _{J}}\left(\int _X |Sf_\omega (x)|^pdx\right)d\omega \\&\quad =\int _{\sigma _{J}}\left(\int _X |\left<Sf(x),\omega \right>|^pdx\right)d\omega \\&\quad =\int _{\sigma _{J}}\left(\int _X |Sf(x)|^p\left|\left<\frac{Sf(x)}{|Sf(x)|},\omega \right>\right|^pdx\right)d\omega \\&\quad =\int _X |Sf(x)|^p\left(\int _{\sigma _{J}}\left|\left<\frac{Sf(x)}{|Sf(x)|},\omega \right>\right|^pd\omega \right) dx=A_{J,p}\int _X |Sf(x)|^p dx, \end{aligned}$$

where $A_{J,p}$ is a positive constant that depends on J and p. We also have

$$\begin{aligned} \int _{\sigma _{J}}\left(\int _X |Sf_\omega (x)|^pdx\right)d\omega&\le ||S||_{p\rightarrow p}^p\int _{\sigma _{J}}\left(\int _X |f_\omega (x)|^pdx\right)d\omega \\&=||S||^p_{p\rightarrow p}\int _{\sigma _{J}}\left(\int _X |f(x)|^p\left|\left<\frac{f(x)}{|f(x)|},\omega \right>\right|^pdx\right)d\omega \\&=||S||^p_{p\rightarrow p}\int _X |f(x)|^p\left(\int _{\sigma _{J}}\left|\left<\frac{f(x)}{|f(x)|},\omega \right>\right|^pd\omega \right) dx\\&=A_{J,p}||S||^p_{p\rightarrow p}\int _X |f(x)|^p dx. \end{aligned}$$

Since the above two relations hold independent of J, we obtain (12). $\square$

3 Proofs of main theorems

3.1 Proof of theorem A

First note that

$$\begin{aligned} \Vert \;|{\sqrt{t}} \nabla (1+t\Delta )^{-m}|\;\Vert _{p \rightarrow p}\sim {C} \quad \forall {0<t\le 1}. \end{aligned}$$

(14)

The proof of the above estimate is a consequence of ideas discussed in Sections 5 and 7 of [16]. The argument is a modification of the proof of [16, Proposition 5.1]. The similar argument is also described in [2, Proposition 5.1], see also [16, Proposition 2.4].

Now, it follows from (14) that it is enough to consider $t>1$. We substitute $t=1/k^2$ and in what follows we consider only $k<1$. Then, using the parametrix for the resolvent $(k^2+\Delta )^{-m}$, we obtain

$$\begin{aligned}{} & {} \Vert \;|{\sqrt{t}} \nabla (1+t\Delta )^{-m}|\;\Vert _{p \rightarrow p}=\Vert \;|k^{2m-1}\nabla (k^2+\Delta )^{-m}|\;\Vert _{p\rightarrow p}\\{} & {} \quad = \Big |\Big |\;\Big |k^{2m-1} \nabla \sum _{j=1}^4H_j(k)\Big |\;\Big |\Big |_{p \rightarrow p}. \end{aligned}$$

To obtain the $L^p$ bounds, we shall analyze all factors $H_j(k)$ for $j=1,\ldots , 4$.

$\bullet$ The $H_1(k)$ term.

From the definition of $H_1(k)$, we need to analyze the boundedness of

$$\begin{aligned} k^{2m-1} \nabla \Big ( (\Delta _{{\mathbb {R}}^{n_i} \times {\mathcal {M}}_i} + k^2)^{-m}(x, y) \phi _i(x) \phi _i(y) \Big ). \end{aligned}$$

(15)

The kernel of $H_1(k)$ can be viewed as an operator acting on ${\mathbb {R}}^{n_i} \times {\mathcal {M}}_i$. We split this kernel into two pieces, according to whether the derivative $\nabla$ hits the function $\phi (x)$ or the resolvent factor.

Case (1): $H_{1,a}$— when the derivative hits the resolvent. That is, we need to compute the $L^p$ norm of

$$\begin{aligned} \phi _i k^{2m-1} \nabla (\Delta _{{\mathbb {R}}^{n_i} \times {\mathcal {M}}_i} + k^2)^{-m} \phi _i. \end{aligned}$$

(16)

Let $R_{1,k}(x,y)= \phi _i(x)k^{2\,m-1}\nabla (\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-m}(x,y)\phi _i(y)$ be the kernel of the above operator. Define the norm

$$\begin{aligned} ||T(x,y)||_{L^1_xL^\infty _y}:=\sup _{y\in {\mathcal {M}}}\int _{\mathcal {M}}|T(x,y)|dx \end{aligned}$$

for some operator T. If we show that

$$\begin{aligned} ||R_{1,k}(x,y)||_{L^1_xL^\infty _y}\le C \end{aligned}$$

and

$$\begin{aligned} ||R_{1,k}(x,y)||_{L^\infty _xL^1_y}\le C \end{aligned}$$

for some constant $C>0$, then the operator in (16) will be bounded on $L^p$ for $p=1$ and $p=\infty$, and hence for all $1\le p\le \infty$, by interpolation. Indeed, consider

$$\begin{aligned}&||R_{1,k}(x,y)||_{L^1_xL^\infty _y}\\&\quad =\sup _{y\in {\mathcal {M}}}\int _{{\mathcal {M}}}|R_{1,k}(x,y)|dx\\&\quad =\sup _{y\in {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}\int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}| \phi _i(x)k^{2m-1}\nabla (\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-m}(x,y)\phi _i(y)|dx\\&\quad \le \sup _{y\in {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i}\int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i}\,| k(d(x,y)^{-(N-1)}+d(x,y)^{-(n_i-1)})e^{-ckd(x,y)}|dx\\&\quad \text { (follows by 2,Proposition 2.2)}\\&\quad \lesssim \int _0^\infty kr^{-(n_i-1)}e^{-ckr} r^{n_i-1}dr\le C<\infty . \end{aligned}$$

The last relation follows by a change of variable, and the second last relation follows from the fact that $\phi _i$ takes the value 1 outside a compact subset $K_i$ of ${\mathbb {R}}^{n_i}\times {\mathcal {M}}_i$ and is zero elsewhere. Since the kernel is symmetric in x and y variables, we have $||R_{1,k}(x,y)||_{L^\infty _xL^1_y}\le C$.

Case (2): $H_{1,b}$— when the derivative hits the function $\phi (x)$. Here we shall show that

$$\begin{aligned} \Vert \nabla \phi _i k^{2m-1} (\Delta _{{\mathbb {R}}^{n_i} \times {\mathcal {M}}_i} + k^2)^{-m}\Vert _{p\rightarrow p } \le C < \infty \end{aligned}$$

for all $1\le p \le n_i$.

Set $D=\{(x,y)\in {\mathcal {M}}^2: d(x,y)\le 1\}$ and let $\chi _{D}$ be the characteristic function of the set D. By the standard argument of [2, Proposition 2.2], we get that

$$\begin{aligned} ||\chi _Dk^{2m-1}(\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-m}||_{p\rightarrow p}\lesssim 1. \end{aligned}$$

Indeed,

$$\begin{aligned} |\chi _Dk^{2m-1}(\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-m}(x,y)| \le Cd(x,y)^{2-N}e^{-ckd(x,y)} \end{aligned}$$

for some constant C and for all $x,y \in {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i$ independent of k. The operator now is bounded because it is estimated by a “convolution” with $L^1({\mathbb {R}}^{n_i}\times {\mathcal {M}}_i)$ function.

Consider now the kernel $R_{2,k}(x,y)$ of the operator $(1-\chi _D)k^{2m-1}(\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-m}$. Then, again using [2, Proposition 2.2] and following as in the $H_{1,a}$ term, we obtain

$$\begin{aligned} ||R_{2,k}(x,y)||_{L^\infty _xL^q_y}&=\sup _{x\in {\mathcal {M}}}\bigg (\int _{\mathcal {M}}|R_{2,k}(x,y)|^qdy\bigg )^{1/q}\\&\lesssim \sup _{x\in {\mathcal {M}}}\bigg (\int _{\mathcal {M}}|kd(x,y)^{2-n_i}e^{-ckd(x,y)}|^qdy\bigg )^{1/q}\\&\simeq \left(\int _{1}^\infty \left| kr^{2-n_i}e^{-ckr}\right| ^{q}r^{n_i-1}dr\right)^{1/q}\\&\simeq \left(\int _k^\infty \frac{k^qr^{(2-n_i)q+n_i-1}}{k^{(2-n_i)q+n_i}}e^{-r}dr\right)^{1/q} \\&\lesssim k^{-1+n_i(1-1/q)}\Big (\int _k^\infty r^{(2-n_i)q+n_i-1}e^{-r}dr\Big )^{1/q}\\&\lesssim k^{-1+n_i(1-1/q)}. \end{aligned}$$

Similarly,

$$\begin{aligned} ||R_{2,k}(x,y)||_{L^q_xL^\infty _y}\lesssim k^{-1+n_i(1-1/q)}. \end{aligned}$$

This implies that this operator is bounded as a map from $L^{q'}\rightarrow L^\infty$ and from $L^1\rightarrow L^q$ with operator norm bounded by $k^{-1+n_i(1-1/q)}$. Interpolating, we find that

$$\begin{aligned} ||(1-\chi _D)k^{2m-1}(\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-m}||_{p\rightarrow q}\lesssim k^{-1+n_i(1/p-1/q)} \end{aligned}$$

for all $p<q$. Hence,

$$\begin{aligned}&||\nabla \phi _i(1-\chi _D)k^{2m-1}(\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-m}||_{p\rightarrow p}\\&\lesssim ||(1-\chi _D)k^{2m-1}(\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-m}||_{p\rightarrow q}\\&\lesssim k^{-1+n_i(1/p-1/q)}\lesssim 1 \end{aligned}$$

for all $p<q$ such that $1/p-1/q\ge 1/n_i$. This, and the fact that $\nabla \phi _i$ is compactly supported, implies that

$$\begin{aligned} ||\nabla \phi _ik^{2m-1}(\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-m}||_{p\rightarrow p}\lesssim 1 \end{aligned}$$

for all $p\le n_i$.

$\bullet$ The $H_2(k)$ term.

From the definition of $G_2(k)$, which occurs in the term $H_2(k)$, it can be noticed that the operators $\{\nabla H_2(k)\}_{k\in (0,1)}$ form a family of pseudodifferential operators of order $1-2m$ with the Schwartz kernel having compact support and depending smoothly on k. At the scale of an individual chart on ${\mathcal {M}}$, it is clear that in local coordinates, the symbol of $\nabla H_2(k)$, denoted by $a_2(k)(x,\xi )$, will satisfy

$$\begin{aligned} |\partial _\xi ^\alpha a_2(k)(x,\xi )|\lesssim (|\xi ^2|+k^2)^{\frac{1-2m-|\alpha |}{2}} \end{aligned}$$

for all multi-indices $\alpha \ge 0$ and $k\in (0,1)$. Thus, from the standard pseudodifferential operator theory, see for instance [23, Section 0.2], we get

$$\begin{aligned} |\nabla _x H_2(k)(x,y)|\lesssim k^{N+1-2m-b}d(x,y)^{-b} \end{aligned}$$

for any $b>N+1-2m$. Setting $b=N-\frac{1}{2}$ then gives

$$\begin{aligned} |k^{2m-1}\nabla _x H_2(k)(x,y)|\lesssim k^{\frac{1}{2}} d(x,y)^{\frac{1}{2}-N}. \end{aligned}$$

Thus, the operator with the $H_2(k)$ term has kernel that is compactly supported and is controlled by $d(x,y)^{\frac{1}{2}-N}$. Therefore, we obtain the boundedness here for all $1\le p \le \infty$.

$\bullet$ The $H_3(k)$ term.

To obtain the boundedness in this case, we consider the kernel $H_3(k)(x,y)$. Now, Proposition 2.1 gives

$$\begin{aligned} |H_3(k)(x,y)|\lesssim k^{-2(m-1)}\omega _2(x,k)\omega _2(y,k) \end{aligned}$$

and

$$\begin{aligned} |\nabla _x H_3(k)(x,y)|\lesssim k^{-2(m-1)}\omega _1(x,k)\omega _2(y,k), \end{aligned}$$

where $\omega _a(x,k)$ is defined in (11) for $a=1,2$.

$$\begin{aligned}&\text {Consider }\hspace{1cm} \Vert k^{2m-1} \nabla H_3(k)f\Vert _{p}\\&\quad =\left(\int _{{\mathcal {M}}}|k^{2m-1}\nabla H_3(k)f(x)|^p dx\right)^{1/p}\\&\quad =\left(\int _{{\mathcal {M}}}\left| k^{2m-1}\int _{{\mathcal {M}}} \nabla _x H_3(k)(x,y)f(y) dy\right| ^p dx\right)^{1/p}\\&\quad \lesssim \left(\int _{{\mathcal {M}}}\left| k\int _{{\mathcal {M}}} \omega _1(x,k)\omega _2(y,k) f(y) dy\right| ^p dx\right)^{1/p}\\&\quad \simeq k ||f||_p\left(\int _{\mathcal {M}}|\omega _1(x,k)|^pdx\right)^{1/p} \left(\int _{\mathcal {M}}|\omega _2(y,k)|^{p'}dy\right)^{1/p'}. \end{aligned}$$

Now, depending on whether x and y are in the compact set or outside, we have the following outcomes from the definition of $\omega _a$

$$\begin{aligned} \left(\int _{\mathcal {M}}|\omega _1(x,k)|^pdx\right)^{1/p}&= \left(\int _{ K_i}1 dx\right)^{1/p}+\left(\int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i}|\omega _1(x,k)|^pdx\right)^{1/p}\\&\simeq \left(\int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i}\left|\left<d(x_i^\circ ,x)\right>^{-(n_i-1)}e^{-ck(d(x_i^\circ ,x)}\right|^{p}dx\right)^{1/p}\\&\simeq \left(\int _{1}^\infty \left| r^{-(n_i-1)}e^{-ckr}\right| ^{p}r^{n_i-1}dr\right)^{1/p}\\&\simeq \left(\int _{1}^\infty r^{-(n_i-1)p+n_i-1}e^{-ckr}dr\right)^{1/p}\\&\simeq \left(\int _k^\infty \frac{r^{-(n_i-1)p+n_i-1}}{k^{-(n_i-1)p+n_i}}e^{-r}dr\right)^{1/p}\\&\simeq \left(k^{-\alpha }\int _k^\infty r^{\alpha -1}e^{-r}dr\right)^{1/p},\;\text { where } \alpha =-(n_i-1)p+n_i. \end{aligned}$$

By simple computations we obtain

$$\begin{aligned} \left( k^{-\alpha }\int _k^\infty r^{\alpha -1}e^{-r}dr\right) ^{1/p}\lesssim {\left\{ \begin{array}{ll} k^{(\frac{n_i}{p'}-1)}, &{}\alpha>0, i.e., p<\frac{n_i-1}{n_i},\\ 1, &{}\alpha <0, i.e., p>\frac{n_i-1}{n_i}. \end{array}\right. } \end{aligned}$$

(17)

Also,

$$\begin{aligned} \left(\int _{\mathcal {M}}|\omega _2(y,k)|^{p'}dy\right)^{1/p'}&= \left(\int _{ K_j}1 dy\right)^{1/p}+ \left(\int _{{\mathbb {R}}^{n_j}\times {\mathcal {M}}_j\setminus K_j}|\omega _2(y,k)|^{p'}dy\right)^{1/p'}\\&\simeq \left(\int _{{\mathbb {R}}^{n_j}\times {\mathcal {M}}_j\setminus K_j}\left|\left<d(x_j^\circ ,x)\right>^{-(n_j-2)}e^{-ck(d(x_j^\circ ,x)}\right|^{p'}dx\right)^{1/p'}\\&\simeq \left(\int _{1}^\infty \left| r^{-(n_j-2)}e^{-ckr}\right| ^{p'}r^{n_j-1}dr\right)^{1/p'}\\&\simeq \left(\int _{1}^\infty r^{-(n_j-2)p'+n_j-1}e^{-ckr}dr\right)^{1/p'}\\&\simeq \left(\int _k^\infty \frac{r^{-(n_j-2)p'+n_j-1}}{k^{-(n_j-2)p'+n_j}}e^{-r}dr\right)^{1/p'}\\&\simeq \left(k^{-\beta }\int _k^\infty r^{\beta -1}e^{-r}dr\right)^{1/p'}, \text { where }\beta =-(n_j-2)p'+n_j. \end{aligned}$$

Set

$$\begin{aligned} \gamma _\beta (k)= {\left\{ \begin{array}{ll} k^{(\frac{n_j}{p}-2)}, &{}\beta>0, i.e., p>\frac{n_j}{2},\\ 1, &{}\beta<0, i.e., p<\frac{n_j}{2}. \end{array}\right. } \end{aligned}$$

This gives

$$\begin{aligned} \left( k^{-\beta }\int _k^\infty r^{\beta -1}e^{-r}dr\right) ^{1/p'}\simeq \gamma _\beta (k). \end{aligned}$$

(18)

We thus have

$$\begin{aligned} \Vert k^{2m-1}\nabla H_{3}(k)\Vert _{p \rightarrow p} \lesssim \max _{n_i,n_j} k&\left(k^{-\alpha }\int _k^\infty r^{\alpha -1}e^{-r}dr\right)^{1/p}\left( k^{-\beta }\int _{k}^\infty s^{\beta -1}e^{-s}ds\right) ^{1/p'} , \end{aligned}$$

where

$$\begin{aligned} k&\left(k^{-\alpha }\int _k^\infty r^{\alpha -1}e^{-r}dr\right)^{1/p}\left( k^{-\beta }\int _{k}^\infty s^{\beta -1}e^{-s}ds\right) ^{1/p'} \nonumber \\&\lesssim k {\left\{ \begin{array}{ll} k^{(\frac{n_i}{p'}-1)+(\frac{n_j}{p}-2)}, &{}\alpha>0,\beta>0 \implies \frac{n_j}{2}<p<\frac{n_i}{n_i-1},\\ k^{(\frac{n_j}{p}-2)}, &{}\alpha<0,\beta>0\implies p>\frac{n_i}{n_i-1}\text { and }p>\frac{n_j}{2},\\ k^{(\frac{n_i}{p'}-1)}, &{}\alpha >0,\beta<0\implies p<\frac{n_i}{n_i-1}\text { and }p<\frac{n_j}{2} , \\ 1, &{} \alpha<0,\beta<0 \implies \frac{n_i}{n_i-1}<p<\frac{n_j}{2}. \end{array}\right. } \end{aligned}$$

(19)

Since $\frac{n_j}{2}\ge \frac{3}{2}$ for all j, and $\frac{n_i}{n_i-1}\le \frac{3}{2}$ for all i, so the first case cannot happen, and from the remaining cases we obtain boundedness for all $1\le p\le n_j$. This is because the exponent $\frac{n_j}{p}-1$ of k in the second case is non-negative only if $p\le n_j$.

$\bullet$ The $H_4(k)$ term.

This term can be treated in the same way as the $H_3(k)$ term, with the difference that it vanishes to an additional order in the right (primed) variable. Indeed, by Proposition 2.2, we have

$$\begin{aligned} |\nabla _x H_4(k)(x,y)|&\lesssim k^{-2(m-1)}\omega _1(x,k)\omega _1(y,k) . \end{aligned}$$

This implies

$$\begin{aligned} ||k^{2m-1}\nabla H_4(k)f||_{p}&=\left(\int _{{\mathcal {M}}}|k^{2m-1}\nabla H_4(k)f(x)|^pdx\right)^{1/p}\\&=\left(\int _{\mathcal {M}}\left| \int _{\mathcal {M}}k^{2m-1}\nabla _x H_4(k)(x,y)f(y)dy\right| ^p dx\right)^{1/p}\\&\;\;\;\times \left(\int _{{\mathcal {M}}}\left| k\int _{{\mathcal {M}}} \omega _1(x,k)\omega _1(y,k) f(y) dy\right| ^p dx\right)^{1/p}\\&\simeq k ||f||_p\left(\int _{\mathcal {M}}|\omega _1(x,k)|^pdx\right)^{1/p} \left(\int _{\mathcal {M}}|\omega _1(y,k)|^{p'}dy\right)^{1/p'}. \end{aligned}$$

Following on similar lines as in the $H_3$ term, we obtain boundedness for all $1\le p\le \infty .$

Now, taking $H_j(k)$ terms together for $j=1,\ldots ,4$, we obtain $||\;|\sqrt{t}\nabla (1+t\Delta )^{-m}|\;||_{p\rightarrow p}\lesssim 1$ for all $t>0$ and all $1\le p\le n^*$, where $n^*=\min _{i}\{n_i\}$. Also, from (19) we obtain $||\;|\sqrt{t}\nabla (1+t\Delta )^{-m}|\;||_{p\rightarrow p}\lesssim (\sqrt{t})^{1-\frac{n^*}{p}}$ for $p\in [n^*,\infty ]$ and $t\ge 1$.

We shall now show that for $m=1$ we also have

$$\begin{aligned} \Vert \;|{\sqrt{t}} \nabla (1+t\Delta )^{-1}|\;\Vert _{p \rightarrow p} \ge {c} (\sqrt{t})^{1-n^*/p} \quad \forall {t\ge 1} \end{aligned}$$

(20)

and for all $p\in [n^*, \infty ]$, where $c>0$ is some constant. It can be noticed that in the previous part of the proof the only parts that did not give the uniform boundedness for the entire range of $p\in [1,\infty ]$ were $H_1(k)$ and $H_3(k)$. Thus, to obtain (20), we only have to look for these two cases. Since we are working with $m=1$, we only need to consider the cases $G_1(k)$ and $G_3(k)$. Therefore, we estimate the following expression

$$\begin{aligned}&\Vert \;| k\nabla (G_1(k)+G_3(k))f(x)|\;\Vert _{p\rightarrow p} \\&\quad = \left|\left|\;\left|k \int _{\mathcal {M}}\nabla [G_1(k)(x,y)+G_3(k)(x,y)]f(y)dy\right|\;\right|\right|_{p\rightarrow p}. \end{aligned}$$

Recalling from the previous part of the proof, the $H_1(k)$ term were divided into two parts, $H_{1,a}$ and $H_{1,b}$ depending on whether the gradient hits the resolvent factor or the function $\phi _i$. The term where the gradient hits the resolvent, $H_{1,a}$, was proved to be uniformly bounded on $L^p$ for all $p\in [1,\infty ]$. It is, therefore, sufficient to verify the lower bounds from (20) for the operator corresponding to the following kernel

$$\begin{aligned}&k\sum _{i = 1}^{l}\int _{\mathcal {M}}\Big |\nabla \phi _{i}(x){(\Delta _{{\mathbb {R}}^{n_{i}}\times {\mathcal {M}}_{i}} + k^{2})}^{-1}(x,y) \phi _{i}(y)f(y)\nonumber \\&\qquad +\nabla u_i(x,k)(\Delta _{{\mathbb {R}}^{n_{i}} \times {\mathcal {M}}_{i}} + k^{2})^{-1}(x_i^\circ ,y)\phi _i(y)f(y)\Big |dy. \end{aligned}$$

(21)

Now, if we compute the first part of the above kernel at $x=x_i^\circ$ for some i, then to verify estimates for (21), it is enough to get lower bounds like (20) for the operator corresponding to the following kernel

$$\begin{aligned} k\left|\nabla \phi _{i}(x)+\nabla u_i(x,k)\right| \int _{\mathcal {M}}\left|(\Delta _{{\mathbb {R}}^{n_{i}}\times {\mathcal {M}}_{i}} + k^{2})^{-1}(x_i^\circ ,y)\phi _i(y)f(y)\right|dy. \end{aligned}$$

(22)

Thus, we are left to show that the $L^p$ norm of the operator with the kernel described in (22) is greater than $ck^{-(1-n^*/p)}$ for some $c>0$ and all $1>k>0$.

Let $a_k(x)=\nabla \phi _{i}(x)+\nabla u_i(x,k)$, then adding and subtracting $\nabla u_i(x,0)$ in $a_k(x)$ we obtain

$$\begin{aligned} a_k(x)=\nabla (u_i(x,k)-u_i(x,0))+(\nabla \phi _i(x)+\nabla u_i(x,0)). \end{aligned}$$

It has been shown in [16] that the first term of $a_k(x)$ is uniformly bounded on $L^p({\mathcal {M}})$ for all $p\in (1,\infty )$. Therefore, the required estimates for (22) will be implied by the lower bounds for

$$\begin{aligned} k\left|\nabla \phi _{i}(x)+\nabla u_i(x,0)\right| \int _{\mathcal {M}}\left|(\Delta _{{\mathbb {R}}^{n_{i}}\times {\mathcal {M}}_{i}} + k^{2})^{-1}(x_i^\circ ,y)\phi _i(y)f(y)\right|dy. \end{aligned}$$

(23)

Let $a(x)=\left|\nabla \phi _{i}(x)+\nabla u_i(x,0)\right|$ and f be a positive function. Also, let

$$\begin{aligned} Rf(x)=k a(x) \int _{\mathcal {M}}(\Delta _{{\mathbb {R}}^{n_{i}}\times {\mathcal {M}}_{i}} + k^{2})^{-1}(x_i^\circ ,y)\phi _i(y)f(y)dy. \end{aligned}$$

Then, by [2, Corollary 2.3], there exists a constant $C>0$ such that

$$\begin{aligned} Rf(x)&\ge C ka(x)\int _{\mathcal {M}}d(x_i^\circ ,y)^{2-n_i}e^{-ckd(x_i^\circ ,y)}\phi _i(y)f(y)dy. \end{aligned}$$

Thus,

$$\begin{aligned} Rf(x)&\gtrsim ka(x)\int _{\mathcal {M}}g(y)f(y)\phi _i(y)dy\simeq ka(x)\int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i} g(y)f(y)dy, \end{aligned}$$

where $g(y)=d(x_i^\circ ,y)^{2-n_i}e^{-ckd(x_i^\circ ,y)}$. Now,

$$\begin{aligned} ||R||_{p\rightarrow p}&\gtrsim k\left(\int _{\mathcal {M}}|a(x)|^pdx\right)^{1/p}||g||_q \gtrsim k||g||_q\text { (since } a(x)\not \equiv 0 )\\&\simeq k \gamma _\beta (k), \text { where }\beta =-(n_i-2)q+n_i\\&\simeq k^{\frac{n_i}{p}-1}. \end{aligned}$$

That is, $||R||_{p\rightarrow p}\gtrsim k^{\frac{n^*}{p}-1}$ for $0<k\le 1$ and $p\in [n^*,\infty ]$, which further implies that (8) holds for all $t\ge 1$ and $p\in [n^*,\infty ]$. This ends the proof of Theorem A.

3.2 Proof of theorem B

The boundedness of the horizontal maximal operator follows from [10, Theorem 3], (10), and the fact that we can write

$$\begin{aligned} t\Delta (1+t\Delta )^{-m}=(1+t\Delta )^{-(m-1)}-(1+t\Delta )^{-m}. \end{aligned}$$

(24)

Therefore, we are interested in obtaining bounds for $M^{res,\nabla }_m$. Clearly, it follows from (7) that

$$\begin{aligned} \Vert M_m^{res,\nabla }f(x)\Vert _{p \rightarrow p } =\infty \end{aligned}$$

for all $p > n^*$.

Again replacing t by $\frac{1}{k^2}$ we obtain

$$\begin{aligned} M_{m}^{res,\nabla } f(x)=\sup _{t>0}\left| \sqrt{t} \nabla (1+t \Delta )^{-m} f(x)\right| =\sup _{k>0}\left| k^{2m-1} \nabla \left( k^{2}+\Delta \right) ^{-m} f(x)\right| . \end{aligned}$$

By the same argument as in Theorem A, we focus on the case when $t>1$, that is, $0<k<1$.

$$\begin{aligned} \left\| M^{res,\nabla }_m f\right\| _{p \rightarrow p}&=\left\| \sup _{0<k<1}\left| k^{2m-1} \nabla \left( k^{2}+\Delta \right) ^{-m} f\right| \;\right\| _{p \rightarrow p} \\&=\left\| \sup _{0<k<1}\left| k^{2m-1} \nabla \sum _{j=1}^{4} H_{j}(k) f\right| \;\right\| _{p \rightarrow p} \\&\le \sum _{j=1}^{4}\left\| \sup _{0<k<1}\left| k^{2m-1} \nabla H_{j}(k) f\right| \;\right\| _{p \rightarrow p}. \end{aligned}$$

We start with the boundedness of the $H_1(k)$ term.

$$\begin{aligned}&\left\| \sup _{0<k<1}\left| k^{2m-1} \nabla H_1(k) f\right| \right\| _{p \rightarrow p}\\&\quad = \left|\left| \sup _{0<k<1}\Big | k^{2m-1} \int _{{\mathcal {M}}} \nabla \Big (\left( \Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_{i}}+k^{2}\right) ^{-m}(\cdot , y) \phi _{i}(\cdot )\phi _i(y)\Big ) f(y) d y\Big |\;\right|\right|_{p\rightarrow p}\\&\quad =\left|\left|\phi _{i}(\cdot ) \sup _{0<k<1}\Big | k^{2m-1} \int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i} \nabla \left( \Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_{i}}+k^{2}\right) ^{-m}(\cdot , y) \phi _i(y) f(y) d y\Big |\;\right|\right|_{p\rightarrow p}\\&\quad \;\;\;+\left|\left|\nabla \phi _i(\cdot ) \sup _{0<k<1}\Big | k^{2m-1} \int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}\left( \Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_{i}}+k^{2}\right) ^{-m}(\cdot , y) \phi _i(y) f(y) d y\Big |\;\right|\right|_{p\rightarrow p}\\&\quad =I_{1}+I_{2}. \end{aligned}$$

Boundedness of $I_1$ holds for all $1<p\le \infty$ and is weak-type (1,1) by Lemma 2.2.

For $I_2$, following on similar lines as in $H_{1,b}$ of Theorem A, we take $\chi _D$ as the characteristic function of the set $D=\{(x,y)\in {\mathcal {M}}^2: d(x,y)\le 1\}$. Then, by [2, Proposition 2.2], we have

$$\begin{aligned} \left|\left|\nabla \phi _i(\cdot )\chi _D(\cdot ,y)\int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}\sup _{k}|k^{2m-1}(\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-m}(\cdot ,y)|\phi _i(y) f(y)dy\right|\right|_{p\rightarrow p}\lesssim ||f||_p. \end{aligned}$$

Consider now

$$\begin{aligned}&\left|\left|\nabla \phi _i(\cdot )(1-\chi _D) \sup _{0<k<1}\Big | k^{2m-1} \int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}\left( \Delta _{\mathbb {R}^{n_i}\times {\mathcal {M}}_{i}}+k^{2}\right) ^{-m}(\cdot , y) \phi _i(y) f(y) d y\Big |\;\right|\right|_{p\rightarrow p}\\&\quad =\left(\int _{{\mathcal {M}}}\left| \nabla \phi _i(x) (1-\chi _D)\sup _{0<k<1}\Big | k^{2m-1} \int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}\left( \Delta _{\mathbb {R}^{n_i}\times {\mathcal {M}}_{i}}+k^{2}\right) ^{-m}(x, y) \phi _i(y) f(y) d y\Big |\right| ^{p} d x\right)^{1/p}\\&\quad \lesssim ||f||_p\left(\int _{{\mathcal {M}}}\Bigg |\nabla \phi _i(x) \sup _{0<k<1}\Big | k \left(\int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i}\left|d(x,y)^{-(n_i-2)}e^{-ckd(x,y)}\right|^{p'}dy\right)^{1/p'}\Big |\Bigg |^p dx\right)^{1/p}\\&\quad \lesssim ||f||_p\left(\int _{\text { supp }\nabla \phi _i}\Bigg |\sup _{0<k<1}\Big | k \left(\int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i}\left|d(x,y)^{-(n_i-2)}e^{-ckd(x,y)}\right|^{p'}dy\right)^{1/p'}\Big |\Bigg |^p dx\right)^{1/p}\\&\quad \lesssim ||f||_p \sup _{0<k<1} k \left(\int _{1}^\infty \left|r^{-(n_i-2)}e^{-ckr}\right|^{p'}r^{n_i-1}dr\right)^{1/p'}\;\text {(since }\nabla \phi _i\text { is compactly supported)}\\&\quad \lesssim ||f||_p\sup _{0<k<1} k \left(\int _{k}^\infty \frac{r^{-(n_i-2)p'+n_i-1}}{k^{-(n_i-2)p'+n_i}}e^{-r}dr\right)^{1/p'}, \end{aligned}$$

where the third inequality follows again by [2, Proposition 2.2] and the fact that $\phi _i$ is supported outside a compact set. The above supremum is finite by the $H_3(k)$ term of Theorem A and thus bounded for $1<p\le n_i$. Hence, $I_2$ is bounded for $1<p\le n_i$. It is also bounded when $p=1$. Indeed,

$$\begin{aligned} I_2&=\int _{{\mathcal {M}}}\left| \nabla \phi _i(x) \sup _{0<k<1}\Big | k \int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}\left( \Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_{i}}+k^{2}\right) ^{-1}(x, y) \phi _i(y) f(y) d y\Big |\right| d x\\&\lesssim ||f||_1\int _{\text { supp }\nabla \phi _i}\Bigg |\sup _{0<k<1}\Big | k \sup _{y\in {\mathcal {M}}}\left|d(x,y)^{-(n_i-2)}e^{-ckd(x,y)}\phi _i(y)\right|\Big |\Bigg | dx\\&\simeq ||f||_1\int _{\text { supp }\nabla \phi _i}\Big | \sup _{y\in {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i}\sup _{0<k<1}\left|k d(x,y)^{-(n_i-2)}e^{-ckd(x,y)}\right|\Big |dx\\&\simeq ||f||_1\int _{\text { supp }\nabla \phi _i} \sup _{y\in {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i}\left| d(x,y)^{-(n_i-1)}\right|dx\\&\lesssim ||f||_1\sup _{x\in \text { supp }\nabla \phi _i} \sup _{y\in {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i}\left| d(x,y)^{-(n_i-1)}\right|\lesssim ||f||_1. \end{aligned}$$

Since the $H_2(k)$ term is again a pseudodifferential operator, its boundedness holds in the same way as in Theorem A for all $1\le p \le \infty$.

For $H_3(k)$ we follow the similar procedure as done in Theorem A.

Case $p>\frac{n_i}{n_i-1}$. We have

$$\begin{aligned}&\left|\left| \sup _{0<k< 1}|k^{2m-1}\nabla H_3(k)f|\;\right|\right|_{p}\nonumber \\&=\left|\left| \sup _{0<k< 1}|k^{2m-1}\nabla \int _{\mathcal {M}}H_3(k)(\cdot ,y)f(y)dy|\;\right|\right|_{p} \nonumber \\&=\left(\int _{{\mathcal {M}}}\sup _{0<k<1}\left| k^{2m-1}\int _{{\mathcal {M}}} \nabla _x H_3(k)(x,y)f(y) dy\right| ^p dx\right)^{1/p}\nonumber \\&\lesssim \left(\int _{{\mathcal {M}}}\sup _{0<k<1}\left| k\int _{{\mathcal {M}}} \omega _1(x,k)\omega _2(y,k) f(y) dy\right| ^p dx\right)^{1/p}\nonumber \\&\lesssim ||f||_p \left(\int _{{\mathcal {M}}}\sup _{0<k<1} |\omega _1(x,k)|^p k^p \left( \int _{{\mathcal {M}}} |\omega _2(y,k)|^{p'} dy\right) ^{p/p'} dx\right)^{1/p}\nonumber \\&\lesssim ||f||_p\left( \int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i} \sup _{0<k<1} \left|\left<d(x_i^\circ ,x)\right>^{-(n_i-1)}e^{-ck(d(x_i^\circ ,x)}\right|^p k^p \right. \nonumber \\&\quad \left. \times \left( \int _{{\mathcal {M}}} |\omega _2(y,k)|^{p'} dy\right) ^{p/p'} dx\right) ^{1/p}\nonumber \\&\lesssim ||f||_p\left(\int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i} \left|\left<d(x_i^\circ ,x)\right>^{-(n_i-1)}\right|^p dx\right)^{1/p} \sup _{0<k<1}\left( k\gamma _\beta (k)\right) , \end{aligned}$$

(25)

where the first integral is obtained by controlling $e^{-ckd(x_i^\circ ,x)}$ by 1 and is finite for all $(n_i-1)p>n_i$, i.e., $p>\frac{n_i}{n_i-1}$. The supremum, on the other hand, is finite by the $H_3(k)$ term of Theorem A. This gives the boundedness for $\frac{n^*}{n^*-1} <p\le n_i$. Hence, the $H_3(k)$ term is bounded here for all $\frac{n^*}{n^*-1} <p\le n^*$.

Case $p=1$. In the next step we shall prove that $H_3(k)$ is of weak-type (1, 1). When $p=1$ we note that

$$\begin{aligned} \sup _k| k\omega _1(x,k)| \le C \omega (x), \end{aligned}$$

where $\omega (x)=\left< d(x_i^\circ ,x)\right>^{-n_i}$ for $x \in {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i$ and $\omega (x)=1$ for $x\in K$. In addition, we note that the $L^\infty$ norm of $\omega _2(y,k)$ is uniformly (independently of k) bounded. Since the function $d(x_i^\circ ,x)^{-n_i}$ is in $L^{1,\infty }$, we obtain that the maximal $H_3(k)$ term is weak-type (1,1). Next, by interpolation it follows that the $H_3$ part is continuous for the whole range $1<p \le n^*$.

The analog of the argument for $H_3$ shows the boundedness of the $H_4(k)$ term but for $1<p\le \infty$. In fact, we obtain the weak-type (1,1) estimates by a straightforward modification of the argument from (25). Thus, the boundedness of the maximal function is obtained on $L^p$ spaces for $1<p\le n^*$ by combining the above $H_j(k)$ terms $(1\le j\le 4)$, and is weak-type (1,1).

We now prove the Fefferman-Stein maximal inequality that is in our case Theorem C.

3.3 Proof of theorem C

First we note that if we consider supremum taken in (4) only over $0<t <1$ then the maximal operator can essentially be discussed in the same way as Proposition 2.3. Therefore, we only consider the range $t\ge 1$ or equivalently $0<k \le 1$.

Similarly, as before, we consider $H_{1,a}$— when the derivative hits the resolvent. That is, we need to compute the $L^p$ norm of

$$\begin{aligned} \phi _i k^{2m-1} \nabla (\Delta _{{\mathbb {R}}^{n_i} \times {\mathcal {M}}_i} + k^2)^{-m} \phi _i. \end{aligned}$$

(26)

The estimates required here for part $H_{1,a}$ follows directly from Proposition 2.3. We discuss the part $H_{1,b}$ later together with $H_3$. Recall that the $H_2$ term can be estimated by

$$\begin{aligned} |k^{2m-1}\nabla _x H_2(k)(x,y)|\lesssim k^{\frac{1}{2}} d(x,y)^{\frac{1}{2}-N}. \end{aligned}$$

It follows that we can replace supremum over $0<k \le 1$ for $H_2$ and so we end up with the linear operator bounded by $d(x,y)^{\frac{1}{2}-N}$. In consequence, we can use the same argument as in the proof of Theorem A and Lemma 2.3 to verify continuity for all $1<p<\infty$ for the $H_2$ term.

First we observe that

$$\begin{aligned} \sup _{k\le 1}k(1+r)^{1-n}e^{-rk} \le (1+r)^{-n}. \end{aligned}$$

Then, assuming that $f(x) \ge 0$ and that $f\in L^1({\mathcal {M}})$, we have

$$\begin{aligned} \sup _k|k^{2m-1}\nabla H_3(k)f(x)|&=\sup _{k}\left|k^{2m-1}\int _{\mathcal {M}}\nabla H_3(k)(x,y)f(y)dy\right|\nonumber \\&\lesssim \sup _k\left|k\omega _1(x,k)\int _{\mathcal {M}}\omega _2(y,k)f(y)dy\right|\nonumber \\&\lesssim \sup _k\left|k\omega _1(x,k)\right|\sup _{y\in {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i} \left|d(x_i^\circ ,y)^{-(n_i-2)}\right|\int _{\mathcal {M}}f(y)dy\nonumber \\&\lesssim \omega (x) \int _{\mathcal {M}}f(y)dy, \end{aligned}$$

(27)

where $\omega (x)=\left<d(x_i^\circ ,x)\right>^{-n_i}$ for $x\in {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i{\setminus } K_i$ and $\omega (x)=1$ for $x\in K$.

Now choose a sequence ${\bar{f}}=\{f_j\}_{j=1}^\infty$. Note that, without loss of generality, we can assume $f_j(x) \ge 0$. Then

$$\begin{aligned} \sum _j\left(\sup _k|k^{2m-1}\nabla H_3(k){f_j}(x)|\right)^2 \le C \omega (x)^2 \sum _j\left|\int _{{\mathcal {M}}} {f_j}dy\right|^2=C \omega (x)^2 \left|\Lambda ( {\bar{f}})\right|^2, \end{aligned}$$

where $\Lambda (f)=\int _{{\mathcal {M}}} f(y)dy$. Now $\Lambda$ is a bounded linear function on $L^1({\mathcal {M}})$ so we can apply Lemma 2.3. This validates weak-type (1, 1).

Observe next that

$$\begin{aligned} \omega _1(x,k) \le C {\tilde{\omega }}(x) \quad \text{ and } \text{ that } \quad \sup _k k\omega _2(x,k) \le C {\tilde{\omega }}(x), \end{aligned}$$

where ${\tilde{\omega }}(x) =\left< d(x_i^\circ ,x)\right>^{1-n_i}$ for $x \in {\mathbb {R}}^{n_i}\times {\mathcal {M}}_i{\setminus } K_i$ and ${\tilde{\omega }}(x)=1$ for $x\in K$. To consider the range $2<p<n^*$, we slightly rearrange (27) and we note that if $f(x) \ge 0$, then

$$\begin{aligned} \sup _k|k^{2m-1}\nabla H_3(k)f(x)|&\lesssim \sup _k\left|\omega _1(x,k)\int _{\mathcal {M}}k\omega _2(y,k)f(y)dy\right| \\&\lesssim {\tilde{\omega }}(x) \int _{\mathcal {M}}{\tilde{\omega }}(y)f(y)dy . \end{aligned}$$

Thus, for ${\bar{f}}=\{f_j\}_{j=1}^\infty$ with each $f_j(x)\ge 0$, we have

$$\begin{aligned} \sum _{j}\left( \sup _k|k^{2m-1}\nabla H_3(k)f_j(x)|\right)^2\lesssim \sum _j\left|{\tilde{\omega }}(x) \int _{\mathcal {M}}{\tilde{\omega }}(y)f_j(y)dy\right|^2\simeq |{\tilde{\Lambda }}({\bar{f}})|^2, \end{aligned}$$

where ${\tilde{\Lambda }}(f)= {\tilde{\omega }}(x) \int _{{\mathcal {M}}} {\tilde{\omega }}(y)f(y)dy$. Note that the operator ${\tilde{\Lambda }}$ acts continuously on $L^p({\mathcal {M}})$ if and only if $(n^*)'< p < n^*$. Using interpolation with weak-type (1, 1) we show boundedness of the $H_3$ term for all $1< p < n^*$. For the $H_{1,b}$ term, observe that

$$\begin{aligned} \nabla \phi _i(x)&\sup _{0<k<1}\left|k^{2m-1}\int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}(\Delta _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i}+k^2)^{-m}(x,y)\phi _i(y)f(y)dy\right|\\&\lesssim \nabla \phi _i(x)\sup _{0<k<1}\left|k\int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i}d(x,y)^{-(n_i-2)}e^{-ckd(x,y)}f(y)dy\right|\\ {}&\lesssim \int _{{\mathbb {R}}^{n_i}\times {\mathcal {M}}_i\setminus K_i} \left<d(x,y)\right>^{-(n_i-1)}f(y)dy,\\ {}&\quad \times \text { for }x\in \text {supp}\nabla \phi _i \text { (since }\nabla \phi _i(x)\text { is compactly supported)}. \end{aligned}$$

Just like in the $H_3$ term, the above integral is also controlled by a bounded linear functional on $L^p$ if and only if $1<p<n^*$, and is weak type (1,1) bounded on $L^1$. Again, for a sequence ${\bar{f}}=\{f_j\}_{j=1}^\infty$, the $H_{1,b}$ term satisfies the Fefferman-Stein inequality. The proof for the $H_4$ term follows on the same lines as $H_3$ and as done in Theorems A and B. However, the boundedness for $H_4$ holds for $1<p<\infty$ with weak type (1,1) at $L^1$. Clubbing the boundedness of each $H_i$ term, we obtain the Fefferman-Stein maximal inequality for the vertical maximal function on $L^p({\mathcal {M}})$ for $1<p<n^*$ with weak type(1,1) estimates at $L^1$. The unboundedness for $p\ge n^*$ is shown in Corollary 4.2.

4 Square functions and R-boundedness

In this section we show that the boundedness of vertical maximal function implies the boundedness of the vertical square function. This is possible with the help of the R-boundedness of the set $\{\sqrt{t}\nabla (1+t\Delta )^{-m}:t>0\}$. Before we jump to our results, we recall some basic definitions and known results required for our proof.

Definition 4.1

( [17, Definition 8.1]) Let X and Y be Banach spaces and let ${\mathcal {T}}\subseteq B(X,Y)$ be a family of bounded operators.

• (a) R-boundedness: ${\mathcal {T}}$ is said to be R-bounded if there exists a constant $C\ge 0$ such that for all finite sequences $(T_j)_{j=1}^J$ in ${\mathcal {T}}$ and $(x_j)_{j=1}^J$ in X,

  $$\begin{aligned} \left|\left|\sum _{j=1}^J\epsilon _j T_j x_j\right|\right|_{L^2(\Omega ,Y)}\le C\left|\left|\sum _{j=1}^J\epsilon _j x_j\right|\right|_{L^2(\Omega ,X)} \end{aligned}$$

  or equivalently,

  $$\begin{aligned} {\mathbb {E}}\left|\left|\sum _{j=1}^J\epsilon _j T_j x_j\right|\right|_{L^2(\Omega ,Y)}\le C{\mathbb {E}}\left|\left|\sum _{j=1}^J\epsilon _j x_j\right|\right|_{L^2(\Omega ,X)}, \end{aligned}$$

  where ${\mathbb {E}}$ is the usual expectation and $(\epsilon _j)_{j\ge 1}$ is a Rademacher sequence on a fixed probability space $(\Omega ,{\mathbb {P}})$.

• (b) $\ell ^2$-boundedness: If X and Y are Banach lattices, then ${\mathcal {T}}$ is said to be $\ell ^2$-bounded if there exists a constant $C>0$ such that for all finite sequences $(T_j)_{j=1}^J$ in ${\mathcal {T}}$ and $(x_j)_{j=1}^J$ in X,

  $$\begin{aligned} \left|\left|\Bigg (\sum _{j=1}^J|T_j x_j|^2\Bigg )^{1/2}\right|\right|_{L^2(\Omega ,Y)}\le C\left|\left|\Bigg (\sum _{j=1}^J| x_j|^2\Bigg )^{1/2}\right|\right|_{L^2(\Omega ,X)}. \end{aligned}$$

The least admissible constants here are called the R-bound and the $\ell ^2$-bound, and are denoted by ${\mathcal {R}}({\mathcal {T}})$ and $\ell ^2({\mathcal {T}})$, respectively.

Remark 4.1

In (a) and (b) of Definition 4.1 we can replace $L^2(\Omega ,Y)$ by $L^p(\Omega ,Y)$, and $L^2(\Omega ,X)$ by $L^q(\Omega ,X)$ for $1\le p,q<\infty$. In such situations we obtain an equivalent definition of the R-boundedness, but with a possibly different constant that we denote by ${\mathcal {R}}_{p,q}({\mathcal {T}})$. For $p=q$, it is convenient to write ${\mathcal {R}}_{p}({\mathcal {T}}):={\mathcal {R}}_{p,p}({\mathcal {T}})$. We also refer to [17, Proposition 8.1.5] for the definition of R-boundedness on $L^p$ spaces for $1\le p<\infty$.

Proposition 4.1

When X and Y are $L^p$ spaces with $p\in [1,\infty )$, then R-boundedness and $\ell ^2$-boundedness are equivalent, see [17, Theorem 8.1.3].

Theorem 4.1

The Fefferman-Stein maximal inequality in (9) implies that the set $\{\sqrt{t}\nabla (1+t\Delta )^{-m}:t>0\}$ is R-bounded on $L^p({\mathcal {M}})$ for $1<p<n^*$.

Proof

Consider the following set

$$\begin{aligned}{} & {} {\mathcal {T}}:=\{T(t)=\sqrt{t}\nabla (1+t\Delta )^{-m}\in {\mathcal {L}}(L^p(\Omega ))\text { for }t>0:\\{} & {} \quad |T(t)f|\le M_m^{res,\nabla }f \text { for all }f\in L^p(\Omega )\}. \end{aligned}$$

By Proposition 4.1, it is enough to show $\ell ^2$ boundedness of the above set to prove its R-boundedness. We have

$$\begin{aligned} \bigg |\bigg |\bigg (\sum _{j=1}^J|T(t_j)f_j|^2\bigg )^{1/2}\bigg |\bigg |_{L^p(\Omega )}&\le \bigg |\bigg |\bigg (\sum _{j=1}^J(M_m^{res,\nabla }f_j)^2\bigg )^{1/2}\bigg |\bigg |_{L^p(\Omega )}\\&\le A_{p,{\mathcal {M}}}\bigg |\bigg |\bigg (\sum _{j=1}^J|f_j|^2\bigg )^{1/2}\bigg |\bigg |_{L^p(\Omega )}. \end{aligned}$$

The last inequality follows from Theorem C. $\square$

Corollary 4.1

The Fefferman-Stein maximal inequality implies the boundedness of the resolvent based vertical square functions $S^{res,\nabla }_m(f):=\left(\int _0^\infty |\sqrt{t}\nabla (1+t\Delta )^{-m}f|^2\frac{dt}{t}\right)^{1/2}$ for any order m and the semigroup based vertical square function $S^{exp,\nabla }(f):=\left(\int _0^\infty |\sqrt{t}\nabla e^{t\Delta }f|^2\frac{dt}{t}\right)^{1/2}$ on $L^p({\mathcal {M}})$ for $1~<~p~<~n*$.

Proof

Theorem 4.1 shows that the Fefferman-Stein maximal inequality of the vertical maximal functions implies R-boundedness of the set $\{\sqrt{t}\nabla (1+t\Delta )^{-m}:t>0\}$ on $L^p$ for $1<p< n^*$. By [6, Proposition 2.2], R-boundedness of the set $\{\sqrt{t}\nabla (1+t\Delta )^{-m}:t>0\}$ is equivalent to R-boundedness of the set $\{\sqrt{t}\nabla e^{t\Delta }:t>0\}$ on $L^p$. Interestingly, from the R-boundedness of the latter set, [6, Theorem 4.1] gives the boundedness of more general square functions which satisfy some decaying conditions. Both $S^{exp,\nabla }$ and $S^{res,\nabla }_m$ satisfy those conditions on $L^p$ spaces, and hence they are bounded for $1<p<n^*$. $\square$

Corollary 4.2

The Fefferman-Stein maximal inequality (9) in Theorem C fails for $p\ge n^*$. The vertical square function operator $S^{exp,\nabla }$ is also unbounded in the same range.

Proof

We know from [2, Theorem 1.1] that the resolvent based square function is unbounded on $L^p({\mathcal {M}})$ for $p \ge n^*$. This and Corollary 4.1 imply that the Fefferman-Stein maximal inequality fails in the same range. Also, since the R-boundedness of the set $\{\sqrt{t}\nabla (1+t\Delta )^{-m}:t>0\}$ is false for $p \ge n^*$ by the same argument, we obtain negative result for $S^{exp,\nabla }$ from [6, Proposition 2.2 and Theorem 4.1]. $\square$

Proposition 4.2

On $L^p({\mathcal {M}})$, the vertical square function operator $S^{exp,\nabla }$ is bounded if and only if $1~<~p~<~n^*$.

Proof

The proof is a direct consequence of Corollaries 4.1 and 4.2. $\square$

Note that at this point we do not have any good estimates for the kernel of $\nabla e^{t\Delta }$ on $L^p({\mathcal {M}})$, and hence we do not have any direct proof to show the boundedness of $S^{exp,\nabla }$ on $L^p({\mathcal {M}})$.

The following theorem is the resolvent based version of the square function in [6, Theorem 3.1]. The proof of it follows exactly in the same way as that of [6, Theorem 3.1], however, the R-boundedness of the resolvent sets require the boundedness of two resolvent based square functions of consecutive exponents. We have included the proof here to give an equivalence argument between the semigroup based square functions and the resolvent based square functions. Note that this theorem works on a general metric measure space $\Omega$. However, on manifolds with ends, the boundedness of square function operators $S^{exp,\nabla }$ and $S^{res,\nabla }_m$ coincide under the assumption that $2\,m+2\le \min {n_i}$. We explain this in Remark 4.2 below.

In the proof of the below theorem we use the notation of the norm with subscript p to denote the norm with subscript $L^p({\mathcal {M}})$ for simplicity.

Theorem 4.2

Let $p\in (1,\infty )$ and $m\ge 1$. If $S^{res,\nabla }_m$ and $S^{res,\nabla }_{m+1}$ are bounded on $L^p({\mathcal {M}})$, then the set $\{t\nabla (1+t^2\Delta )^{-m}, t>0\}$ is R-bounded on $L^p({\mathcal {M}}).$

Proof

The proof follows on the same lines of [6, Theorem 3.1]. We shall show if $S^{res,\nabla }_m$ and $S^{res,\nabla }_{m+1}$ are bounded on $L^p({\mathcal {M}})$ for some m then $\{t\nabla (1+t^2\Delta )^{-m}, t>0\}$ is R-bounded on $L^p({\mathcal {M}})$. Let $t_j\in (0,\infty )$ and $f_j\in L^p({\mathcal {M}})$ for $j=1,\ldots ,J$. We begin by estimating the quantity $I:={\mathbb {E}}|\sum _j\epsilon _jt_j\nabla (1+t_j^2\Delta )^{-m}f_j|^2$. Using the fact that the Rademacher variables are independent we obtain

$$\begin{aligned} I&=-\int _0^\infty \frac{d}{dt}{\mathbb {E}}|\nabla (1+t^2\Delta )^{-m}\sum _j\epsilon _jt_j(1+t_j^2\Delta )^{-m}f_j|^2dt\\&=2m\int _0^\infty {\mathbb {E}}\Bigg [\Big (\nabla (1+t^2\Delta )^{-2m-1}(2t\Delta )\sum _j\epsilon _jt_j(1+t_j^2\Delta )^{-m}f_j\Big )\\&\quad \cdot \Big (\nabla \sum _j\epsilon _jt_j(1+t_j^2\Delta )^{-m}f_j\Big )\Bigg ]dt\\&=2m\int _0^\infty {\mathbb {E}}\Bigg [\Big (\nabla (1+t^2\Delta )^{-m}\sum _j\epsilon _jt_j(1+t_j^2\Delta )^{-m}f_j\Big )\\&\quad \cdot \Big (\nabla (1+t^2\Delta )^{-m-1}(2t\Delta )\epsilon _jt_j(1+t_j^2\Delta )^{-m}f_j\Big )\Bigg ]dt\\&=4m\int _0^\infty {\mathbb {E}}\Bigg [\Big (\nabla t(1+t^2\Delta )^{-m}\sum _j\epsilon _j(1+t_j^2\Delta )^{-m}f_j\Big )\\&\quad \cdot \Big (\nabla t(1+t^2\Delta )^{-(m+1)}\epsilon _jt_j^2\Delta (1+t_j^2\Delta )^{-m}f_j\Big )\Bigg ]\frac{dt}{t}\\&=4m\int _0^\infty {\mathbb {E}}\Bigg [\Big (\nabla t(1+t^2\Delta )^{-m}\sum _j\epsilon _j(1+t_j^2\Delta )^{-m}f_j\Big )\\&\quad \cdot \Big (\nabla t(1+t^2\Delta )^{-(m+1)}\sum _j\epsilon _jt_j^2\Delta (1+t_j^2\Delta )^{-m}f_j\Big )\Bigg ]dt. \end{aligned}$$

By the Cauchy-Schwartz inequality,

$$\begin{aligned} I&\le 4m\int _0^\infty \Bigg ({\mathbb {E}}|t\nabla (1+t^2\Delta )^{-m}\sum _j\epsilon _j(1+t_j^2\Delta )^{-m}f_j|^2\Bigg )^{1/2}\\&\quad \times \Bigg ({\mathbb {E}}|t\nabla (1+t^2\Delta )^{-(m+1)} \sum _j\epsilon _jt_j^2\Delta (1+t_j^2\Delta )^{-m}f_j|^2\Bigg )^{1/2}\frac{dt}{t}\\&\le 2m\Bigg (\int _0^\infty {\mathbb {E}}|t\nabla (1+t^2\Delta )^{-m}\sum _j\epsilon _j(1+t_j^2\Delta )^{-m}f_j|^2 \frac{dt}{t}\\&\quad +\int _0^\infty {\mathbb {E}}|t\nabla (1+t^2\Delta )^{-(m+1)}\sum _j\epsilon _jt_j^2\Delta (1+t_j^2\Delta )^{-m}f_j|^2\frac{dt}{t}\Bigg ). \end{aligned}$$

Since both $S^{res,\nabla }_m$ and $S^{res,\nabla }_{m+1}$ are bounded, we get

$$\begin{aligned} I&\lesssim _{m}&{\mathbb {E}}\Bigg [\Big (S^{res,\nabla }_m\Big (\sum _j\epsilon _j(1+t_j^2\Delta )^{-m}f_j\Big )\Big )^2\Bigg ]\\{} & {} +{\mathbb {E}}\Bigg [\Big (S^{res,\nabla }_{m+1}\Big (\sum _j\epsilon _jt_j^2\Delta (1+t_j^2\Delta ))^{-m}f_j\Big )\Big )^2\Bigg ]. \end{aligned}$$

Looking at $S^{res,\nabla }_m$ and $S^{res,\nabla }_{m+1}$ as the norm in $L^2((0,\infty ),\frac{dt}{t})$ we obtain

$$\begin{aligned}{} & {} {\mathbb {E}}\Bigg [\Big (S^{res,\nabla }_m\Big (\sum _j\epsilon _j(1+t_j^2\Delta )^{-m}f_j\Big )\Big )^2\Bigg ]\\{} & {} \quad ={\mathbb {E}}\Bigg |\Bigg |\sum _j\epsilon _jt\nabla (1+t^2\Delta )^{-m}(1+t_j^2\Delta )^{-m}f_j\Bigg |\Bigg |^2_{L^2((0,\infty ),\frac{dt}{t})} \end{aligned}$$

and

$$\begin{aligned}{} & {} {\mathbb {E}}\Bigg [\Big (S^{res,\nabla }_{m+1}\Big (\sum _j\epsilon _j(1+t_j^2\Delta )^{-m}f_j\Big )\Big )^2\Bigg ]\\{} & {} \quad ={\mathbb {E}}\Bigg |\Bigg |\sum _j\epsilon _jt\nabla (1+t^2\Delta )^{-(m+1)}\\{} & {} \quad \sum _j\epsilon _jt_j^2\Delta (1+t_j^2\Delta )^{-m}f_j\Bigg |\Bigg |^2_{L^2((0,\infty ),\frac{dt}{t})}. \end{aligned}$$

Now, by the Kahane inequality

$$\begin{aligned}{} & {} c_{p,m}\sqrt{I}\le \Bigg |{\mathbb {E}}\Bigg [\Big (S^{res,\nabla }_m\Big (\sum _j\epsilon _j(1+t_j^2\Delta )^{-m}f_j\Big )\Big )^p\Bigg ]\Bigg |^{1/p}\\{} & {} \quad +\Bigg |{\mathbb {E}}\Bigg [\Big (S^{res,\nabla }_{m+1}\Big (\sum _j\epsilon _jt_j^2\Delta (1+t_j^2\Delta )^{-m}f_j\Big )\Big )^p\Bigg ]\Bigg |^{1/p} \end{aligned}$$

for some constant $c_{p,m}>0$. Now, using the assumption that $S^{res,\nabla }_m$ and $S^{res,\nabla }_{m+1}$ are bounded on $L^p({\mathcal {M}})$, we obtain

$$\begin{aligned} \Big |\Big |\sqrt{I}\Big |\Big |_p&\lesssim _{p,m}\bigg |{\mathbb {E}}\bigg |\bigg |\sum _j\epsilon _j(1+t_j^2\Delta )^{-m}f_j\bigg |\bigg |_p^p\bigg |^{1/p}+\bigg |{\mathbb {E}}\bigg |\bigg |\sum _j\epsilon _jt_j^2\Delta (1+t_j^2\Delta )^{-m}f_j\bigg |\bigg |_p^p\bigg |^{1/p}\\&\lesssim _{p,m}{\mathbb {E}}\bigg |\bigg |\sum _j\epsilon _j(1+t_j^2\Delta )^{-m}f_j\bigg |\bigg |_p+{\mathbb {E}}\bigg |\bigg |\sum _j\epsilon _jt_j^2\Delta (1+t_j^2\Delta )^{-m}f_j\bigg |\bigg |_p, \end{aligned}$$

where we again used the Kahane inequality. We can also see from the Kahane inequality that $||\sqrt{I}||_p$ is equivalent to ${\mathbb {E}}||\sum _j\epsilon _jt_j\nabla (1+t_j^2\Delta )^{-m}f_j||_p$. Since $\Delta$ has a bounded holomorphic functional calculus on $L^p({\mathcal {M}})$, it follows from [17, Theorem 10.3.4] that $\{(1+t^2\Delta )^{-m},t>0\}$ and $\{t^2\Delta (1+t^2\Delta )^{-m},t>0\}$ are R-bounded on $L^p({\mathcal {M}})$. This, together with previous estimates, gives

$$\begin{aligned} {\mathbb {E}}\bigg |\bigg |\sum _j\epsilon _jt_j\nabla (1+t_j^2\Delta )^{-m}f_j\bigg |\bigg |_p\lesssim {\mathbb {E}}\bigg |\bigg |\sum _j\epsilon _jf_j\bigg |\bigg |_p \end{aligned}$$

with a constant independent of $t_j$ and $f_j$. This shows that the set $\{t\nabla (1+t^2\Delta )^{-m}, t>0\}$ is R-bounded on $L^p({\mathcal {M}})$. $\square$

Remark 4.2

The converse of the above theorem holds true by [6, Theorem 4.1]. Nevertheless, it is true for $1<p< n^*$. This is because by [2, Theorem 1.1] $S^{res,\nabla }_m$ is not bounded for $p\ge n^*$ under the assumption that $2m< n^*$. Now, R-boundedness of the sets $\{\sqrt{t}\nabla (1+t\Delta )^{-m}:t>0\}$ and $\{\sqrt{t}\nabla e^{t\Delta }:t>0\}$ is equivalent by [6, Proposition 2.2], and Cometx-Ouhabaz have shown the equivalence between the R-boundedness of $\{\sqrt{t}\nabla e^{t\Delta }:t>0\}$ with the boundedness of the square function $S^{exp,\nabla }$ in [6, Theorem 3.1 and Theorem 4.1]. On combining this together with Theorem 4.2 we obtain equivalence of the boundedness of both $S^{exp,\nabla }$ and $S^{res,\nabla }_m$ on $L^p({\mathcal {M}})$ for $1<p< n^*$ under the assumption that $2m+2\le n^*$.

4.1 Open problems

Here we mention some open problems that one could think of solving on manifolds with ends or on a more general metric measure space.

• (1) We have shown in this article the boundedness of the maximal operator $M^{res,\nabla }_m$ on $L^p({\mathcal {M}})$. The problem to check the boundedness of $M^{exp,\nabla }$ on $L^p({\mathcal {M}})$ is still open. The boundedness of these maximal operators is also open on other metric measure spaces. We believe that this problem is solvable if one can find Grigor’yan and Saloff-Coste type estimates [14] on the gradient of the heat kernel, $\nabla e^{t\Delta }$, on manifolds with ends.

• (2) It is natural to expect that weak type (1, 1) can be included in the statement of Proposition 4.2, but we do not know how to verify it.