Endpoint estimates for spherical maximal operators acting on radial functions in hyperbolic spaces

Introduction

In the 1970s, Stein [18] introduced the spherical maximal operators

of complex order $\alpha$, with a priori for $\operatorname{Re}\alpha>0$, on $\mathbb{R}^{n}$. This definition can be analytically continued to $\operatorname{Re}\alpha>(1-n)/2$. In particular, when $\alpha=0$, $\mathfrak{M}^{\alpha}$ recovers the standard spherical maximal operator $\mathfrak{M}$ defined by

where $d\omega$ is the canonical surface measure of the unit sphere in $\mathbb{R}^{n}$. Stein ([17 pp. 518–519] & [18 Theorem 2]) showed that for $n\ge2$, $1<p<\infty$ and $\alpha\in\mathbb{R}$ with $\alpha>(1-n)/2$, if

then $\mathfrak{M}^{\alpha}$ is bounded on $L^{p}(\mathbb{R}^{n})$, and that if $\mathfrak{M}^{\alpha}$ is bounded on $L^{p}(\mathbb{R}^{n})$ for $1<p\le2$, then we must have

Stein’s result with $\alpha=0$ indicates that for $n\geq3$ and $p>n/(n-1)$, $\mathcal{M}$ is bounded on $L^p(\mathbb{R}^n)$, and that for $n\geq2$, $\mathcal{M}$ is bounded on $L^p(\mathbb{R}^n)$ only if $p>n/(n-1)$. When $n=2$, the same sufficient condition $p>2=2/(2-1)$ for the $L^p$-boundedness of $\mathcal{M}$ was obtained by Bourgain [2]. At the endpoint $p=n/(n-1)$, Bourgain [1] showed that $\mathcal{M}^{\alpha}$ is bounded from $L^{p,1}(\mathbb{R}^n)$ to $L^{p,\infty}(\mathbb{R}^n)$ when $n\geq3$, and that the same restricted weak-type $(p,p)$ estimate holds for radial functions when $n=2$. However, this 2-dimensional restricted weak-type estimate fails for general functions, as shown by Seeger, Tao and Wright [15]. For further results concerning $L^p$-boundedness of the operators $\mathcal{M}^{\alpha}$, one can refer to [11, 12, 13, 20] and the references therein.

In the last decade, Ciaurri et al. [5, 6, 14] investigated $\mathcal{M}^{\alpha}$ acting on radial functions. They established the sharp range of $\alpha$ for $\mathcal{M}^{\alpha}$ to be bounded on $L^p_{\mathrm{rad}}(\mathbb{R}^n)$, the radial subspace of $L^p(\mathbb{R}^n)$, as well as the corresponding endpoint result in terms of restricted weak-type estimates. More precisely, they showed that for $n\geq2$, $1<p\leq\infty$ and $\alpha\in\mathbb{R}$ with $\alpha>(1-n)/2$, $\mathcal{M}^{\alpha}$ is bounded on $L^p_{\mathrm{rad}}(\mathbb{R}^n)$ if and only if

and that for $1<p\leq2n/(n-1)$ and

$\mathcal{M}^{\alpha}$ is bounded from $L^{p,1}_{\mathrm{rad}}(\mathbb{R}^n)$ to $L^{p,\infty}_{\mathrm{rad}}(\mathbb{R}^n)$. See [14 Corollaries 1.5 and 1.6(b)].

In the 1980s, El Kohen [7] introduced on the hyperbolic space $\mathbb{H}^n$ the spherical maximal operators

of complex order $\alpha$ with $\operatorname{Re}\alpha>0$, where

is the spherical averaging operators of order $\alpha$. Here $d(\cdot,\cdot)$ is the geodesic distance, $dx$ is the Riemannian measure and $B(x,t)$ is the geodesic ball centered at $x$ with radius $t$ in $\mathbb{H}^n$. The definition can be analytically continued to $\operatorname{Re}\alpha>(1-n)/2$. In particular, at order zero,

where $S(x,t)$ is the geodesic sphere in $\mathbb{H}^n$ and $\operatorname{Avg}_{S(x,t)}$ denotes its normalized surface average. El Kohen [7 Theorem 3] showed that for $n\geq2$, $1<p<\infty$ and $\alpha\in\mathbb{R}$ with $\alpha>(1-n)/2$, if

then $\mathfrak{m}^{\alpha}$ is bounded on $L^p(\mathbb{H}^n)$.

This result with $\alpha=0$ yields that for $n\geq3$ and $p>n/(n-1)$, $\mathrm{m}^{0}$ is bounded on $L^{p}(\mathbb{H}^{n})$. The case of $n=2$ was later resolved by Ionescu [10]. More recently, Chen, Shen, Yan and the named author [4] gave the necessary condition for the $L^{p}$-boundedness of $\mathrm{m}^{\alpha}$, and improved the range of $\alpha$ in [7 Theorem 3] for $p>2$. They showed that for $n\geq2$, $1<p<\infty$ and $\alpha\in\mathbb{R}$ with $\alpha>(1-n)/2$, if $\mathrm{m}^{\alpha}$ is bounded on $L^{p}(\mathbb{H}^{n})$, then we must have

and that if $p>2$ and

with

then $\mathrm{m}^{\alpha}$ is bounded on $L^{p}(\mathbb{H}^{n})$. See [4 Theorems 1.1 and 1.2]. In addition, Zhang and the named author [19] investigated the related problems of maximal operators with the supremum restricted to lacunary radii.

In this article we focus on $\mathrm{m}^{\alpha}$ acting on radial functions, which depend only on $d(x,0)$ for $0$ being the origin in $\mathbb{H}^{n}$. Our main result is the following.

The relations between $\alpha$ and $p$ in Theorem 1.1 are indicated in Figure 1 below.

We would like to mention that according to the arguments in [4], the threshold of the range of $\alpha$ in Theorem 1.1(i) corresponds to both the threshold of the Lebesgue index and the growth in time in the local smoothing estimates for wave equation with radial initial datum, obtained by Chen, Duong, Yan and the named author [3]. In addition, both the strong-type and the restricted weak-type boundedness coincide with its Euclidean counterpart [14 Corollaries 1.5 and 1.6(b)], indicating that the geometry at infinity induced by the negative curvature of $\mathbb{H}^{n}$ imposes no additional restriction on the $L_{\mathrm{rad}}^{p}$-boundedness. Based on this observation, we conjecture that the sufficient condition for the general $L^{p}$-boundedness of $\mathbf{m}^{\alpha}$ in [4 Theorems 1.1 and 1.2] can be sharpened so that there is no gap between them.

To show Theorem 1.1, we first derive the explicit radial kernel expressions of the operator $\mathcal{M}_{t}^{\alpha}$ in terms of the Gauss hypergeometric function. We then decompose the operator $\mathbf{m}^{\alpha}$ into local and global parts, with supremums taken over all small and large radii, respectively. The local part can be essentially dominated by the maximal operators with positive Euclidean radial kernels studied in [5, 6, 14]. For the global part, we continue splitting the supremum into unit intervals and use the exponential decay of the kernel to obtain summable estimates.

The rest of this article is organized as follows. Section 2 recalls basic facts about hyperbolic spaces and establishes the explicit radial kernel expressions and estimates. With these preparations, in Sections 3 we prove Theorem 1.1.

Preliminaries

In this section, we recall basic facts about the hyperbolic space $\mathbb{H}^{n}$ and obtain the explicit radial kernel expressions of the spherical maximal operators.

Hn as the hyperboloid

On $\mathbb{R}^{1+n}$ we define the Minkowski metric

The hyperboloid model is

For $x \in\mathbb{H}^{n}$, its tangent space is

The Riemannian metric $\mathbf{g}$ is given by the bilinear form

This metric gives the complete simply connected space of constant sectional curvature $-1$. Its geodesic distance is given by

Take $\mathbf{0} := (1,0,\ldots,0) \in\mathbb{H}^{n}$. The geodesic polar coordinates at $\mathbf{0}$ are

In these coordinates the Riemannian metric reads

The induced Riemannian measure is therefore given by

If $x = (\operatorname{ch} R,\operatorname{sh} R\theta)$ and $y = (\operatorname{ch} r,\operatorname{sh} r\omega)$, then we have the hyperbolic cosine law:

Radial functions on $\mathbb{H}^{n}$ depend on $R(x) := d(x,\mathbf{0})$ only. In other words, for such a function $f$, there exists another function $h$, namely the radial profile of $f$, defined on $(0,\infty)$, such that

In the sequel, we always identify $f$ with its radial profile. Note that in view of (2.1),

where $\nu_{n-1} = |S^{n-1}|$.

Explicit kernel expressions for $M_t^\alpha$

For $\operatorname{Re} c > \operatorname{Re} b > 0$ and $|z| < 1$, let

be the Gauss hypergeometric function. We further define

We need the following analytic facts about the functions $F$ and $\mathbf{F}$.

Proof. (1) follows readily from (7) and Morera’s Theorem.

To show (2), it follows from [8 (2) and (5) on pp. 56–57] that

Since for each $j=0,1,\ldots$, $1/\Gamma(\cdot+j)$ is entire, for $|z|<1$ the above function is entire in $c$.

To show (3), we apply [8 (2) and (5) on pp. 56–57] to derive

This series is dominated by the desired estimate, so are $F(a,b;c;z)$ and $\mathbf{F}(a,b;c;z)$. □

Recall that for $\operatorname{Re}\alpha>0$, the spherical means $M_t^\alpha$ on $\mathbb{H}^n$ are defined by (1). We have the following explicit expression.

(iii) If $t>R+r$, then

Moreover, for $\operatorname{Re}\alpha>(1-n)/2$, the values on $t=|R-r|$ and $t=R+r$ may be assigned arbitrarily.

Proof. Assume first $\operatorname{Re}\alpha>0$. In view of (1), (6) and (4), for $x=(\operatorname{ch}R,\operatorname{sh}R\theta)$,

where $(A)_+^{\alpha-1}$ denotes $A^{\alpha-1}\mathbf{1}_{A>0}$. Hence

Now suppose $\operatorname{Re}\alpha>(1-n)/2$. (i) follows from analytic continuation and the fact that if $\operatorname{ch}t<\operatorname{ch}(R-r)$ then the integrand in (9) is zero. To see (ii), note that if $\operatorname{ch}(R-r)<\operatorname{ch}t<\operatorname{ch}(R+r)$, letting

in the integral in (9), we have

In view of (7),

It remains to invoke (8) to obtain the desired expression. Then one uses Lemma 2.1(2) to analytically continue $K_t^\alpha$ to $\operatorname{Re}\alpha>(1-n)/2$.

To see (iii), note that if $\operatorname{ch}t>\operatorname{ch}(R+r)$, letting

in the integral in (9), we have

In view of (7), we obtain the desired expression. Then one uses Lemma 2.1(1) to analytically continue $K_t^\alpha$ to $\operatorname{Re}\alpha>(1-n)/2$.

Finally, for $\operatorname{Re}\alpha>(1-n)/2$, the expression in (ii) as well as Lemma 2.1(3) indicates that $K_t^\alpha$ is a locally integrable function on $(0,\infty)\times(0,\infty)$. Thus the values on these surfaces can be assigned arbitrarily. □

Lemmas 2.1(3) and 2.2(ii)(iii) immediately yield the following crucial estimates.

We conclude this section by showing the following result, which holds for general $C_c^\infty(\mathbb{H}^n)$ functions.

Lemma 2.4. For $\alpha> (1-n)/2$, $\delta> 0$ and $f \in C^\infty_c(\mathbb{H}^n)$,

Proof. In view of the proof of [7 Lemma 1],

where the function $\tau(t,u)\geq0$ is determined by

Note that $\sinh\tau(t,u)\lesssim u^{1/2}\sinh t$, whence

as desired. □

Proof of Theorem 1.1

In this section, we decompose the maximal operator $\mathrm{m}^\alpha$ into the local part

as well as the global parts

We then separately consider their $L^p_{\mathrm{rad}}$-boundedness.

The local part $\mathrm{m}_0^\alpha$

The aim of this subsection is to show the following result.

Proposition 3.1. Suppose $1<p<\infty$, $\alpha<1/p$ and

Then $\mathrm{m}_0^\alpha$ is bounded from $L_{\mathrm{rad}}^{p,1}(\mathbb{H}^n)$ to $L_{\mathrm{rad}}^{p,\infty}(\mathbb{H}^n)$. Furthermore, if $p>2n/(n-1)$ and $\alpha=(1-n)/p$, then $\mathrm{m}_0^\alpha$ is bounded on $L_{\mathrm{rad}}^p(\mathbb{H}^n)$.

To prove Proposition 3.1, we define the Euclidean radial kernel $E_t^\alpha(R,r)$ as follows: (i) If $t<|R-r|$, then $E_t^\alpha(R,r)=0$. (ii) If $|R-r|<t<R+r$, then

(iii) If $t>R+r$, then

We further define the associated maximal function

The following result concerning the $L^p_{\mathrm{rad}}$-boundedness of $\mathcal{E}^\alpha$ serves as a key ingredient in the proof of Proposition 3.1.

Proof. This follows from the proofs of sufficiency part of [14 Theorems A and B(b)], see [14 Sections 3–4]. □

Now we present the

Proof of Proposition 3.1. Fix a radial function $f$. We write

where the second equality follows from Lemma 2.2(i).

We begin by estimating $\mathfrak{I}_1^\alpha(f)$. Since

Lemma 2.3 yields that for $0<t\leq1$, $R\leq2$ and $r\leq3$, we have

Therefore, in view of Lemma 3.2, $\mathfrak{I}_1^\alpha$ satisfies the desired boundedness properties.

Fix $j=2,3,\ldots$. It remains to estimate $\mathfrak{I}_j^\alpha(f)$. For $0<t\leq1$, $R\in(j,j+1]$ and $r\in(j-1,j+2]$, we have

Thus Lemma 2.3 yields

whence

It follows from [14 Theorems A and B(b)] that $I_j^\alpha$ is bounded from $L^p((j-1,j+2],\mathrm{d}r)$ to $L^p((j,j+1],\mathrm{d}R)$ when $\alpha>1/p-(n-1)/2$, and from $L^{p,1}((j-1,j+2],\mathrm{d}r)$ to $L^{p,\infty}((j,j+1],\mathrm{d}R)$. These estimates in turn imply

and

By the disjointness of $(j,j+1]$ and finite overlapping of $(j-1,j+2]$, we sum over $j=2,3,\ldots$ to obtain

while in view of [9 Proposition 1.4.9]

According to (13), these estimates yield the desired boundedness of $m_0^\alpha$.

The global parts $m_{T}^{\alpha}$ ($T=1,2,\ldots$)

The aim of this subsection is to show the following result.

Proof. We first write

• Estimates for $J_{1}^{\alpha}(f)$. We shall show that for $\alpha\geq1/p-(n-1)/2$

and that for $\alpha>1/p-(n-1)/2$ the above estimate holds with $L_{\mathrm{rad}}^{p}$ in place of $L_{\mathrm{rad}}^{p,1}$.

We first claim that for $R\geq1/10$ and $\alpha\geq1/p-(n-1)/2$,

and that the same estimates holds with $L^{p}$ in place of $L^{p',\infty}$ for $\alpha>1/p-(n-1)/2$.

Once these estimates are prove, it follows from Lemma 2.2(i) and the Lorentz Hölder inequality (cf. [9], the proof of (v) on p. 58]) that for $\alpha\geq1/p-(n-1)/2$

and for $\alpha>1/p-(n-1)/2$

Note that these estimates imply, for $j=0,1,\ldots$,

Arguing analogously to (14) and (15) to sum over $j$, we obtain the desired estimate (17).

To verify the claimed estimates (18) and its strong version, we apply Lemma 2.3 and the identity

to obtain for $R,r\ge1/10$ and $T\le t\le T+1$

where

For $\alpha>(1-n)/2$,

In addition, for $\alpha\ge1/p-(n-1)/2$,

and the above estimate holds with $L^{p'}$ in place of $L^{p',\infty}$ for $\alpha>1/p-(n-1)/2$. Therefore, the desired estimates (17) and its strong version are proved.

• Estimates for $J_{2}^{\alpha}(f)$. We shall show that for $\alpha<1/p$ such that (12) holds,

and that for $p>2n/(n-1)$ and $\alpha=(1-n)/p$, the above estimate holds with $L_{\mathrm{rad}}^{p'}$ in place of $L_{\mathrm{rad}}^{p',\infty}$ and $L_{\mathrm{rad}}^{p,1}$.

In view of Lemma 2.2(i), we further decompose

For $|t-R|<1/2$ and $r\le1/10$, we have

$\operatorname{sh} R\operatorname{sh} r\sim r e^{T}$,

which implies

according to Lemma 2.3 and the definition of $E_t^\alpha$. Therefore, in view of Lemma 3.2, for $\alpha<1/p$ satisfying (12)

and that for $p>2n/(n-1)$ and $\alpha=(1-n)/p$, the above estimate holds with $L_{\mathrm{rad}}^p$ in place of $L_{\mathrm{rad}}^{p,\infty}$ and $L_{\mathrm{rad}}^{p,1}$.

In addition, for $t-R\geq1/2$ and $r\leq1/10$, it follows from Lemma 2.3 that

which implies for all $1\leq p\leq\infty$ and $\alpha>(1-n)/2$

Therefore, the desired estimates (19) and its strong version are proved.

• Estimates for $J_3^\alpha(f)$. We shall show that for $\alpha<1/p$ such that (12) holds,

and that for $p>2n/(n-1)$ and $\alpha=(1-n)/p$, the above estimate holds with $L_{\mathrm{rad}}^p$ in place of $L_{\mathrm{rad}}^{p,\infty}$ and $L_{\mathrm{rad}}^{p,1}$.

In view of Lemma 2.2(i), we further decompose

For $R\leq1/10$ and $\lvert t-r\rvert<1/2$, we argue as in the estimate for $J_{2,1}^\alpha(f)$ above to obtain

Therefore, in view of Lemma 3.2, for $\alpha<1/p$ satisfying (12)

and that for $p>2n/(n-1)$ and $\alpha=(1-n)/p$, the above estimate holds with $L_{\mathrm{rad}}^p$ in place of $L_{\mathrm{rad}}^{p,\infty}$ and $L_{\mathrm{rad}}^{p,1}$.

In addition, for $R\leq1/10$ and $t-r>1/2$, it follows from Lemma 2.3 that

which implies for all $1\leq p\leq\infty$ and $\alpha>(1-n)/2$

Therefore, the desired estimates (20) and its strong version are proved. Finally, we combine (16) with estimates (17)–(20) and there strong versions to conclude the proof of Proposition 3.3. □

Conclusion of the proof of Theorem 1.1

Proof of Theorem 1.1 (sufficiency part). Note that

Since for $1 < p < \infty$ and sufficiently small $\varepsilon> 0$, $\sum_{T=1}^{\infty}(T+1)e^{(\varepsilon-cp)T}<\infty$, Propositions 3.1 and 3.3 imply the $L_{\mathrm{rad}}^{p,1} \to L_{\mathrm{rad}}^{p,\infty}$-boundedness of $\mathbf{m}^{\alpha}$ whenever $\alpha< 1/p$ and (12) holds, as well as the $L_{\mathrm{rad}}^{p}$-boundedness of $\mathbf{m}^{\alpha}$ when $p > 2n/(n-1)$ and $\alpha= (1-n)/p$. Theorem 1.1(ii) is therefore proved.

To show the sufficiency part of Theorem 1.1(i), we fix $1 < p < \infty$,

and choose $1 < p_{0} < p < p_{1} < \infty$ sufficiently close to $p$ such that $\alpha$ satisfies (21) as well as $\alpha< 1/p$ at $p_{0}$ and $p_{1}$. Interpolation between the $L_{\mathrm{rad}}^{q,1} \to L_{\mathrm{rad}}^{q,\infty}$-boundedness of $\mathbf{m}^{\alpha}$ at $q=p_{0}$ and $q=p_{1}$ (cf. [9 Theorem 1.4.19]) yields the $L_{\mathrm{rad}}^{p}$-boundedness of $\mathbf{m}^{\alpha}$. Thus all the desired $L_{\mathrm{rad}}^{p}$-boundedness of $\mathbf{m}^{\alpha}$ for $1 < p < \infty$ under the additional restriction $\alpha< 1/p$ is proved. At $p=\infty$, it follows directly from (2) that $\mathbf{m}^{0}$ is bounded on $L^{\infty}(\mathbb{H}^{n})$. Finally, we apply Lemma 2.4 to remove the additional restriction $\alpha< 1/p$ for $1 < p < \infty$ and $\alpha=0$ at $p=\infty$. This proves the sufficiency part of Theorem 1.1(i), and thus the whole sufficiency part of Theorem 1.1 is concluded. □

Proof of Theorem 1.1 (necessity part). We shall construct three counterexamples to show that the necessity conditions for the $L_{\mathrm{rad}}^{p}$-boundedness of $\mathbf{m}^{\alpha}$ are $\alpha> 1-n+n/p$, $\alpha> 1/p-(n-1)/2$ and $\alpha\ge(1-n)/p$, respectively.

• The condition $\alpha> 1-n+n/p$. Let $0 < \delta< 1/100$ be a fixed sufficiently small number to be determined later. For $0 < \varepsilon< 1/100$ we set

For $5\varepsilon\le R \le1/10$, we choose $t=R-\varepsilon$. If $r\in\operatorname{supp}f_{\varepsilon}$, then $|R-r|<t<R+r$,

Since $F(a,b;c;0)=1$, by taking $\delta=\delta_{\alpha,n}$ sufficiently small such that the hypergeometric function term in Lemma 2.2(ii) is uniformly positive and in view of Lemma 2.2(ii), we have

whence for $R$ as above

This, together with the fact that $\|f_\varepsilon\|_{L^p_{\mathrm{rad}}(\mathbb{H}^n)} \sim\varepsilon^n$, implies

Thus $\alpha>1-n+n/p$ is necessary for the $L^p_{\mathrm{rad}}$-boundedness of $\mathbf{m}^{\alpha}$.

• The condition $\alpha>1/p-(n-1)/2$. Let $\delta>0$ be a fixed sufficiently small number to be determined later. For $0<\varepsilon<\delta/2$ we set

If $r\in\operatorname{supp}f_\varepsilon$, $5/4\leq R\leq3/2$ and $t=R-1$, then $|R-r|<t<R+r$,

Since $F(a,b;c;0)=1$, by taking $\delta=\delta_{\alpha,n}$ sufficiently small such that the hypergeometric function term in Lemma Lemma 2.2(ii) is uniformly positive and in view of Lemma Lemma 2.2(ii), we have

whence for $R$ as above and $\alpha\leq1/p-(n-1)/2$

On the other hand, $\|f_\varepsilon\|_{L^p_{\mathrm{rad}}(\mathbb{H}^n)}\lesssim1$ uniformly in $\varepsilon$. Thus $\alpha>1/p-(n-1)/2$ is necessary for the $L^p_{\mathrm{rad}}$-boundedness of $\mathbf{m}^{\alpha}$.

• The condition $\alpha\geq(1-n)/p$. Let $0<\delta<1/100$ be a fixed sufficiently small number to be determined later. For $0<\varepsilon<\delta/2$ we set

For $\varepsilon\leq R\leq2\varepsilon$, we choose $t=1-R+2\delta\varepsilon$. If $r\in\operatorname{supp}f_\varepsilon$, then $|R-r|<t<R+r$,

Since $F(a,b;c;0)=1$, by taking $\delta=\delta_{\alpha,n}$ sufficiently small such that the hypergeometric function term in Lemma Lemma 2.2(ii) is uniformly positive and in view of Lemma Lemma 2.2(ii), we have

whence for $R$ as above

This, together with the fact that $\|f_\varepsilon\|_{L^p_{\mathrm{rad}}(\mathbb{H}^n)}\sim\varepsilon^{1/p}$, implies

When $p = \infty$, the above estimates are understood by the convention $1/\infty= 0$. Thus $\alpha\ge(1-n)/p$ is necessary for the $L^p_{\mathrm{rad}}$-boundedness of $\mathbf{m}^{\alpha}$.

Declaration of AI usage

During preparation of the manuscript, the author used GPT-6 Astra for assistance with concrete computations and language editing. The author takes responsibility for the mathematical arguments and the final content of the paper.

Acknowledgements

The author would like to thank Naijia Liu and Professor Lixin Yan for helpful discussions.

References

YUNXIANG WANG, DEPARTMENT OF MATHEMATICS, SUN YAT-SEN UNIVERSITY, GUANGZHOU, 510275, P.R. CHINA

Email address: [email protected]