---
title: Manifold-Adapted Maximal Operators
url: https://www.emergentmind.com/topics/manifold-adapted-maximal-operators
type: topic
---

# Manifold-Adapted Maximal Operators

Manifold-adapted maximal operators are maximal constructions whose averaging, spectral, or oscillatory mechanism is tied to an intrinsic manifold, to a submanifold, or to a geometry induced by an operator rather than to Euclidean translation invariance alone. In current work this includes spectral maximal operators \(f\mapsto \sup_{t>0}|m(tL)f|\), vertical resolvent maximal operators on manifolds with ends, geodesic-sphere maximal operators on compact manifolds, oscillatory spectral maximal operators defined by Laplace eigenfunction expansions, and directional or configuration-space maximal operators whose parameter sets lie on algebraic varieties or curved manifolds [2410.01164][2303.17721][2401.16955][2403.04996][1807.08255].

## 1. Geometric and operator-theoretic scope

A central feature of the subject is that “maximal” need not mean Hardy–Littlewood maximality. On manifolds with ends, the natural maximal operators studied in detail are spectral-analytic and resolvent-based,
\[
M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad
M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,
\]
and their mapping properties are controlled not by a uniform doubling dimension but by
\[
n^*=\min_i n_i
\]
for a connected sum
\[
M=(\mathbb R^{n_1}\times M_1)\#\cdots\#(\mathbb R^{n_l}\times M_l).
\]
In this setting the geometry is typically non-doubling when the \(n_i\) are not all equal, and the paper emphasizes that classical Hardy–Littlewood maximal theory and Calderón–Zygmund techniques are not directly available [2303.17721].

A complementary intrinsic framework arises from maximally subelliptic operators on compact manifolds. Here the ambient space is a connected compact \(C^\infty\) manifold \(X\) with a strictly positive smooth density \(\mu\), together with Hörmander vector fields with formal degrees \((W,d)\), the Carnot–Carathéodory metric
\[
\rho(x,y)=\inf\{\delta>0:\ y\in B(x,\delta)\},
\]
and a non-negative self-adjoint extension \(L\) of a symmetric nonnegative differential operator \(L_0\). Maximal subellipticity implies discrete spectrum, short-time non-isotropic Gaussian heat kernel bounds, dyadic spectral pieces \(\phi_j(2^{-2\kappa j}L)\) that are elementary operators, and Besov, Triebel–Lizorkin, and Sobolev scales intrinsically adapted to \(L\). The same work explicitly states that it does **not** prove estimates for
\[
f\mapsto \sup_{t>0}|m(tL)f|,\qquad
f\mapsto \sup_{t>0}|e^{-tL}f|,\qquad
f\mapsto \sup_{t>0}|\phi(tL)f|,
\]
but provides the dyadic and kernel structure usually used to study such maximal operators [2302.07086].

## 2. Spectral multiplier maximal operators on homogeneous-type spaces

A broad operator-theoretic theory is developed for a metric measure space \((X,d,\mu)\) with doubling measure and a nonnegative self-adjoint operator \(L\) whose heat kernel satisfies the Gaussian upper bound
\[
|T_t(x,y)|\le \frac{C}{\mu(B(x,\sqrt t))}\exp\!\Big(-c\frac{d(x,y)^2}{t}\Big).
\]
The key additional hypothesis is the existence of a positive \(L\)-harmonic function \(h\) such that, after the Doob transform \(f\mapsto h^{-1}f\), the transformed semigroup kernel
\[
\widetilde T_t(x,y)=\frac{T_t(x,y)}{h(x)h(y)}
\]
satisfies two-sided Gaussian bounds, conservation, and Hölder kernel regularity. In that setting the paper proves a sharp finite-family maximal estimate: if bounded Borel multipliers \(m_i\) satisfy
\[
\sup_{t>0}\|\phi\,m_i(t\cdot)\|_{W_s^q}+|m_i(0)|\le B
\]
for some \(s>n/r\), then
\[
\Big\|\sup_{1\le i\le N}|m_i(L)f|\Big\|_{L^p(d\mu)}
\le C_{p,r}\,B\,\sqrt{\log(1+N)}\,\|f\|_{L^p(d\mu)}
\]
for \(p\in(r,\infty)\) under the weight condition \(h^{p-2}\in A_{p/r}(h^2\,d\mu)\). The same framework yields sufficient conditions for the dyadic maximal operator
\[
M_{m,L}^{\mathrm{dyad}}f(x)=\sup_{k\in\mathbb Z}|m(2^kL)f(x)|
\]
and the continuous-scale maximal operator
\[
M_{m,L}f(x)=\sup_{t>0}|m(tL)f(x)|.
\]
Applications include Schrödinger operators with inverse square potential, scattering operators, Bessel operators, and Laplace–Beltrami operators. For complete Riemannian manifolds satisfying doubling and Poincaré, one may take \(h\equiv 1\), so the manifold case is recovered directly [2410.01164].

Within the compact maximally subelliptic setting, the multiplier operator \(m(L)\) acquires a singular-integral structure under full Mihlin estimates,
\[
\sup_{\lambda>0}\lambda^{-t}\big|(\lambda\partial_\lambda)^k m(\lambda)\big|<\infty,
\]
and the paper proves the equivalence
\[
\mathcal F_{p,q}^t(X,(W,d))=\mathcal F_{p,q}^t(X,L),\qquad
\mathcal B_{p,q}^t(X,(W,d))=\mathcal B_{p,q}^t(X,L)
\]
with the same topology. A plausible implication is that maximal operators defined spectrally can be transferred to geometry-adapted function spaces and conversely, although no direct maximal theorem is stated there [2302.07086].

## 3. Noncompact manifolds and end geometry

For manifolds with ends
\[
M=(\mathbb R^{n_1}\times M_1)\#\cdots\#(\mathbb R^{n_l}\times M_l),
\]
with \(n_i\ge 3\) and \(n^*=\min_i n_i\), the main object is the vertical resolvent family
\[
\mathcal T_t:=\sqrt t\,\nabla(1+t\Delta)^{-m},\qquad t>0,
\]
and the associated maximal operator
\[
M_m^{res,\nabla}f(x)=\sup_{t>0}|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)|.
\]
The paper proves a sharp closed-end condition: the family \(\{\mathcal T_t\}_{t>0}\) is uniformly bounded on \(L^p(M)\) for
\[
1\le p\le n^*,
\]
while for \(p\in[n^*,\infty]\) and \(t\ge 1\),
\[
\|\sqrt t\,\nabla(1+t\Delta)^{-m}\|_{L^p\to L^p}
\lesssim (\sqrt t)^{1-n^*/p},
\]
and for \(m=1\) the same growth rate is sharp from below. Consequently,
\[
\|M_m^{res,\nabla}f\|_{L^p}\le C\|f\|_{L^p}
\quad\Longleftrightarrow\quad
1<p\le n^*,
\]
with weak type \((1,1)\) at \(p=1\). In contrast, the horizontal maximal operator
\[
M_m^{res,\Delta}f(x)=\sup_{t>0}|t\Delta(1+t\Delta)^{-m}f(x)|
\]
is bounded on all \(1<p\le\infty\) and is also of weak type \((1,1)\) [2303.17721].

The proof is built from a low-energy resolvent parametrix
\[
(\Delta+k^2)^{-1}=\sum_{j=1}^4 G_j(k),
\]
weight functions
\[
\omega_a(x,k)=
\begin{cases}
1, & x\in K,\\[1mm]
\langle d(x_i^\circ,x)\rangle^{-(n_i-a)}e^{-ck\,d(x_i^\circ,x)}, & x\in \mathbb R^{n_i}\times M_i\setminus K_i,
\end{cases}
\]
and endwise decay estimates such as
\[
|u(x,k)| \le C\|v\|_\infty \langle d(x_i^\circ,x)\rangle^{-(n_i-2)}e^{-k d(x_i^\circ,x)},\qquad
|\nabla u(x,k)| \le C\|v\|_\infty \langle d(x_i^\circ,x)\rangle^{-(n_i-1)}e^{-k d(x_i^\circ,x)}.
\]
These formulas make explicit that the smallest Euclidean factor dimension governs the admissible \(L^p\)-range. The same paper also proves a Fefferman–Stein type vector-valued inequality for \(M_m^{res,\nabla}\) only for
\[
1<p<n^*,
\]
with weak type \((1,1)\), and states that it is false for \(p\ge n^*\) [2303.17721].

## 4. Compact manifolds, geodesic spheres, and oscillatory maximal operators

On compact manifolds, one major line of work treats maximal operators through Hardy spaces for Fourier integral operators. For a smooth compact boundaryless Riemannian manifold \((M,g)\), geodesic spheres
\[
S_t(x)=\partial B_t(x),\qquad t\in (0,\inj(M)),
\]
give rise to the intrinsic averaging operator
\[
A_{t,g}f(x):=\int_{S_t(x)} f(y)\,\sqrt{dV_g}.
\]
If \(I\subset (0,\inj(M))\) is compact, \(p\in[1,\infty]\), and
\[
s>d(p)+\frac1p-\frac{n-1}{2},
\]
then
\[
\Big\|\sup_{t\in I}|A_{t,g}f|\Big\|_{L^p(M)}
\le C \|f\|_{\mathcal H^{s,p}_{FIO}(M)}.
\]
The same FIO framework yields
\[
\Big\|\sup_{t\in I}|e^{it\sqrt{-\Delta_g}}f|\Big\|_{L^p(M)}
\le C \|f\|_{\mathcal H^{s,p}_{FIO}(M)}
\]
for \(s>d(p)+1/p\), and the paper states that the maximal function bounds are essentially sharp for all
\[
p\in[1,2]\cup \Big[\frac{2(n+1)}{n-1},\infty\Big]
\]
for every such hypersurface and on every manifold [2401.16955].

A different compact-manifold theory concerns oscillatory spectral maximal operators. For an \(n\)-dimensional compact connected manifold without boundary, with Laplacian \(\mathcal L\) and eigenvalues \(\lambda_j^2\), the oscillatory operator
\[
T_{\alpha,\beta,t}(f)(x)=
\sum_j
\frac{e^{i|t\lambda_j|^\alpha}}{|t\lambda_j|^\beta}
\Phi(|t\lambda_j|)
\langle f,e_j\rangle e_j(x),
\qquad 0<\alpha<1,\ \beta>0,
\]
has maximal version
\[
T_{\alpha,\beta}^*(f)(x)=\sup_{0<t\le \sigma}|T_{\alpha,\beta,t}(f)(x)|.
\]
For \(0<p<1\), the paper proves the sharp Hardy-space weak-type theorem
\[
T_{\alpha,\beta}^*:H^p(\mathbb M)\to L^{p,\infty}(\mathbb M)
\quad\text{iff}\quad
\beta\ge n\alpha\Big(\frac1p-\frac12\Big),
\]
and the strong-type statement
\[
T_{\alpha,\beta}^*:H^p(\mathbb M)\to L^p(\mathbb M)
\quad\text{if}\quad
\beta> n\alpha\Big(\frac1p-\frac12\Big).
\]
The same threshold governs maximal Riesz means for the Schrödinger-type group \(e^{is\mathcal L^{\alpha/2}}\), almost everywhere convergence, and convergence rates for combinations of fractional Schrödinger evolutions. The authors state that all results are even new on the \(n\)-torus \(T^n\) [2403.04996].

A third compact theory concerns regularity rather than \(L^p\)-improving. For maximal operators of convolution type associated with elliptic and parabolic equations, including Poisson and heat maximal operators on \(\mathbb T^d\) and \(\mathbb S^d\), the common mechanism is that the maximal function is subharmonic on the detachment set
\[
A=\{x:u^*(x)>u_0(x)\}.
\]
This yields variation-diminishing results such as
\[
V(u^*)\le V(u_0)
\]
in one dimension, and higher-dimensional contraction statements including
\[
\|\nabla u^*\|_{L^2(\mathbb S^d)}\le \|\nabla u_0\|_{L^2(\mathbb S^d)}.
\]
The same work notes, however, that these arguments rely heavily on symmetry and do **not** automatically extend to arbitrary compact manifolds [1512.02715].

## 5. Algebraic varieties, moment curves, and configuration manifolds

A large class of manifold-adapted maximal operators is parameterized not by intrinsic manifolds \(M\) but by submanifolds of Euclidean space. For directional averages,
\[
\langle f\rangle_v(x):=\int_{-1}^1 f(x-tv)\,dt,\qquad
M_V f(x):=\sup_{v\in V}|\langle f\rangle_v(x)|,
\]
with \(V\subset Z_m\cap A_n(1)\) lying on an \(m\)-dimensional real algebraic variety, the sharp \(L^2\) growth is governed by the dimension of the variety. If \(\#V\le N^m\), then
\[
\|M_V\|_{L^2(\mathbb R^n)}\le \Theta\, N^{\frac{m-1}{2}+\eta}
\]
for all \(\eta>0\), uniformly over transverse complete intersections of bounded degree, and this is sharp up to the arbitrarily small \(N^\eta\)-loss. For \(V\subset S^2\) with \(\#V\le N^2\), the paper improves the best known bound to
\[
\|M_V\|_{L^2(\mathbb R^3)}\le \Theta_k\, N^{1/2}\log N\,\log^{[k]}N.
\]
These estimates imply Kakeya-type maximal bounds for tubes constrained to polynomial direction sets,
\[
\|M_{Z_m,\delta}\|_{L^2(\mathbb R^n)}\le \Theta\,\delta^{-\frac{m-1}{2}-\eta},
\]
and are proved through an iterated polynomial partitioning scheme on varieties [1807.08255].

The moment curve gives a different one-dimensional manifold parameter. In maximal Fourier restriction, with
\[
\gamma(t)=\Bigl(t,\frac12 t^2,\dots,\frac1d t^d\Bigr),\qquad
\Gamma=\gamma(\mathbb R),
\]
the operator
\[
M_r h(x)=\Bigl(\sup_{s>0}\dashint_{B(x,s)}|h|^r\Bigr)^{1/r}
\]
is restricted to \(\Gamma\), and the paper proves
\[
\|M_r\widehat f\|_{L^q(\Gamma)}\le C_{p,r}\|f\|_{L^p(\mathbb R^d)}
\]
in a substantial range of exponents, with the case \(r=2\) reaching the full Drury restriction range. It also proves the necessary condition
\[
r\le p'
\]
for any such estimate. A related 2025 note studies averages over affine copies
\[
H(\mathbf x,r)=\{\mathbf x+r\gamma(t):t\in[-1,1]\}
\]
of the standard moment curve \(\gamma(t)=(t,t^2,\dots,t^d)\), defining
\[
M_\delta^s f(\underline{\mathbf x}_{s'-1},r)
=
\sup_{\mathbf x\in\mathbb R^s}
\frac{1}{\mathcal L^d(H_\delta(\mathbf x,r))}
\int_{H_\delta(\mathbf x,r)}|f(\mathbf y)|\,d\mathbf y.
\]
For \(p\ge p_d\) it proves
\[
\|M_\delta^s f\|_{L^p(I_1^{s'-1}\times I_2)}
\le C_\varepsilon \delta^{-\alpha(s)-\varepsilon}\|f\|_{L^p(\mathbb R^d)},
\]
where
\[
\alpha(s)=\frac{s-2}{p}\quad (s=2,\dots,d),\qquad \alpha(1)=0,
\]
and for \(s=d\) the threshold \(p\ge 3\) is sharp. The associated dimension bound for \(s'\)-parameter moment BRK-type sets is
\[
\dim(S_{s'})\ge \min\{s'+1,d\}.
\]
This shows that manifold-adapted maximality can be driven either by restriction geometry or by BRK-type averaging over affine curve families [2102.09025][2509.01173].

Configuration manifolds introduce multilinear and discrete variants. For bilinear convolutions
\[
L_t(f,g)(x)=\int f(x-ty)\,g(x-tz)\,d\mu(y,z),
\]
with \(\mu\) supported on a curved manifold in \(\mathbb R^{2d}\), including the sphere \(S^{2d-1}\) and the triangle manifold
\[
\mathcal I=\{(y,z)\in\mathbb R^{2d}: |y|=|z|=|y-z|=1\},
\]
sparse domination is obtained for lacunary and full maximal operators from \(L^p\)-improving and continuity estimates. In the discrete setting, the equilateral-triangle maximal operator
\[
T^*(f,g)(x)=\sup_{\lambda\in\mathbb N}|T_\lambda(f,g)(x)|,\qquad
T_\lambda(f,g)(x)=\lambda^{3-d}\sum_{(u,v)\in V_\lambda} f(x-u)g(x-v),
\]
with
\[
V_\lambda=\{(u,v)\in \mathbb Z^d\times\mathbb Z^d:\ |u|^2=|v|^2=|u-v|^2=\lambda\},
\]
is a discrete analogue of averaging over a codimension-\(3\) configuration manifold. For \(d>9\) and
\[
p>p_0(d)=\max\!\left(\frac{32}{d+8},\frac{d+4}{d-2}\right),
\]
the paper proves type \((p,\infty;p)\) bounds and a major-arc approximation
\[
T_\lambda f=M_\lambda f+E_\lambda f
\]
with dyadic decay for the error term [2110.08928][2006.09968].

## 6. Endpoints, vector-valued theory, and unresolved ranges

Endpoint behavior is often the most distinctive feature of manifold-adapted maximal operators. On manifolds with ends, the scalar vertical maximal operator \(M_m^{res,\nabla}\) is bounded up to the closed endpoint \(p=n^*\), while the Fefferman–Stein vector-valued inequality fails already at \(p=n^*\) and holds only for
\[
1<p<n^*.
\]
In the spectral multiplier setting on doubling spaces, the finite-family inequality
\[
\Big\|\sup_{1\le i\le N}|m_i(L)f|\Big\|_{L^p}
\lesssim \sqrt{\log(1+N)}\,\|f\|_{L^p}
\]
is explicitly identified as optimal, by transfer of the sharp Euclidean rate to the operator setting [2303.17721][2410.01164].

Several sharpness statements coexist with substantial open ranges. The FIO-Hardy-space theory states that its maximal function bounds are essentially sharp for
\[
p\in[1,2]\cup \Big[\frac{2(n+1)}{n-1},\infty\Big],
\]
but says that sharpness is not known for
\[
2<p<\frac{2(n+1)}{n-1}.
\]
The oscillatory compact-manifold theory gives a sharp threshold
\[
\beta\ge n\alpha\Big(\frac1p-\frac12\Big)
\]
for
\[
T_{\alpha,\beta}^*:H^p(\mathbb M)\to L^{p,\infty}(\mathbb M),
\]
whereas the moment-curve BRK theory proves the sharp \(\delta\)-exponent for all \(s\) and the sharp \(p\)-range only for \(s=d\); for \(1\le s\le d-1\) it proves \(p\ge 4d-2\) but only shows the necessary condition
\[
p\ge 2s,
\]
and explicitly remarks that this is probably not sharp in general. In maximal restriction on the moment curve, the author states that the \(r\)-range is not believed to be sharp [2401.16955][2403.04996][2509.01173][2102.09025].

A persistent structural theme is that foundational dyadic and kernel theory often precedes direct maximal theorems. For maximally subelliptic operators on compact manifolds, the available results include elementary-operator kernel bounds, a Calderón reproducing identity,
\[
\phi_j(2^{-2\kappa j}L)
=
\phi_j(2^{-2\kappa j}L)\sum_{|l-j|\le 1}\phi_l(2^{-2\kappa l}L),
\]
vector-valued almost-orthogonality,
\[
\|\mathcal T_k\|_{L^p(X;\ell^q)\to L^p(X;\ell^q)}\lesssim 2^{-N|k|},
\]
and equivalence between \(L\)-adapted and geometry-adapted Besov and Triebel–Lizorkin spaces, but no direct maximal-function characterization. This suggests that many future manifold-adapted maximal estimates will continue to be built by combining intrinsic dyadic decompositions, kernel decay, and geometry-specific square-function or Fourier-integral-operator techniques [2302.07086].

Source: https://www.emergentmind.com/topics/manifold-adapted-maximal-operators