Manifold-Adapted Maximal Operators

• Manifold-adapted maximal operators are advanced constructions defined via intrinsic geometries that integrate spectral, oscillatory, and resolvent techniques.
• They extend classical maximal operator theory by addressing non-doubling measures and unique geometric structures on both compact manifolds and manifolds with ends.
• Applications include sharp L^p bounds, vector-valued inequalities, and refined averaging methods over curved and algebraic varieties.

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 (Chen et al., 2024, Sharma et al., 2023, Ghosh et al., 2024, Liu et al., 2024, Plinio et al., 2018).

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 (Sharma et al., 2023).

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 $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$0 of a symmetric nonnegative differential operator $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$1. Maximal subellipticity implies discrete spectrum, short-time non-isotropic Gaussian heat kernel bounds, dyadic spectral pieces $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$2 that are elementary operators, and Besov, Triebel–Lizorkin, and Sobolev scales intrinsically adapted to $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$3. The same work explicitly states that it does not prove estimates for

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$4

but provides the dyadic and kernel structure usually used to study such maximal operators (Zhang, 2023).

2. Spectral multiplier maximal operators on homogeneous-type spaces

A broad operator-theoretic theory is developed for a metric measure space $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$5 with doubling measure and a nonnegative self-adjoint operator $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$6 whose heat kernel satisfies the Gaussian upper bound

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$7

The key additional hypothesis is the existence of a positive $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$8-harmonic function $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$9 such that, after the Doob transform $n^*=\min_i n_i$0, the transformed semigroup kernel

$n^*=\min_i n_i$1

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 $n^*=\min_i n_i$2 satisfy

$n^*=\min_i n_i$3

for some $n^*=\min_i n_i$4, then

$n^*=\min_i n_i$5

for $n^*=\min_i n_i$6 under the weight condition $n^*=\min_i n_i$7. The same framework yields sufficient conditions for the dyadic maximal operator

$n^*=\min_i n_i$8

and the continuous-scale maximal operator

$n^*=\min_i n_i$9

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 $$M=(\mathbb R^{n_1}\times M_1)\#\cdots\#(\mathbb R^{n_l}\times M_l).$$0, so the manifold case is recovered directly (Chen et al., 2024).

Within the compact maximally subelliptic setting, the multiplier operator $$M=(\mathbb R^{n_1}\times M_1)\#\cdots\#(\mathbb R^{n_l}\times M_l).$$1 acquires a singular-integral structure under full Mihlin estimates,

$$M=(\mathbb R^{n_1}\times M_1)\#\cdots\#(\mathbb R^{n_l}\times M_l).$$2

and the paper proves the equivalence

$$M=(\mathbb R^{n_1}\times M_1)\#\cdots\#(\mathbb R^{n_l}\times M_l).$$3

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 (Zhang, 2023).

3. Noncompact manifolds and end geometry

For manifolds with ends

$$M=(\mathbb R^{n_1}\times M_1)\#\cdots\#(\mathbb R^{n_l}\times M_l).$$4

with $$M=(\mathbb R^{n_1}\times M_1)\#\cdots\#(\mathbb R^{n_l}\times M_l).$$5 and $$M=(\mathbb R^{n_1}\times M_1)\#\cdots\#(\mathbb R^{n_l}\times M_l).$$6, the main object is the vertical resolvent family

$$M=(\mathbb R^{n_1}\times M_1)\#\cdots\#(\mathbb R^{n_l}\times M_l).$$7

and the associated maximal operator

$$M=(\mathbb R^{n_1}\times M_1)\#\cdots\#(\mathbb R^{n_l}\times M_l).$$8

The paper proves a sharp closed-end condition: the family $$M=(\mathbb R^{n_1}\times M_1)\#\cdots\#(\mathbb R^{n_l}\times M_l).$$9 is uniformly bounded on $n_i$0 for

$n_i$1

while for $n_i$2 and $n_i$3,

$n_i$4

and for $n_i$5 the same growth rate is sharp from below. Consequently,

$n_i$6

with weak type $n_i$7 at $n_i$8. In contrast, the horizontal maximal operator

$n_i$9

is bounded on all $C^\infty$0 and is also of weak type $C^\infty$1 (Sharma et al., 2023).

The proof is built from a low-energy resolvent parametrix

$C^\infty$2

weight functions

$C^\infty$3

and endwise decay estimates such as

$C^\infty$4

These formulas make explicit that the smallest Euclidean factor dimension governs the admissible $C^\infty$5-range. The same paper also proves a Fefferman–Stein type vector-valued inequality for $C^\infty$6 only for

$C^\infty$7

with weak type $C^\infty$8, and states that it is false for $C^\infty$9 (Sharma et al., 2023).

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 $X$0, geodesic spheres

$X$1

give rise to the intrinsic averaging operator

$X$2

If $X$3 is compact, $X$4, and

$X$5

then

$X$6

The same FIO framework yields

$X$7

for $X$8, and the paper states that the maximal function bounds are essentially sharp for all

$X$9

for every such hypersurface and on every manifold (Ghosh et al., 2024).

A different compact-manifold theory concerns oscillatory spectral maximal operators. For an $\mu$0-dimensional compact connected manifold without boundary, with Laplacian $\mu$1 and eigenvalues $\mu$2, the oscillatory operator

$\mu$3

has maximal version

$\mu$4

For $\mu$5, the paper proves the sharp Hardy-space weak-type theorem

$\mu$6

and the strong-type statement

$\mu$7

The same threshold governs maximal Riesz means for the Schrödinger-type group $\mu$8, almost everywhere convergence, and convergence rates for combinations of fractional Schrödinger evolutions. The authors state that all results are even new on the $\mu$9-torus $(W,d)$0 (Liu et al., 2024).

A third compact theory concerns regularity rather than $(W,d)$1-improving. For maximal operators of convolution type associated with elliptic and parabolic equations, including Poisson and heat maximal operators on $(W,d)$2 and $(W,d)$3, the common mechanism is that the maximal function is subharmonic on the detachment set

$(W,d)$4

This yields variation-diminishing results such as

$(W,d)$5

in one dimension, and higher-dimensional contraction statements including

$(W,d)$6

The same work notes, however, that these arguments rely heavily on symmetry and do not automatically extend to arbitrary compact manifolds (Carneiro et al., 2015).

5. Algebraic varieties, moment curves, and configuration manifolds

A large class of manifold-adapted maximal operators is parameterized not by intrinsic manifolds $(W,d)$7 but by submanifolds of Euclidean space. For directional averages,

$(W,d)$8

with $(W,d)$9 lying on an $\rho(x,y)=\inf\{\delta>0:\ y\in B(x,\delta)\},$0-dimensional real algebraic variety, the sharp $\rho(x,y)=\inf\{\delta>0:\ y\in B(x,\delta)\},$1 growth is governed by the dimension of the variety. If $\rho(x,y)=\inf\{\delta>0:\ y\in B(x,\delta)\},$2, then

$\rho(x,y)=\inf\{\delta>0:\ y\in B(x,\delta)\},$3

for all $\rho(x,y)=\inf\{\delta>0:\ y\in B(x,\delta)\},$4, uniformly over transverse complete intersections of bounded degree, and this is sharp up to the arbitrarily small $\rho(x,y)=\inf\{\delta>0:\ y\in B(x,\delta)\},$5-loss. For $\rho(x,y)=\inf\{\delta>0:\ y\in B(x,\delta)\},$6 with $\rho(x,y)=\inf\{\delta>0:\ y\in B(x,\delta)\},$7, the paper improves the best known bound to

$\rho(x,y)=\inf\{\delta>0:\ y\in B(x,\delta)\},$8

These estimates imply Kakeya-type maximal bounds for tubes constrained to polynomial direction sets,

$\rho(x,y)=\inf\{\delta>0:\ y\in B(x,\delta)\},$9

and are proved through an iterated polynomial partitioning scheme on varieties (Plinio et al., 2018).

The moment curve gives a different one-dimensional manifold parameter. In maximal Fourier restriction, with

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$00

the operator

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$01

is restricted to $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$02, and the paper proves

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$03

in a substantial range of exponents, with the case $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$04 reaching the full Drury restriction range. It also proves the necessary condition

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$05

for any such estimate. A related 2025 note studies averages over affine copies

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$06

of the standard moment curve $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$07, defining

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$08

For $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$09 it proves

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$10

where

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$11

and for $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$12 the threshold $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$13 is sharp. The associated dimension bound for $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$14-parameter moment BRK-type sets is

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$15

This shows that manifold-adapted maximality can be driven either by restriction geometry or by BRK-type averaging over affine curve families (Jesurum, 2021, Wang, 1 Sep 2025).

Configuration manifolds introduce multilinear and discrete variants. For bilinear convolutions

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$16

with $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$17 supported on a curved manifold in $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$18, including the sphere $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$19 and the triangle manifold

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$20

sparse domination is obtained for lacunary and full maximal operators from $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$21-improving and continuity estimates. In the discrete setting, the equilateral-triangle maximal operator

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$22

with

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$23

is a discrete analogue of averaging over a codimension-$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$24 configuration manifold. For $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$25 and

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$26

the paper proves type $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$27 bounds and a major-arc approximation

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$28

with dyadic decay for the error term (Palsson et al., 2021, Anderson et al., 2020).

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 $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$29 is bounded up to the closed endpoint $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$30, while the Fefferman–Stein vector-valued inequality fails already at $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$31 and holds only for

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$32

In the spectral multiplier setting on doubling spaces, the finite-family inequality

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$33

is explicitly identified as optimal, by transfer of the sharp Euclidean rate to the operator setting (Sharma et al., 2023, Chen et al., 2024).

Several sharpness statements coexist with substantial open ranges. The FIO-Hardy-space theory states that its maximal function bounds are essentially sharp for

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$34

but says that sharpness is not known for

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$35

The oscillatory compact-manifold theory gives a sharp threshold

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$36

for

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$37

whereas the moment-curve BRK theory proves the sharp $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$38-exponent for all $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$39 and the sharp $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$40-range only for $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$41; for $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$42 it proves $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$43 but only shows the necessary condition

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$44

and explicitly remarks that this is probably not sharp in general. In maximal restriction on the moment curve, the author states that the $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$45-range is not believed to be sharp (Ghosh et al., 2024, Liu et al., 2024, Wang, 1 Sep 2025, Jesurum, 2021).

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,

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$46

vector-valued almost-orthogonality,

$$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$47

and equivalence between $$M_m^{res,\nabla}f(x)=\sup_{t>0}\big|\sqrt t\,\nabla(1+t\Delta)^{-m}f(x)\big|,\qquad M_m^{res,\Delta}f(x)=\sup_{t>0}\big|t\Delta(1+t\Delta)^{-m}f(x)\big|,$$48-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 (Zhang, 2023).