Helical Maximal Operator in ℝ³

Updated 6 July 2026

Helical maximal operator is defined as the supremum of averages over dilated, compact segments of a helix, characterized by nonvanishing curvature and torsion.
It serves as a canonical model in ℝ³ for maximal averages over nondegenerate space curves, with sharp Lᵖ boundedness established precisely for p>3 using microlocal and square function techniques.
Weighted and lacunary variants extend the theory by leveraging local smoothing, dyadic decompositions, and Fourier-analytic methods to control anisotropic spatial behaviors.

The helical maximal operator is the maximal averaging operator associated to dilates of a compactly cut off helix segment in $\mathbb R^3$ . For a smooth curve $\gamma:I\to\mathbb R^3$ and a cutoff $\chi\in C_c^\infty(I)$ , the basic averaging operators are

$A_t f(x)=\int_I f(x-t\gamma(s))\,\chi(s)\,ds,$

and the corresponding maximal operator is

$Mf(x)=\sup_{t>0}|A_t f(x)|.$

For the standard helix

$\gamma(s)=(a\cos s,\ a\sin s,\ bs),\qquad a,b\neq 0,$

this operator is a model case of maximal averages over non-degenerate space curves, and its sharp unweighted $L^p(\mathbb R^3)$ boundedness range is $p>3$ (Ko et al., 2021). Subsequent work has developed its local $L^p\to L^q$ theory, weighted theory, and lacunary theory (Beltran et al., 2023, Ghosh et al., 20 Feb 2026).

1. Geometric definition and nondegeneracy

The helical maximal operator is usually studied in the broader class of smooth space curves satisfying

$|\det(\gamma'(s),\gamma''(s),\gamma'''(s))|\ge c_0>0,$

or equivalently having nonvanishing curvature and nonzero torsion. This condition is the natural nondegeneracy hypothesis for averages over space curves in $\gamma:I\to\mathbb R^3$ 0, and the helix is the model example (Ko et al., 2021).

For the standard helix,

$\gamma:I\to\mathbb R^3$ 1

and

$\gamma:I\to\mathbb R^3$ 2

Its curvature and torsion are constant: $\gamma:I\to\mathbb R^3$ 3 Thus the helix lies squarely in the class of smooth non-degenerate space curves treated in the sharp theory (Ko et al., 2021).

The cutoff $\gamma:I\to\mathbb R^3$ 4 is compactly supported in the curve parameter $\gamma:I\to\mathbb R^3$ 5, so the averages are taken over a compact arc of the helix rather than over the entire infinite helix. Geometrically, $\gamma:I\to\mathbb R^3$ 6 averages $\gamma:I\to\mathbb R^3$ 7 over the translated and dilated copy $\gamma:I\to\mathbb R^3$ 8; for the model helix $\gamma:I\to\mathbb R^3$ 9, this is a truncated helix of radius $\chi\in C_c^\infty(I)$ 0 and pitch proportional to $\chi\in C_c^\infty(I)$ 1 (Ghosh et al., 20 Feb 2026).

Three related maximal operators occur in the literature. The full maximal operator is

$\chi\in C_c^\infty(I)$ 2

the local maximal operator is

$\chi\in C_c^\infty(I)$ 3

and the lacunary maximal operator is

$\chi\in C_c^\infty(I)$ 4

The global theory is reduced to the local slab $\chi\in C_c^\infty(I)$ 5 by Littlewood–Paley decomposition and scaling (Ko et al., 2021).

2. Sharp unweighted $\chi\in C_c^\infty(I)$ 6 boundedness

The sharp diagonal theorem states that if $\chi\in C_c^\infty(I)$ 7 is smooth with nonvanishing curvature and torsion and $\chi\in C_c^\infty(I)$ 8 is a nontrivial, nonnegative, smooth function supported in $\chi\in C_c^\infty(I)$ 9, then

$A_t f(x)=\int_I f(x-t\gamma(s))\,\chi(s)\,ds,$ 0

for all $A_t f(x)=\int_I f(x-t\gamma(s))\,\chi(s)\,ds,$ 1 if and only if $A_t f(x)=\int_I f(x-t\gamma(s))\,\chi(s)\,ds,$ 2 (Ko et al., 2021). For the helix, this means precisely that the helical maximal operator is bounded on $A_t f(x)=\int_I f(x-t\gamma(s))\,\chi(s)\,ds,$ 3 for $A_t f(x)=\int_I f(x-t\gamma(s))\,\chi(s)\,ds,$ 4, and boundedness fails for $A_t f(x)=\int_I f(x-t\gamma(s))\,\chi(s)\,ds,$ 5.

This result sharpened the previous known bounds for the helical maximal function from $A_t f(x)=\int_I f(x-t\gamma(s))\,\chi(s)\,ds,$ 6 to the sharp range $A_t f(x)=\int_I f(x-t\gamma(s))\,\chi(s)\,ds,$ 7. One formulation established the sharp $A_t f(x)=\int_I f(x-t\gamma(s))\,\chi(s)\,ds,$ 8 boundedness of the helical maximal function by introducing a new microlocal smoothing estimate for averages along dilates of the helix, obtained via a square function analysis (Beltran et al., 2021). A parallel formulation placed the helix inside the full class of smooth space curves with nonvanishing curvature and torsion and identified the same threshold $A_t f(x)=\int_I f(x-t\gamma(s))\,\chi(s)\,ds,$ 9 as both sufficient and necessary (Ko et al., 2021).

The necessity for $Mf(x)=\sup_{t>0}|A_t f(x)|.$ 0 is proved by adapting Stein’s classical example. In coordinates adapted to the frame

$Mf(x)=\sup_{t>0}|A_t f(x)|.$ 1

which is a basis because of the nonvanishing determinant condition, one constructs $Mf(x)=\sup_{t>0}|A_t f(x)|.$ 2 such that

$Mf(x)=\sup_{t>0}|A_t f(x)|.$ 3

on a nonempty open set. For the helix, the construction applies verbatim because

$Mf(x)=\sup_{t>0}|A_t f(x)|.$ 4

for every $Mf(x)=\sup_{t>0}|A_t f(x)|.$ 5 (Ko et al., 2021).

The theorem is therefore sharp in the literal sense: for compactly cut off helical arcs in $Mf(x)=\sup_{t>0}|A_t f(x)|.$ 6,

$Mf(x)=\sup_{t>0}|A_t f(x)|.$ 7

3. Fourier-analytic structure and proof mechanism

The proof of sufficiency for $Mf(x)=\sup_{t>0}|A_t f(x)|.$ 8 proceeds by reducing the maximal estimate to a frequency-localized spacetime estimate on $Mf(x)=\sup_{t>0}|A_t f(x)|.$ 9. If $\gamma(s)=(a\cos s,\ a\sin s,\ bs),\qquad a,b\neq 0,$ 0 is supported in the dyadic annulus

$\gamma(s)=(a\cos s,\ a\sin s,\ bs),\qquad a,b\neq 0,$ 1

it suffices to prove

$\gamma(s)=(a\cos s,\ a\sin s,\ bs),\qquad a,b\neq 0,$ 2

In fact, a stronger estimate is established against $\gamma(s)=(a\cos s,\ a\sin s,\ bs),\qquad a,b\neq 0,$ 3-dimensional measures $\gamma(s)=(a\cos s,\ a\sin s,\ bs),\qquad a,b\neq 0,$ 4 in $\gamma(s)=(a\cos s,\ a\sin s,\ bs),\qquad a,b\neq 0,$ 5: $\gamma(s)=(a\cos s,\ a\sin s,\ bs),\qquad a,b\neq 0,$ 6 This weighted spacetime formulation is one of the mechanisms by which maximal estimates are recovered by linearization (Ko et al., 2021).

A central normalization reduces each short curve segment to a perturbation of the model cubic curve

$\gamma(s)=(a\cos s,\ a\sin s,\ bs),\qquad a,b\neq 0,$ 7

For each small interval around $\gamma(s)=(a\cos s,\ a\sin s,\ bs),\qquad a,b\neq 0,$ 8, the curve is renormalized by

$\gamma(s)=(a\cos s,\ a\sin s,\ bs),\qquad a,b\neq 0,$ 9

and one studies

$L^p(\mathbb R^3)$ 0

Because $L^p(\mathbb R^3)$ 1, each short helical arc becomes, after affine normalization, a member of a fixed compact class of perturbations of $L^p(\mathbb R^3)$ 2 (Ko et al., 2021).

On the Fourier side, the difficult geometry is organized by the critical point $L^p(\mathbb R^3)$ 3 determined by

$L^p(\mathbb R^3)$ 4

together with

$L^p(\mathbb R^3)$ 5

The set $L^p(\mathbb R^3)$ 6 is a cone-like surface in frequency space. The multiplier is then split into a nondegenerate region, a near-conic region $L^p(\mathbb R^3)$ 7, an error region $L^p(\mathbb R^3)$ 8, and a main conic region $L^p(\mathbb R^3)$ 9. In the nondegenerate region, estimates from Pramanik–Seeger combined with Bourgain–Demeter decoupling for the cone provide the required bounds. The harder contribution is the degenerate near-conic portion, where a new induction-on-scales argument is developed directly for the maximal operator (Ko et al., 2021).

The multilinear heart of the argument is a quadrilinear estimate for separated intervals $p>3$ 0: $p>3$ 1 for $p>3$ 2 and $p>3$ 3-dimensional weights $p>3$ 4. To reach this, the near-conic pieces are expressed as sums of adjoint restriction operators with phase functions $p>3$ 5, and a version of the Bennett–Carbery–Tao multilinear restriction theorem is proved for $p>3$ 6 hypersurfaces. A technical point is that the phase surfaces associated to $p>3$ 7 are only $p>3$ 8, not $p>3$ 9, so the multilinear restriction input is adapted to that regularity (Ko et al., 2021).

The broad–narrow decomposition is encoded in the pointwise estimate

$L^p\to L^q$ 0

with separated $L^p\to L^q$ 1. The single-interval term is handled by rescaling and induction, and the quadrilinear term by multilinear restriction. The resulting recursive inequality closes when $L^p\to L^q$ 2, which is exactly where the threshold $L^p\to L^q$ 3 appears in the sufficiency argument (Ko et al., 2021). The helical-specific formulation in (Beltran et al., 2021) emphasizes that the new microlocal smoothing estimate is established via square function analysis.

4. Off-diagonal local estimates

The local helical maximal function,

$L^p\to L^q$ 4

has an essentially complete off-diagonal $L^p\to L^q$ 5 theory, except for endpoints (Beltran et al., 2023). The admissible region is the triangle

$L^p\to L^q$ 6

and the main theorem states that

$L^p\to L^q$ 7

for all $L^p\to L^q$ 8, where $L^p\to L^q$ 9 is the half-open diagonal segment joining $|\det(\gamma'(s),\gamma''(s),\gamma'''(s))|\ge c_0>0,$ 0 and $|\det(\gamma'(s),\gamma''(s),\gamma'''(s))|\ge c_0>0,$ 1, with the endpoint $|\det(\gamma'(s),\gamma''(s),\gamma'''(s))|\ge c_0>0,$ 2 excluded (Beltran et al., 2023).

Sharpness is expressed by three necessary conditions: $|\det(\gamma'(s),\gamma''(s),\gamma'''(s))|\ge c_0>0,$ 3 The first comes from Hörmander’s theorem on translation-invariant operators. The second is a Knapp-type example using the anisotropic box

$|\det(\gamma'(s),\gamma''(s),\gamma'''(s))|\ge c_0>0,$ 4

and the third is a dimensional or fixed-time example based on the $|\det(\gamma'(s),\gamma''(s),\gamma'''(s))|\ge c_0>0,$ 5-neighborhood $|\det(\gamma'(s),\gamma''(s),\gamma'''(s))|\ge c_0>0,$ 6 of the curve (Beltran et al., 2023).

The maximal theorem is derived from a local smoothing estimate for oscillatory multipliers. At the critical vertex $|\det(\gamma'(s),\gamma''(s),\gamma'''(s))|\ge c_0>0,$ 7, the decisive estimate is

$|\det(\gamma'(s),\gamma''(s),\gamma'''(s))|\ge c_0>0,$ 8

The analysis combines two geometries in spacetime Fourier space: a $|\det(\gamma'(s),\gamma''(s),\gamma'''(s))|\ge c_0>0,$ 9-dimensional cone $\gamma:I\to\mathbb R^3$ 00, associated to first-order stationarity, and a $\gamma:I\to\mathbb R^3$ 01-dimensional cone $\gamma:I\to\mathbb R^3$ 02, associated to simultaneous first- and second-order degeneracy. The paper’s key new input is a localised trilinear restriction theorem,

$\gamma:I\to\mathbb R^3$ 03

whose gain $\gamma:I\to\mathbb R^3$ 04 comes from localisation near the more degenerate cone (Beltran et al., 2023).

This multilinear estimate is combined with stationary phase normal forms, an $\gamma:I\to\mathbb R^3$ 05 estimate from decoupling,

$\gamma:I\to\mathbb R^3$ 06

the trivial bound

$\gamma:I\to\mathbb R^3$ 07

and a Bourgain–Guth broad–narrow decomposition. The outcome is the optimal off-diagonal local mapping region, up to endpoint issues (Beltran et al., 2023).

5. Weighted and lacunary variants

The weighted theory studies boundedness on power-weighted spaces

$\gamma:I\to\mathbb R^3$ 08

For the full helical maximal operator, the necessary conditions are

$\gamma:I\to\mathbb R^3$ 09

and the same necessity already holds for the local maximal operator. For each fixed time $\gamma:I\to\mathbb R^3$ 10, however, the complete range is larger: $\gamma:I\to\mathbb R^3$ 11 A pointwise estimate central to this fixed-time theory is

$\gamma:I\to\mathbb R^3$ 12

for $\gamma:I\to\mathbb R^3$ 13 if and only if

$\gamma:I\to\mathbb R^3$ 14

and together with Jawerth’s factorization theorem this yields the full fixed-time range $\gamma:I\to\mathbb R^3$ 15 (Ghosh et al., 20 Feb 2026).

For the full maximal operator, the main weighted theorem distinguishes two $\gamma:I\to\mathbb R^3$ 16-ranges. If $\gamma:I\to\mathbb R^3$ 17, then

$\gamma:I\to\mathbb R^3$ 18

and this is optimal except possibly at the endpoint $\gamma:I\to\mathbb R^3$ 19. If $\gamma:I\to\mathbb R^3$ 20, then

$\gamma:I\to\mathbb R^3$ 21

Thus the upper endpoint $\gamma:I\to\mathbb R^3$ 22 is the necessary one throughout, while the lower range for $\gamma:I\to\mathbb R^3$ 23 remains incomplete (Ghosh et al., 20 Feb 2026).

The proof for negative weights is driven by dyadic local maximal bounds of the form

$\gamma:I\to\mathbb R^3$ 24

At the borderline $\gamma:I\to\mathbb R^3$ 25, the central step is

$\gamma:I\to\mathbb R^3$ 26

The proof decomposes space into the tiny ball $\gamma:I\to\mathbb R^3$ 27 and annuli $\gamma:I\to\mathbb R^3$ 28, then exploits localized gains such as

$\gamma:I\to\mathbb R^3$ 29

For $\gamma:I\to\mathbb R^3$ 30, an endpoint-type estimate at $\gamma:I\to\mathbb R^3$ 31,

$\gamma:I\to\mathbb R^3$ 32

combined with Stein–Weiss interpolation, yields the range

$\gamma:I\to\mathbb R^3$ 33

This explains why the method closes exactly at $\gamma:I\to\mathbb R^3$ 34 and why a gap remains for $\gamma:I\to\mathbb R^3$ 35 (Ghosh et al., 20 Feb 2026).

The lacunary maximal operator has a complete weighted characterization: $\gamma:I\to\mathbb R^3$ 36 In particular, the lacunary operator has exactly the same weighted range as the fixed-time averages $\gamma:I\to\mathbb R^3$ 37, unlike the full maximal operator, whose range loses $\gamma:I\to\mathbb R^3$ 38 powers on the upper end: $\gamma:I\to\mathbb R^3$ 39 The lacunary proof uses square functions, $\gamma:I\to\mathbb R^3$ 40-decay from Fourier decay and orthogonality, weak $\gamma:I\to\mathbb R^3$ 41-type with $\gamma:I\to\mathbb R^3$ 42 loss via randomization and a Calderón–Zygmund kernel difference estimate, and interpolation (Ghosh et al., 20 Feb 2026).

The helical maximal operator belongs to the genuinely spatial, nonplanar theory of averages over space curves. The decisive geometric feature is torsion: the helix is not merely curved, but has nonvanishing curvature and torsion everywhere. The sharp theorems do not extend, as stated, to degenerate curves where curvature or torsion vanishes. The papers also contrast space curves with planar curves, noting that in the plane maximal averages over curved arcs are older and use methods such as incidence geometry and local smoothing, whereas in space the slower Fourier decay makes the problem harder (Ko et al., 2021).

A basic analytic manifestation of this difficulty is the decay of the fixed-time multiplier

$\gamma:I\to\mathbb R^3$ 43

This decay is too weak by itself to control the full maximal operator sharply. The weighted theory therefore relies on local smoothing estimates, dyadic frequency decompositions, decomposition near geometric singular sets such as

$\gamma:I\to\mathbb R^3$ 44

and delicate spatial localization near the origin (Ghosh et al., 20 Feb 2026).

One common misconception is to treat the “helical maximal operator” as averaging over the entire infinite helix. The sharp theorem described above is for compactly cut off helical arcs, because $\gamma:I\to\mathbb R^3$ 45 is compactly supported in the curve parameter $\gamma:I\to\mathbb R^3$ 46. The global maximal parameter is $\gamma:I\to\mathbb R^3$ 47, but the curve itself is truncated in $\gamma:I\to\mathbb R^3$ 48 (Ko et al., 2021).

A distinct but related theme appears in the Heisenberg-group literature. The paper on the circular maximal operator on Heisenberg radial functions studies a reduced Euclidean maximal operator arising from the Heisenberg group law, where the reduced averaging curves have a circular lift with oscillatory vertical component. That work is relevant to helical-type geometry because the vertical coordinate varies sinusoidally with the angular parameter, but it does not study averages over literal Euclidean helices $\gamma:I\to\mathbb R^3$ 49. Its theorem is that for $\gamma:I\to\mathbb R^3$ 50,

$\gamma:I\to\mathbb R^3$ 51

for Heisenberg radial functions, and the paper emphasizes that the reduced operator fails both the usual rotational curvature and cinematic curvature conditions (Beltran et al., 2019). This suggests that “helical” phenomena in maximal averaging can also arise from group-geometric lifts rather than from fixed Euclidean helices.

In the Euclidean setting, the modern theory therefore presents the helical maximal operator as a canonical non-degenerate space-curve maximal operator. Its sharp diagonal theorem is $\gamma:I\to\mathbb R^3$ 52, its local off-diagonal theory is optimal up to endpoints, and its weighted theory is now well understood for $\gamma:I\to\mathbb R^3$ 53 and partially understood for $\gamma:I\to\mathbb R^3$ 54, with the endpoint $\gamma:I\to\mathbb R^3$ 55 for the full maximal operator remaining open (Beltran et al., 2023, Ghosh et al., 20 Feb 2026).