Kolmogorov–Riesz Compactness Theorem

• The Kolmogorov–Riesz compactness theorem is a criterion that characterizes relative compactness in L^p spaces by controlling spatial tightness and limiting small translations.
• It ensures precompactness by preventing mass escape to infinity and uncontrolled oscillations, using conditions like tightness and translation continuity.
• Modern extensions apply the theorem to vector-valued, weighted, and non-Euclidean spaces by replacing translations with localized averages, truncations, or other regularization techniques.

Searching arXiv for recent and foundational papers on the Kolmogorov–Riesz compactness theorem and extensions. {"query":"Kolmogorov Riesz compactness theorem extensions L^p vector-valued weighted metric measure asymptotic L_p", "max_results": 10} The Kolmogorov–Riesz compactness theorem is the classical criterion for relative compactness, equivalently total boundedness, of subsets of $L^p$-spaces in terms of uniform control of tails and small translations. In the modern literature, the same compactness principle is extended far beyond the Euclidean scalar setting: literal translations may be replaced by local averages, conditional expectations, truncations, oscillation restrictions, or microlocal Sobolev decay, but the theorem continues to identify precompactness by ruling out escape to infinity and uncontrolled oscillation in the geometry appropriate to the space under consideration (Hanche-Olsen et al., 2017, Askoura, 2020, Liu et al., 2021, Pilipović et al., 2024).

1. Classical formulation in $L^p(\mathbb R^n)$

In its standard form, for $1\le p<\infty$, a bounded set $F\subset L^p(\mathbb R^n)$ is relatively compact if and only if it satisfies two further conditions: tightness in space and uniform translation continuity. One formulation requires that for every $\varepsilon>0$ there exists $R>0$ such that

$$\int_{|x|>R}|f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }f\in F,$$

and that for every $\varepsilon>0$ there exists $r>0$ such that

$$\int_{\mathbb R^n} |\tau_y f(x)-f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }|y|<r,\ f\in F,$$

where $L^p(\mathbb R^n)$0 (Alves, 20 Jul 2025). Equivalent formulations use $L^p(\mathbb R^n)$1 and describe the same criterion as boundedness, tightness, and uniform translation-continuity (Koshino, 2022).

Because $L^p(\mathbb R^n)$2 is complete, total boundedness and relative compactness coincide. Several modern accounts therefore state the theorem as a characterization of totally bounded subsets of $L^p(\mathbb R^n)$3 rather than of relatively compact ones (Hanche-Olsen et al., 2017).

A recurring refinement concerns boundedness. In the classical three-condition statement, boundedness is listed explicitly, but several papers note that in the Euclidean $L^p(\mathbb R^n)$4 setting it is redundant. For $L^p(\mathbb R^n)$5, Sudakov’s improvement shows that tightness and translation equicontinuity alone already imply total boundedness (Hanche-Olsen et al., 2017). The same redundancy is emphasized in other modern treatments of the classical theorem and its $L^p(\mathbb R^n)$6-based generalizations (Mitkovski et al., 2022, Alves, 20 Jul 2025).

2. Compactness mechanism and equivalent viewpoints

The theorem encodes two obstructions to compactness. Tightness excludes loss of $L^p(\mathbb R^n)$7-mass to spatial infinity, while translation continuity excludes fine-scale oscillation. In the language used in one abstract reformulation, compactness requires that a family cannot “spread out” either spatially or by oscillation (Mitkovski et al., 2022).

In $L^p(\mathbb R^n)$8, the theorem admits a frequency-side analogue. A bounded set $L^p(\mathbb R^n)$9 is precompact if and only if

$1\le p<\infty$0

and

$1\le p<\infty$1

This formulation replaces translation continuity by uniform control of the Fourier tails, and it is obtained by choosing compact operators that truncate both the spatial and frequency domains (Mitkovski et al., 2022).

A second equivalent viewpoint is operator-theoretic. Mazur’s criterion states that if $1\le p<\infty$2 are compact operators on a Banach space with $1\le p<\infty$3 for every $1\le p<\infty$4, then a bounded set $1\le p<\infty$5 is precompact if and only if

$1\le p<\infty$6

In Hilbert spaces this can be weakened to a quadratic form condition. This recasts the Kolmogorov–Riesz theorem as a statement about approximation by compact regularizing operators rather than solely about translations (Mitkovski et al., 2022).

Concrete proofs of the classical theorem and its refinements typically use smoothing or averaging. One short proof of the Sudakov improvement uses repeated small translations to derive uniform $1\le p<\infty$7-boundedness from tightness and translation continuity, and then approximates the family by compact convolution operators built from Steklov averages (Hanche-Olsen et al., 2017). In weighted mixed Lebesgue spaces, a corresponding Fréchet–Kolmogorov proof uses a step-function operator $1\le p<\infty$8 built from averages on a finite grid of cubes, with the finite-dimensional image providing total boundedness (Carro et al., 16 Mar 2025).

3. Replacements for translation in non-Euclidean settings

When the ambient space has no Euclidean translation structure, the theorem persists only after the translation condition is replaced by the correct local regularity surrogate.

Setting	Compactness mechanism	Source
Doubling metric measure $1\le p<\infty$9	Uniform approximation by averages $F\subset L^p(\mathbb R^n)$0 and tightness on bounded sets	(Koshino, 2022)
$F\subset L^p(\mathbb R^n)$1 on locally compact Hausdorff groups	$F\subset L^p(\mathbb R^n)$2-equicontinuity under small left and right translations, plus $F\subset L^p(\mathbb R^n)$3-equivanishing	(Krukowski, 2018)
Stieltjes space $F\subset L^p(\mathbb R^n)$4	Kolmogorov–Riesz on $F\subset L^p(\mathbb R^n)$5 and $F\subset L^p(\mathbb R^n)$6-tail control on jumps $F\subset L^p(\mathbb R^n)$7	(Fernández et al., 2022)
Laguerre and Bessel half-line settings	Equicontinuity under Laguerre or Bessel translations and weighted tail control	(Horváth, 2021)

On a doubling metric measure space $F\subset L^p(\mathbb R^n)$8, the absence of translations is addressed by the average function

$F\subset L^p(\mathbb R^n)$9

Under the standing assumptions that $\varepsilon>0$0 is a Borel-regular Borel metric measure space, every open ball of positive radius has positive and finite measure, $\varepsilon>0$1 is doubling, and $\varepsilon>0$2 as $\varepsilon>0$3, a bounded family $\varepsilon>0$4 is relatively compact if and only if it is uniformly approximable by averages $\varepsilon>0$5 as $\varepsilon>0$6 and uniformly tight outside a bounded set (Koshino, 2022).

For locally compact Hausdorff groups $\varepsilon>0$7 with left Haar measure, the natural replacement is two-sided translation control. A family $\varepsilon>0$8 is relatively compact if and only if it is $\varepsilon>0$9-bounded, $R>0$0-equicontinuous under small left and right translations, and $R>0$1-equivanishing. In $R>0$2, boundedness is again redundant (Krukowski, 2018).

The Stieltjes setting is structurally different because the measure may have atoms. If $R>0$3 is nondecreasing and left-continuous, with jump set $R>0$4, then

$R>0$5

where $R>0$6 is the continuous part of $R>0$7 and $R>0$8 is its pseudoinverse. The Kolmogorov–Riesz theorem in $R>0$9 therefore splits into an ordinary $$\int_{|x|>R}|f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }f\in F,$$0-criterion for $$\int_{|x|>R}|f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }f\in F,$$1 and an $$\int_{|x|>R}|f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }f\in F,$$2-criterion for the jump sequence $$\int_{|x|>R}|f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }f\in F,$$3 (Fernández et al., 2022).

In Laguerre and Bessel harmonic analysis on $$\int_{|x|>R}|f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }f\in F,$$4, ordinary translations are replaced by generalized translation operators $$\int_{|x|>R}|f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }f\in F,$$5 and $$\int_{|x|>R}|f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }f\in F,$$6. The corresponding compactness theorems state that a bounded family is precompact if and only if it is tight in the weighted half-line $$\int_{|x|>R}|f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }f\in F,$$7 norm and equicontinuous under the relevant generalized translations on bounded $$\int_{|x|>R}|f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }f\in F,$$8-ranges (Horváth, 2021).

4. Vector-valued, weighted, and variable-exponent extensions

A decisive vector-valued extension is the compactness criterion in the Bochner space $$\int_{|x|>R}|f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }f\in F,$$9, where $\varepsilon>0$0 is a finite measure space and $\varepsilon>0$1 is a Banach space. A subset $\varepsilon>0$2 is relatively norm compact if and only if three conditions hold: the Fréchet oscillation condition, integral tightness, and $\varepsilon>0$3-uniform integrability. Integral tightness requires that for every measurable set $\varepsilon>0$4,

$\varepsilon>0$5

is relatively norm compact in $\varepsilon>0$6. The Fréchet oscillation condition requires a finite partition and small essential oscillation on each cell after removal of a set of small measure, and $\varepsilon>0$7-uniform integrability is defined by uniform integrability of $\varepsilon>0$8 (Askoura, 2020). This replaces Euclidean translations by conditional expectations on finite measurable partitions and local oscillation control.

Weighted and matrix-weighted settings retain the same tripartite structure but alter the regularity condition. For $\varepsilon>0$9 and a matrix weight $r>0$0, a subset $r>0$1 is totally bounded if and only if it is bounded, vanishes uniformly at infinity, and satisfies

$r>0$2

Here the averaging operator $r>0$3 replaces literal translation, and the $r>0$4 hypothesis is used through the boundedness of the Christ–Goldberg maximal operator and a matrix-weighted Lebesgue differentiation theorem (Liu et al., 2021).

Weighted mixed Lebesgue spaces on $r>0$5 satisfy an exact Fréchet–Kolmogorov theorem of the familiar form: relative compactness in $r>0$6 is equivalent to boundedness, vanishing at infinity, and joint translation equicontinuity in the mixed norm (Carro et al., 16 Mar 2025). In weighted variable Lebesgue spaces $r>0$7, the translation condition is replaced by a local averaged difference in $r>0$8: $r>0$9 together with uniform boundedness and uniform vanishing at infinity (Kakaroumpas et al., 26 May 2026).

Weighted variable exponent amalgam and Sobolev spaces use yet another equivalent regularization device: uniform mollifier approximation. In $$\int_{\mathbb R^n} |\tau_y f(x)-f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }|y|<r,\ f\in F,$$0 and in $$\int_{\mathbb R^n} |\tau_y f(x)-f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }|y|<r,\ f\in F,$$1, relative compactness is characterized by boundedness, weighted tightness, and

$$\int_{\mathbb R^n} |\tau_y f(x)-f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }|y|<r,\ f\in F,$$2

with derivative-wise versions for Sobolev spaces (Aydin et al., 2019).

5. Beyond Banach $$\int_{\mathbb R^n} |\tau_y f(x)-f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }|y|<r,\ f\in F,$$3: asymptotic and microlocal compactness

In asymptotic $$\int_{\mathbb R^n} |\tau_y f(x)-f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }|y|<r,\ f\in F,$$4 spaces $$\int_{\mathbb R^n} |\tau_y f(x)-f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }|y|<r,\ f\in F,$$5, the classical theorem changes because the $$\int_{\mathbb R^n} |\tau_y f(x)-f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }|y|<r,\ f\in F,$$6-norm

$$\int_{\mathbb R^n} |\tau_y f(x)-f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }|y|<r,\ f\in F,$$7

is not homogeneous. A family $$\int_{\mathbb R^n} |\tau_y f(x)-f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }|y|<r,\ f\in F,$$8 is totally bounded if and only if it satisfies three conditions: tightness at infinity in the $$\int_{\mathbb R^n} |\tau_y f(x)-f(x)|^p\,dx<\varepsilon^p \qquad\text{for all }|y|<r,\ f\in F,$$9-norm, translation continuity in the $L^p(\mathbb R^n)$00-norm, and almost equiboundedness,

$L^p(\mathbb R^n)$01

The additional almost equiboundedness condition is essential because truncation at level $L^p(\mathbb R^n)$02 suppresses large amplitudes (Alves, 20 Jul 2025).

On an arbitrary measure space $L^p(\mathbb R^n)$03, the asymptotic theory becomes purely truncation-based. In $L^p(\mathbb R^n)$04, total boundedness is equivalent to uniform approximability by truncations and total boundedness in ordinary $L^p(\mathbb R^n)$05 of every truncated family: $L^p(\mathbb R^n)$06 This is presented as a measure-theoretic counterpart to the Kolmogorov–Riesz theorem (Alves, 21 Apr 2026).

A microlocal extension appears in spaces of distributions with constrained Sobolev wave front set. For a smooth manifold $L^p(\mathbb R^n)$07, a vector bundle $L^p(\mathbb R^n)$08, and a closed conic set $L^p(\mathbb R^n)$09, the space

$L^p(\mathbb R^n)$10

admits a compactness criterion in terms of the Sobolev compactness wave front set. If $L^p(\mathbb R^n)$11 is bounded, then

$L^p(\mathbb R^n)$12

Here compactness is controlled by uniform Sobolev decay away from the forbidden microlocal set $L^p(\mathbb R^n)$13, and the proof uses Gérard’s idea together with a topological embedding into a product of $L^p(\mathbb R^n)$14-spaces where the classical Kolmogorov–Riesz theorem can be applied factorwise (Pilipović et al., 2024).

6. Applications and conceptual significance

The theorem functions both as a compactness criterion and as a proof engine. In $L^p(\mathbb R^n)$15 and $L^p(\mathbb R^n)$16, a Fréchet–Kolmogorov argument applied to the bounded transform

$L^p(\mathbb R^n)$17

yields a new proof of compactness that avoids Korn or Poincaré–Korn inequalities. The crucial step is a uniform translation estimate for $L^p(\mathbb R^n)$18, after which the classical $L^p(\mathbb R^n)$19 criterion gives strong compactness of the transformed sequence (Almi et al., 2021).

Operator theory supplies further applications. In the Bessel setting $L^p(\mathbb R^n)$20, a new Fréchet–Kolmogorov theorem underlies the characterization

$L^p(\mathbb R^n)$21

for $L^p(\mathbb R^n)$22 (Duong et al., 2016). In matrix-weighted analysis, the compactness of commutators of rough singular integrals is proved by verifying the three matrix-weighted Kolmogorov–Riesz conditions—boundedness, vanishing at infinity, and translation equicontinuity—for the image of a bounded set under a truncated commutator (Laukkarinen et al., 7 Mar 2025).

The compactness principle also supports extrapolation frameworks. In weighted mixed Lebesgue spaces, a Fréchet–Kolmogorov theorem is combined with Rubio de Francia-type extrapolation to obtain uniform compactness results for commutators of Calderón–Zygmund operators and for pseudodifferential operators (Carro et al., 16 Mar 2025). In weighted variable Lebesgue spaces, a weighted Riesz–Kolmogorov theorem is used to prove interpolation and extrapolation of compactness for multilinear operators, with applications to commutators of multilinear $L^p(\mathbb R^n)$23-Calderón–Zygmund operators, multilinear fractional integrals, and multilinear Fourier multipliers (Kakaroumpas et al., 26 May 2026).

A common misconception is that the Kolmogorov–Riesz theorem is intrinsically a theorem about Euclidean translations. The contemporary literature shows a broader picture. In some settings translations remain literal; in others they become averages over balls, conditional expectations on partitions, truncation operators, generalized Bessel or Laguerre translations, or uniform microlocal decay conditions. Another common misconception is that boundedness is always an independent hypothesis; several papers show that in the classical Euclidean theorem, and in closely related settings, it can be derived from the other compactness conditions (Hanche-Olsen et al., 2017, Krukowski, 2018, Mitkovski et al., 2022).

Across these extensions, the invariant content remains stable: compactness is equivalent to the impossibility of losing mass or oscillation in the directions relevant to the ambient topology. In Euclidean $L^p(\mathbb R^n)$24, those directions are spatial infinity and small translations; in vector-valued, weighted, asymptotic, and microlocal settings, the same compactness philosophy is expressed through the corresponding replacement structures (Askoura, 2020, Pilipović et al., 2024, Alves, 21 Apr 2026).