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Quantitative $\Gamma$-Convergence: A Variational Extrapolation Framework

Oscar Dominguez, Mario Milman

arXiv:2610.06645v1math.FAcs.NAmath.APmath.NA

TL;DR

Γ-convergence provides stability of minimizers and convergence of energies, but does not intrinsically give approximation rates. The paper develops variational K-functionals that connect Γ-limits to Gagliardo relative completions and apply this quantitative framework to discrete ROF inverse regularity. It also identifies a limit on exact formulae: in general, the extrapolation results are expressed only up to equivalence constants.

  • Problem

    De Giorgi’s Γ-convergence does not intrinsically supply rates of approximation, while classical K-functionals are not directly suited to evolving sequences of functional penalties.

    Source: §1.1 · p. 2

  • Method

    The paper combines De Giorgi’s variational framework with Gagliardo relative completions using variational K-functionals that replace fixed norms with sequences of functionals.

    Source: §1.1 · p. 2

  • Results

    For discrete ROF approximations, rates of convergence of minimizers and minimum energies are controlled by a variational K-functional of the observed image, which can be estimated by a classical K-functional.

    Source: §1.2 · p. 3

  • Takeaways & Limitations

    Brezis–Van Schaftingen–Yung spaces provide a natural framework for error bounds in discretizations of total variation in the ROF denoising model.

    Source: §1.2 · p. 4

    In most settings, extrapolation results are expressed only up to equivalence constants because K-functionals are rarely computable exactly.

    Source: §1.8 · p. 9

Abstract

from arXiv · show

This paper introduces a new extrapolation framework within the Calculus of Variations. We develop a robust quantitative theory of the $\Gamma$-convergence approach of De Giorgi by combining it with Gagliardo's work on relative completions in interpolation theory. A central new tool is the extrapolation of \emph{variational $K$-functionals}, which associates to an arbitrary family of functionals a dynamic penalty method that simultaneously characterizes their $\Gamma$-limits and yields explicit rates of convergence for minimizers. As a technical foundation we prove compactness extrapolation theorems also involving Gagliardo completions. To illustrate the mechanism, we apply the technique to variational models arising in image processing, notably the Rudin--Osher--Fatemi (ROF) and Chan--Esedoglu models. Our main result in this setting is a contribution to the \emph{inverse regularity problem} associated to discrete approximations of the ROF model: we show that the required regularity of the observed image that guarantees a prescribed optimal rate of convergence is measured in terms of the decay of variational $K$-functionals. Surprisingly, our characterization is explicitly captured by using the recently introduced spaces of Brezis, Van Schaftingen, and Yung. Moreover, our methods unify, and considerably extend, classical pointwise convergence and $\Gamma$-versions of the Bourgain--Brezis--Mironescu and Maz'ya--Shaposhnikova formulae, as well as related results in a variety of settings that involve BMO-type functionals, heat kernels, Sobolev spaces on metric measure spaces, ... By bridging De Giorgi's variational convergence with Gagliardo's completion through variational $K$-functionals and extrapolation of compactness, this work establishes a quantitative approach to the Calculus of Variations.

1. Introduction

The introduction presents variational K-functionals as a quantitative extension of Γ-convergence, addressing the lack of intrinsic convergence rates in De Giorgi’s theory. It applies this framework to the inverse regularity problem for discrete ROF approximations, relating minimizer rates to regularity measured through K-functionals and Brezis–Van Schaftingen–Yung spaces.

1.1. Overview.

  • Γ-convergence ensures stability of minimizers and convergence of energies but does not intrinsically provide approximation rates.Classical K-functionals are not directly suited to sequences whose penalization changes with the approximation parameter.Source: §1.1 · p. 2
  • The variational K-functional combines De Giorgi’s framework with Gagliardo relative completions to provide a dynamic, quantitative characterization of Γ-limits.The paper also develops compactness extrapolation and analyzes rates of variational limits.Source: §1.1 · p. 2

1.2. Image processing: the inverse regularity problem.

  • For discrete ROF approximations, optimal rates of convergence of minimizers and minimum energies are controlled by a variational K-functional of the observed image.That variational K-functional can in turn be estimated by a classical K-functional.Source: §1.2 · p. 3
  • The ROF inverse regularity question asks what regularity of the observed image guarantees a prescribed minimizer rate O(n^-α), for α in (0, 1).The paper identifies Brezis–Van Schaftingen–Yung spaces as a natural framework for error bounds in ROF total-variation discretization.Source: §1.2 · pp. 3–4
  • Outside the Hilbert setting, the approach based on the Hilbert-space identity does not extend because a suitable analogue is lacking.The paper instead invokes a characterization of the relevant interpolation spaces using Brezis–Van Schaftingen–Yung spaces.Source: §1.2 · p. 4

1.3. Gagliardo completions and extrapolation.

The section connects Gagliardo relative completions and extrapolation through K-functionals, then points to applications in asymptotic formulae for Sobolev norms.

1.3. Gagliardo completions and extrapolation.

  • The interplay between Gagliardo relative completions and K-functional formulations is central to connecting those completions with extrapolation.The paper describes this connection through pointwise limits associated with relative completions.Source: §1.3 · p. 5
  • For the pair (Lp(RN), Wdot 1,p(RN)), the extrapolation approach yields new approaches to asymptotic Bourgain–Brezis–Mironescu and Maz’ya–Shaposhnikova formulae.The limit case p = 1 also holds when the gradient norm is replaced by the BV seminorm.Source: §1.3 · p. 6

1.4. A new key idea: the connection De Giorgi–Gagliardo.

This section proposes a dynamic penalty method to connect De Giorgi Γ-convergence with Gagliardo extrapolation. The classical K-functional is too rigid for this purpose, motivating a version that incorporates a sequence of functionals.

1.4. A new key idea: the connection De Giorgi–Gagliardo.

  • The proposed connection treats Γ-convergence as a structural link between De Giorgi variational limits and Gagliardo relative completions.The paper notes that the setting of normed spaces suffices for its purposes, although Γ-convergence also has a metric-space formulation.Source: §1.4 · pp. 6–7
  • The section asks whether a parameterized penalty method can characterize Γ-convergence while also extracting explicit rates for minimizers.Classical K-functionals use two fixed endpoint spaces, whereas the target sequence of functionals changes with the approximation parameter.Source: §1.4 · p. 7
  • The construction replaces one endpoint of the classical K-functional with a sequence of functionals, allowing the penalty to evolve dynamically.This is motivated by the fact that the sequence whose Γ-limit is sought is itself part of the controlled K-functional.Source: §1.4 · p. 7

1.5. Γ-convergence as Gagliardo extrapolation of variational K-functionals.

The paper defines a variational K-functional by pairing distance in X with a parameter-weighted functional penalty. It shows that Γ-convergence corresponds to Gagliardo completion and thereby brings Γ-convergence into a quantitative K-functional framework.

1.5. Γ-convergence as Gagliardo extrapolation of variational K-functionals.

  • The variational K-functional dynamically replaces fixed target norms with an evolving sequence of functional penalties.The absence of norm properties makes its analysis more delicate than that of the classical K-functional.Source: §1.5 · p. 7
  • Theorem 1.3 characterizes Γ-convergence through an extrapolation formula for variational K-functionals.The formula’s infimum ranges over all infinitesimal sequences.Source: §1.5 · p. 7
  • Under this characterization, Γ-convergence corresponds to Gagliardo completion and enters the quantitative setting provided by K-functionals.The paper presents this as combining the De Giorgi and Gagliardo theories.Source: §1.5 · p. 7

1.6. Variational K-functionals and extrapolation of Γ-convergence.

The section illustrates Γ-convergence extrapolation by applying Theorem 1.3 to a family of functionals indexed by θ. Two inequalities relate that family’s variational K-functional to the classical K-functional.

1.6. Variational K-functionals and extrapolation of Γ-convergence.

  • The Γ-version of (1.15) is obtained as an application of Theorem 1.3 to the case θ → 1−.The section treats θ → 0+ by symmetry.Source: §1.6 · p. 7
  • The proof considers functionals Fθ on A0 that take the value +∞ outside the interpolation space (A0, A1)θ,p.This specifies the family to which the variational K-functional method is applied.Source: §1.6 · p. 8
  • For f ∈ A0, the proof uses inequalities comparing K(t, f; A0, Fθ) with K(t1/θ, f; A0, A1).The displayed inequality gives K(t1/θ, f; A0, A1) ≤ K(t, f; A0, Fθ) for every t > 0.Source: §1.6 · p. 8

1.7. Some applications of extrapolation of Γ-convergence: a unified perspective to BBM and MS formulae.

The section applies Γ-convergence extrapolation across function-space settings and connects it to compactness and stability of minimizers. Its applications include BBM- and MS-type limits, while exact formulae can require an additional normalization step.

1.7. Some applications of extrapolation of Γ-convergence: a unified perspective to BBM and MS formulae.

  • The applications include BBM formulae for rearrangement-invariant Sobolev spaces on R^N and metric measure spaces, plus BMO-type and heat-content energies and MS extensions.The metric-space result includes the approach to Sobolev spaces on Cheeger spaces proposed in.Source: §1.7 · pp. 8–9
  • The method uses a carefully chosen pair of spaces and control of its K-functionals, despite technical differences among applications.The section presents this as a unified starting procedure.Source: §1.7 · p. 9

1.8. Computing exact limit formulae.

  • Exact K-functional computation is rare, so asymptotic formulae stated through equivalent expressions generally hold only up to multiplicative constants.The section describes working directly with those expressions and normalizing them as a route to exact limits.Source: §1.8 · p. 9

1.9. Equi-coercivity via extrapolation of compactness.

  • With compact embedding A1 ↪ A0 and θn → 1, the compactness extrapolation theorem yields a subsequence converging in A0.The theorem’s sequence assumptions are introduced in the supplied passage, while the compact embedding is its stated space assumption.Source: §1.9 · p. 9
  • The compactness result combines Cwikel’s one-sided compactness theorem [36] with special reiteration formulae and unifies multiple coercivity results.The authors state that results from different settings follow by specifying the pair (A0, A1).Source: §1.9 · p. 9

1.10. Variational extrapolation.

  • Under the stated Banach-space and uniform-convexity assumptions, near-minimizers of Eθ converge in B, up to a subfamily, to a minimizer of the Gagliardo functional.The theorem assumes B ↪ A0 and compactness of A0 ∩ A1 ↪ A0.Source: §1.10 · pp. 9–10
  • The proof of the variational stability result uses both Γ-convergence extrapolation and compactness extrapolation.The text identifies these as basic ingredients of Theorem 1.6.Source: §1.10 · p. 10

1.11. Convergence of filters in image processing: from non-local to local models.

The paper treats convergence from non-local image filters to local variational models as an instance of extrapolation with Gagliardo completions. It also situates this work among earlier studies of BBM and MS limits and interpolation endpoint behavior.

1.11. Convergence of filters in image processing: from non-local to local models.

  • With B = L2, A0 = L1, and A1 = ˙W 1,1, the interpolated functionals yield L2-stability of minimizers converging to those of the ROF model as θ → 1−.The cited passage identifies the functionals and their L2-stability; the accompanying passage names the ROF limit.Source: §1.11 · p. 10
  • With B = L1, A0 = L1, and A1 = ˙W 1,1, Theorem 1.6 gives L1-stability of minimizers for the Chan–Esedoglu denoising model as θ → 1−.The model is identified as Chan–Esedoglu.Source: §1.11 · p. 10
  • Theorem 1.6 places convergence results for image-processing filters within a general phenomenon based on extrapolation and Gagliardo completions.Earlier work established convergence of non-local to local filters, including settings studied by Aubert–Kornprobst and Brezis–Nguyen.Source: §1.11 · pp. 10–11

1.12. Brief interlude: historical context and references.

  • Earlier studies examined pointwise limits of real interpolation norms at endpoints and their role in deriving BBM and MS formulae.The passage describes this work as building a bridge between interpolation theory and endpoint behavior.Source: §1.12 · p. 11

2. Interpolation of normed pairs and Gagliardo completions

This section reviews real interpolation via K-functionals and the endpoint role of Gagliardo relative completions. The Gagliardo pair retains the original K-functional structure and has a stated limiting behavior as the interpolation parameter approaches an endpoint.

2. Interpolation of normed pairs and Gagliardo completions

  • Real interpolation spaces (A0, A1)θ,p are constructed using K-functionals for a compatible pair of normed spaces.The passage introduces the spaces as intermediate spaces relative to the pair.Source: §2 · p. 11
  • Each Ai embeds continuously into its Gagliardo completion Gi(A0, A1), with the completion norm bounded by the Ai norm.This is the stated embedding and norm estimate.Source: §2 · p. 12
  • The Gagliardo pair preserves the K-functional: K(t, f; G0, G1) = K(t, f; A0, A1).The passage states this identity for the two pairs.Source: §2 · p. 12
  • As θ approaches 1, the stated limiting behavior of the interpolation norms is the norm of the corresponding Gagliardo completion.The proposition gives this endpoint behavior for i ∈ {0, 1} and f ∈ Ai∗.Source: §2 · p. 12

3. Variational K-functionals and Γ-convergence

The section extends Γ-convergence through variational K-functionals, linking De Giorgi’s framework with Gagliardo completions. It defines generalized Γ-limits and a dynamic penalty functional, then states an equivalence connecting these limits to the completions.

3. Variational K-functionals and Γ-convergence

  • This extension creates a bridge between De Giorgi’s variational convergence and Gagliardo’s methodology.When G_n is the ambient norm, the generalized definition reduces to classical Γ-convergence.Source: §3 · pp. 12–13
  • The section introduces generalized upper and lower Γ-limits for sequences of functionals paired with sequences of functionals G_n.The pair Γ-converges when these limits coincide for every f in X.Source: §3 · p. 12
  • The variational K-functional extends classical K-functionals by replacing fixed norms with general functionals.It takes the infimum over decompositions f = g + h of G(g) + tF(h).Source: §3 · p. 13
  • Theorem 3.3 states a core equivalence relating generalized Γ-limits to Gagliardo completions.Its accompanying condition takes an infimum over positive sequences t_n tending to zero.Source: §3 · p. 13

4. Γ-extrapolation

The section applies Γ-extrapolation to interpolation norms, relating their limiting behavior to Gagliardo completions at the endpoints. It also treats the variational K-functional as a parameterized family whose Γ-limits can be computed.

4. Γ-extrapolation

  • As the interpolation parameter approaches either endpoint, the interpolation norms converge in the Γ-sense to Gagliardo completions.The theorem states this endpoint result for p in [1, ∞] and f in either member of the pair.Source: §4 · p. 14
  • The variational K-functional of a normed pair can be viewed as a family of functionals indexed by its penalty parameter, with computable Γ-limits.The section gives the limit as t tends to infinity for f in A1.Source: §4 · p. 15
  • For f in A0, the functionals F_n converge to the Gagliardo completion norm G1(A0,A1).This is stated as a byproduct of the endpoint extrapolation argument.Source: §4 · p. 16

5. Extrapolation of Γ-convergence: applications and examples

This section applies a unified extrapolation framework to Γ-convergence and pointwise limits for varied functionals, beginning with compatible spaces and their K-functionals. It covers Sobolev characterizations on metric measure spaces, rearrangement-invariant norms, and heat-type energies, including fractional heat kernels and connections to classical BBM results.

5.1. Γ-convergence to Sobolev norms.

  • For rearrangement-invariant spaces, the framework extends Γ-convergence to higher-order Sobolev norms using Besov seminorms.Theorem 5.1 treats order k and q ∈ [1, ∞].Source: §5.1 · p. 17

5.2. Γ-convergence to fractional Sobolev norms via Butzer seminorms.

  • For p ∈ (1, ∞) and t > 0, Butzer seminorms yield Γ-convergence to fractional Sobolev norms.Theorem 5.2 applies to f ∈ H^{t,p}(R^N), addressing non-local fractional differential operators.Source: §5.1 · p. 19, §5.2 · p. 20

5.3. Γ-convergence to the total variation.

  • For f ∈ BV^k(R^N), the method extends Γ-convergence to the total variation using Besov norms and higher-order variants.Theorem 5.3 covers k ∈ N and q ∈ [1, ∞].Source: §5.3 · p. 22

5.4. Γ-convergence of BMO-type functionals to the total variation.

  • For BMO-type functionals, extrapolation yields a limit-superior bound by the total variation seminorm.The estimate is limsup_{ε→0} κε(f) ≲ ∥f∥_{BV̇( R^N)}.Source: §5.4 · p. 23

5.5. Γ-convergence to Sobolev norms in metric measure spaces.

  • For a Cheeger or PI space satisfying (5.36) and (5.37) for some p ≥ 1, Korevaar–Schoen type energy functionals are defined for f ∈ Lp(X).Theorem 5.5 takes p ∈ [1, ∞), q ∈ [1, ∞], and f ∈ Lp(X) ∩ KS1,p(X).Source: §5.5 · p. 24
  • The Γ-extrapolation method yields the characterization: f∈KS1,p(X) if and only if Ep(f)<∞.The characterization is based on a sequence tn→0 and is presented as a consequence of Γ-extrapolation.Source: §5.5 · p. 25

5.6. Γ-convergence to rearrangement invariant norms.

  • For r.i. spaces X on RN, Theorem 5.7 extends via Γ-extrapolation to f∈W^kX, with q∈[1,∞] and k∈N.The authors describe it as the counterpart of Theorem 5.1 as s→0+.Source: §5.6 · p. 26

5.7. Γ-convergence of heat-type energies.

  • The heat-energy Γ-extrapolation approach is self-contained, independent of BBM formulae, and extends to Sobolev and Lebesgue spaces.The passage describes the approach as proving (5.53) and points to Theorem 5.9 for the extensions.Source: §5.7 · pp. 27, 30, 31
  • The approach also treats fractional heat kernels: Theorem 5.10 applies for s∈(0,1), p∈[1,∞), 2s≥p, and f∈W^1,p; this result was previously obtained in [54].The paper gives a new proof using Γ-extrapolation.Source: §5.7 · p. 32
  • The discussion focuses on 2s ≥ p; for 2s < p, expected limit formulas are noted by reference to [54, Theorem 1.7].Source: §5.7 · p. 32
  • The argument establishes the desired result for heat-type energies in the limiting case 2s = p.Source: §5.7 · p. 33
  • Formula (5.89) is a special case of the general BBM formulation, while extending (5.90) via non-standard interpolation weights requires further technical work and separate study.Source: §5.7 · p. 33

6. Exact formulae for Γ-limits

The section addresses why exact extrapolation results may yield only equivalence-class descriptions when explicit K-functional formulas are unavailable. A refinement tracks constants to recover exact interpolation norms and their limiting behavior.

6. Exact formulae for Γ-limits

  • Explicit K-functional formulas are available for only a few classical pairs, so exact limiting procedures generally characterize target norms only up to multiplicative constants.The passage names (L1, L∞), (C, C1), and (L∞, L∞(1/t)) as classical cases.Source: §6 · p. 34
  • Reversing the approximate K-functional characterization lets the method treat computed quantities as limits of expressions built from explicit computable formulas.The passage presents this refinement as a way to overcome the limitation of approximate characterizations.Source: §6 · p. 34
  • Tracking equivalence constants through the argument allows normalization to recover exact interpolation norms and their limiting behavior.The method repeats the earlier arguments with the explicit expression G(t, f) in place of the true K-functional.Source: §6 · p. 34
  • For Lp(RN), p ∈ (1, ∞), and k = 1, the formulas recover the classical fractional Sobolev formula; the method also applies at p = 1 in ˙BV(RN).The passage also states that the methodology can handle the limit θ → 0+.Source: §6 · p. 37
  • Extensions to other settings are left for future work when pointwise limits are known for smooth functions but not necessarily for general functions.A full development of those settings is stated to fall outside this paper’s scope.Source: §6 · p. 37

7. Extrapolation of compactness

The section extends compactness results through extrapolation, using endpoint operator assumptions and criteria involving K-functionals. Applications yield subsequential convergence results in function spaces and metric measure settings.

7. Extrapolation of compactness

  • When an operator is bounded on one endpoint and compact on the other, compactness extends to extrapolated spaces as the interpolation parameter approaches 1.Theorem 7.2 gives a subsequential limit in an extrapolated target space for every η ∈ (0, 1).Source: §7 · p. 38
  • A K-functional criterion gives a subsequence whose operator images converge in B0 + B1 to a limit in B1.This is the stated conclusion of Theorem 7.3.Source: §7 · p. 39

7.1. Applications.

  • Applications include local W1,p convergence for p > 1, BVloc convergence for p = 1, and strong Lp compactness on compact metric measure spaces.The metric-space application states that the sequence contains a subsequence converging strongly in Lp(X).Source: §7.1 · pp. 41–42

8. Variational problems via Γ-extrapolation

The section proves convergence of near-minimizers for variational limits, then applies the result to fractional energies and a K-functional variant. Strong convergence in the data space follows under additional assumptions.

8.1. Proof of Theorem 1.6.

  • Γ-convergence yields a subsequence of near-minimizers converging to a minimizer of E1.The proof first establishes convergence in A0.Source: §8.1 · p. 44
  • If B is uniformly convex, the convergence upgrades to B; for general Banach function spaces, the proof guarantees convergence only in A0.An alternative argument using the Brezis–Lieb lemma is given for certain spaces including L1.Source: §8.1 · p. 45

8.2. Applications of Theorem 1.6.

  • For fractional energies, near-minimizers converge in B as s → 1− under the stated uniform-convexity and embedding assumptions, with a limit minimizing the limiting functional.The Lp, k = 1 specialization identifies the limiting regularizer as ∥∇u∥Lp for p > 1 and ∥u∥˙BV for p = 1.Source: §8.2 · p. 46
  • The available characterizations of the relevant interpolation quantities are only up to equivalence, so the proof instead uses exact formulas while tracking constants.The passage explains this as a way around the possible mismatch between near-minimizers of equivalent functionals.Source: §8.2 · p. 46

8.3. A variant of Theorem 1.6 involving K-functionals.

  • A variant based on K-functionals also gives subsequential B-convergence of near-minimizers to a minimizer of E1 as ε → 0+.The theorem additionally identifies the limiting energy with limε→0+ Fε(ūε).Source: §8.3 · p. 47

9. Applications to image processing

This section applies variational convergence and K-functional estimates to image-denoising models, connecting non-local models with local limits and discrete ROF approximations with minimizer-rate estimates. It also relates image-approximation error bounds to interpolation regularity described by Brezis–Van Schaftingen–Yung spaces.

9. Applications to image processing

  • The Chan–Esedoglu model addresses ROF’s lack of contrast invariance, but its L1 fidelity term does not ensure a unique solution.ROF uses L2 fidelity and total variation regularization; the Chan–Esedoglu model instead uses L1 fidelity.Source: §9 · p. 47

9.1. From the non-local to the local setting: asymptotic analysis.

  • As s → 1−, a subfamily of non-local near-minimizers converges in Lr to a minimizer of the corresponding local problem.The result applies to f ∈ Lr for r ∈ [1, ∞) and positive q and λ.Source: §9.1 · p. 48

9.2.1. Proof of Theorem 1.1.

  • The proof combines strong-convexity and projection estimates to obtain the discrete-to-continuum minimizer estimate.It also uses a self-improving K-functional inequality to control the approximation.Source: §9.2.1 · pp. 49, 51, 52

9.2. From the discrete to the continuous setting: rates of convergence via K-functionals.

  • The analysis establishes the K-functional estimate K(t, ū; L2, Ẇ1,2)^2 ≲ ||ū||L∞ K(t, ū; L1, BV̇), relating the L2–Ẇ1,2 and L1–BV̇ K-functionals.Source: §9.2.1 · p. 52

9.2.2. Some consequences of Theorem 1.1.

  • ∥ū − ūn∥L2 ≲ K(n−1, f; L2, Ẇ1,2)1/2 min{n−1/4, K(n−1, f; L2, Ẇ1,2)1/2} gives a K-functional-based error bound.The theorem’s estimates are stated to be optimal, with a comparison constant independent of n.Source: §9.2.2 · p. 52
  • The sharpness argument treats α ≠ 1/2 separately, using an extremal local singularity when α ∈ (1/2, 1).The proof also constructs data fn for which gn is the unique minimizer, supporting the sharpness analysis.Source: §9.2.2 · pp. 52–54
  • For f in ˙W^s,p ∩ L∞ with p ∈ [1, 2), Corollary 9.3 gives extrapolation estimates for minimizer convergence rates with constants independent of n and s.Taking s → 1^- and applying the BBM formula yields a further consequence, including an L∞ bound when p = 1.Source: §9.2.2 · p. 55

9.2.3. Error bounds via Brezis–Van Schaftingen–Yung spaces. ¨

  • Brezis–Van Schaftingen–Yung spaces characterize the interpolation regularity governing the K-functional growth and thereby provide error bounds, including for functions in ˙W^β,q.The interpolation formula applies when α ≥ p/[4(p−1)]; the Sobolev-to-Besov embedding yields the stated Sobolev case.Source: §9.2.3 · p. 57
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