Hölder–Zygmund Spaces

• Hölder–Zygmund spaces are a rigorously defined family of function spaces that interpolate between classical Hölder continuity and stronger differentiability, capturing fine regularity properties.
• They are equivalently characterized through methods like difference quotients, wavelet decompositions, and Poisson extension estimates, aligning with Besov space frameworks.
• Their generalizations to weighted, anisotropic, and non-Euclidean settings underpin applications in harmonic analysis, PDEs, hyperbolic dynamics, and multifractal analysis.

Hölder–Zygmund spaces provide a rigorous scale of function spaces capturing fine regularity properties interpolating between classical Hölder continuity and more rigid differentiability conditions. These spaces are fundamental in harmonic analysis, partial differential equations, approximation theory, hyperbolic dynamics, operator theory, and multifractal analysis, and are characterized equivalently by difference quotients, wavelet coefficients, and Poisson-extension (hyperbolic gradient) estimates. Moreover, these spaces admit meaningful generalizations on domains, manifolds, discrete lattices, Lie groups, and in weighted, anisotropic, or vector-valued settings.

1. Classical Definitions and Fundamental Characterizations

For $0 < s < 2$, the (nonhomogeneous) Hölder–Zygmund space $\Lambda_s(\mathbb R^n)$ comprises all bounded functions $f$ equipped with equivalent norms:

• Hölder-type norm:

$$\|f\|_{L^\infty} + \sup_{0 < |h| \leq 1} \frac{|f(x+h)-f(x)|}{|h|^s} < \infty$$

for $0 < s \leq 1$. The case $s=1$ defines the classical Zygmund space.

• Second-difference characterization:

$$\omega_s(f) := \sup_{h \neq 0} \frac{\|\Delta^2_h f\|_{L^\infty}}{|h|^s}$$

where $\Delta^2_h f(x) = f(x+h) - 2f(x) + f(x-h)$. The norm

$\|f\|_{\Lambda_s} = \|f\|_{L^\infty} + \omega_s(f)$

is equivalent to the usual Hölder norm for $0 < s < 2$ and recovers the Zygmund norm for $s = 1$.

• Littlewood–Paley and wavelet characterizations: For an orthonormal wavelet basis $\{\varphi_Q\} \cup \{\psi_{j,k}\}$ of regularity $>s$,

$$f(x) = \sum_Q d_Q \varphi_Q(x) + \sum_{j \ge 0} \sum_k c_{j,k} \psi_{j,k}(x)$$

with

$$\|f\|_{\Lambda_s} \simeq \sup_Q |d_Q| + \sup_{j,k} 2^{j(n/2+s)} |c_{j,k}|$$

so that decay of wavelet coefficients encodes the regularity in the Zygmund scale (Saksman et al., 2020, Hu, 2015, Charina et al., 2018).

• Poisson extension / hyperbolic gradient: Take $F(x,y) = P_y * f(x)$, with $P_y$ the Poisson kernel. Then

$$\|f\|_{\Lambda_s} \simeq \|f\|_{L^\infty} + \sup_{(x,y) \in \mathbb R^{n+1}_+} y^{2-s} |\partial^2_y F(x,y)|$$

or using the hyperbolic gradient $y^{1-s}|\nabla_h F|$ (Saksman et al., 2020).

• Besov equivalence: $$\Lambda_s(\mathbb R^n) = B^s_{\infty,\infty}(\mathbb R^n)$$, so the Zygmund scale interpolates between $C^{k,\alpha}$ spaces and Sobolev/Besov spaces (Magaña, 24 Oct 2024, Bonthonneau et al., 2020).
• Integer (Zygmund) and fractional cases: For integer $s = m$, the space matches the $m$-th order Zygmund class, characterized by boundedness of the $(m+1)$-th difference or of the second difference of $D^{m-1}f$. For non-integer $s = m + \alpha$, one recovers $C^{m,\alpha}$ (Bonthonneau et al., 2020, Saksman et al., 2020, Rainer, 2022).

2. Generalizations and Structural Properties

Hölder–Zygmund spaces admit generalizations to

• General moduli of continuity $\omega(r)$ that interpolate between classical Hölder and Zygmund scales, yielding spaces $\mathcal C_\omega$ (Vasin, 2017). With suitable growth and Dini-type conditions on $\omega$, these spaces possess equivalent characterizations via difference quotients and Littlewood–Paley decompositions.
• Weighted and anisotropic versions: Spaces $C^{\alpha,L}(\mathbb R^n)$, with $L$ a slowly varying function, further refine the scale of regularity and admit equivalent definitions using weighted difference quotients or wavelet transform seminorms (Pilipovic et al., 2011).
• Spaces on domains, manifolds, Lie groups, and discrete settings: All core properties and equivalences transfer (with technical adjustments) to
  • Bounded or polynomial domains in $\mathbb R^n$ (Vasin, 2017)
  • Smooth manifolds, via coordinate charts and pseudodifferential calculus (Bonthonneau et al., 2020)
  • Stratified Lie groups, where dilations and quasimetric structure are used (Hu, 2015)
  • Discrete lattices ($\mathbb Z$), via iterated finite differences or semigroups of the discrete Laplacian (Abadias et al., 2021)
• Vector-valued and Banach-space-valued extensions: Zygmund-Hölder classes extend to Banach-valued functions, with compositional stability properties and curve-testing characterizations preserved (Rainer, 2022).

3. Core Analytical and Functional Properties

• Algebra properties: For $s > 0$, the space $\Lambda_s$ is a Banach algebra under pointwise multiplication, with Moser-type estimates for products and nonlinear maps:

$$\|fg\|_{\Lambda^s} \leq C (\|f\|_\infty\|g\|_{\Lambda^s} + \|g\|_\infty\|f\|_{\Lambda^s})$$

(Magaña, 24 Oct 2024).

• Composition and inversion: Zygmund spaces are stable under composition with suitable diffeomorphisms and—more strongly—under inversion in the little Zygmund space, which is the closure of $C_c^\infty$ in the Zygmund norm. The vanishing of seminorms at small scales (the little Zygmund property) is crucial for continuity of nonlinear operations (Magaña, 24 Oct 2024).
• Embeddings: There are continuous, but strict, embeddings among the classical and Zygmund scales

$$C^{m+1} \subset C^{m,1} \subset Z^{m,1} \subset C^{m,\alpha}$$

for $0 < \alpha < 1$ (Magaña, 24 Oct 2024, Rainer, 2022, Bonthonneau et al., 2020).

• Operator theory and PDEs: Calderón–Zygmund operators act boundedly on Hölder–Zygmund spaces on domains of sufficient regularity (Vasin, 2017, Magaña, 24 Oct 2024). For even kernels and polynomial domains, only the test on the characteristic function is needed, due to extra cancellation properties (the T(P) theorem).

4. Quantitative Estimates and Approximation

• Distance to Bessel-potential lifts of BMO: For each $f \in \Lambda_s$, its distance to the subspace $J_s(\mathbf{bmo})$ (the image under the Bessel operator of non-homogeneous BMO) can be evaluated equivalently through:
  1. Carleson-measure conditions on second differences,
  2. Carleson-measure norm of significant wavelet coefficients,
  3. Carleson-measure norm of large Poisson-extension hyperbolic gradients.

Explicitly,

$$\operatorname{dist}_{\Lambda_s}(f, J_s(\mathbf{bmo})) \simeq \sup_{h\neq 0} \frac{\|\Delta^2_h f\|_{L^\infty}}{|h|^s}$$

(Saksman et al., 2020).

• Semigroup and spectral characterizations: On the discrete lattice, $\Lambda^1$ (Zygmund) coincides with the class of bounded sequences with second differences/semigroup decay as $t^{-1/2}$. Heat and Poisson semigroups provide direct characterization and sharp kernel bounds for discrete and continuous cases (Abadias et al., 2021, Hu, 2015).
• Wavelet tight frame criteria and numerical regularity detection: For semi-regular refinable functions, the Zygmund regularity exponent is obtained precisely by a wavelet tight-frame decomposition. The ratio test based on frame coefficient decay yields rapid and sharp estimation of regularity, outperforming linear regression (Charina et al., 2018).

5. Non-Euclidean and Weighted Frameworks

• Lie groups: For a stratified (Carnot) Lie group $G$, $\mathcal{C}^\sigma(G)$ is defined via difference operators induced by the group structure and satisfies

$$\|f\|_{\mathcal{C}^\sigma(G)} \simeq \sup_{j \ge 1} 2^{j \sigma} \|\Delta_j f\|_{L^\infty(G)}$$

with dyadic blocks adapted to the group dilations (Hu, 2015).

• Weighted and anisotropic spaces: For a slowly varying weight $L(r)$ (e.g. logarithmic corrections), weighted Hölder–Zygmund spaces $C^{\alpha,L}(\mathbb R^n)$ have difference or wavelet norms weighted by $L(|h|)$, enabling finer discrimination of pointwise regularity and function space embeddings (Pilipovic et al., 2011).
• General growth/anisotropy: Using a modulus $\omega(r)$, quasi-Banach spaces $\mathcal{C}^\omega(D)$ extend the theory to domains and anisotropic/inhomogeneous settings, interpolating all intermediate Hölder–Zygmund scales and supporting operator-theoretic extensions such as T(P) theorems and cancellation for special kernels (Vasin, 2017).

6. Applications and Structural Impact

• PDE well-posedness: The structure of Hölder–Zygmund and especially their "little" subspaces supports continuity of PDE solution maps, notably for active scalar equations with nonlocal velocities and nonlinear transport structures (Magaña, 24 Oct 2024).
• Hyperbolic dynamics and rigidity: These spaces play a distinguished role in geometric and dynamical rigidity theorems; for example, stability of solutions to cohomological equations and length spectrum rigidity on manifolds is analyzed in the Zygmund scale via pseudodifferential calculus and microlocal estimates (Bonthonneau et al., 2020).
• Superposition operators: Regularity properties of nonlinear maps like $f(g(x))$ in Zygmund spaces are comprehensively controlled in terms of Zygmund regularity of the outer function $f$; precise requirements and sharp operator regularity are established via chain-rule-type difference formulas and composition testing along curves (Rainer, 2022).
• Multifractal and fine regularity analysis: The Zygmund and weighted Hölder–Zygmund scales are crucial in multifractal formalisms where local regularity exponents are detected via wavelet or difference-quotient behavior (Pilipovic et al., 2011).

7. Key Characterization Table

Characterization Type	Formula/Property	Applicability
Difference Quotient	$$\sup_{h\ne 0} \frac{\|\Delta_h^m f\|_{L^\infty}}{|h|^s}$$	Euclidean/Manifold/Group
Wavelet/Littlewood–Paley	$\sup_{j} 2^{js} \|\Delta_j f\|_{L^\infty}$	Euclidean/Group/Frames
Poisson Extension	$\sup_{(x,y)} y^{2-s}|\partial^2_y F(x,y)|$	Euclidean/Manifold
Weighted (log, etc.)	Norms with $L(|h|)$ factor	Finer/Multifractal regularity

These equivalences and their generalizations are foundational for modern harmonic analysis and the paper of regularity structures, influencing operator theory, semigroup methods, and the analysis of singular integrals and nonlocal PDEs (Saksman et al., 2020, Vasin, 2017, Bonthonneau et al., 2020, Hu, 2015, Charina et al., 2018, Abadias et al., 2021, Magaña, 24 Oct 2024, Rainer, 2022, Pilipovic et al., 2011).