H0,H1-Smoothness: Operator Analysis & Testing

• The (H0, H1)-smoothness condition is a dual criterion that governs regularity in operator differences and density functions using Kato smoothness and Schatten-class norms.
• It facilitates sharp spectral estimates via double operator integrals, enabling precise control in functional calculus and perturbation theory.
• Applied in Schrödinger operator perturbations and wavelet-based density testing, the framework underpins critical endpoint phenomena and robust hypothesis tests.

The $(H_0, H_1)$-smoothness condition is a technical criterion central to spectral analysis, perturbation theory, and hypothesis testing in the context of smoothness and functional calculus for operators. It formalizes a type of dual smoothness requirement for data or operator pairs $(H_0, H_1)$, underpinning Schatten-class norm estimates, operator perturbations, and density regularity testing. Contemporary applications range from double operator integral representations of functions of operators to wavelet-based smoothness tests for probability densities.

1. Definition and Framework of $(H_0, H_1)$-Smoothness

$(H_0, H_1)$-smoothness arises in multiple domains, notably Hilbert space operator theory and nonparametric function testing. In the operator-theoretic context, with $H_0$ and $H_1$ self-adjoint operators on Hilbert spaces, a factorization

$H_1 - H_0 = G_1^* G_0$

is considered, where $G_0$ and $G_1$ are auxiliary operators (often bounded, sometimes belonging to a Schatten class), and each is evaluated in terms of Kato smoothness with respect to the corresponding operator: $$G_0 \in \mathrm{Smooth}(H_0), \qquad G_1 \in \mathrm{Smooth}(H_1)$$ The pair $(G_0, G_1)$ is said to be $(H_0, H_1)$-smooth if this joint property holds. Norms and estimates typically use the product

$$A := \|G_0\|_{\mathrm{Smooth}(H_0)} \cdot \|G_1\|_{\mathrm{Smooth}(H_1)}$$

This framework naturally generalizes to Schatten $p$-class smoothness, with $G \in \mathrm{Smooth}_p(H)$ meaning that the $L^2 \to \mathcal K$ map $\varphi \mapsto G\,\varphi(H)$ lands in $\mathbf S_p(\mathcal H, \mathcal K)$ (Frank et al., 2019).

In statistical smoothness testing, the $(H_0, H_1)$-condition specifically refers to testing hypotheses about the differentiability order of an unknown density $f$, where:

• $H_0$: $f \in C^\mu(\mathbb{R})$
• $H_1$: $f \in C^{\mu+\varepsilon}(\mathbb{R})$, for some $\varepsilon>0$ Here, the test encodes the regularity distinction as a "smoothness index", operationalized via wavelet projections and Besov space membership (Ćmiel et al., 2018).

2. Classical and Schatten-Class Kato Smoothness for Operators

For a self-adjoint operator $H$ on Hilbert space $\mathcal H$, the Kato smoothness of $G$ is characterized by several equivalent criteria: $$G \in \mathrm{Smooth}(H) \iff \sup_{x \in \mathbb{R},\, \varepsilon > 0} \| G R_H(x+i\varepsilon) \| < \infty \iff \sup_{a < b} \frac{ \| G E_H((a, b)) \| }{ \sqrt{b - a} } < \infty$$ where $R_H(z) = (H - z)^{-1}$ and $E_H(\cdot)$ is the spectral measure.

The Schatten-class extension ($p > 0$) introduces

$$G \in \mathrm{Smooth}_p(H) \iff \sup_{a < b} \frac{ \| G E_H((a, b)) \|_{\mathbf S_p} }{ \sqrt{b - a} } < \infty$$

with corresponding norm $\|G\|_{\mathrm{Smooth}_p(H)}$ (Frank et al., 2019).

For the joint $(H_0, H_1)$ framework, the critical estimates for spectral functional calculus take the form: $$\|f(H_1) - f(H_0)\|_{\mathcal B} \leq 2\pi\,\|G_0\|_{\mathrm{Smooth}(H_0)} \|G_1\|_{\mathrm{Smooth}(H_1)} \|f\|_{\mathrm{BMO}(\mathbb{R})}$$ and for Schatten $p$-norms: $$\|f(H_1) - f(H_0)\|_{\mathbf S_p} \leq 2\pi C_1(p) \|G_0\|_{\mathrm{Smooth}_q(H_0)} \|G_1\|_{\mathrm{Smooth}_r(H_1)} \|f\|_{B_{p,p}^{1/p}(\mathbb{R})}$$ where $1/p = 1/q + 1/r$ (Frank et al., 2019).

3. Double Operator Integrals and Fundamental Operator Estimates

The $(H_0, H_1)$-smoothness condition plays a central role in double operator integrals (DOI) and Birman–Solomyak representations. For suitable functions $f$,

$D(f):=f(H_1)-f(H_0)=\mathrm{DOI}\bigl(f(x,y)\bigr)$

where $f(x, y) = \frac{f(x) - f(y)}{x - y}$ forms the kernel in the DOI framework. Norms of these differences are controlled by the $(H_0, H_1)$-smoothness of $G_0$, $G_1$, and the regularity (e.g., Besov class) of $f$. The operator norm and Schatten norm estimates follow directly from the DOI machinery, with sharp bounds derived in the case of Schrödinger operators and abstract perturbations (Frank et al., 2019, Frank et al., 2019).

The joint smoothness in the Schatten context determines which classes of functions $f$ allow $D(f)$ to belong to $\mathbf S_p$:

• For $f \in B_{p,p}^{1/p}(\mathbb{R})$, $D(f) \in \mathbf S_p$ provided $G_0$ and $G_1$ satisfy suitable smoothness-$p$ criteria.
• The critical index is $s=1/p$ in the Besov scale; for $p>1$, $f$ may admit certain unbounded singularities (cusps), but not jumps (Frank et al., 2019).

4. Applications in Schrödinger Operator Perturbation and Quasicommutators

The $(H_0, H_1)$-smoothness condition facilitates sharp Schatten-class estimates for resolvent and spectral function differences under perturbations:

• Schrödinger operators: For $H_0 = -\Delta$ and $H_1 = -\Delta + V$ with potential decay $|V(x)| \leq C(1 + |x|)^{-\rho}$, $D(f) \in \mathbf S_p$ under the precise rate-of-decay and Besov smoothness of $f$, with endpoint exponents determined solely by $\rho$, $d$, and $p$ (Frank et al., 2019).
• Quasicommutators: For expressions of the form $f(H_1)J - Jf(H_0)$, the DOI and $(H_0, H_1)$-smoothness framework yields operator and Schatten-class bounds, with $H_1J - JH_0$ factorizable as $G_1^* G_0$ (Frank et al., 2019).

These results encapsulate both functional analytic regularity and spectral perturbation theory, and they apply broadly to short-range perturbations and spectral calculations.

5. Wavelet-Based Hypothesis Testing for Density Smoothness

In statistical applications, the $(H_0, H_1)$-smoothness condition formalizes hypothesis tests for the differentiability order of densities. The null and alternative are encoded as: $$H_0: \text{density } f \text{ has at most } \mu \text{ derivatives}$$

$$H_1: \text{density } f \text{ has more than } \mu \text{ derivatives}$$

This canonically translates, via wavelet projections and Besov regularity, into critical smoothness index comparison: $$H_0: s_2^*(f) \leq \mu + \tfrac{1}{2} \qquad H_1: s_2^*(f) \geq \mu + 1 + \tfrac{1}{2}$$ where $s_2^*(f) = \sup \{ s : f \in B^s_{2,\infty} \}$ and $\text{id}(f)$ tracks the maximal order of differentiability at all points except possible defects (Ćmiel et al., 2018).

The test employs wavelet U-statistics as estimators for projections onto detail spaces, yielding

$$L_{n, j} = \frac{2}{n(n-1)} \sum_{1 \leq i < \ell \leq n} G_j(X_i, X_\ell)$$

where $G_j$ is a wavelet kernel. The smoothness estimator

$$\widehat{s}_{2, n} = -\frac{\log_2 L_{n, j(n)}}{2j(n)}$$

is shown to converge almost surely to $s_2^*(f)$, with test level and power explicitly characterized under enrichment procedures and regularity conditions (Ćmiel et al., 2018).

6. Theoretical Sharpness and Endpoint Phenomena

The $(H_0, H_1)$-smoothness condition delineates sharp and endpoint regimes:

• For operator differences, the critical Besov index $s = 1/p$ is strict: jumps in $f$ are not admissible, but logarithmic singularities may be for $p > 1$. At $p=1$, continuity is also required (Frank et al., 2019).
• For density regularity testing, convergence rates and derivative growth conditions are nearly optimal: omitting either fast decay or derivative growth control yields counterexamples (Ćmiel et al., 2018).
• In all contexts, endpoint constants (e.g., for model Hilbert-transform perturbations) are realized in explicit models (Frank et al., 2019).

These sharpness properties ensure that the $(H_0, H_1)$-smoothness framework provides not merely sufficient, but in most cases necessary, conditions for smoothness estimates and hypothesis testing.

7. Connections to Related Regularity Criteria and Further Directions

While $(H_0, H_1)$-smoothness is prominent in operator theory and nonparametric testing, other frameworks for regularity and smoothness exist. For example, distributional regularity, characterizations via Hölder–Zygmund spaces, and rate-order regularization via convolution with approximate identities all provide two-parameter control schemes, but may not reference the $(H_0, H_1)$ formalism directly (Pilipovic et al., 2012).

A plausible implication is that the general principle underlying $(H_0, H_1)$-smoothness—simultaneous control over approximation rate and growth, or dual operator smoothness—anchors a broad class of regularity criteria in mathematical analysis, functional calculus, and applied statistics. Future research can elaborate connections to interpolation theory, non-self-adjoint perturbations, and ill-posed inverse problems.