Algebras of Integrable Functions

• Algebras of Integrable Functions are Banach algebras on LCA groups defined via convolution and summable Fourier transforms, enabling uniform norm-controlled inversion.
• Advanced techniques, including spectral gap estimates and the power-trick, are employed to derive explicit bounds for the inverse elements in these algebras.
• The framework contrasts with the classical Wiener algebra by providing enhanced stability and practical implications in abstract harmonic analysis and operator theory.

An algebra of integrable functions refers to a Banach algebra structure formed from functions defined on a locally compact abelian (LCA) group $G$, equipped with Haar measure. These algebras are central in abstract harmonic analysis, particularly in the study of convolution operators and spectral synthesis. Recent advancements have focused on norm-controlled inversion properties, which concern estimates for the norm of the inverse element in the algebra under spectral constraints. Specifically, algebras incorporating summability of Fourier transforms, such as the Fourier-regular ideals $A_p(G)$ and the unitized $L^p(G)_1$, demonstrate improved inversion behavior relative to classical algebras like $L^1(G)$.

1. Classes of Algebras of Integrable Functions

Two principal families of Banach convolution algebras are considered:

• Fourier-Regular Ideal $A_p(G)$:

$$A_p(G) = \{f \in L^1(G) : \widehat{f} \in L^p(\widehat{G})\}$$

with norm

$$\|f\|_{A_p} = \|f\|_{L^1(G)} + \|\widehat{f}\|_{L^p(\widehat{G})}$$

The convolution product is defined as

$(f * g)(x) = \int_G f(y) g(x-y) dy$

and $A_p(G)$ is closed under convolution, with the norm submultiplicative. The unitization is

$$A_p(G)^1 = A_p(G) \oplus \mathbb{C}1,\quad \|a+\lambda 1\| = \|a\|_{A_p} + |\lambda|$$

• Unitized $L^p$ Algebra $L^p(G)_1$:

$L^p(G)_1 = L^p(G) \oplus \mathbb{C}1$

with norm

$$\|\lambda\,1 + f\|_{L^p_1} = |\lambda| + \|f\|_{L^p(G)}$$

In both cases, the Gelfand space is the one-point compactification of $\widehat{G}$, and the Gelfand transform at character $\gamma$ is given by $$\widehat{x}(\gamma) = \lambda + \widehat{f}(\gamma)$$.

2. Norm-Controlled Inversion Problem and Main Results

Norm-controlled inversion asks: given an element $x$ with bounded norm and spectrum bounded away from zero,

$$x \in A,\quad \|x\|_A \leq 1,\quad \inf_{\gamma \in \widehat{G}} |\widehat{x}(\gamma)| \geq \delta > 0,$$

can one bound $\|x^{-1}\|_A$ in terms of $\delta$ alone? In $A_p(G)^1$ and $L^p(G)_1$, Ohrysko’s main theorem establishes that such control is possible for all $\delta > 0$, in contrast to the classical Wiener algebra $L^1(G)^1$, where uniform control fails once $\delta \leq 1/2$.

For $1 \leq p \leq 2$, if $G$ is compact abelian and $x = \lambda 1 + f \in A_p(G)^1$ with $\|x\|_{A_p^1} \leq 1$, the explicit bound is

$$\|x^{-1}\|_{A_p^1} \leq \frac{1}{\delta} + \frac{2(1-\delta)}{\delta^2}$$

Extensions to $p > 2$ use a “power-trick,” with similarly explicit bounds for $L^p(G)_1$ (Ohrysko, 22 Jan 2026).

3. Proof Strategy and Analytical Techniques

The argument leverages several key analytic components:

• Spectral Decay via Riemann–Lebesgue Lemma: On compact $G$, $\widehat{f} \to 0$, yielding control over $|\lambda|$.
• Nikolski’s Splitting: For spectral gap $\delta > 1/2$, elementary bounds $(2\delta - 1)^{-1}$ arise.
• Hausdorff–Young Inequality: For $p \leq 2$, this positions the problem within a Hilbert space framework, facilitating norm estimates.
• Structural Decomposition:
  • Set $x = \lambda 1 + f$, consider $a = x * x^*$, with $$\widehat{a}(\gamma) = |\widehat{x}(\gamma)|^2 \geq \delta^2$$.
  • Decompose $a = |\lambda|^2 1 + k$, with $\|k\| \leq 1 - |\lambda|^2$.
  • Use higher convolution powers and polynomial identity for the inverse,

  $x^{-1} = Q_n(k) * y_n^{-1}$

  where

  $$y_n = |\lambda|^{2n} 1 + k^{*n},\quad Q_n(k) = \sum_{j=0}^{n-1} (-1)^j |\lambda|^{2(n-1-j)} k^{*j}$$

For unitized $L^p$ algebras, inverse Hausdorff–Young is employed to manage Fourier coefficients.

4. Comparison with Classical $L^1(G)$ Framework

Contrasts with the classical Wiener algebra $L^1(G)^1$ are significant. In $L^1(G)^1$, norm-controlled inversion fails uniformly for $\delta \leq 1/2$, as proved by Nikolski—here, the inversion constant diverges. The summability condition $\widehat{f} \in L^p$ in $A_p(G)^1$ or $L^p(G)_1$ enforces stronger decay, closing the gap and enabling control for all $\delta > 0$. No analogue of these explicit uniform bounds exists in pure $L^1(G)$.

5. Impact, Significance, and Broader Perspective

The demonstration that membership in $L^1(G)$ together with the summability condition $\widehat{f} \in L^p(\widehat{G})$ suffices for uniform norm-controlled inversion—with no restriction on the visible spectrum—is a notable improvement over classical results. It extends the “invisible spectrum” principle from Nikolski to broader classes, providing explicit quantitative bounds potentially useful in operator theory and sampling theory. The closed-form estimates for norm of the inverse enhance applicability and theoretical understanding of Banach convolution algebras with integrable functions and summable Fourier transforms (Ohrysko, 22 Jan 2026).

6. Applications and Directions

While explicit applications are not detailed, a plausible implication is enhanced inversion estimates in spectral synthesis, abstract harmonic analysis, and the analysis of convolution operators. The norm-control results also bear relevance for practical implementations in operator theory, particularly when invertibility with quantitative bounds is required. Further exploration may address analogous phenomena in non-abelian groups or in non-commutative settings.