Adams–Moser–Trudinger Inequalities

• Adams–Moser–Trudinger inequalities are sharp functional bounds that extend classical Sobolev embeddings by providing precise exponential integrability limits in both first and higher-order spaces.
• They reveal critical threshold phenomena where the existence of extremals depends on exact constants (e.g., 32π²) and refined variational methods using rearrangement and concentration techniques.
• These inequalities are essential in analyzing elliptic PDEs and geometric problems, underpinning results on regularity, blow-up behavior, and symmetry in diverse mathematical settings.

The Adams–Moser–Trudinger inequalities constitute the sharp borderline for exponential-type integrability in critical and higher-order Sobolev spaces, extending the classical Sobolev embedding and Trudinger–Moser inequalities. They form a foundational analytic tool for the study of extremal behavior in variational problems, sharp regularity of PDEs, and geometric analysis in a variety of settings, including Euclidean (ℝⁿ), manifold, weighted, product, and non-Euclidean spaces. The theory is marked by explicit threshold phenomena for the existence of extremals, symmetry results via rearrangement techniques, and a delicate balance between exponential growth and norm constraints.

1. Sharp Forms of the Inequalities

Fundamental cases of the Adams–Moser–Trudinger inequalities are established in both first-order and higher-order Sobolev spaces. In the classical case, the Moser–Trudinger inequality for the critical Sobolev space $W^{1,n}(\Omega)$ for $n\geq2$, $\Omega\subset\mathbb{R}^n$ bounded, provides optimal exponential integrability: $$\sup_{u\in W_0^{1,n}(\Omega),\ \|\nabla u\|_{L^n}\leq1} \int_{\Omega} \exp\bigl(\alpha |u|^{n/(n-1)}\bigr)\,dx<\infty$$ if and only if $\alpha\leq \alpha_n := n\,|S^{n-1}|^{1/(n-1)}$ (Lam et al., 2015).

The Adams inequality extends to higher-order spaces. In the prototypical second-order case ($m=2, n=4$), the sharp inequality is

$$\sup_{u\in W^{2,2}(\mathbb{R}^4),\ \|u\|_{W^{2,2}}\leq1} \int_{\mathbb{R}^4} \left( \exp\bigl(32\pi^2|u|^2\bigr) - 1 \right) dx < \infty$$

with $32\pi^2$ being the sharp constant (Yang, 2011). On bounded domains $\Omega\subset\mathbb{R}^n$, the critical exponent and sharp constants are explicitly determined from the geometry and order $m$ (DelaTorre et al., 2017).

Weighted, singular, and exact growth extensions are also available, incorporating weights such as $n\geq2$0 and more general Orlicz classes (Lam et al., 2011, Morpurgo et al., 2022).

2. Existence, Nonexistence, and Threshold Phenomena for Extremals

A major advance is the identification of sharp threshold phenomena in the existence of extremals for these inequalities in the Euclidean setting. Consider the “critical Adams inequality” on $n\geq2$1: $n\geq2$2 where $n\geq2$3. For $n\geq2$4, $n\geq2$5 is finite. The existence of maximizers (extremals) depends on whether $n\geq2$6 is below an explicitly computed threshold $n\geq2$7. For $n\geq2$8, $n\geq2$9 is attained; for $\Omega\subset\mathbb{R}^n$0, there is no extremal and $\Omega\subset\mathbb{R}^n$1 (Chen et al., 2018). The threshold is estimated sharply in terms of the best Gagliardo–Nirenberg constant: $\Omega\subset\mathbb{R}^n$2 with $\Omega\subset\mathbb{R}^n$3.

This transition is mirrored in the first-order Trudinger–Moser case and for higher- and fractional-order versions, where vanishing and concentration (blow-up) phenomena regulate attainability (Macedo et al., 30 May 2025).

3. Symmetry, Rearrangement, and Analytical Structure

Extremal functions for the Adams–Moser–Trudinger inequalities exhibit specific structural properties; for instance, every maximizer (assuming existence) is real-valued, nonnegative, and up to translation, radially symmetric. This is rigorously achieved using sharp Fourier rearrangement principles: under such symmetrization, the Laplacian (or higher gradient norms) do not increase, enforcing symmetry of extremals. These techniques, which are essential in both existence theory and a priori analysis, avoid reliance on classical rearrangement or Pólya–Szegő inequalities, particularly on the full space $\Omega\subset\mathbb{R}^n$4 where such tools are unavailable (Chen et al., 2018).

Additionally, compactness and concentration–compactness principles play a critical role in characterizing the lack of compactness due to translation invariance (bubbling and vanishing), distinguished from dichotomy by symmetrization arguments (Chen et al., 2018, Macedo et al., 30 May 2025).

4. Generalizations: Domains, Singular, and Product Spaces

The theory encompasses sharp inequalities on unbounded domains, with boundary, product spaces, and for singular weights. For example, in $\Omega\subset\mathbb{R}^n$5 with a singular weight $\Omega\subset\mathbb{R}^n$6: $\Omega\subset\mathbb{R}^n$7 holds for $\Omega\subset\mathbb{R}^n$8 and this is sharp (Lam et al., 2011, Yang, 2011).

Product versions (e.g., intersections of two Sobolev spaces) reveal new threshold phenomena and have applications to systems of PDEs (e.g., Kirchhoff–Choquard type equations) (Arora et al., 2019).

On noncompact Riemannian manifolds, Heisenberg groups, hyperbolic spaces, or general metric-measure spaces, the critical threshold is modulated by geometric parameters (injectivity radius, curvature), with the sharp constant corresponding to the Euclidean value under certain geometric conditions (Ngô et al., 2016, Fontana et al., 2024, Morpurgo et al., 2022, Yang, 2011).

Fractional Adams–Moser–Trudinger inequalities hold in Bessel potential and Lorentz spaces, further broadening the analytic framework (Martinazzi, 2015, Fontana et al., 2017, Fontana et al., 2015, Fontana et al., 2019).

5. Analytical Techniques and Proof Strategies

The analysis leverages advanced symmetrization (Fourier or Schwarz), blow-up profile analysis, and variational methods. Central technical ingredients include:

• Polyharmonic truncation: Replacing a function near a candidate blow-up point by an $\Omega\subset\mathbb{R}^n$9-harmonic function matching data on annuli, preserving Sobolev norms but removing the blow-up core. This enables precise capacity–energy comparison and is central to recent proofs of extremal existence for higher-order cases (Chen et al., 2018, DelaTorre et al., 2017, Macedo et al., 30 May 2025).
• Optimal Gagliardo–Nirenberg constants: The optimal constants for the corresponding power-type inequalities control the threshold for attainability and facilitate the expansion of the exponential (Chen et al., 2018, Macedo et al., 30 May 2025).
• Blow-up and concentration–compactness: Maximizing (or Palais–Smale) sequences either converge to extremals, vanish (energy escapes to infinity), or concentrate (“bubble”) at points, captured through scaled rescalings and matched expansions with explicit bubbles solving limit Liouville-type equations (Chen et al., 2018, DelaTorre et al., 2017, Macedo et al., 30 May 2025).
• Truncation-on-level-sets and Poincaré inequalities: These tools are used to extend local inequalities to the global settings on infinite-volume domains (Fontana et al., 2024, Morpurgo et al., 2022).

6. Applications to PDEs and Further Directions

The Adams–Moser–Trudinger framework is essential for proving existence and multiplicity of solutions for elliptic PDEs with exponential or critical nonlinearities, particularly in fourth-order and Kirchhoff-type equations, often under variational/mountain-pass schemes–the sharp inequalities directly control the nonlinear growth permitted by the energy functional (Yang, 2011, Arora et al., 2019, Morpurgo et al., 2022).

Open problems include extending sharp threshold phenomena to higher even dimension $$\sup_{u\in W_0^{1,n}(\Omega),\ \|\nabla u\|_{L^n}\leq1} \int_{\Omega} \exp\bigl(\alpha |u|^{n/(n-1)}\bigr)\,dx<\infty$$0, fractional-order Sobolev spaces, and general non-Euclidean settings, as well as refining asymptotics of threshold constants and sharpness for more general classes of inequalities (Chen et al., 2018, Martinazzi, 2015, Morpurgo et al., 2022).

7. Tabular Summary of Key Properties

Setting / Space	Sharp Exponential Constant	Threshold Structure
$$\sup_{u\in W_0^{1,n}(\Omega),\ \|\nabla u\|_{L^n}\leq1} \int_{\Omega} \exp\bigl(\alpha |u|^{n/(n-1)}\bigr)\,dx<\infty$$1	$$\sup_{u\in W_0^{1,n}(\Omega),\ \|\nabla u\|_{L^n}\leq1} \int_{\Omega} \exp\bigl(\alpha |u|^{n/(n-1)}\bigr)\,dx<\infty$$2	Threshold for extremal existence
$$\sup_{u\in W_0^{1,n}(\Omega),\ \|\nabla u\|_{L^n}\leq1} \int_{\Omega} \exp\bigl(\alpha |u|^{n/(n-1)}\bigr)\,dx<\infty$$3	$$\sup_{u\in W_0^{1,n}(\Omega),\ \|\nabla u\|_{L^n}\leq1} \int_{\Omega} \exp\bigl(\alpha |u|^{n/(n-1)}\bigr)\,dx<\infty$$4	$$\sup_{u\in W_0^{1,n}(\Omega),\ \|\nabla u\|_{L^n}\leq1} \int_{\Omega} \exp\bigl(\alpha |u|^{n/(n-1)}\bigr)\,dx<\infty$$5 extremals iff $$\sup_{u\in W_0^{1,n}(\Omega),\ \|\nabla u\|_{L^n}\leq1} \int_{\Omega} \exp\bigl(\alpha |u|^{n/(n-1)}\bigr)\,dx<\infty$$6
$$\sup_{u\in W_0^{1,n}(\Omega),\ \|\nabla u\|_{L^n}\leq1} \int_{\Omega} \exp\bigl(\alpha |u|^{n/(n-1)}\bigr)\,dx<\infty$$7	Adams constant $$\sup_{u\in W_0^{1,n}(\Omega),\ \|\nabla u\|_{L^n}\leq1} \int_{\Omega} \exp\bigl(\alpha |u|^{n/(n-1)}\bigr)\,dx<\infty$$8 (explicit)	Subcritical/critical dichotomy
Weighted/Singular (e.g., $$\sup_{u\in W_0^{1,n}(\Omega),\ \|\nabla u\|_{L^n}\leq1} \int_{\Omega} \exp\bigl(\alpha |u|^{n/(n-1)}\bigr)\,dx<\infty$$9)	$\alpha\leq \alpha_n := n\,|S^{n-1}|^{1/(n-1)}$0 times unweighted constant	Damped critical constant
Product Spaces (e.g. $\alpha\leq \alpha_n := n\,|S^{n-1}|^{1/(n-1)}$1)	$\alpha\leq \alpha_n := n\,|S^{n-1}|^{1/(n-1)}$2	Joint norm governs threshold
Manifold / Non-Euclidean	Geometry-dependent, proportional to Euclidean	Local-to-global; sharpness under geometric hypotheses

The theory of Adams–Moser–Trudinger inequalities is thus characterized by a deep interplay between analysis, geometry, and the underlying functional inequalities, with a robust structure of symmetry, concentration, and sharp partition of attainability regimes (Chen et al., 2018, Macedo et al., 30 May 2025, Yang, 2011, DelaTorre et al., 2017, Fontana et al., 2024, Ngô et al., 2016, Morpurgo et al., 2022).