Young inequality
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| Young inequality | |
|---|---|
 | |
| Name | Thomas Young |
| Birth date | 1773–1829 |
| Nationality | British |
| Field | Royal Society-era science |
| Known for | Tension in mathematics, optics, linguistics |
Young inequality Young inequality is a family of fundamental inequalities in mathematical analysis connecting products and sums, integrals and convolutions, and exponents in functional spaces. Originating from work in the 19th century by Thomas Young and later developed in the context of Émile Picard-era analysis and Lebesgue integration theory, these inequalities play central roles in Fourier analysis, convexity, and the theory of Banach spaces. Multiple named variants appear across harmonic analysis, measure theory, and partial differential equations, each providing tight bounds that underlie many classical estimates.
Introduction
Young-type inequalities form bridges between algebraic operations and analytic norms encountered in the work of Thomas Young, Hardy–Littlewood collaborators, and later contributors such as G. H. Hardy, John Littlewood, and Oskar Perron. They are often introduced alongside the Hölder inequality, the Minkowski inequality, and the Jensen inequality within treatments of Lebesgue integral theory and L^p space geometry. In harmonic analysis texts referencing Stein, Weiss, or Grafakos, Young inequalities are presented as indispensable tools for estimating convolutions and products in Fourier transform arguments and for establishing continuity of bilinear maps between Banach or Hilbert spaces.
Statement and variants
Several distinct but related statements bear the Young name. The algebraic exponent form used in elementary analysis says that for positive real numbers a, b and conjugate exponents p, q > 1 with 1/p + 1/q = 1, a b ≤ a^p/p + b^q/q; this is taught alongside Hölder inequality and Young–Fenchel transform discussions in convex analysis texts citing Fenchel and Moreau. The convolution form used in harmonic analysis asserts that for f ∈ L^p(ℝ^n) and g ∈ L^q(ℝ^n) with 1 ≤ p, q, r ≤ ∞ satisfying 1 + 1/r = 1/p + 1/q, the convolution f * g belongs to L^r(ℝ^n) with a bound ‖f * g‖_r ≤ ‖f‖_p ‖g‖_q; treatments by Elias Stein and Rudin present this variant in analysis curricula. Other variants include integral forms for kernels, discrete analogues for sequences encountered in Hardy and Littlewood sums, and operator versions in the context of Young measures and the calculus of variations, often referenced alongside Tonelli and Fubini theorems.
Proofs and methods
Proofs of Young-type inequalities exploit convexity, duality, interpolation, or Fourier-analytic methods. The elementary exponent inequality follows from convexity of the exponential map and the Fenchel–Legendre transform argument as in convex analysis expositions tied to Rockafellar and Fenchel. The convolution inequality is often proved via Hölder inequality combined with translation invariance and Fubini-type arguments; alternative proofs use Minkowski inequality or interpolation theorems of Riesz–Thorin and Marcinkiewicz. In Fourier analysis, proofs use the Hausdorff–Young inequality and multiplier theorems familiar from works by Władysław Orlicz and Salem; operator-theoretic proofs rely on Schur test variants and Young operator estimates in C*-algebra contexts.
Equality cases and sharpness
Equality conditions vary by variant and are tied to extremal functions or numbers that saturate convexity bounds. For the exponent form, equality holds when a^p and b^q are proportional, a condition also appearing in Young–Fenchel convex duality discussions. In convolution inequalities, sharpness involves extremizing functions often realized by Gaussians or approximations thereof in Euclidean settings, a theme that connects to the Beckner and Brascamp–Lieb inequalities and to sharp constants discovered in works by E. H. Lieb and William Beckner. Determining sharp constants and characterizing extremizers invokes rearrangement inequalities as in Hardy–Littlewood–Pólya and uses symmetry or scaling arguments familiar from the study of Sobolev inequality extremals and concentration-compactness principles introduced by Pierre-Louis Lions.
Applications and consequences
Young inequalities underpin regularity and well-posedness results in partial differential equation theory, such as estimates in the study of Navier–Stokes equations and nonlinear dispersive equations treated in literature by Terence Tao and J. Bourgain. They are used to bound nonlinear terms in energy estimates appearing in the analysis of evolution equations associated with Eulerian and Hamiltonian systems; researchers cite their use in semigroup theory involving Hille–Yosida and in perturbation theory of Schrödinger operators. In harmonic analysis, Young convolution estimates yield continuity of convolution operators on L^p spaces and appear in proofs of multiplier theorems connected to Calderón–Zygmund theory and Littlewood–Paley decompositions. In probability, discrete Young-type bounds help control moments and convolutions of measures related to limit theorems linked to Kolmogorov, Lévy, and Khinchin.
Generalizations and related inequalities
Generalizations include the Young–Fenchel or convex conjugate inequality in convex analysis, the multidimensional convolution bounds with optimal constants studied by Bennett and Carbery, and nonlinear analogues embodied in the Brascamp–Lieb inequality and its stability results due to Barthe and Bennett–Carbery–Wright. Related inequalities that interact with Young-type estimates include Hölder inequality, Minkowski inequality, Hardy inequality, and the Sobolev embedding theorem, all central in modern analysis literature by Adams, Evans, and Maz'ya. Operator and noncommutative extensions appear in work on quantum information and von Neumann algebra analysis, connecting to results by Lieb and others in mathematical physics.
Category:Mathematical inequalities