Moser-Trudinger inequality

An inequality arising as a limiting Sobolev inequality (cf. also Sobolev space). Let $ M $ be a finite-volume manifold of dimension $ n $ and denote by $ D $ the corresponding gradient defined on $ C ^ {1} ( M ) $. Then for $ n \geq 2 $,

$$ \int\limits _ { M } {\left | {Df } \right | ^ {n} } {dx } \leq 1, $$

and the normalization conditions

$$ f \mid _ {\partial M } = 0 \textrm{ for a manifold with boundary } $$

or

$$ \int\limits _ { M } f {dx } = 0 \textrm{ for a manifold without boundary } , $$

there exist positive constants $ \alpha $ and $ C $, depending only on $ n $ and $ M $, such that

$$ \int\limits _ { M } {e ^ {\alpha \left | f \right | ^ {q} } } {dx } \leq C { \mathop{\rm vol} } ( M ) , $$

where $ q = {n / {n - 1 } } $.

This estimate and its consequences appear naturally in two contexts:

1) as a limiting Sobolev inequality corresponding to end-point phenomena for the Sobolev imbedding on $ \mathbf R ^ {n} $:

$$ \left \| f \right \| _ {L ^ {r} ( \mathbf R ^ {n} ) } \leq A \left \| {\nabla f } \right \| _ {L ^ {p} ( \mathbf R ^ {n} ) } , $$

where $ {1 / r } = {1 / p } - {1 / n } $, $ 1 < p < n $;

2) as the determining estimate (linearized form) used to study conformal deformation on manifolds and the problem of prescribing Gaussian curvature.

This inequality was first demonstrated by N. Trudinger [a8] when $ M $ is a bounded domain in $ \mathbf R ^ {n} $, and the sharp value

$$ \alpha _ {n} = n \left [ { \frac{2 \pi ^ {n/2 } }{\Gamma ( n/2 ) } } \right ] ^ { {1 / {( n - 1 ) } } } $$

was found by J. Moser [a6]. The functional is finite for all values of $ \alpha $, but for values greater than $ \alpha _ {n} $ the constant $ C $ depends on the function $ f $. Moser also found the sharp value $ \alpha = 4 \pi $ for the two-dimensional sphere $ S ^ {2} $. For the $ n $- dimensional sphere $ S ^ {n} $, the sharp value of $ \alpha $ is the same as for a bounded domain in $ \mathbf R ^ {n} $( see [a2]). L. Carleson and S.-Y.A. Chang [a4] established the existence of an extremal function when the domain $ M $ is the unit ball in $ \mathbf R ^ {n} $. D.R. Adams [a1] extended this class of inequalities on bounded domains in $ \mathbf R ^ {n} $ to higher-order gradients. Generalizations of the Carleson–Chang theorem have been given for general domains in $ \mathbf R ^ {n} $, and a development of the Moser framework has also been studied for Kähler manifolds (cf. also Kähler manifold). For boundary values of analytic functions in the unit disc with finite Dirichlet integral, a similar estimate holds based on methods of A. Beurling [a5].

Two essential techniques that have been applied to understanding this variational inequality (cf. also Variational equations) are: symmetric rearrangement of functions and conformal invariance. For geometric applications, the linearized Moser–Trudinger inequality has been the critical result: in dimension two with normalized surface measure on $ S ^ {2} $[a6]:

$$ { \mathop{\rm ln} } \int\limits _ {S ^ {2} } {e ^ {F} } {d \xi } \leq \int\limits _ {S ^ {2} } F {d \xi } + { \frac{1}{4} } \int\limits _ {S ^ {2} } {\left | {\nabla F } \right | ^ {2} } {d \xi } + K, $$

and for the unit disc in $ \mathbf R ^ {2} $[a4]:

$$ { \mathop{\rm ln} } \left [ { \frac{1} \pi } \int\limits _ {\left | x \right | \leq 1 } {e ^ {2f } } {dx } \right ] \leq 1 + { \frac{1}{4 \pi } } \int\limits _ {\left | x \right | \leq 1 } {\left | {\nabla f } \right | ^ {2} } {dx } . $$

Here, the constant $ K $ was shown to be zero by E. Onofri [a7] and the functional inequality implies, using the Polyakov action, that the determinant of the Laplacian under conformal deformation with fixed area is maximized by the standard metric. For dimension two these linearized inequalities are equivalent. W. Beckner [a3] generalized this result on the sphere to higher dimensions, where on $ S ^ {n} $ the inequality has the form

$$ { \mathop{\rm ln} } \int\limits _ {S ^ {n} } {e ^ {F} } {d \xi } \leq \int\limits _ {S ^ {n} } F {d \xi } + { \frac{1}{2n! } } \int\limits _ {S ^ {n} } {F ( P _ {n} F ) } {d \xi } . $$

Here, $ P _ {n} $ is a conformally invariant pseudo-differential operator which acts on spherical harmonics of degree $ k $ in dimension $ n $:

$$ P _ {n} Y _ {k} = { \frac{\Gamma ( n + k ) }{\Gamma ( k ) } } Y _ {k} , $$

and $ P _ {n} $ can be represented as an algebraic function of the Laplacian:

$$ P _ {n} ( - \Delta ) = \left \{ \begin{array}{l} {\prod _ { {\mathcal l} = 0 } ^ { {( } n - 2 ) /2 } [ - \Delta + {\mathcal l} ( n - 1 - {\mathcal l} ) ] , \ n \textrm{ even, } } \\ {[ - \Delta + ( { \frac{n - 1 }{2} } ) ^ {2} ] ^ {1/2 } \times \ } \\ {\times \prod _ { {\mathcal l} = 0 } ^ { {( } n - 3 ) /2 } [ - \Delta + {\mathcal l} ( n - 1 - {\mathcal l} ) ] , \ n \textrm{ odd } . } \end{array} \right . $$

In dimension four this is the Paneitz operator $ - \Delta ( - \Delta + 2 ) $. The one-dimensional result corresponds to Szegő's theorem on determinants of Toeplitz operators. Equality is attained above only for functions of the form

$$ F ( \xi ) = - n { \mathop{\rm ln} } \left | {1 - \zeta \cdot \xi } \right | + C, \left | \zeta \right | < 1. $$

From the Orlicz duality between the classes $ e ^ {L} $ and $ L { \mathop{\rm ln} } L $, the exponential-class Moser–Trudinger inequality is equivalent to a logarithmic fractional-integral inequality written in terms of the fundamental solution or Green's function for the operator $ P _ {n} $:

$$ - n \int\limits _ {S ^ {n} \times S ^ {n} } {F ( \xi ) { \mathop{\rm ln} } \left | {\xi - \eta } \right | ^ {2} G ( n ) } {d \xi d \eta } \leq $$

$$ \leq - n \int\limits _ {S ^ {n} } { { \mathop{\rm ln} } \left | {\xi - \eta } \right | ^ {2} } {d \xi } + \int\limits _ {S ^ {n} } {F { \mathop{\rm ln} } F } {d \xi } + \int\limits _ {S ^ {n} } {G { \mathop{\rm ln} } G } {d \xi } , $$

where $ F $ and $ G $ are probability densities on the sphere. Equality is attained only for functions of the form

$$ F ( \xi ) = G ( \xi ) = A \left | {1 - \zeta \cdot \xi } \right | ^ {- n } , \left | \zeta \right | < 1. $$

Using conformal equivalence, this inequality can be reformulated on $ \mathbf R ^ {n} $ for probability densities $ f $ and $ g $:

$$ - n \int\limits _ {\mathbf R ^ {n} \times \mathbf R ^ {n} } {f ( x ) { \mathop{\rm ln} } \left | {x - y } \right | ^ {2} g ( y ) } {dx dy } \leq $$

$$ \leq C _ {n} + \int\limits _ {\mathbf R ^ {n} } {f { \mathop{\rm ln} } f } {dx } + \int\limits _ {\mathbf R ^ {n} } {g { \mathop{\rm ln} } g } {dx } , $$

where the relation with the sharp Hardy–Littlewood–Sobolev inequality is evident. This inequality, defined by the mapping

$$ f \in L ^ {p} ( \mathbf R ^ {n} ) \rightarrow \int\limits _ {\mathbf R ^ {n} } {\left | {x - y } \right | ^ {- \lambda } f ( y ) } {dy } \in L ^ {p ^ \prime } ( \mathbf R ^ {n} ) $$

for $ \lambda = { {2n } / {p ^ \prime } } $, $ 1 < p < 2 $ and $ {1 / p } + {1 / {p ^ \prime } } = 1 $, implies both the linearized Moser–Trudinger inequality and the logarithmic fractional-integral inequality [a3].

The logarithmic inequalities above can be interpreted as variational problems for the free energy with fixed entropy in a statistical mechanics setting. There are various applications of the Moser–Trudinger inequalities to extremal problems for determinants and zeta-functions under conformal deformation of metric. For example, on the four-dimensional sphere the determinant of the conformal Laplacian is extremized under conformal deformation with fixed area by the standard metric (see [a3]). The most important aspect of the Moser–Trudinger inequality has been its connection to the Polyakov–Onofri log determinant variation formula and its subsequent development in terms of conformal geometry and geometric analysis of conformally invariant operators on higher-dimensional manifolds.

References

[a1]	D.R. Adams, "A sharp inequality of J. Moser for higher order derivatives" Ann. of Math. , 128 (1988) pp. 385–398
[a2]	T. Aubin, "Nonlinear analysis on manifolds. Monge–Ampère equations" , Springer (1982) MR0681859 Zbl 0512.53044
[a3]	W. Beckner, "Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality" Ann. of Math. , 138 (1993) pp. 213–242
[a4]	L. Carleson, S.-Y.A. Chang, "On the existence of an extremal function for an inequality of J. Moser" Bull. Sc. Math. , 110 (1986) pp. 113–127
[a5]	S.-Y.A. Chang, D.E. Marshall, "On a sharp inequality concerning the Dirichlet integral" Amer. J. Math. , 107 (1985) pp. 1015–1033
[a6]	J. Moser, "A sharp form of an inequality by N. Trudinger" Indiana Math. J. , 20 (1971) pp. 1077–1092
[a7]	E. Onofri, "On the positivity of the effective action in a theory of random surfaces" Comm. Math. Phys. , 86 (1982) pp. 321–326
[a8]	N. Trudinger, "On imbeddings into Orlicz spaces and some applications" J. Math. Mech. , 17 (1967) pp. 473–483