View source for Moser-Trudinger inequality

An inequality arising as a limiting Sobolev inequality (cf. also [[Sobolev space|Sobolev space]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m1102101.png" /> be a finite-volume manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m1102102.png" /> and denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m1102103.png" /> the corresponding gradient defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m1102104.png" />. Then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m1102105.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m1102106.png" /></td> </tr></table>

and the normalization conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m1102107.png" /></td> </tr></table>

or

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m1102108.png" /></td> </tr></table>

there exist positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m1102109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021010.png" />, depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021012.png" />, such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021013.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021014.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021015.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021016.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021018.png" />;

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 [[#References|[a8]]] when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021019.png" /> is a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021020.png" />, and the sharp value

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021021.png" /></td> </tr></table>

was found by J. Moser [[#References|[a6]]]. The functional is finite for all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021022.png" />, but for values greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021023.png" /> the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021024.png" /> depends on the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021025.png" />. Moser also found the sharp value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021026.png" /> for the two-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021027.png" />. For the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021028.png" />-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021029.png" />, the sharp value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021030.png" /> is the same as for a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021031.png" /> (see [[#References|[a2]]]). L. Carleson and S.-Y.A. Chang [[#References|[a4]]] established the existence of an extremal function when the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021032.png" /> is the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021033.png" />. D.R. Adams [[#References|[a1]]] extended this class of inequalities on bounded domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021034.png" /> to higher-order gradients. Generalizations of the Carleson–Chang theorem have been given for general domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021035.png" />, and a development of the Moser framework has also been studied for Kähler manifolds (cf. also [[Kähler manifold|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 [[#References|[a5]]].

Two essential techniques that have been applied to understanding this variational inequality (cf. also [[Variational equations|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021036.png" /> [[#References|[a6]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021037.png" /></td> </tr></table>

and for the unit disc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021038.png" /> [[#References|[a4]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021039.png" /></td> </tr></table>

Here, the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021040.png" /> was shown to be zero by E. Onofri [[#References|[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 [[#References|[a3]]] generalized this result on the sphere to higher dimensions, where on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021041.png" /> the inequality has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021042.png" /></td> </tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021043.png" /> is a conformally invariant [[Pseudo-differential operator|pseudo-differential operator]] which acts on spherical harmonics of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021044.png" /> in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021045.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021046.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021047.png" /> can be represented as an algebraic function of the Laplacian:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021048.png" /></td> </tr></table>

In dimension four this is the Paneitz operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021049.png" />. The one-dimensional result corresponds to Szegő's theorem on determinants of Toeplitz operators. Equality is attained above only for functions of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021050.png" /></td> </tr></table>

From the Orlicz duality between the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021052.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021053.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021054.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021055.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021057.png" /> are probability densities on the sphere. Equality is attained only for functions of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021058.png" /></td> </tr></table>

Using conformal equivalence, this inequality can be reformulated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021059.png" /> for probability densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021061.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021062.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021063.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021064.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110210/m11021067.png" />, implies both the linearized Moser–Trudinger inequality and the logarithmic fractional-integral inequality [[#References|[a3]]].

The logarithmic inequalities above can be interpreted as variational problems for the free energy with fixed [[Entropy|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 [[#References|[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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.R. Adams,   "A sharp inequality of J. Moser for higher order derivatives"  ''Ann. of Math.'' , '''128'''  (1988)  pp. 385–398</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Aubin,   "Nonlinear analysis on manifolds. Monge–Ampère equations" , Springer  (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Beckner,   "Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality"  ''Ann. of Math.'' , '''138'''  (1993)  pp. 213–242</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S.-Y.A. Chang,   D.E. Marshall,   "On a sharp inequality concerning the Dirichlet integral"  ''Amer. J. Math.'' , '''107'''  (1985)  pp. 1015–1033</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Moser,   "A sharp form of an inequality by N. Trudinger"  ''Indiana Math. J.'' , '''20'''  (1971)  pp. 1077–1092</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E. Onofri,   "On the positivity of the effective action in a theory of random surfaces"  ''Comm. Math. Phys.'' , '''86'''  (1982)  pp. 321–326</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  N. Trudinger,   "On imbeddings into Orlicz spaces and some applications"  ''J. Math. Mech.'' , '''17'''  (1967)  pp. 473–483</TD></TR></table>