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The origin of Garding's inequality was an effort to solve Dirichlet's problem for linear elliptic operators of high even order.Let $$P(x,D)= \sum a_{\alpha}(x)D^{\alpha}$$ with principal part $$P_{2m}(D)= \sum_{\alpha=2m} a_{\alpha}(x)D^{\alpha}$$ have the property that $$ReP_{2m}(x,\xi)\geq c|\xi|^{2m}$$ for some $c>0$.It is further assumed that the coefficients $a_{\alpha}(x)$ have continious and bounded derivatives.Under this circumstances,one have for all $\epsilon>0$,$u\in C_{0}^{m}$,$$Re(Pu,u)\geq (c-\epsilon)\|u\|^{m,2}-b_{\epsilon}\|u\|^{2}$$. The original Garding's inequality seems to have been used for the first time by Leray in his 1954 lectures for an existence proof for Cauchy's problem for strongly hyperbolic systems.It was used again by Garding(1956) for a proof of the same result using only functional analysis. There is a sharp form of this inequality due to Hormander which says that if the symbol $P(x,\xi)\in S^{2m}$,$ReP(x,\xi) \geq 0$. Then $$Re(P(x,D)u,u)\geq-b\|u\|_{m-\frac{1}{2}}^{2}$$. A much more precise statement in the scalar case is known as the Fefferman-Phong inequality. Here my first question is what's the application of the sharp Garding's inequality? I once saw that Fefferman had wrote a paper named "sharp Garding's inequality and the uncertainty principle". I'm really surprised to see that this two inequalities can have connections.so i become more interested in inequalities with lower bounds.Can anyone list some inequalities appeared in PDE or other banches of math with a lower bound?Maybe some of them can have interesting links.

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Fefferman-Phong (FP) inequality may be not precise is some cases, see Hormander Vol.3 and V. Ivrii "Microlocal Analysis and Precise Spectral Asymptotics", for some remarks about FP and Melin inequalities –  user36539 Sep 16 '13 at 11:47
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2 Answers

Let us start with a short review of the so-called G{\aa}rding's inequality. Let $a\in S^m$ be nonnegative. Then with $a^w$ standing for the Weyl quantization of the Hamiltonian $a$, $$ (1)\quad m=1\quad \exists C,\ a^w +C\ge 0. $$ $$ (2)\quad m=2\quad \exists C,\ a^w +C\ge 0. $$ $$ (3)m=2, a_j,j=1,2\ \text{homogeneous degree $j$}. \quad\text{If }a_2\ge 0\quad a_{1}+\frac12trace_+(a_2'')\ge 0,$$ $$\text{then}\quad a_2^w+a_1^w+C\ge 0. $$ Of course (2) is (much) stronger than (1), but (1) is true for systems, that is symbols valued in symmetric matrices and even infinite-dimensional self-adjoint operators in a Hilbert space. (2) is the Fefferman-Phong inequality is true only for (nonnegative) scalar symbols. (3) is essentially Melin inequality ($trace_+(a_2'')$ is a symplectic invariant of the Hessian).

The third inequality is closely related to the uncertainty principle since, $$ -\Delta_{\mathbb R^d}+\frac{\vert x\vert^2}{4}-\frac d2\ge 0 $$ is a consequence of (3): the symbol $\vert \xi\vert^2+{\vert x\vert^2}/{4}-\frac d2$ takes nonpositive values but the associated operator is nevertheless nonnegative. Heuristically you could say that the set of nonpositivity of the symbol is symplectically small and thus unimportant.

Another example is Hardy's inequality for $d\ge 3$ $$ -\Delta_{\mathbb R^d}-\frac{(d-2)^2}{4}\frac{1}{\vert x\vert^2}\ge 0. $$ Here the set $S_{-}$ of nonpositivity of the symbol has infinite volume, but is still symplectically small in the sense that no canonical image of the unit cube can be embedded in $S_-$.

Summing-up a (second-order) nonnegative symbol gives rise to a semi-bounded (from below) operator, but an operator can be semi-bounded from below with a symbol taking negative values. The important matter is to check the symplectic relevance of the set $S_{-}$ where the symbol is negative. Of course, this is linked to the uncertainty principle.

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This short survey is very useful. The two example are interesting. Does there exists other references, except Hormander's book(it's hard for me to follow, :) ), exploiting this topic in detail? –  Wang Ming May 8 '12 at 3:01
    
In the scalar case.I'm toid that the ferfferman-Phong inequality can't be improved.Are their some examples to illustrate this? I also wonder how the Garding types inequalities make a role in the existence and regularity of the solutions of some equation (say $a^{w}(x,D)u=f$)with $Rea(x,\xi)\geq 0$ –  user23078 May 8 '12 at 7:53
    
@user23078: see the papers by Parmegianni available I think on google. For thes second question, the book of Grub "distribution and Operators". –  user36539 Sep 16 '13 at 11:39
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Daniel Tataru obtained sharp and strong results (with restricted regularity of the coefficients of the operator)about Garding and Fefferman-Phong inequalities whith the use of the FBI (Fourier-Bros-Iagolnitzer) transform without any suplementary condition on the principal and subprincipal symbols (trace+ condition). Notice that the condition of positivity is on the principal symbol for Garding inequality but it is on the complete symbol for the Fefferman-Phong inequality. There is also a refinement of Melin inequality (see Mohamed, "Etude spectrale d'operateurs hypoelliptiques a caracteristiques multiples I et II") whith a genralization of the trace+ condition. Also there is some generalizations of the Melin-Hormander inequality obtained by the italian school (Parenti, Parmeggiani).

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