Ngô Quốc Anh

February 25, 2015

Continuous functions on subsets can be extended to the whole space: The Kirzbraun-Pucci theorem

Filed under: Uncategorized — Ngô Quốc Anh @ 1:22

Let f be a continuous function defined on a set E \subset \mathbb R^N with values in \mathbb R and with modulus of continuity

\displaystyle \omega_f (s) := \sup_{|x-y|\leqslant s,x,y\in E} |f(x) - f(y)| \quad s>0.

Obviously, the function s \mapsto \omega_f(s) is nonnegative and nondecreasing in [0,+\infty).

Our first assumption is that \omega_f is bounded from above in [0, \infty) by some increasing, affine function; that is to say there exists some a,b \in \mathbb R^+ such that

\displaystyle \omega_f (s) \leqslant a s +b \quad \forall s \geqslant 0.

Associated with \omega_f having the above first assumption is the concave modulus of continuity of f, i.e. some smallest concave function c_f lies above \omega_f. Such the function c_f can be easily constructed using the following

\displaystyle c_f (s) = \inf_\ell \{\ell(s) : \ell \text{ is affine and } \ell \geqslant \omega_f \text{ in } [0,+\infty)\}.

As can be easily seen, once \omega_f can be bounded from above by some affine function, the concave modulus of continuity of f exists and is well-defined.

By definition and the monotonicity of \omega_f, we obtain

\displaystyle |f(x)-f(y)| \leqslant \omega_f (|x-y|) \leqslant c_f (|x-y|).

In this note, we prove the following extension theorem.

Theorem (Kirzbraun-Pucci). Let f be a real-valued, uniformly continuous function on a set E \subset \mathbb R^N with modulus of continuity \omega_f satisfying the first assumption. There exists a continuous function \widetilde f defined on \mathbb R^N that coincides with f on E. Moreover, f and \widetilde f have the same concave modulus of continuity c_f and

\displaystyle \sup_{\mathbb R^N} \widetilde f = \sup_E f, \quad \inf_{\mathbb R^N} \widetilde f = \inf_E f.

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February 22, 2015

The conditions (NN), (P), (NN+) and (P+) associated to the Paneitz operator for 3-manifolds

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 18:54

Of recent interest is the prescribed Q-curvature on closed Riemannian manifolds since it involves high-order differential operators.

In a previous post, I have talked about prescribed Q-curvature on 4-manifolds. Recall that for 4-manifolds, this question is equivalent to finding a conformal metric \widetilde g =e^{2u}g for which the Q-curvature of \widetilde g equals the prescribed function \widetilde Q? That is to solving

\displaystyle P_gu+2Q_g=2\widetilde Q e^{4u},

where for any g, the so-called Paneitz operator P_g acts on a smooth function u on M via

\displaystyle {P_g}(u) = \Delta _g^2u - {\rm div}\left( {\frac{2}{3}{R_g} - 2{\rm Ric}_g} \right)du

which plays a similar role as the Laplace operator in dimension two and the Q-curvature of \widetilde g is given as follows

\displaystyle Q_g=-\frac{1}{12}(\Delta\text{Scal}_g -\text{Scal}_g^2 +3|{\rm Ric}_g|^2).

Sometimes, if we denote by \delta the negative divergence, i.e. \delta = - {\rm div}, we obtain the following formula

\displaystyle {P_g}(u) = \Delta _g^2u + \delta \left( {\frac{2}{3}{R_g} - 2{\rm Ric}_g} \right)du.

Generically, for n-manifolds, we obtain

\displaystyle Q_g=-\frac{1}{2(n-1)} \Big(\Delta\text{Scal}_g - \frac{n^3-4n^2+16n-16}{4(n-1)(n-2)^2} \text{Scal}_g^2+\frac{4(n-1)}{(n-2)^2} |{\rm Ric}_g|^2 \Big)

and

\displaystyle {P_g}(u) = \Delta _g^2u + {\rm div}\left( { a_n {R_g} + b_n {\rm Ric}_g} \right)du + \frac{n-4}{2} Q_g u,

where a_n = -((n-2)^2+4)/2(n-1)(n-2) and b_n =4/(n-2).

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