# Ngô Quốc Anh

## May 20, 2012

### The Wolff potential

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 19:26

The Wolff potential probably first appeared in a joint paper between L.I. Hedberg and Th.H. Wolff in 1983 in relation to the spectral synthesis problem for Sobolev spaces. Generally speaking, it is defined for any non-negative Borel measure $\mu$ as follows $\displaystyle {\mathbf W_{\beta ,\gamma }}\mu (x) = \int_0^\infty {{{\left[ {\frac{{\mu ({B_t}(x))}}{{{t^{n - \beta \gamma }}}}} \right]}^{\frac{1}{{\gamma - 1}}}}\frac{{dt}}{t}}$

where $1<\gamma<\infty$, $\beta>0$, $\beta \gamma, and $B_t(x)$ is the ball of radius $t$ centered at the point $x$.

If $d\mu=f dx$ with $f \geqslant 0$ and $f \in L^1_{loc}(\mathbb R^n)$, we write $\displaystyle {\mathbf W_{\beta ,\gamma }}(f)(x) = \int_0^\infty {{{\left[ {\frac{{\int_{{B_t}(x)} {f(y)dy} }}{{{t^{n - \beta \gamma }}}}} \right]}^{\frac{1}{{\gamma - 1}}}}\frac{{dt}}{t}} .$

There are several cases

• If $\beta=1$ and $\gamma=2$, we have $\displaystyle {\mathbf W_{1,2}}(f)(x) = \int_0^\infty {\left( {\int_{{B_t}(x)} {f(y)dy} } \right)\frac{{dt}}{{{t^{n - 3}}}}} .$

Clearly, this is the well-known Newton potential. Indeed, by exchanging the order of the integral variables, we have

## May 15, 2012

### An example of compact embedding between two Banach spaces

Filed under: PDEs — Ngô Quốc Anh @ 3:09

Assume $V : \mathbb R^N \to \mathbb R$ is continuous and satisfies the following two conditions $V(x) \geqslant V_0 >0$ in $\mathbb R^N$ for some $V_0 >0$

and the function $\displaystyle \frac{1}{V(x)} \in L^\frac{1}{N-1}(\mathbb R^N).$

Let us define the following space $\displaystyle E=\left\{ u \in W^{1,N}(\mathbb R^N): \int_{\mathbb R^N}V(x)|u|^N dx<+\infty\right\}.$

On $E$, we can use the following norm $\displaystyle {\left\| u \right\|_E} = {\left( {\int_{{\mathbb{R}^N}} {(|\nabla u{|^N} + V|u{|^N})dx} } \right)^{\frac{1}{N}}}.$

In this note, we prove that $E$ is compactly embedded in $L^q(\mathbb R^N)$ for all $q \geqslant 1$. This is Lemma 2.4 in a paper by Y. Yang published in JFA this 2012 although the technique used is standard.

Indeed, by the first condition on $V$, the standard Sobolev embedding theorem implies that the following embedding $\displaystyle E \hookrightarrow W^{1,N}(\mathbb R^N)\hookrightarrow L^q(\mathbb R^N)$

is obviously continuous for any $q \geqslant N$. It follows from the second condition on $V$ and the Holder inequality that $\displaystyle\int_{{\mathbb{R}^N}} {|u|dx} \leqslant {\left( {\int_{{\mathbb{R}^N}} {\frac{{dx}}{{{V^{\frac{1}{{N - 1}}}}}}} } \right)^{1 - \frac{1}{N}}}{\left( {\int_{{\mathbb{R}^N}} {V|u{|^N}dx} } \right)^{\frac{1}{N}}} \leqslant {\left( {\int_{{\mathbb{R}^N}} {\frac{{dx}}{{{V^{\frac{1}{{N - 1}}}}}}} } \right)^{1 - \frac{1}{N}}}{\left\| u \right\|_E}.$

## May 6, 2012

### A note on the Sobolev trace inequality

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 11:38

The purpose of this note is to talk about the following so-called Sobolev trace inequality $\displaystyle {\left( {\int_{\partial M} {|u{|^{\frac{{2(n - 1)}}{{n - 2}}}}d{s_g}} } \right)^{\frac{{n - 2}}{{n - 1}}}} \leqslant (S + \varepsilon )\int_M {|\nabla u{|^2}d{v_g}} + A(\varepsilon )\int_{\partial M} {{u^2}d{s_g}}$

where $(M,g)$ is a smooth $n$-dimensional, compact, Riemannian manifold with a smooth boundary $\partial M$ with $n \geqslant 3$ and $\varepsilon >0$. The constant $S$ appearing from the above inequality is called the best constant.

In fact, this is just a weak type of the true Sobolev trace inequality, which can be stated as follows $\displaystyle {\left( {\int_{\partial M} {|u{|^{\frac{{2(n - 1)}}{{n - 2}}}}d{s_g}} } \right)^{\frac{{n - 2}}{{n - 1}}}} \leqslant S\int_M {|\nabla u{|^2}d{v_g}} + A\int_{\partial M} {{u^2}d{s_g}}$

where $A$ and $S$ are positive constant. It is know that in order to prove the above Sobolev inequality, a weaker version is needed.

We now talk about its motivation. First, we start with the standard Sobolev inequality appearing when we talk about the the following embedding $\displaystyle H^1(M) \hookrightarrow L^\frac{2n}{n-2}(M).$

More precise, the following $\displaystyle {\left( {\int_M {|u{|^{\frac{{2n}}{{n - 2}}}}d{v_g}} } \right)^{\frac{{n - 2}}{{n }}}} \leqslant S\int_M {|\nabla u{|^2}d{v_g}} + B\int_M {{u^2}d{v_g}}$