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<n, 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

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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}.

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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}}

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