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This is the standard Hardy-Littlewood-Sobolev inequality(or the theorem of fractional integration).A more direct approach is write $$ \int{f(x-y)|y|^{\alpha-n}dy}=\int_{|y|<R}+\int_{|y|\ge R} $$ Fo …

The closed range theorem tells us that given two banach spaces X,Y,and a closed densely defined linear operator T：$X \to Y$. We have the following equivalence $R(T)$ is closed in $Y \iff R(T^{*})$ is …

According to the basic rules of symbolic caculus,$[a(x,D),x_{j}]=-ia^{j}[x,D]$.So we have $[(1-\triangle)^{\frac{1}{2}},x_i]=\partial_i(1-\triangle)^{-\frac{1}{2}}$ which is $L^2$ bounded. It's also t …

Consider the dispersive estimates for the Schrödinger flow $$ e^{itH}P_{c},\quad H=-\Delta+V \quad \text{on}\quad \mathbb{R}^n,n\ge 1 $$ where $P_{c}$ is the projection onto the continuous spectrum o …

When we have a distribuion $u\in \mathcal{D}'(R^n)$,and the restriction to $R^{n}\backslash{0}$ is homogeneous of degree a,we have $u \in \varphi'$ and $\widehat u$ is of degree(-n-a) in $R^{n}\backsl …

Assume that $P \in \Psi^{m}(X)$ (X is a $C^{\infty}$ manifold)is properly supported and has a real principal part p which is homogeneous of degree m.I'm interested in the existence theorem(at least lo …

The related topic here is the homogeneous distribution on $\mathbb{R}^n\0$ and its extension to $\mathbb{R}^n$. In your case $T_{f}$ is a homogeneous distribution on $\mathbb{R}^n\0$ of degreee $-n$. …

Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost …

Let $K$ be a compact subset in $\mathbb{R}^n$ with $m(K)=0$, Suppose $supp\hat{u}\subset K$ for some $u\in L^p$,where $2\leq p\leq \frac{2n}{n-1}$,can we get $u\equiv 0$ ? Motivation: If $K$ is a com …

In the theory of Pseudo-differential operators,when a symbol $a(x,\xi)\in S^{0}$,then the operator $a(x,D)$ defined by$$a(x,D)u=\int{e^{ix\xi}a(x,\xi)\widehat{u}}d \xi$$ is $L^2$ bounded.$ $ My ques …

Let $T=(-\Delta)^{1/2}$. Can we have estimates, similar to the one below $$ \| T^{\alpha}(fg)-(T^{\alpha}f)g-f(T^{\alpha}g) \|_p \leq \|T^{\alpha-1}f\|_p \|T^{\alpha-1}g\|_p, $$ hold in $L^p$, where …

Denote by $D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$,then we have $$\|D^{\alpha}(f\cdot g)\| \leq C(\|D^{\alpha+s}(f)\|_{p_1}\|D^{-s}(g)\|_{q_1}+\|D^{\alpha+t}(f)\|_{p_2}\|D^{-t}(f)\|_{q_2})$$ where …

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 …

In fact, when $0<\alpha<1$, the following estimate has been shown: $$G_{\alpha}(t,x)\leq \frac{t}{{t^{1/\alpha}+x^2}^{n/2+\alpha}}.$$ When $\alpha>1$, a similar result holds by a similar proof. See Mi …