記號, 函數空間及不等式

 

 

** $$\bex \p_i=\f{\p}{\p x_i},\quad \n=(\p_1,\cdots,\p_n),\quad \lap=\p_1\p_1+\cdots+\p_n\p_n. \eex$$

 

** Young inequality (Young 不等式) $$\bex 1<p,q<\infty,\quad \f{1}{p}+\f{1}{q}=1,\quad a,b>0\ra ab\leq \f{1}{p}a^p+\f{1}{q}a^q. \eex$$

 

** Lebesgue space $$\bex L^p(\Om)=\sed{f; \sen{f}_{L^p}=\sez{\int_\Om |f(x)|^p\rd x}^\f{1}{p}<\infty},\quad 1\leq p<\infty; \eex$$ $$\bex L^\infty(\Om) =\sed{ f; \sen{f}_{L^\infty}=\inf_{E\subset \Om: mE=0} \sup_{x\in E} |f(x)|<\infty}. \eex$$

 

** H\"older inequality $$\bex 1\leq p,q\leq\infty,\quad \f{1}{p}+\f{1}{q}=1\ra \int fg\rd x\leq \sen{f}_{L^p} \sen{g}_{L^q}. \eex$$

 

** Minkowski inequality $$\bex 1\leq p\leq \infty\ra \sen{f+g}_{L^p}\leq \sen{f}_{L^p}+\sen{g}_{L^p}. \eex$$

 

** Interpolation inequality $$\bex 1\leq p\leq r\leq q\leq\infty,\quad \f{1}{r}=\f{1-\tt}{p}+\f{\tt}{q} \ra \sen{f}_{L^r}\leq \sen{f}_{L^p}^{1-\tt} \sen{f}_{L^q}^\tt. \eex$$

 

** Fourier multiplier (Fourier 乘子) $$\bex m(D)f(x)=\calF^{-1}(m(\cdot)\hat f(\cdot))(x). \eex$$

 

** 分數階 Laplacian $$\bex \vLm=(-\lap)^\f{1}{2}:\ \vLm f(x)=\calF^{-1}(|\cdot|\calF f(\cdot))(x); \eex$$ $$\bex \vLm^s =(-\lap)^{\f{s}{2}}:\ \vLm^s f(x)=\calF^{-1}(|\cdot|^s\calF f(\cdot))(x). \eex$$

 

** (Homogeneous Sobolev spaces) Let $s$ be a real number and $f$ a tempered distribution such that $\hat f\in L^1_{loc}$. We say that $f$ belongs to the homogeneous Sobolev space $\dot H^s$ if $$\bex \sen{f}_{\dot H^s} =\sex{\int |\xi|^{2s}|\hat f(\xi)|^2\rd \xi}^\f{1}{2}<\infty. \eex$$

 

** (BMO spaces) A distribution $f\in \calD'(\bbR^n)$ is said to belong to the space $BMO(\bbR^n)$ if $f$ is locally integrable and $\dps{\sup_{B\in \calB}\f{1}{|B|} \int_B|f-m_Bf|\rd x<\infty}$ where $\calB$ is the collection of all open balls in $\bbR^n$ and $\dps{m_Bf=\f{1}{|B|}\int_B f(x)\rd x}$. When seen as a distribution space modulo the constants, $BMO(\bbR^n)$ is a Banach space for the norm $\dps{\sen{f}_{BMO}=\sup_{B\in \calB}\f{1}{|B|} \int_B|f-m_Bf|\rd x}$. The space $L^\infty$ is a embedded in $BMO$.

 

** Commutator estimate (交換子估計) $$\bex \sen{\vLm^s(fg)-f\vLm^s g}_{L^p} \leq C\sez{ \sen{\n f}_{L^{p_1}} \sen{\vLm^{s-1}g}_{L^{p_2}} +\sen{\vLm^s f}_{L^{p_3}} \sen{g}_{L^{p_4}} }, \eex$$ if $$\bex s>0,\quad 1<p,p_2,p_3<\infty,\quad 1\leq p_1,p_4\leq\infty,\quad \f{1}{p}=\f{1}{p_1}+\f{1}{p_2}=\f{1}{p_3}+\f{1}{p_4}. \eex$$ See [Kato, Tosio; Ponce, Gustavo. Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41 (1988), no. 7, 891--907] Lemma X1.

 

** $$\bex \sen{\vLm^s(fg)}_{L^p} \leq C\sex{ \sen{f}_{L^{p_1}}\sen{\vLm^s g}_{L^{p_2}} +\sen{\vLm^s f}_{L^{p_3}} \sen{g}_{L^{p_4}} }, \eex$$ if $$\bex s>0,\quad 1<p,p_2,p_3<\infty,\quad 1\leq p_1,p_4\leq\infty,\quad \f{1}{p}=\f{1}{p_1}+\f{1}{p_2}=\f{1}{p_3}+\f{1}{p_4}. \eex$$ See [Kato, Tosio; Ponce, Gustavo. Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41 (1988), no. 7, 891--907] Lemma X4.