** $$ex p_i=f{p}{p x_i},quad =(p_1,cdots,p_n),quad lap=p_1p_1+cdots+p_np_n. eex$$
** Young inequality (Young 不等式) $$ex 1<p,q<infty,quad f{1}{p}+f{1}{q}=1,quad a,b>0 a ableq f{1}{p}a^p+f{1}{q}a^q. eex$$
** Lebesgue space $$ex L^p(Om)=sed{f; sen{f}_{L^p}=sez{int_Om |f(x)|^p d x}^f{1}{p}<infty},quad 1leq p<infty; eex$$ $$ex L^infty(Om) =sed{ f; sen{f}_{L^infty}=inf_{Esubset Om: mE=0} sup_{xin E} |f(x)|<infty}. eex$$
** H"older inequality $$ex 1leq p,qleqinfty,quad f{1}{p}+f{1}{q}=1 a int fg d xleq sen{f}_{L^p} sen{g}_{L^q}. eex$$
** Minkowski inequality $$ex 1leq pleq infty a sen{f+g}_{L^p}leq sen{f}_{L^p}+sen{g}_{L^p}. eex$$
** Interpolation inequality $$ex 1leq pleq rleq qleqinfty,quad f{1}{r}=f{1- t}{p}+f{ t}{q} a sen{f}_{L^r}leq sen{f}_{L^p}^{1- t} sen{f}_{L^q}^ t. eex$$
** Fourier multiplier (Fourier 乘子) $$ex m(D)f(x)=calF^{-1}(m(cdot)hat f(cdot))(x). eex$$
** 分数阶 Laplacian $$ex vLm=(-lap)^f{1}{2}: vLm f(x)=calF^{-1}(|cdot|calF f(cdot))(x); eex$$ $$ex vLm^s =(-lap)^{f{s}{2}}: vLm^s f(x)=calF^{-1}(|cdot|^scalF f(cdot))(x). eex$$
** (Homogeneous Sobolev spaces) Let $s$ be a real number and $f$ a tempered distribution such that $hat fin L^1_{loc}$. We say that $f$ belongs to the homogeneous Sobolev space $dot H^s$ if $$ex sen{f}_{dot H^s} =sex{int |xi|^{2s}|hat f(xi)|^2 d xi}^f{1}{2}<infty. eex$$
** (BMO spaces) A distribution $fin calD'(bR^n)$ is said to belong to the space $BMO(bR^n)$ if $f$ is locally integrable and $dps{sup_{Bin calB}f{1}{|B|} int_B|f-m_Bf| d x<infty}$ where $calB$ is the collection of all open balls in $bR^n$ and $dps{m_Bf=f{1}{|B|}int_B f(x) d x}$. When seen as a distribution space modulo the constants, $BMO(bR^n)$ is a Banach space for the norm $dps{sen{f}_{BMO}=sup_{Bin calB}f{1}{|B|} int_B|f-m_Bf| d x}$. The space $L^infty$ is a embedded in $BMO$.
** Commutator estimate (交换子估计) $$ex sen{vLm^s(fg)-fvLm^s g}_{L^p} leq Csez{ sen{ f}_{L^{p_1}} sen{vLm^{s-1}g}_{L^{p_2}} +sen{vLm^s f}_{L^{p_3}} sen{g}_{L^{p_4}} }, eex$$ if $$ex s>0,quad 1<p,p_2,p_3<infty,quad 1leq p_1,p_4leqinfty,quad f{1}{p}=f{1}{p_1}+f{1}{p_2}=f{1}{p_3}+f{1}{p_4}. eex$$ 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.
** $$ex sen{vLm^s(fg)}_{L^p} leq Csex{ sen{f}_{L^{p_1}}sen{vLm^s g}_{L^{p_2}} +sen{vLm^s f}_{L^{p_3}} sen{g}_{L^{p_4}} }, eex$$ if $$ex s>0,quad 1<p,p_2,p_3<infty,quad 1leq p_1,p_4leqinfty,quad f{1}{p}=f{1}{p_1}+f{1}{p_2}=f{1}{p_3}+f{1}{p_4}. eex$$ 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.