|
11
|
|
|
Denote by $D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$,then we have $$\|D^{\alpha}(f\cdot g)\|\leq g)\| \leq C(\|D^{\alpha+s}(f)\|_{p_1}\|D^{-s}(g)\|_{q_1}+\|D^{\alpha+t}(f)\|_{p_2}\|D^{-t}(f)\|_{q_2})$$ where $\alpha$,s,t are positive real numbers,and $\frac{1}{p}=\frac{1}{p_{i}}+\frac{1}{q_{i}}$ with $i=1,2$.
The proof can be seen in [Exact smoothing properties of Schrödinger semigroups]（ http://www.jstor.org/stable/10.2307/25098514.）
|
|
|
|
|
10
|
|
|
|
|
|
|
|
|
9
|
|
|
Denote by $J^{\alpha}=(1-\triangle)^{\frac{\alpha}{2}}$,and by $D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$,then we have $$\|J^{\alpha}(fg)\|{p}\leq C(\|J^{\alpha+s}(f)\|{p_1}\|J^{-s}(g)\|_{q_1}+\|J^{\alpha+t}(f)\|_{p_2}\|J^{-t}(f)\|_{q_2})$$,where $\|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})$$ where $1{p}\leq C(\|D^{\alpha+s}(f)\|{p_1}\|D^{-s}(g)\|_{q_1}+\|D^{\alpha+t}(f)\|_{p_2}\|D^{-t}(f)\|_{q_2})$$
\alpha$,s,t are positive real numbers,and $\frac{1}{p}=\frac{1}{p_{i}}+\frac{1}{q_{i}}$ with $i=1,2$.
The proof can be seen in [Exact smoothing properties of Schrödinger semigroups]（ http://www.jstor.org/stable/10.2307/25098514.）
|
|
|
|
8
|
|
|
Denote by $J^{\alpha}=(1-\triangle)^{\frac{\alpha}{2}}$ and J^{\alpha}=(1-\triangle)^{\frac{\alpha}{2}}$,and by $D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$,suppose $1{p}\leq C(\|J^{\alpha+s}(f)\|{p_1}\|J^{-s}(g)\|_{q_1}+\|J^{\alpha+t}(f)\|_{p_2}\|J^{-t}(f)\|_{q_2})$$
and the homogeneous version D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$,then we have $$\|D^{\alpha}(fg)\|$\|J^{\alpha}(fg)\|{p}\leq C(\|D^{\alpha+s}(f)\|C(\|J^{\alpha+s}(f)\|{p_1}\|D^{-s}(g)\|_{q_1}+\|D^{\alpha+t}(f)\|_{p_2}\|D^{-t}(f)\|_{q_2})$$
p_1}\|J^{-s}(g)\|_{q_1}+\|J^{\alpha+t}(f)\|_{p_2}\|J^{-t}(f)\|_{q_2})$$,where $1{p}\leq C(\|D^{\alpha+s}(f)\|{p_1}\|D^{-s}(g)\|_{q_1}+\|D^{\alpha+t}(f)\|_{p_2}\|D^{-t}(f)\|_{q_2})$$
The proof can be seen in [Exact smoothing properties of Schrödinger semigroups]: （ http://www.jstor.org/stable/10.2307/25098514.）
|
|
|
|
|
7
|
|
|
|
|
|
|
|
|
6
|
|
|
Denote by $J^{\alpha}=(1-\triangle)^{\frac{\alpha}{2}}$ and by $D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$,suppose $1{p}\leq C(||J^{\alpha+s}(f)||{p_1}||J^{-s}(g)||_{q_1}+||J^{\alpha+t}(f)||_{p_2}||J^{-t}(f)||_{q_2})$$
C(\|J^{\alpha+s}(f)\|{p_1}\|J^{-s}(g)\|_{q_1}+\|J^{\alpha+t}(f)\|_{p_2}\|J^{-t}(f)\|_{q_2})$$
and the homogeneous version $$||D^{\alpha}(fg)||$\|D^{\alpha}(fg)\|{p}\leq C(||D^{\alpha+s}(f)||{p_1}||D^{-s}(g)||_{q_1}+||D^{\alpha+t}(f)||{L^{pC(\|D^{\alpha+s}(f)\|{2}}}||D^{-t}(f)||_{q_2})$$
p_1}\|D^{-s}(g)\|_{q_1}+\|D^{\alpha+t}(f)\|_{p_2}\|D^{-t}(f)\|_{q_2})$$
The proof can be seen in [Exact smoothing properties of Schrödinger semigroups]: http://www.jstor.org/stable/10.2307/25098514.
|
|
|
|
|
5
|
|
|
Denote by $J^{\alpha}=(1-\triangle)^{\frac{\alpha}{2}}$ and by $D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$, $\frac{1}{p}=\frac{1}{p_{i}}+\frac{1}{q_{i}}$ with $i=1,2$ , then we have D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$,suppose $$||J^{\alpha}(fg)||{L^{p}}\leq C(||J^{\alpha+s}(f)||{L^{p_{1}}}||J^{-s}(g)||{L^{q{1}}}+||J^{\alpha+t}(f)||{L^{p{2}}}||J^{-t}(f)||{L^{q{2}}})$$1{p}\leq C(||J^{\alpha+s}(f)||{p_1}||J^{-s}(g)||_{q_1}+||J^{\alpha+t}(f)||_{p_2}||J^{-t}(f)||_{q_2})$$
and the homogeneous version $$||D^{\alpha}(fg)||{L^{p}}\leq p}\leq C(||D^{\alpha+s}(f)||{L^{p_{1}}}||D^{-s}(g)||{L^{q{1}}}+||D^{\alpha+t}(f)||p_1}||D^{-s}(g)||_{q_1}+||D^{\alpha+t}(f)||{L^{p{2}}}||D^{-t}(f)||{L^{q{2}}})$$ 2}}}||D^{-t}(f)||_{q_2})$$
The proof can be seen in [Exact smoothing properties of Schrödinger semigroups]: http://www.jstor.org/stable/10.2307/25098514.
|
|
|
| |
|
Post Undeleted by Shanlin Huang
|
|
|
|
|
|
| |
|
Post Deleted by Shanlin Huang
|
|
|
|
|
|
|
|
4
|
|
|
Denote by $J^{\alpha}=(1-\triangle)^{\frac{\alpha}{2}}$ and by $D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$,suppose $1<p<\infty$,$\alpha,s,t\geq 0$,and D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$, $\frac{1}{p}=\frac{1}{p_{i}}+\frac{1}{q_{i}}$ with $i=1,2$ , then we have $$||J^{\alpha}(fg)||_{L^{p}}\leq C(||J^{\alpha+s}(f)||_{L^{p_{1}}}||J^{-s}(g)||_{L^{q_{1}}}+||J^{\alpha+t}(f)||_{L^{p_{2}}}||J^{-t}(f)||_{L^{q_{2}}})$$$||J^{\alpha}(fg)||{L^{p}}\leq C(||J^{\alpha+s}(f)||{L^{p_{1}}}||J^{-s}(g)||{L^{q{1}}}+||J^{\alpha+t}(f)||{L^{p{2}}}||J^{-t}(f)||{L^{q{2}}})$$
and the homogeneous version $$||D^{\alpha}(fg)||_{L^{p}}\leq C(||D^{\alpha+s}(f)||_{L^{p_{1}}}||D^{-s}(g)||_{L^{q_{1}}}+||D^{\alpha+t}(f)||_{L^{p_{2}}}||D^{-t}(f)||_{L^{q_{2}}})$$$||D^{\alpha}(fg)||{L^{p}}\leq C(||D^{\alpha+s}(f)||{L^{p_{1}}}||D^{-s}(g)||{L^{q{1}}}+||D^{\alpha+t}(f)||{L^{p{2}}}||D^{-t}(f)||{L^{q{2}}})$$
|
|
|
|
|
3
|
|
|
Denote by $J^{\alpha}=(1-\triangle)^{\frac{\alpha}{2}}$ and by $D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$,suppose $1{L^{p}}\leq C(||J^{\alpha+s}(f)||{L^{p_{1}}}||J^{-s}(g)||{L^{q{1}}}+||J^{\alpha+t}(f)||{L^{p{2}}}||J^{-t}(f)||{L^{q{2}}})$$
1<p<\infty$,$\alpha,s,t\geq 0$,and $\frac{1}{p}=\frac{1}{p_{i}}+\frac{1}{q_{i}}$ with $i=1,2$ ,then we have $$||J^{\alpha}(fg)||_{L^{p}}\leq C(||J^{\alpha+s}(f)||_{L^{p_{1}}}||J^{-s}(g)||_{L^{q_{1}}}+||J^{\alpha+t}(f)||_{L^{p_{2}}}||J^{-t}(f)||_{L^{q_{2}}})$$
and the homogeneous version $$||D^{\alpha}(fg)||{L^{p}}\leq C(||D^{\alpha+s}(f)||{L^{p_{1}}}||D^{-s}(g)||{L^{q{1}}}+||D^{\alpha+t}(f)||{L^{p{2}}}||D^{-t}(f)||{L^{q{2}}})$$ $||D^{\alpha}(fg)||_{L^{p}}\leq C(||D^{\alpha+s}(f)||_{L^{p_{1}}}||D^{-s}(g)||_{L^{q_{1}}}+||D^{\alpha+t}(f)||_{L^{p_{2}}}||D^{-t}(f)||_{L^{q_{2}}})$$
|
|
|
|
|
2
|
|
|
Denote by $J^{\alpha}=(1-\triangle)^{\frac{\alpha}{2}}$ and by $D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$,suppose $1{L^{p}}\leq C(||J^{\alpha+s}(f)||{L^{p_{1}}}||J^{-s}(g)||{L^{q{1}}}+||J^{\alpha+t}(f)||{L^{p{2}}}||J^{-t}(f)||{L^{q{2}}})$$
and the homogeneous version $$||D^{\alpha}(fg)||{L^{p}}\leq C(||D^{\alpha+s}(f)||{L^{p_{1}}}||D^{-s}(g)||{L^{q{1}}}+||D^{\alpha+t}(f)||{L^{p{2}}}||D^{-t}(f)||{L^{q{2}}})$$
The proof can be seen in {Exact smoothing properties of schrodinger semigroup}[http://www.jstor.org/stable/25098514?&Search=yes&searchText=Exact&searchText=schrodinger&searchText=smoothing&searchText=semigroup&searchText=properties&list=hide&searchUri=%2Faction%2FdoBasicSearch%3FQuery%3D%2BExact%2Bsmoothing%2Bproperties%2Bof%2Bschrodinger%2Bsemigroup%26acc%3Don%26wc%3Don&prevSearch=&item=1&ttl=14&returnArticleService=showFullText],which the generalized chain rule had also been treated.
|
|
|
|
|
1
|
|
|
Denote by $J^{\alpha}=(1-\triangle)^{\frac{\alpha}{2}}$ and by $D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$,suppose $1{L^{p}}\leq C(||J^{\alpha+s}(f)||{L^{p_{1}}}||J^{-s}(g)||{L^{q{1}}}+||J^{\alpha+t}(f)||{L^{p{2}}}||J^{-t}(f)||{L^{q{2}}})$$
and the homogeneous version $$||D^{\alpha}(fg)||{L^{p}}\leq C(||D^{\alpha+s}(f)||{L^{p_{1}}}||D^{-s}(g)||{L^{q{1}}}+||D^{\alpha+t}(f)||{L^{p{2}}}||D^{-t}(f)||{L^{q{2}}})$$
The proof can be seen in {Exact smoothing properties of schrodinger semigroup}[http://www.jstor.org/stable/25098514?&Search=yes&searchText=Exact&searchText=schrodinger&searchText=smoothing&searchText=semigroup&searchText=properties&list=hide&searchUri=%2Faction%2FdoBasicSearch%3FQuery%3D%2BExact%2Bsmoothing%2Bproperties%2Bof%2Bschrodinger%2Bsemigroup%26acc%3Don%26wc%3Don&prevSearch=&item=1&ttl=14&returnArticleService=showFullText],which the generalized chain rule had also been treated.
|
|
|
