Symbol class

Symbol classes are mathematical objects from the field of partial differential equations. They were introduced by Lars Hörmander and are therefore sometimes also called Hörmander classes . Its elements are a generalization of the symbol of a differential operator .

Symbol classes

If one would like to consider generalizations of differential operators such as pseudo differential operators or Fourier integral operators , one can also use or investigate symbols of real degree. For this purpose, the symbol classes were introduced by Lars Hörmander.

definition

Let be natural numbers, an open subset, and real numbers with and . Then one understands by the set of all smooth functions , so that for every compact set and all the inequality ${\displaystyle n,N\in \mathbb {N} }$ ${\displaystyle X\subset \mathbb {R} ^{n}}$ ${\displaystyle m,\rho ,\delta }$ ${\displaystyle 0<\rho \leq 1}$ ${\displaystyle 0\leq \delta <1}$ ${\displaystyle S_{\rho ,\,\delta }^{m}(X\times \mathbb {R} ^{N})}$ ${\displaystyle a\in C^{\infty }(X\times \mathbb {R} ^{N})}$ ${\displaystyle K\subset X}$ ${\displaystyle \alpha ,\beta \in \mathbb {N} \cup \{0\}}$

${\displaystyle \left|{\frac {\partial ^{\beta }}{\partial x^{\beta }}}{\frac {\partial ^{\alpha }}{\partial \xi ^{\alpha }}}a(x,\xi )\right|\leq C_{\alpha ,\beta ,K}(1+|\xi |)^{m-\rho |\alpha |+\delta |\beta |}}$

is satisfied for a constant . The elements of are called symbols of order and type . In addition, the symbol classes and by ${\displaystyle C_{\alpha ,\beta ,K}}$ ${\displaystyle S_{\rho ,\,\delta }^{m}}$ ${\displaystyle m}$ ${\displaystyle (\rho ,\delta )}$ ${\displaystyle S^{-\infty }}$ ${\displaystyle S^{\infty }}$

${\displaystyle {\begin{aligned}S^{-\infty }&:=\bigcap _{m\in \mathbb {R} }S_{\rho ,\delta }^{m}\\S_{\rho ,\delta }^{\infty }&:=\bigcup _{m\in \mathbb {R} }S_{\rho ,\delta }^{m}\end{aligned}}}$

Are defined.

Topologization

The best constants of the inequality

${\displaystyle \left|{\frac {\partial ^{\beta }}{\partial x^{\beta }}}{\frac {\partial ^{\alpha }}{\partial \xi ^{\alpha }}}a(x,\xi )\right|\leq C_{\alpha ,\beta ,K}(1+|\xi |)^{m-\rho |\alpha |+\delta |\beta |}}$

that is, the constants

${\displaystyle p_{K,\alpha ,\beta }(a):=\sup _{x\in K;\ \xi \in \mathbb {R} ^{n}}{\frac {\partial ^{\beta }}{\partial x^{\beta }}}{\frac {\partial ^{\alpha }}{\partial \xi ^{\alpha }}}a(x,\xi )(1+|\xi |)^{-m+\rho |\alpha |-\delta |\beta |}}$

are semi-norms . These turn the rooms into Fréchet rooms . Since and the countable average of Fréchet spaces is a Fréchet space again, it is also a Fréchet dream. ${\displaystyle S^{m}(X\times \mathbb {R} ^{n})}$ ${\displaystyle \textstyle S^{-\infty }:=\bigcap _{m\in \mathbb {R} }S_{\rho ,\delta }^{m}=\bigcap _{m\in \mathbb {Z} }S_{\rho ,\delta }^{m}}$ ${\displaystyle S^{-\infty }}$

Examples

Be an open subset. ${\displaystyle X\subset \mathbb {R} ^{n}}$

• If one identifies the space of real numbers with the space of constant functions, then this is a subspace of . ${\displaystyle \mathbb {R} }$ ${\displaystyle S_{1,0}^{0}(X\times \mathbb {R} ^{N})}$
• Be

${\displaystyle p(x,\xi )=\sum _{|\alpha |\leq k}a_{\alpha }(x)\xi ^{\alpha }}$

with coefficient functions a symbol of a differential operator of order . Then applies . ${\displaystyle a_{\alpha }\in C^{\infty }(X)}$ ${\displaystyle k\in \mathbb {N} _{0}}$ ${\displaystyle p\in S_{1,0}^{k}(X\times \mathbb {R} ^{N})}$

• Be with . Then applies . ${\displaystyle p(\xi )=(1+|\xi |^{2})^{m/2}}$ ${\displaystyle -\infty <m<\infty }$ ${\displaystyle p\in S_{1,0}^{m}(X\times \mathbb {R} ^{N})}$

properties

• The symbol classes are for everyone , and Montel spaces . ${\displaystyle S_{\rho ,\delta }^{m}(X\times \mathbb {R} ^{N})}$ ${\displaystyle m\in \mathbb {R} \cup \{-\infty ,\infty \}}$ ${\displaystyle 0<\rho \leq 1}$ ${\displaystyle 0\leq \delta <1}$
• Differentiating the symbol by the second variable improves (i.e. reduces) the order. Precisely this means that the figure

${\displaystyle p\mapsto {\frac {\partial ^{\beta }}{\partial x^{\beta }}}{\frac {\partial ^{\alpha }}{\partial \xi ^{\alpha }}}p(x,\xi )\colon S_{\rho ,\delta }^{m}(X\times \mathbb {R} ^{N})\to S_{\rho ,\delta }^{m-\rho |\alpha |}(X\times \mathbb {R} ^{N})}$

is linear and continuous .

• Multiplying two symbols results in another symbol, namely it applies

${\displaystyle (p,{\tilde {p}})\mapsto p(x,\xi ){\tilde {p}}(x,\xi )\colon S_{\rho ,\delta }^{m}(X\times \mathbb {R} ^{N})\times S_{\rho ,\delta }^{m'}(X\times \mathbb {R} ^{N})\to S_{\rho ,\delta }^{m+m'}(X\times \mathbb {R} ^{N})\,.}$

This mapping is bilinear and continuous.

• For true . ${\displaystyle m\leq m'}$ ${\displaystyle S_{1,0}^{m}\subset S_{1,0}^{m'}}$
• Let be positive homogeneous of degree m for , that is ${\displaystyle a\in C^{\infty }(X\times \mathbb {R} ^{N})}$ ${\displaystyle |\xi |\geq 1}$

${\displaystyle a(x,\lambda \xi )=\lambda ^{m}a(x,\xi )}$

for and . Then applies . ${\displaystyle \lambda \geq 1}$ ${\displaystyle |\xi |\geq 1}$ ${\displaystyle a\in S_{1,0}^{m}(X\times \mathbb {R} ^{N})}$

• Be open and . On bounded subsets of , the topology induced by is the topology of point-wise convergence . ${\displaystyle X\subset \mathbb {R} ^{N}}$ ${\displaystyle m<m'}$ ${\displaystyle S_{1,0}^{m}(X\times \mathbb {R} ^{N})}$ ${\displaystyle S_{1,0}^{m'}(X\times \mathbb {R} ^{N})}$
• Be . Then the topology is tight in . ${\displaystyle m<m'}$ ${\displaystyle S^{-\infty }(X\times \mathbb {R} ^{N})}$ ${\displaystyle S_{1,0}^{m'}}$ ${\displaystyle S_{1,0}^{m}(X\times \mathbb {R} ^{N})}$

Asymptotic development of a symbol

definition

Be a symbol. Exist with ${\displaystyle a\in S_{\rho ,\delta }^{m}(X\times \mathbb {R} ^{N})}$ ${\displaystyle a_{i}\in S_{\rho ,\delta }^{m_{i}}(X\times \mathbb {R} ^{N})}$

${\displaystyle m=m_{0}>m_{1}>\ldots >m_{i}\to -\infty \quad i\to \infty \,,}$

so that

${\displaystyle a-\sum _{j=0}^{N-1}a_{j}\in S_{\rho ,\delta }^{m_{N}}(X\times \mathbb {R} ^{N})}$

holds for every positive number . The formal series is an asymptotic expansion of and one writes ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle \textstyle \sum _{j=0}^{\infty }a_{j}}$ ${\displaystyle a}$

${\displaystyle a\sim \sum _{j=0}^{\infty }a_{j}\,.}$

Uniqueness

The asymptotic expansion of a symbol is clearly modulo symbols of the class . To put it precisely this means: ${\displaystyle S^{-\infty }(X\times \mathbb {R} ^{N})}$

Be a decomposition with and be . Then there is a symbol such that ${\displaystyle m_{0}>m_{1}>\ldots >m_{i}\to \infty }$ ${\displaystyle i\to \infty }$ ${\displaystyle a_{i}\in S_{\rho ,\delta }^{m_{i}}(X\times \mathbb {R} ^{N})}$ ${\displaystyle a\in S_{\rho ,\delta }^{m_{0}}(X\times \mathbb {R} ^{N})}$

${\displaystyle a\sim \sum _{j=0}^{\infty }a_{j}}$

applies. If there is another symbol with the same asymptotic expansion, then applies . ${\displaystyle b}$ ${\displaystyle a-b\in S^{-\infty }(X\times \mathbb {R} ^{N})}$

Classic symbol

A classic symbol is a special case of a symbol from space. These prove to be easier to handle in connection with pseudo-differential operators. This class of functions was introduced by the mathematicians Joseph Kohn and Louis Nirenberg . ${\displaystyle S_{1,0}^{m}.}$

A symbol is called a classical symbol and one writes for it if there is a peeling function and functions so that each is positively homogeneous of the order in the variable . So it has to ${\displaystyle a\in S_{1,0}^{m}(X\times \mathbb {R} ^{N})}$ ${\displaystyle a\in S_{cl}^{m}(X\times \mathbb {R} ^{N})}$ ${\displaystyle \phi \in C^{\infty }(\mathbb {R} ^{N})}$ ${\displaystyle a_{j}\in C^{\infty }(X\times (\mathbb {R} ^{N}\backslash \{0\}))}$ ${\displaystyle a_{j}}$ ${\displaystyle m-j}$ ${\displaystyle \xi }$

${\displaystyle a_{j}(x,t\xi )=t^{m-j}a_{j}(x,\xi )\qquad \forall (x,t,\xi )\in X\times \mathbb {R} ^{N}\times \mathbb {R} _{+}}$

apply and also must

${\displaystyle a(x,\xi )-\sum _{j=0}^{k-1}\phi (x)a_{j}(x,\xi )\in S^{m-k}(X\times \mathbb {R} ^{N})}$

apply to all . This gives an asymptotic expansion of . ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle a}$

Individual evidence

1. ↑ Alain Grigis & Johannes Sjöstrand: Microlocal analysis for differential operators: an introduction , Cambridge University Press, 1994, ISBN 0-521-44986-3 , page 40.
2. ^ MA Shubin: Pseudo-differential operator . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
3. ↑ ^{a b} Man Wah Wong: An introduction to pseudo-differential operator . World Scientific, River Edge, NJ 1999, ISBN 978-981-02-3813-1 , pp. 29 .
4. ↑ Man Wah Wong: An introduction to pseudo-differential operator . World Scientific, River Edge, NJ 1999, ISBN 978-981-02-3813-1 , pp. 33 .
5. ↑ Man Wah Wong: An introduction to pseudo-differential operator . World Scientific, River Edge, NJ 1999, ISBN 978-981-02-3813-1 , pp. 33-36 .
6. ↑ JJ Kohn, L. Nirenberg: On the algebra of pseudo-differential operators , Comm. Pure Appl. Math. 18: 269-305 (1965).

literature

• Hörmander, Lars - The analysis of linear partial differential operators 1. Distribution theory and fourier analysis, 2. Edition, Springer-Verlag, 1990, ISBN 3-540-52345-6
• Hörmander, Lars - The analysis of linear partial differential operators 3 .. Pseudo-differential operators, Springer-Verlag, Berlin, 1994, ISBN 978-3-540-49937-4