Hardy room

In function theory , a Hardy space is a function space of holomorphic functions on certain subsets of . Hardy spaces are the equivalent of the spaces in functional analysis . They are named after Godfrey Harold Hardy , who introduced them in 1914. ${\displaystyle H^{p}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle L^{p}}$

definition

Usually two classes of Hardy spaces are defined, depending on the area in the complex plane on which their functions are defined. ${\displaystyle D\subset \mathbb {C} }$

Hardy spaces on the unit disk

Let be the unit disk in . Then for the Hardy space consists of all holomorphic functions for which holds ${\displaystyle \mathbb {D} =\{z\in \mathbb {C} :|z|<1\}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle p>0}$ ${\displaystyle H^{p}(\mathbb {D} )}$ ${\displaystyle F:\mathbb {D} \to \mathbb {C} }$

${\displaystyle \sup _{0<r<1}\left({\frac {1}{2\pi }}\int _{0}^{2\pi }\left|F(re^{i\theta })\right|^{p}\;{\rm {d}}\theta \right)^{1/p}<\infty .}$

The value of the term on the left of this inequality is called the " norm" of , in symbols . ${\displaystyle H^{p}}$ ${\displaystyle F}$ ${\displaystyle \|F\|_{H^{p}}}$

For is set and defined as the space of all holomorphic functions for which this value is finite. ${\displaystyle p=\infty }$ ${\displaystyle \textstyle \|F\|_{H^{\infty }(\mathbb {D} )}=\sup _{z\in \mathbb {D} }|F(z)|}$ ${\displaystyle H^{\infty }(\mathbb {D} )}$ ${\displaystyle F:\mathbb {D} \to \mathbb {C} }$

Hardy rooms on the upper half-level

Let the upper half-plane be in . Then for the Hardy space consists of all holomorphic functions for which holds ${\displaystyle \mathbb {H} =\{x+iy\in \mathbb {C} :y>0\}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle p>0}$ ${\displaystyle H^{p}(\mathbb {H} )}$ ${\displaystyle F:\mathbb {H} \to \mathbb {C} }$

${\displaystyle \sup _{y>0}\left(\int _{0}^{\infty }\left|F(x+iy)\right|^{p}\;{\rm {d}}x\right)^{1/p}<\infty .}$

The value of the term to the left of this inequality is also referred to as the " norm" of , in symbols . ${\displaystyle H^{p}}$ ${\displaystyle F}$ ${\displaystyle \|F\|_{H^{p}(\mathbb {H} )}}$

For is set and defined as the space of all holomorphic functions for which this value is finite. ${\displaystyle p=\infty }$ ${\displaystyle \textstyle \|F\|_{H^{\infty }(\mathbb {H} )}=\sup _{z\in \mathbb {H} }|F(z)|}$ ${\displaystyle H^{\infty }(\mathbb {H} )}$ ${\displaystyle F:\mathbb {H} \to \mathbb {C} }$

When Hardy spaces are mentioned in general , it is usually clear which of the two classes is meant (whether or not ); usually it is the space of functions on the unit disk . ${\displaystyle H^{p}}$ ${\displaystyle D=\mathbb {D} }$ ${\displaystyle D=\mathbb {H} }$ ${\displaystyle H^{p}(\mathbb {D} )}$ ${\displaystyle \mathbb {D} }$

Factorization

For each function can be written as a product , in which there is an outer function and an inner function . ${\displaystyle p\geq 1}$ ${\displaystyle f\in H^{p}}$ ${\displaystyle f=Gh}$ ${\displaystyle G}$ ${\displaystyle h}$

For on the unit disk, for example, an internal function is exactly if and is the limit value on the unit disk ${\displaystyle H^{p}=H^{p}(\mathbb {D} )}$ ${\displaystyle h}$ ${\displaystyle |h(z)|\leq 1}$

${\displaystyle \lim _{r\rightarrow 1^{-}}h(re^{i\theta })}$

exists for almost all and its absolute amount is 1. is an external function when ${\displaystyle \theta }$ ${\displaystyle G}$

${\displaystyle G(z)=\exp \left(i\phi +{\frac {1}{2\pi }}\int _{0}^{2\pi }{\frac {e^{i\theta }+z}{e^{i\theta }-z}}g(e^{i\theta }){\rm {d}}\theta \right)}$

for a real value and a real-valued function integrable on the unit circle . ${\displaystyle \phi }$ ${\displaystyle g}$

Other properties

• For the rooms are Banach rooms . ${\displaystyle 1\leq p\leq \infty }$ ${\displaystyle H^{p}}$
• For true and . ${\displaystyle p>1}$ ${\displaystyle H^{p}(\mathbb {D} )\cong L^{p}([0,1])}$ ${\displaystyle H^{p}(\mathbb {H} )\cong L^{p}(\mathbb {R} )}$

• For true . All of these inclusions are real. ${\displaystyle 1<p<q<\infty }$ ${\displaystyle H^{\infty }(\mathbb {D} )\subset H^{q}(\mathbb {D} )\subset H^{p}(\mathbb {D} )\subset H^{1}(\mathbb {D} )}$

Real Hardy rooms

Elias Stein and Guido Weiss developed the theory of real Hardy spaces from the Hardy spaces in the upper half -level . ${\displaystyle {\mathcal {H}}^{p}(\mathbb {R} ^{n})}$

definition

Let be a Schwartz function on and a Dirac sequence for t> 0 . If a distribution is tempered , the radial maximum function and the non-tangential maximum function are defined by ${\displaystyle \phi \in S}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \phi _{t}(x)=t^{-n}\phi (t^{-1}x)}$ ${\displaystyle f\in S'}$ ${\displaystyle m_{\phi }(f)}$ ${\displaystyle M_{\phi }(f)}$

${\displaystyle {\begin{aligned}m_{\phi }(f)(x)=&\sup _{t>0}|f*\phi _{t}(x)|,\\M_{\phi }(f)(x)=&\sup _{|y-x|<t<\infty }|f*\phi _{t}(y)|.\end{aligned}}}$

This describes the folding between a temperature-controlled distribution and a Schwartz function. ${\displaystyle *}$

Charles Fefferman and Elias M. Stein proved for and that the following three conditions are equivalent: ${\displaystyle f\in S'(\mathbb {R} ^{n})}$ ${\displaystyle 0<p\leq \infty }$

1. ${\displaystyle m_{\phi }(f)\in L^{p}}$for a with , ${\displaystyle \phi \in S}$ ${\displaystyle \int _{\mathbb {R} ^{n}}\phi \neq 0}$
2. ${\displaystyle M_{\phi }(f)\in L^{p}}$for a with , ${\displaystyle \phi \in S}$ ${\displaystyle \int _{\mathbb {R} ^{n}}\phi \neq 0}$
3. ${\displaystyle M_{\phi }(f)\in L^{p}}$for each and is in a suitable subset equally limited in . ${\displaystyle \phi \in S}$ ${\displaystyle M_{\phi }(f)}$ ${\displaystyle U\subset S}$ ${\displaystyle \phi }$

The real Hardy space is defined as the space that contains all tempered distributions that meet the above conditions. ${\displaystyle {\mathcal {H}}^{p}(\mathbb {R} ^{n})}$

Atomic decomposition

In particular, functions have the property that they can be broken down into a series of "small" functions of so-called atoms. An -atom is for a function such that: ${\displaystyle {\mathcal {H}}^{1}(\mathbb {R} ^{n})}$ ${\displaystyle {\mathcal {H}}^{p}}$ ${\displaystyle p\leq 1}$ ${\displaystyle a}$

1. ${\displaystyle a}$has its bearer in a ball ; ${\displaystyle B}$
2. ${\displaystyle |a|\leq \mu (B)^{-1/p}}$almost everywhere; and
3. ${\displaystyle \int _{B}x^{\beta }a(x)\mathrm {d} \mu (x)\,=\,0}$for everyone with . ${\displaystyle \beta }$ ${\displaystyle |\beta |\leq n(p^{-1}-1)}$

Requirements 1 and 2 guarantee the inequality and requirement 3 brings the stronger inequality ${\displaystyle \textstyle \int _{\mathbb {R} ^{n}}|a(x)|^{p}\mathrm {d} x\leq 1}$

${\displaystyle \int _{\mathbb {R} ^{n}}(M_{\Phi }(a)(x))^{p}\mathrm {d} x\leq c}$.

The theorem about atomic decomposition now says for with can as a series of -atoms ${\displaystyle f\in {\mathcal {H}}^{p}(\mathbb {R} ^{n})}$ ${\displaystyle p\leq 1}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {H}}^{p}}$ ${\displaystyle a_{k}}$

${\displaystyle f=\sum _{k=1}^{\infty }\lambda _{k}a_{k}}$

to be written. Here is a sequence of complex numbers with . The series converges in the distribution sense and it still holds ${\displaystyle (\lambda _{k})_{k}}$ ${\displaystyle \textstyle \sum _{k=1}^{\infty }|\lambda _{k}|^{p}<\infty }$ ${\displaystyle \textstyle \sum _{k=1}^{\infty }\lambda _{k}a_{k}}$

${\displaystyle \|f\|_{{\mathcal {H}}^{p}}\leq c\left(\sum _{k=1}^{\infty }|\lambda _{k}|^{p}\right)^{1/p}}$.

Connection to the Hardy rooms

As mentioned above, the real Hardy spaces have been developed out of the Hardy spaces of function theory. This is explained in the following section, but we limit ourselves to the case here . The interesting case is also dealt with and for one receives the whole range . ${\displaystyle 1-1/n<p<\infty }$ ${\displaystyle p=1}$ ${\displaystyle n=1}$ ${\displaystyle 0<p<\infty }$

Be

${\displaystyle u_{0},u_{1},\ldots ,u_{n}:\mathbb {R} _{+}^{n+1}\to \mathbb {R} }$

Functions on the upper half-plane , which the generalized Cauchy-Riemann's differential equations

${\displaystyle \sum _{j=0}^{n}{\frac {\partial u_{j}}{\partial x_{j}}}=0}$ and

${\displaystyle {\frac {\partial u_{j}}{\partial x_{k}}}={\frac {\partial u_{k}}{\partial x_{j}}}}$

for meet. ${\displaystyle 0\leq j,k\leq n}$

So every function is a harmonic function and in this case the generalized Cauchy-Riemann differential equations correspond exactly to the normal Cauchy-Riemann equations. So there is a holomorphic function with respect to the variables . ${\displaystyle u_{j}}$ ${\displaystyle n=1}$ ${\displaystyle f=u_{0}+iu_{1}}$ ${\displaystyle x_{1}+ix_{0}}$

According to a further theorem by Fefferman and Stein, a harmonic function fulfills one of the three equivalent -conditions if and only if there is a function which satisfies the generalized Cauchy-Riemann differential equations and which is -bounded, what ${\displaystyle u}$ ${\displaystyle H^{p}}$ ${\displaystyle F=(u,u_{1},\ldots ,u_{n})}$ ${\displaystyle L^{p}}$

${\displaystyle \sup _{x_{0}>0}\int _{\mathbb {R} ^{n}}|F(x,x_{0})|^{p}\mathrm {d} x<\infty }$

means.

Other properties

• The same applies to . So the real Hardy spaces can also be identified for these p with the corresponding spaces. ${\displaystyle 1<p\leq \infty }$ ${\displaystyle {\mathcal {H}}^{p}(\mathbb {R} ^{n})\cong L^{p}(\mathbb {R} ^{n})}$ ${\displaystyle L^{p}(\mathbb {R} ^{n})}$
• In this case , it can be understood as a real subset of . ${\displaystyle p=1}$ ${\displaystyle {\mathcal {H}}^{1}(\mathbb {R} ^{n})}$ ${\displaystyle L^{1}(\mathbb {R} ^{n})}$
• ${\displaystyle {\mathcal {H}}^{p}(\mathbb {R} ^{n})\cap L_{loc}^{1}(\mathbb {R} ^{n})}$lies for tight in . ${\displaystyle 0<p<\infty }$ ${\displaystyle {\mathcal {H}}^{p}(\mathbb {R} ^{n})}$
• The Hardy space is not reflexive , the function space BMO is its dual space . ${\displaystyle {\mathcal {H}}^{1}(\mathbb {R} ^{n})}$

Applications

Hardy spaces are used in functional analysis itself, but also in control theory and scattering theory . They also play a fundamental role in signal processing . The analytic signal is assigned to a real-valued signal , which is of finite energy for all , so that . Is , so is and ${\displaystyle f}$ ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle F}$ ${\displaystyle f(t)=\Re F(t)}$ ${\displaystyle f\in L^{2}(\mathbb {R} )}$ ${\displaystyle F\in H^{2}(\mathbb {H} )}$

${\displaystyle F(t)=f(t)+ig(t).}$

(The function is the Hilbert transform of ). For example, for a signal whose associated analytical signal is given by. ${\displaystyle g}$ ${\displaystyle f}$ ${\displaystyle f(t)=A(t)\cos \varphi (t)}$ ${\displaystyle F\in H^{2}(\mathbb {H} )}$ ${\displaystyle F(t)=A(t)e^{i\varphi (t)}}$

literature

• Joseph A. Cima and William T. Ross: The Backward Shift on the Hardy Space. American Mathematical Society 2000, ISBN 0-8218-2083-4 .
• Peter Colwell: Blaschke Products - Bounded Analytic Functions. University of Michigan Press, Ann Arbor 1985, ISBN 0-472-10065-3 .
• Peter Duren : Theory of Spaces. ${\displaystyle H^{p}}$Academic Press, New York 1970.
• Kenneth Hoffman : Banach spaces of analytic functions. Dover Publications, New York 1988, ISBN 0-486-65785-X .
• Javier Duoandikoetxea: Fourier Analysis. American Mathematical Society, Providence, Rhode Island 2001, p. 126, ISBN 0-8218-2172-5 .
• Elias M. Stein : Harmonic Analysis: Real-Variable Mathods, Orthogonality and Oscillatory Integrals , Princeton University Press 1993, ISBN 0-691-03216-5

Individual evidence

1. ^ GF Hardy: The mean value of the modulus of an analytic function . Proc. London Math. Soc. 14, pp. 269-277 (1914).