Subharmonic function: Difference between revisions

Revision as of 17:46, 15 December 2016

Revision as of 15:09, 18 July 2017

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One can show that a real-valued, continuous function <math>\varphi</math> of a complex variable (that is, of two real variables) defined on a set <math>G\subset \mathbb{C}</math> is subharmonic if and only if for any closed disc <math>D(z,r) \subset G</math> of center <math>z</math> and radius <math>r</math> one has

:<math> \varphi(z) \leq \frac{1}{2\pi} \int_0^{2\pi} \varphi(z+ r \mathrm{e}re^{i\theta}) \, d\theta. </math>

Intuitively, this means that a subharmonic function is at any point no greater than the [[arithmetic mean|average]] of the values in a circle around that point, a fact which can be used to derive the [[maximum principle]].

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=== Subharmonic functions in the unit disc. Radial maximal function ===

Let ''φ'' be subharmonic, continuous and non-negative in an open subset ''Ω'' of the complex plane containing the closed unit disc ''D''(0,&nbsp;1). The ''radial maximal function'' for the function ''φ'' (restricted to the unit disc) is defined on the unit circle by

:<math> (M \varphi)(\mathrm{e}^{\mathrm{i} \theta}) = \sup_{0 \le r < 1} \varphi(r \mathrm{e}re^{\mathrm{i} \theta}). </math>

If ''P''_''r'' denotes the [[Poisson kernel]], it follows from the subharmonicity that

:<math> 0 \le \varphi(r \mathrm{e}re^{\mathrm{i} \theta}) \le \frac{1}{2\pi} \int_0^{2\pi} P_r\left(\theta- t\right) \varphi\left(\mathrm{e}^{\mathrm{i} tit}\right) \, \mathrm{d} t, \ \ \ r < 1.</math>

It can be shown that the last integral is less than the value at e^{&nbsp;i''θ''} of the [[Hardy–Littlewood maximal function]] ''φ''^∗ of the restriction of ''φ'' to the unit circle '''T''',

:<math> \varphi^*(\mathrm{e}^{\mathrm{i} \theta}) = \sup_{0 < \alpha \le \pi} \frac{1}{2 \alpha} \int_{\theta - \alpha}^{\theta + \alpha} \varphi\left(\mathrm{e}^{\mathrm{i} tit}\right) \, \mathrm{d}t,</math>

so that 0&nbsp;≤ ''M''&nbsp;''φ''&nbsp;≤ ''φ''^∗. It is known that the Hardy–Littlewood operator is bounded on [[Lp space|''L''^''p''('''T''')]] when 1&nbsp;< ''p''&nbsp;< ∞.

It follows that for some universal constant ''C'',

::<math> \|M \varphi\|_{L^2(\mathbf{T})}^2 \le C^2 \, \int_0^{2\pi} \varphi(\mathrm{e}^{\mathrm{i} \theta})^2 \, \mathrm{d}\theta.</math>

If ''f'' is a function holomorphic in ''Ω'' and 0&nbsp;< ''p''&nbsp;< ∞, then the preceding inequality applies to ''φ''&nbsp;= |''f''^&nbsp;|^{&nbsp;''p''/2}. It can be deduced from these facts that any function ''F'' in the classical Hardy space ''H^p'' satisfies

::<math> \int_0^{2\pi} \Bigl( \sup_{0 \le r < 1} |F(r \mathrm{e}re^{\mathrm{i} \theta})| \Bigr)^p \, \mathrm{d}\theta \le C^2 \, \sup_{0 \le r < 1} \int_0^{2\pi} |F(r \mathrm{e}re^{\mathrm{i} \theta})|^p \, \mathrm{d}\theta.</math>

With more work, it can be shown that ''F'' has radial limits ''F''(e^{&nbsp;i''θ''}) almost everywhere on the unit circle, and (by the [[dominated convergence theorem]]) that ''F_r'', defined by ''F_r''(e^{&nbsp;i''θ''})&nbsp;= ''F''(''r''^&nbsp;e^{&nbsp;i''θ''}) tends to ''F'' in ''L''^''p''('''T''').