Sharp inequalities for the Beurling-Ahlfors transform on radial functions. (English) Zbl 1277.60076

Bañuelos, Rodrigo; Osȩkowski, Adam

Duke Math. J. 162, No. 2, 417-434 (2013).

For \(p\in[1,2]\), the authors discover the weak \((p,p)\) norm, and recover the strong one, of the Beurling-Ahlfors transform on radial functions, where it essentially reduces to the Hardy operator \[ \Lambda f(t)=t^{-1}\int_0^t f(u)du-f(t). \] They first apply a modification of Burkholder’s method – i.e., calculus of special auxiliary functions – to prove a sharp martingale inequality \[ \operatorname{P}((\Delta X)^*\geq 1)\leq c_p\|X\|_p^p, \] where \((\Delta X)^*\) is the maximal function of the jump process \(\Delta X\) of the continuous-time martingale \(X\). Finally, they obtain \[ |\Lambda f(t)|=|\Delta X_t| \] for the (reverse) martingale \(X_t=\operatorname{E}[f|\mathcal{F}_t]\) relative to \(\mathcal{F}_t=\sigma([0,t),\mathcal{B}[t,\infty))\).

Reviewer: Tuomas Hytönen (Helsinki)

MSC:

60G44	Martingales with continuous parameter
42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Keywords:

weak-type inequality; sharp constant; Beurling-Ahlfors transform; martingale; jump process; Hardy operator; Iwaniec conjecture

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