Sharp inequalities for differentially subordinate harmonic functions and martingales (Q2909645)

Let \(D\subset\mathbb{R}^N\) be open and connected and let \(u\), \(v\) be real-valued harmonic functions on \(D\). Following \textit{D. L. Burkholder} [Lect. Notes Math. 1384, 1--23 (1989; Zbl 0675.31003)], \(v\) is called differentially subordinate to \(u\) if for \(x\in D\), NEWLINE\[NEWLINE|\nabla v(x)|\leq|\nabla u(x)|. \tag{1}NEWLINE\]NEWLINE The functions \(u\), \(v\) are called orthogonal if NEWLINE\[NEWLINE\nabla u\cdot\nabla v= 0\quad\text{on}\quad D.\tag{2}NEWLINE\]NEWLINE (The dot denotes the standard scalar product on \(\mathbb{R}^N\).) Let \(D_0\) be a bounded connected subdomain of \(D\) and let \(\xi\in D\) be such that \(\xi\in D_0\subset D_0\cup\partial D_0\subset D\). Assume that \(u\), \(v\) satisfy NEWLINE\[NEWLINE|v(\xi)|\leq|u(\xi)|.\tag{3}NEWLINE\]NEWLINE Let \(\mu^\xi_{D_0}\) denote the harmonic measure on \(\partial D_0\) with respect to \(\xi\). For \(1\leq p<\infty\) let NEWLINE\[NEWLINE\| u\|_p:= \Biggl[\sup_{D_0}\, \int_{\partial D_0} |u(x)|^p\,d\mu^\xi_{D_0}(x)\Biggr]^{1/p}.NEWLINE\]NEWLINE Two of the main results in the present paper are the following.NEWLINENEWLINE NEWLINELet \(u\), \(v\) satisfy (1)--(3). Then, for \(1< p<\infty\), NEWLINENEWLINE\[NEWLINE\| v\|_p\leq C_{p,\infty}\| u\|_\infty,NEWLINE\]NEWLINE NEWLINEwhere the optimal constant is given by NEWLINENEWLINE\[NEWLINEC_{p,\infty}:= \begin{cases} 1\quad &\text{if }1<p\leq 2,\\\Big[{2^{p+2}\over p+1} \Gamma(p+1) \sum^\infty_{k=0} {(-1)^k\over (2k+1)^{p+ 1}}\Big]^{1/p}\quad &\text{if }p> 2.\end{cases}\tag{\(*\)}NEWLINE\]NEWLINE NEWLINEThis has a martingale analogue given by the following theorem:NEWLINENEWLINE Let \(X\), \(Y\) be continuous-time orthogonal martingales such that \(Y\) is differentially subordinate to \(X\). Then, for \(1<p<\infty\), NEWLINE\[NEWLINE\| Y\|_p\leq C_{p,\infty}\| X\|_\infty,NEWLINE\]NEWLINE where \(C_{p,\infty}\) given by \((*)\) is again optimal.