An analogue for harmonic functions of Kolmogorov's law of the iterated logarithm
The authors prove the ``law'' in the title in its analysis setting. This result is in turn a consequence of the following ``restricted version''. Let \(L(t)=(2t^ 2\log \log t)^{1/2}.\) In the upper half-plane \(D=\{(x,y):\quad y>0\},\) let \(\Gamma^ 1_{\alpha}(x)\) be the non- tangential cone of vertex (x,0), of half-aperture \(\tan^{-1}\alpha\) and truncated at \(y=1\). For any f continuous in D, let \(A^ 1_{\beta}(f)(x)\) be the square root of the integral in \(\Gamma^ 1_{\beta}(x)\) of \((\partial f/\partial \xi)^ 2\); further, let \(A^ 1_{\beta}(f)(x,\epsilon)\) denote the same integral on the same cone but translated to vertex (x,\(\epsilon)\) and truncated at \(y=1\). Then, given \(0<\alpha <\beta\), \(0<\gamma <1\), there exists a constant C such that, for every u harmonic in D satisfying \(\int^{\infty}_{-\infty}| u(x,y)| dx<\infty\) \((y>0)\), we have that, for almost every x, the condition \(A^ 1_{\beta}(u)(x)<\infty\) implies sup\(\{\) \(| u(\xi,\eta)|:\) \((\xi,\eta)\in \Gamma^ 1_{\alpha}(x)\}<\infty\), while \(A^ 1_{\beta}(u)(x)=\infty\) implies \[ \limsup | u(\xi,\eta)| /L(A^ 1_{\beta}(u)(\xi,\gamma \eta))<C(1+\alpha^{- 3/2})\quad as\quad (\xi,\eta)\to (x,0) \] from inside \(\Gamma^ 1_{\alpha}(x)\).
scientific article; zbMATH DE number 4089874
scientific article