By Rodrigo Banuelos, Charles N. Moore

Harmonic research and likelihood have lengthy loved a together worthwhile dating that has been wealthy and fruitful. This monograph, aimed toward researchers and scholars in those fields, explores numerous elements of this courting. the first concentration of the textual content is the nontangential maximal functionality and the world functionality of a harmonic functionality and their probabilistic analogues in martingale conception. The textual content first supplies the considered necessary historical past fabric from harmonic research and discusses identified effects about the nontangential maximal functionality and sector functionality, in addition to the relevant and crucial function those have performed within the improvement of the field.The booklet subsequent discusses extra refinements of conventional effects: between those are sharp good-lambda inequalities and legislation of the iterated logarithm regarding nontangential maximal capabilities and sector services. Many purposes of those effects are given. all through, the consistent interaction among chance and harmonic research is emphasised and defined. The textual content includes a few new and lots of fresh effects mixed in a coherent presentation.

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Also, since f(re i9 ) ::; C aN a f(e i9 ) and the latter function is in £1, we may apply Lebesgue's dominated convergence theorem to conclude that as desired. As the proof showed, the case p = I distinguishes analytic functions in HP from harmonic functions which satisfy a condition like that in the definition of HP. 3. 3 is as follows: Let f E HP and let B denote the Blaschke product of the zeros of f· Then 9 = has no zeros in D and since IB(z)1 = I for z E aD then IlgllHP = IlfIIHP. In the interior of D, IBI ::; I so If I ::; Igl.

Stein [Stl] showed the inequalities for the area integral that we've shown here. Fllp ::; CaiIAaFllp. FIIHP ~ IIAaFIIp" Gasper [Gas] showed a version of this for functions in HP of the unit ball in JR n . Let f E V(JR), 1 < p < 00, Set u(x, y) = Py * f(x) and let v(x, y) be a harmonic conjugate to u on JR~ with v(x, y) - t 0 as y - t 00. (So U + iv is analytic on JR~). 8(b) imply that v is the Poisson integral of a function in LP (JR); this is usually denoted as H f and is called the Hilbert transform of f.

We want to do the same here. 3. ) Now in higher dimensions, the same will be true: the case p > 1 will be a consequence of results on harmonic functions but the case p = 1 will need an estimate of the nontangential maximal function. ) We do this estimate now. 26 1. 10 Suppose F E HP(IR++ 1 ) , n~l < p ::; 00 and a > O. Then NaF E LP(lRn). In fact, there exists a constant Ca,n, depending only on a and n, such that IINaFllp ::; Ca,nIIFIIHP. 3 although we need to take a little more care here because of the noncompactness of 1R++1 .