By Rodrigo Banuelos, Charles N. Moore

Harmonic research and chance have lengthy loved a together useful dating that has been wealthy and fruitful. This monograph, geared toward researchers and scholars in those fields, explores numerous facets of this courting. the first concentration of the textual content is the nontangential maximal functionality and the realm functionality of a harmonic functionality and their probabilistic analogues in martingale idea. The textual content first provides the needful history fabric from harmonic research and discusses recognized effects about the nontangential maximal functionality and quarter functionality, in addition to the relevant and crucial function those have performed within the improvement of the field.The publication subsequent discusses additional refinements of conventional effects: between those are sharp good-lambda inequalities and legislation of the iterated logarithm regarding nontangential maximal services and zone features. 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 plenty of fresh effects mixed in a coherent presentation.

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**Example text**

With this definition of HP (~~+1 ), a proof similar to that given above shows that this one-to-one correspondence is valid on the entire range 0 < p < 00. See Fefferman and Stein [FS] for details. Burkholder and Gundy [BG2] generalized Fefferman and Stein's result relating the V norms of the area function and nontangential maximal function. Consider a nondecreasing continuous function IP defined on [0,00] with IP(O) = 0, IP not identically zero and which satisfies the growth condition: 1P(2A) :S CIP(A) for every A > 0, where c is a fixed constant.

00 then u(x,y) = Py * f(x) for some f E Ll(JRn). e. and hence it is not the Poisson extension of its nontangentiallimit. It is not the case that if SUPy>o Iluylll Proof: As we have noted above, one implication of (a) and the first sentence of (b) have already been shown. Suppose u is harmonic on JR~+1 and has SUPy>o Iluyll p < 00. Let (x, y) E JR~+1 and set B = B((x, y), y) ~ JR~+1. 5) and Jensen's inequality we have: Therefore, lu(x, y)1 :::; Cy-'f;. Let k be a positive integer. 9 (Dirichlet problem for JR~+1), u(x, y + i) = JlRn Py(x - s)u(s, i)ds.

We have: r Py(x-s)f(s)ds+ r Py(x-s)dv(s). e. Part (b) will be completed if we show that the nontangentiallimit of JRn Py(x - s)dv(s) is 0 almost everywhere. To see this, note that IPy * v(x)1 :s: Py * Ivl(x), and further note that Ivl is singular with respect to Lebesgue measure since v is. 2 (Fatou's Theorem) If), is a finite positive Borel measure on ]Rn and if D),(x) = lim ~<:c(x,lP = 0, then the nontangentiallimit of Pt * ),(s) at x is O. f L t P roOJ: e e 0 we can choose ro so tha t > 0. (B(x,r» IB(x,r)1 = for every r :s: roo Fix a > 0, and write u = U1 + U2 where U1(S, t) = J Pt(s - rJ)d),(rJ) Rn\B(x,ro) and U2(S, t) = f B(x,ro) Pt(s - rJ)d)'(rJ)· '>'(B(x,r ) B(x,r)