Nontangential limits in 𝒫^t(μ)-spaces and the index of invariant subgroups

Abstract

Let µ be a finite positive measure on the closed disk D in the complex plane, let 1 ≤ t < ∞, and let P t (µ) denote the closure of the analytic polynomials in L t (µ). We suppose that D is the set of analytic bounded point evaluations for P t (µ), and that P t (µ) contains no nontrivial characteristic functions. It is then known that the restriction of µ to ∂D must be of the form h|dz|. We prove that every function f ∈ P t (µ) has nontangential limits at h|dz|-almost every point of ∂D, and the resulting boundary function agrees with f as an element of L t (h|dz|).Our proof combines methods from James E. Thomson's proof of the existence of bounded point evaluations for P t (µ) whenever P t (µ) = L t (µ) with Xavier Tolsa's remarkable recent results on analytic capacity. These methods allow us to refine Thomson's results somewhat. In fact, for a general compactly supported measure ν in the complex plane we are able to describe locations of bounded point evaluations for P t (ν) in terms of the Cauchy transform of an annihilating measure.As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We show that for 1 < t < ∞ dim M/zM = 1 for every nonzero invariant subspace M of P t (µ) if and only if h = 0.We also investigate the boundary behaviour of the functions in P t (µ) near the points z ∈ ∂D where h(z) = 0. In particular, for 1 < t < ∞ we show that there are interpolating sequences for P t (µ) that accumulate nontangentially almost everywhere on {z : h(z) = 0}.

Density of the polynomials in Hardy and Bergman spaces of slit domains

“…We cannot duplicate the result of Section 3 here since the graph of any continuous real-valued function on [1,2] has zero two-dimensional Lebesgue measure. The arc we construct in this context must have positive twodimensional Lebesgue measure in any neighborhood of any of its points; and this alone is by no means sufficient to ensure that P is dense in A t (E Γ ).…”

“…By our hypothesis, g can be uniformly approximated on [1, 2] by a sequence of step functions. Thus, via a piecewise analysis, we can reduce to the case that g is constant on [1,2]; indeed, that g≡0 on [1,2]. Now, in order for a Brownian motion path starting at − 3 2 to reach a point in E γ ∩T (g, δ) before it exits E γ it must successfully navigate down one of 2k corridors delineated by portions of γ, each of length a−δ and of maximum width 1/k.…”

“…Notice that f a,k (1)=f a,k (2)=0 and that f a,k has period 1/k and amplitude a on [1,2]. For any function g : [1, 2]→[−1, 1] that is continuous on [1,2], with g(1)=g(2)=0, and for any δ, 0<δ<1, let T (g, δ)={x+iy:1≤x≤2 and |y−g(x)|<δ}. [1,2] and that g(1)=g(2)=0.…”

“…For any function g : [1, 2]→[−1, 1] that is continuous on [1,2], with g(1)=g(2)=0, and for any δ, 0<δ<1, let T (g, δ)={x+iy:1≤x≤2 and |y−g(x)|<δ}. [1,2] and that g(1)=g(2)=0. If 0<a<1, ε>0 and 0<δ<a, then, provided k is sufficiently large, γ :={x+i(g(x)+f a,k (x)):1≤x≤2} has the property: The probability that a Brownian motion path starting at − 3 2 will reach a point in E γ ∩T (g, δ) before it exits E γ is less than ε; and hence ω γ (T (g, δ)) < ε.…”

“…Further, let ω Γ denote harmonic measure on ∂E Γ for evaluation at − 3 2 and let ν Γ denote two-dimensional Lebesgue measure restricted to E Γ . In this paper we first produce a continuous real-valued function f : [1,2]→[−1, 1], where f (1)=f (2)=0, such that Γ:={x+if (x):1≤x≤2} has the property that P is dense in the Hardy space H t (E Γ ); see Theorem 3.3. We then produce a Jordan arc Γ with endpoints 1 and 2 such that Γ\{1, 2}⊆E and with the property that the polynomials are dense in A t (E Γ ); see Theorem 4.1.…”