The Lusin area function and admissible convergence of harmon. non-homogeneous trees

Transcription

1 The Lusin area function and admissible convergence of harmonic functions on non-homogeneous trees Mathematics Department, University of Roma Tor Vergata Bardonecchia, June 15th, 2009

2 We prove admissible convergence to the boundary of functions that are harmonic on subsets of a non-homogeneous tree equipped with a transition operator that satisfies uniform bounds suitable for transience. The approach is based on a discrete Green formula, suitable estimates for the Green and Poisson kernel and an analogue of the Lusin area function. Estimates of the area function by means of the nontangential maximal function were used by F.Di Biase and this speaker (Zeitscrift 1995) to deal with the H p theory on trees, with an approach based on good lambda inequalities: what we introduce here is an alternative, more differential approach that yields the admissible convergence.

3 To start, let us review the continuous case: the Lusin area theorem in the half-plane (or equivalently the disc). Let f be harmonic in the half-plane H +, for instance. ω R generic point in the real axis. Γ α (ω) coneofwidthα in H + with vertex in ω (note: in the hyperbolic distance of H + this is a tube with axis {ω + iτ, τ > 0}) Area function: A α (ω) := f 2 dx dy Γ α(ω)

4 To start, let us review the continuous case: the Lusin area theorem in the half-plane (or equivalently the disc). Let f be harmonic in the half-plane H +, for instance. ω R generic point in the real axis. Γ α (ω) coneofwidthα in H + with vertex in ω (note: in the hyperbolic distance of H + this is a tube with axis {ω + iτ, τ > 0}) Area function: A α (ω) := f 2 dx dy Γ α(ω)

5 To start, let us review the continuous case: the Lusin area theorem in the half-plane (or equivalently the disc). Let f be harmonic in the half-plane H +, for instance. ω R generic point in the real axis. Γ α (ω) coneofwidthα in H + with vertex in ω (note: in the hyperbolic distance of H + this is a tube with axis {ω + iτ, τ > 0}) Area function: A α (ω) := f 2 dx dy Γ α(ω)

6 Theorem (Lusin area theorem in H + ) Let E R measurable, f harmonic on H +. Then the following are equivalent: (i) f is non-tangentially bounded at a.e. ω E: α = α(ω) such that, E almost everywhere, f is bounded on Γ α (ω); (ii) f has non-tangential limit at almost every ω E; (iii) α = α(ω) such that A αω f (ω) < for almost every ω E; (iv) for every fixed α 0, A α f (ω) < for a.e. in E; (v) fixed α 0, f is bounded on Γ α (ω) ω a.e. on E.

7 Moreover, A α f p N β f p,forallp < and all α, β > 0, where N β f (ω) =sup Γβ (ω) f is the non-tangential maximal function.

8 Sketch of proof. Up to negligible measure E is a union of intervals. For simplicity let us assume E is an interval and prove (i) (iv). We are assuming that f is bounded on almost all cones with vertex in E. For the moment, assume more: that f L.Also,forthe time being assume α costant. Consider the conical extension W E of E. Truncate at heights τ <τ + (τ near the boundary) to get a trapeze WE τ.

9 Sketch of proof. Up to negligible measure E is a union of intervals. For simplicity let us assume E is an interval and prove (i) (iv). We are assuming that f is bounded on almost all cones with vertex in E. For the moment, assume more: that f L.Also,forthe time being assume α costant. Consider the conical extension W E of E. Truncate at heights τ <τ + (τ near the boundary) to get a trapeze WE τ.

10 Sketch of proof. Up to negligible measure E is a union of intervals. For simplicity let us assume E is an interval and prove (i) (iv). We are assuming that f is bounded on almost all cones with vertex in E. For the moment, assume more: that f L.Also,forthe time being assume α costant. Consider the conical extension W E of E. Truncate at heights τ <τ + (τ near the boundary) to get a trapeze WE τ.

11 τ + y τ - c y α W E τ E

12 Now for ω E and τ =[τ,τ + ] consider the truncated cone Γ τ ω and compute the integral I := f 2 dx dy dω E Γ τ ω When the vertices of two cones are translated along E, their sections at height y overlap by a quantity w(y) proportional to y: But so, w is harmonic on H +! Since also f is harmonic, w(y) = {ω : y Γ(ω)} = c α y f 2 = f 2 +2f f = (f 2 ) so we can now compute the integral I via the Green identities and the Green formula.

13 Now for ω E and τ =[τ,τ + ] consider the truncated cone Γ τ ω and compute the integral I := f 2 dx dy dω E Γ τ ω When the vertices of two cones are translated along E, their sections at height y overlap by a quantity w(y) proportional to y: But so, w is harmonic on H +! Since also f is harmonic, w(y) = {ω : y Γ(ω)} = c α y f 2 = f 2 +2f f = (f 2 ) so we can now compute the integral I via the Green identities and the Green formula.

14 I = = W τ E W τ E w (f 2 )= w n f 2 f 2 W τ E w (f 2 ) f 2 w n w Now remember that f L and observe the following:

15 n f 2 =2 f n f C f 2 by elliptic estimates (or simply because for a bounded harmonic function nf f by Harnack s inequality...) w = c α y is proportional to the Green function of H +, hence n w w harmonic measure on boundary. Hence: w is bounded, and W τ E {y=τ + } independently of τ, WE τ {y=τ } w ν(e) (ν istheharmonicmeasure,i.e.,lebesguemeasureonr)

16 n f 2 =2 f n f C f 2 by elliptic estimates (or simply because for a bounded harmonic function nf f by Harnack s inequality...) w = c α y is proportional to the Green function of H +, hence n w w harmonic measure on boundary. Hence: w is bounded, and W τ E {y=τ + } independently of τ, WE τ {y=τ } w ν(e) (ν istheharmonicmeasure,i.e.,lebesguemeasureonr)

17 n f 2 =2 f n f C f 2 by elliptic estimates (or simply because for a bounded harmonic function nf f by Harnack s inequality...) w = c α y is proportional to the Green function of H +, hence n w w harmonic measure on boundary. Hence: w is bounded, and W τ E {y=τ + } independently of τ, WE τ {y=τ } w ν(e) (ν istheharmonicmeasure,i.e.,lebesguemeasureonr)

18 Moreover, the integral of w on the side facets is bounded. So, by letting τ 0 + and τ +,wehave lim I = w (f 2 ) W E = w n f 2 f 2 n w c f 2 W τ E hence, for a.a. ω, A α (ω) 2 c f 2 This proves the theorem if f is bounded. If not, uniformization:

19 Moreover, the integral of w on the side facets is bounded. So, by letting τ 0 + and τ +,wehave lim I = w (f 2 ) W E = w n f 2 f 2 n w c f 2 W τ E hence, for a.a. ω, A α (ω) 2 c f 2 This proves the theorem if f is bounded. If not, uniformization:

20 Moreover, the integral of w on the side facets is bounded. So, by letting τ 0 + and τ +,wehave lim I = w (f 2 ) W E = w n f 2 f 2 n w c f 2 W τ E hence, for a.a. ω, A α (ω) 2 c f 2 This proves the theorem if f is bounded. If not, uniformization:

21 Assume f unbounded, but bounded on almost all cones Γ α (ω). Claim: we can choose E N E such that f is bounded on the conical extension W E N. Indeed, E k := {ω : f < k on Γ α (ω)} E N := k N E k Since f is bounded on almost each cone, for N large we have ν(e \ E N ) <ε and on W E N one has f < N.

22 Assume f unbounded, but bounded on almost all cones Γ α (ω). Claim: we can choose E N E such that f is bounded on the conical extension W E N. Indeed, E k := {ω : f < k on Γ α (ω)} E N := k N E k Since f is bounded on almost each cone, for N large we have ν(e \ E N ) <ε and on W E N one has f < N.

23 Finally, let us uniformize again to remove the extra assumption that α be costant (this step will not be necessary in a tree). Choose j N, letβ =min{α(ω), j}, Γ j (ω) :=Γ β (ω) and E j,k := {ω : f < k on Γ J (ω)} Now let E M,N := j M k N E j,k For every ε, ν(e \ E M,N ) <εfor large M, N, and f < N on E M,N.

24 Finally, let us uniformize again to remove the extra assumption that α be costant (this step will not be necessary in a tree). Choose j N, letβ =min{α(ω), j}, Γ j (ω) :=Γ β (ω) and E j,k := {ω : f < k on Γ J (ω)} Now let E M,N := j M k N E j,k For every ε, ν(e \ E M,N ) <εfor large M, N, and f < N on E M,N.

25 Finally, let us uniformize again to remove the extra assumption that α be costant (this step will not be necessary in a tree). Choose j N, letβ =min{α(ω), j}, Γ j (ω) :=Γ β (ω) and E j,k := {ω : f < k on Γ J (ω)} Now let E M,N := j M k N E j,k For every ε, ν(e \ E M,N ) <εfor large M, N, and f < N on E M,N.

26 Finally, let us uniformize again to remove the extra assumption that α be costant (this step will not be necessary in a tree). Choose j N, letβ =min{α(ω), j}, Γ j (ω) :=Γ β (ω) and E j,k := {ω : f < k on Γ J (ω)} Now let E M,N := j M k N E j,k For every ε, ν(e \ E M,N ) <εfor large M, N, and f < N on E M,N.

27 Definition (Notation) A tree T is a connected, simply connected, locally finite graph. We shall also write T for its set of vertices. We do not assume that T is homogeneous (but our forthcoming assumptions imply that the number od edges at each vertex is bounded). Natural distance d(x, y): length of the direct path from x to y. Write x y if x, y are neighbors: distance 1. We fix a reference vertex o T and call it the origin. The choice of o induces a partial ordering in T : x y if x belongs to the geodesic from o to y. Length: x = d(o, x). For any vertex x and k x, x k is the vertex of length k in the geodesic [o, x].

28 Definition (Notation) A tree T is a connected, simply connected, locally finite graph. We shall also write T for its set of vertices. We do not assume that T is homogeneous (but our forthcoming assumptions imply that the number od edges at each vertex is bounded). Natural distance d(x, y): length of the direct path from x to y. Write x y if x, y are neighbors: distance 1. We fix a reference vertex o T and call it the origin. The choice of o induces a partial ordering in T : x y if x belongs to the geodesic from o to y. Length: x = d(o, x). For any vertex x and k x, x k is the vertex of length k in the geodesic [o, x].

29 Definition (Notation) A tree T is a connected, simply connected, locally finite graph. We shall also write T for its set of vertices. We do not assume that T is homogeneous (but our forthcoming assumptions imply that the number od edges at each vertex is bounded). Natural distance d(x, y): length of the direct path from x to y. Write x y if x, y are neighbors: distance 1. We fix a reference vertex o T and call it the origin. The choice of o induces a partial ordering in T : x y if x belongs to the geodesic from o to y. Length: x = d(o, x). For any vertex x and k x, x k is the vertex of length k in the geodesic [o, x].

30

31 Definition (Boundary) Let Ω be the set of infinite geodesics starting at o. ω n is the vertex of length n in the geodesic ω. For x T the interval U(x) Ω, generated by x, isthesetu(x) = { } ω Ω:x = ω x.thesets U(ω n ), n N, formanopenbasisatω Ω. Equipped with this topology Ω is compact and totally disconnected. For every vertex x, the set of vertices v > x form the sector S(x). Call closed sector the set S(x) :=S(x) U(x) T Ω. The closed sectors induce on T Ω a compact topology.

32 Definition (Boundary) Let Ω be the set of infinite geodesics starting at o. ω n is the vertex of length n in the geodesic ω. Forx T the interval U(x) Ω, generated by x, isthesetu(x) = { } ω Ω:x = ω x.thesets U(ω n ), n N, formanopenbasisatω Ω. Equipped with this topology Ω is compact and totally disconnected. For every vertex x, the set of vertices v > x form the sector S(x). Call closed sector the set S(x) :=S(x) U(x) T Ω. The closed sectors induce on T Ω a compact topology.

33 Definition (Boundary) Let Ω be the set of infinite geodesics starting at o. ω n is the vertex of length n in the geodesic ω. Forx T the interval U(x) Ω, generated by x, isthesetu(x) = { } ω Ω:x = ω x.thesets U(ω n ), n N, formanopenbasisatω Ω. Equipped with this topology Ω is compact and totally disconnected. For every vertex x, the set of vertices v > x form the sector S(x). Call closed sector the set S(x) :=S(x) U(x) T Ω. The closed sectors induce on T Ω a compact topology.

34 On the vertices of T we consider operators P with transition probabilities p(u, v) thatsatisfy: Definition (Very regular transition operators) (H1) P is a nearest neighbor operator, that is, p(u, v) =0unless the vertices u and v are neighbors; (H2) for some constant δ, with0<δ< 1 2 and all neighbors u and v the following inequality holds: δ p(u, v) 1 2 δ.

35 p(u, v) δ number of neighbors 1 δ. p(u, v) 1 2 δ probability of moving forward larger than probability of moving backwards transience. The sets {U(x) : x = n} generate a σ algebraonω.onthis σ algebra, balayage theory yields the harmonic measure, that is the measure that reconstructs the value of harmonic functions at o from their boundary values (see later): Definition (Harmonic measure) ν(e) =Pr[X E].

36 p(u, v) δ number of neighbors 1 δ. p(u, v) 1 2 δ probability of moving forward larger than probability of moving backwards transience. The sets {U(x) : x = n} generate a σ algebraonω.onthis σ algebra, balayage theory yields the harmonic measure, that is the measure that reconstructs the value of harmonic functions at o from their boundary values (see later): Definition (Harmonic measure) ν(e) =Pr[X E].

37 p(u, v) δ number of neighbors 1 δ. p(u, v) 1 2 δ probability of moving forward larger than probability of moving backwards transience. The sets {U(x) : x = n} generate a σ algebraonω.onthis σ algebra, balayage theory yields the harmonic measure, that is the measure that reconstructs the value of harmonic functions at o from their boundary values (see later): Definition (Harmonic measure) ν(e) =Pr[X E].

38 Definition (Laplace operator) The Laplace operator is Δ = P I. If p (x, y) =p(y, x), then the conjugate Laplacian is the transpose operator Δ = P I. Definition (Harmonic functions) f : T R is harmonic at x T if Pf (x) y x p(x, y) f (y) =f (x). f harmonic on T Δf =0. f conjugate harmonic if Δ f =0.

39 Definition (Laplace operator) The Laplace operator is Δ = P I. If p (x, y) =p(y, x), then the conjugate Laplacian is the transpose operator Δ = P I. Definition (Harmonic functions) f : T R is harmonic at x T if Pf (x) y x p(x, y) f (y) =f (x). f harmonic on T Δf =0. f conjugate harmonic if Δ f =0.

40 Definition (Laplace operator) The Laplace operator is Δ = P I. If p (x, y) =p(y, x), then the conjugate Laplacian is the transpose operator Δ = P I. Definition (Harmonic functions) f : T R is harmonic at x T if Pf (x) y x p(x, y) f (y) =f (x). f harmonic on T Δf =0. f conjugate harmonic if Δ f =0.

41 For σ Λdenotebyb(σ) the beginning vertex of σ and by e(σ) the ending vertex: σ =[b(σ), e(σ)]. Definition (Gradient) For any function f : T R, thegradient f :Λ R is For x T, f (σ) =f (e(σ)) f (b(σ)). f (x) 2 y x p([x, y]) f ([x, y]) 2.

42 For σ Λdenotebyb(σ) the beginning vertex of σ and by e(σ) the ending vertex: σ =[b(σ), e(σ)]. Definition (Gradient) For any function f : T R, thegradient f :Λ R is For x T, f (σ) =f (e(σ)) f (b(σ)). f (x) 2 y x p([x, y]) f ([x, y]) 2.

43 For σ Λdenotebyb(σ) the beginning vertex of σ and by e(σ) the ending vertex: σ =[b(σ), e(σ)]. Definition (Gradient) For any function f : T R, thegradient f :Λ R is For x T, f (σ) =f (e(σ)) f (b(σ)). f (x) 2 y x p([x, y]) f ([x, y]) 2.

44 x T, ω Ω, set d(x,ω)=min j N d(x,ω j ). Definition Let α 0 be an integer. The tube Γ α (ω) around the geodesic ω Ωis Γ α (ω) ={x T : d(x,ω) α}. Definition (Area function) A α f (ω) = x Γ α(ω) 1 2 f (x) 2.

45 x T, ω Ω, set d(x,ω)=min j N d(x,ω j ). Definition Let α 0 be an integer. The tube Γ α (ω) around the geodesic ω Ωis Γ α (ω) ={x T : d(x,ω) α}. Definition (Area function) A α f (ω) = x Γ α(ω) 1 2 f (x) 2.

46 x T, ω Ω, set d(x,ω)=min j N d(x,ω j ). Definition Let α 0 be an integer. The tube Γ α (ω) around the geodesic ω Ωis Γ α (ω) ={x T : d(x,ω) α}. Definition (Area function) A α f (ω) = x Γ α(ω) 1 2 f (x) 2.

47 Observe that, if f L 1 (T ), A α f (ω) < for every α, ω. Definition (Non-tangential limits and boundedness) A function f on T has non-tangential limit at ω Ω if, for every integer α 0, lim f (x) existsas x and x Γ α (ω). We say that f has non-tangential limit up to width β if the above limit exists for all 0 α β. A function f on T is non-tangentially bounded at ω Ω if, for some M > 0, one has f (x) M for x Γ 0 (ω).

48 Observe that, if f L 1 (T ), A α f (ω) < for every α, ω. Definition (Non-tangential limits and boundedness) A function f on T has non-tangential limit at ω Ω if, for every integer α 0, lim f (x) existsas x and x Γ α (ω). We say that f has non-tangential limit up to width β if the above limit exists for all 0 α β. A function f on T is non-tangentially bounded at ω Ω if, for some M > 0, one has f (x) M for x Γ 0 (ω).

49 Observe that, if f L 1 (T ), A α f (ω) < for every α, ω. Definition (Non-tangential limits and boundedness) A function f on T has non-tangential limit at ω Ω if, for every integer α 0, lim f (x) existsas x and x Γ α (ω). We say that f has non-tangential limit up to width β if the above limit exists for all 0 α β. A function f on T is non-tangentially bounded at ω Ω if, for some M > 0, one has f (x) M for x Γ 0 (ω).

50 Definition (Conical - better, tubular - extension of a boundary set) The cone (actually, tube) W α (E) over a measurable subset E of Ωis W α (E) = ω E Γ α (ω). For any integer s > 0letW s α (E) = ω E Γ α (ω) [s; ). The main goal of this paper is the following extension of the Lusin area theorem to non-homogeneous trees. This theorem has been proved for homogeneous trees By L. Atanasi and this speaker (TAMS 2008).

51 Main Theorem (The Lusin Area Theorem) Let E be a measurable subset of Ω and f a harmonic function on T. Then the following are equivalent: (i) f is non-tangentially bounded at almost every ω E; (ii) f has non-tangential limit at almost every ω E; (iii) A 0 f (ω) < for almost every ω E; (iv) for every fixed α 0, A α f (ω) < for almost every ω E.

52 The same statement holds if f is harmonic on a connected subset of T whose boundary contains E, or more precisely on some tube W β (E), provided that α β in (iv) and, in (ii), f is assumed to be non-tangentially bounded up to width β. Observe that the definition of non-tangential boundedness is equivalent to say that, for some non-negative integer α = α(ω), f is bounded on a tube of width α around the geodesic Γ 0 (ω) (bya different constant depending on α), and, in this discrete setting, condition (iii) in the theorem is equivalent to the more familiar statement that for almost every ω there is some integer α such that A α f is finite at ω.

53 The same statement holds if f is harmonic on a connected subset of T whose boundary contains E, or more precisely on some tube W β (E), provided that α β in (iv) and, in (ii), f is assumed to be non-tangentially bounded up to width β. Observe that the definition of non-tangential boundedness is equivalent to say that, for some non-negative integer α = α(ω), f is bounded on a tube of width α around the geodesic Γ 0 (ω) (bya different constant depending on α), and, in this discrete setting, condition (iii) in the theorem is equivalent to the more familiar statement that for almost every ω there is some integer α such that A α f is finite at ω.

54 Hitting probability from u to v: F (u, v) :=Pr[ n > 0:X n = v, X j v j < n X 0 = u]. Nearest neighbor random walk stopping time argument multiplicativity rule: F (u 0, u n )= n F (u i 1, u i ). (1) i=1 Definition (Green kernel) The Green kernel G(u, v) is the expected number of visits to v of the random walk starting at u: G (u, v) = P n (u, v). (2) n=0

55 Remarks G (u, v) =F (u, v)g (v, v). (3) For each v T, the Green kernel G (x, v) is harmonic with respect to x at every vertex x v, and conjugate harmonic with respect to v at every v x.

56 Given two positive functions f and g, wewritef g if f < Cg and g < Cf for some constant C. Proposition (A. Korányi, M. A. Picardello, M. H. Taibleson, 1988) If P is very regular, then for x T ν(u(x)) F (o, x) G (o, x).

57 Corollary (Uniform estimate for harmonic measure) If the transition operator P is very regular, with δ as in (H2), then for all vertices u, v at distance n, ( ) n δ n δ < F (u, v) < δ. Therefore, for some 0 <ε<1 and for all vertices x in T and v in the sector S(x) at distance d(v, x) from x, ν(u(v)) < (1 ε) d(v,x) ν(u(x)).

58 Definition (Poisson kernel) For every x, v T the Martin kernel K(x, v) sdefinedas K(x, v) G (x, v) F (x, v) = G (o, v) F (o, v). For every x T, ω ΩthePoisson kernel K(x,ω)isdefinedas G (x, v) K(x,ω) lim v G (o, v) = lim F (x, v) v F (o, v). For every ω Ω, K(,ω)isharmoniconT.

59 Definition (Poisson kernel) For every x, v T the Martin kernel K(x, v) sdefinedas K(x, v) G (x, v) F (x, v) = G (o, v) F (o, v). For every x T, ω ΩthePoisson kernel K(x,ω)isdefinedas G (x, v) K(x,ω) lim v G (o, v) = lim F (x, v) v F (o, v). For every ω Ω, K(,ω)isharmoniconT.

60 Corollary (Estimates for the Poisson kernel) (i) If ω U(x) then δ x ( δ ) x < K(x, v) <δ x. The same inequalities are satisfied by K (u,ω) if ω U(x). (ii) If x < yandω/ U(x) then ( ) K(y,ω) 1 d(x,y) K(x,ω) < 2 + δ.

61 Remarks (Poisson integral representation) The Poisson kernel, being a locally constant function on Ω, belongs to L p (Ω) for every x T, for 1 p. The Poisson integral of a function h in L 1 (Ω) is defined by Kh(x) = h(ω)k(x,ω)dν. Ω

62 Remarks (Poisson integral representation) The Poisson kernel, being a locally constant function on Ω, belongs to L p (Ω) for every x T, for 1 p. The Poisson integral of a function h in L 1 (Ω) is defined by Kh(x) = h(ω)k(x,ω)dν. Ω

63 Definition (Variants of the gradient) σ =[u, v] edge,setp(σ) =p(u, v). Gradient on functions on vertices: f (σ) =(f e f b)(σ). Gradient on functions h on edges: f (σ) =f (σ ) f (σ), where σ =[e(σ), b(σ)] (opposite orientation). Moreover, f (σ) =p(σ) f (σ), Df (σ) =(Pf e f b)(σ), D f = P f e f b.

64 The Green formula, well known in the continuous setup, has been extended to the discrete context of trees by F. Di Biase and this speaker (Zeitschrift, 1995). Proposition (The Green formula) f and h functions on T, Q Tfinite, (hδf f Δ h)= (h bdf f bd h) Q Q = (h b f f b h h bf b P). Q Proposition (The Green identity) Δ(f 2 )(x) = f (x) 2 +2f (x)δf (x).

65 Let us see a sketch of the same part of the proof that we gave for the half-plane. For simplicity, let us deal with a homogeneous tree with isotropic transition operator: say, homogeneity degree q, that is q + 1 neighbors at each vertex. Up to a perfectly analogous uniformization procedure, we need to show that boundedness of f on cones implies A α f < for some α: sincea 0 f A α f this is equivalent to show: Theorem (Lusin theorem, (i) (iii)) f L (T ) Af < C f

66 Let us see a sketch of the same part of the proof that we gave for the half-plane. For simplicity, let us deal with a homogeneous tree with isotropic transition operator: say, homogeneity degree q, that is q + 1 neighbors at each vertex. Up to a perfectly analogous uniformization procedure, we need to show that boundedness of f on cones implies A α f < for some α: sincea 0 f A α f this is equivalent to show: Theorem (Lusin theorem, (i) (iii)) f L (T ) Af < C f

67 Let us see a sketch of the same part of the proof that we gave for the half-plane. For simplicity, let us deal with a homogeneous tree with isotropic transition operator: say, homogeneity degree q, that is q + 1 neighbors at each vertex. Up to a perfectly analogous uniformization procedure, we need to show that boundedness of f on cones implies A α f < for some α: sincea 0 f A α f this is equivalent to show: Theorem (Lusin theorem, (i) (iii)) f L (T ) Af < C f

68 Sketch of proof. o reference vertex, y predecessor of y T, (y o). A 0 f (ω) 2 = f (y) f (y ) 2 y vertex in ω K Aut(T ) stability subgroup of o, compact.k can be regarded as a group od automorphisms of Ω. ν K invariant measure on Ω.

69 Sketch of proof. o reference vertex, y predecessor of y T, (y o). A 0 f (ω) 2 = f (y) f (y ) 2 y vertex in ω K Aut(T ) stability subgroup of o, compact.k can be regarded as a group od automorphisms of Ω. ν K invariant measure on Ω.

70 Sketch of proof. o reference vertex, y predecessor of y T, (y o). A 0 f (ω) 2 = f (y) f (y ) 2 y vertex in ω K Aut(T ) stability subgroup of o, compact.k can be regarded as a group od automorphisms of Ω. ν K invariant measure on Ω.

71 Choose reference point ω Ω. Then A 0 f (ω) 2 dν = f (y) f (y ) 2 dk Ω = n>0 := n>0 K y kω f (y) f (y ) 2 {k K : kω n = y} y =n f (y) f (y ) 2 w(n) y =n K n := stabilizer of ω n in K. The group K n fixes the chain o ω 1 ω 2 ω n.

72 Choose reference point ω Ω. Then A 0 f (ω) 2 dν = f (y) f (y ) 2 dk Ω = n>0 := n>0 K y kω f (y) f (y ) 2 {k K : kω n = y} y =n f (y) f (y ) 2 w(n) y =n K n := stabilizer of ω n in K. The group K n fixes the chain o ω 1 ω 2 ω n.

73 But K acts transitively: so w(n) =[K : K n ]= 1 { y = n} = 1 (q +1)q n 1 (remember that q is the homogeneity degree of T ). Now, this w is exactly the Green function of the homogeneous tree with singularity at o: Pw = w off o. Indeed, the average of w at any vertex y o is again w:

74 But K acts transitively: so w(n) =[K : K n ]= 1 { y = n} = 1 (q +1)q n 1 (remember that q is the homogeneity degree of T ). Now, this w is exactly the Green function of the homogeneous tree with singularity at o: Pw = w off o. Indeed, the average of w at any vertex y o is again w:

75 y =n, w(y)=w(n) y - =n-1, w(y) = q w(n) + y =n+1, w(y) = w(n)/q Figure: Harmonicity away from 0 of the isotropic Green function w: w(y )+ y + w(y + )=qw(y)+qw(y)/q =(q +1)w(y)

76 So w is harmonic away from o. The rest of the proof is now exactly as in the continuous case, via the discrete Green formula and Green identities.

77 So w is harmonic away from o. The rest of the proof is now exactly as in the continuous case, via the discrete Green formula and Green identities.

78 L. Atanasi, M. A. Picardello The Lusin area function and local admissible convergence of harmonic functions on homogeneous trees, Trans. Amer. Math. Soc. 360 (2008), P. Cartier, Fonctions harmoniques sur un arbre, Symp. Math. 9 (1972), F. Di Biase, M. A. Picardello, The Green formula and H p Spaces on trees, Math. Z. 218 (1995), A. Korányi, M. A. Picardello, Boundary behaviour of eigenfunctions of the Laplace Operator on trees, Annali Scuola Normale Superiore - Pisa (Serie IV), 13 (1986), 3, A. Korányi,R.B.Putz, Local Fatou theorem and area theorem for symmetric spaces of rank one, Trans. Amer. Math. Soc. 224 (1976), 1,

79 A. Korányi,M.A.Picardello,M.H.Taibleson,Hardy spaces on non-homogeneous trees, Symp. Math. 29 (1988), F. Mouton, Comportement asymptotique des fonctions harmoniques sur les arbres, Séminaire de probabilités (Strasbourg) 34 (2000),