$p$ Harmonic Measure in Simply Connected Domains

Let $\Omega\subset\mathbf{R}^{n}$ be a bounded domain and recall that the continuous Dirichlet problem for Laplace’s equation in $\Omega$ can be stated as follows. Given a continuous function $f$ on $\partial\Omega$, find a harmonic function $u$ in $\Omega$ which is continuous in $\overline{\Omega}$, with $u=f$ on $\partial\Omega$. Although such a classical solution may not exist, it follows from a method of Perron-Wiener-Brelot that there is a unique bounded harmonic function $u$ with continuous boundary values equal to $f$, outside a set of capacity zero (logarithmic capacity for $n=2$ and Newtonian capacity for $n>2$). The maximum principle and Riesz representation theorem yield, for each $x\in\Omega$, the existence of a Borel measure $\omega^{x}$ with $\omega^{x}(\partial\Omega)=1,$ and

$$u(x)=\int_{\partial\Omega}f(y)d\omega^{x}(y)\quad\mbox{whenever $x\in\Omega$.}$$

Then, $\omega=\omega^{x}$ is referred to as the harmonic measure at $x$ associated with the Laplace operator.

Let also $g=g(\cdot)=g(\cdot,x)$ be the Green function for $\Omega$ with pole at $x\in\Omega$ and extend $g$ to $\mathbf{R}^{n}\setminus\Omega$ by putting $g\equiv 0$ on $\mathbf{R}^{n}\setminus\Omega$. Then $\omega$ is the Riesz measure associated to $g$ in the sense that

$\displaystyle{\displaystyle\int}\langle\nabla g,\nabla\phi\rangle\,dx=-{\displaystyle\int}\,\phi\,d\omega\mbox{ whenever $\phi\in C_{0}^{\infty}(\mathbf{R}^{n}\setminus\{x\})$}.$

We define the Hausdorff dimension of $\omega$, denoted $\mbox{H-dim}\;\omega$, by

$$\mbox{H-dim}\;\omega\,={\displaystyle\inf}\{\alpha:\mbox{ there exists $E$ Borel $\subset\partial\Omega$ with }H^{\alpha}(E)=0\mbox{ and }\omega(E)=\omega(\partial\Omega)\},$$

where $H^{\alpha}(E)$, for $\alpha\in\mathbf{R}_{+}$, is the $\alpha$-dimensional Hausdorff measure of $E$ defined below. In the past twenty years a number of remarkable results concerning $\mbox{H-dim}\;\omega$ have been established in planar domains, $\Omega\subset\mathbf{R}^{2}$. In particular, Carleson [C85] showed that $\mbox{H-dim}\;\omega=1$ when $\partial\Omega$ is a snowflake and that $\mbox{H-dim}\;\omega\leq 1$ for any self similar Cantor set. Later Makarov [Mak85] proved that $\mbox{H-dim}\;\omega=1$ for any simply connected domain in the plane. Furthermore, Jones and Wolff [JW88] proved that $\mbox{H-dim}\;\omega\leq 1$ whenever $\Omega\subset\mathbf{R}^{2}$ and $\omega$ exists and Wolff [W93] strengthened [JW88] by showing that $\omega$ is concentrated on a set of s finite $H^{1}$-measure. We also mention results of Batakis [Ba96], Kaufmann-Wu [KW85], and Volberg [V93] who showed, for certain fractal domains and domains whose complements are Cantor sets, that

Hausdorff dimension of $\partial\Omega$ = $\inf\{\alpha:\ H^{\alpha}(\partial\Omega)=0\}>\mbox{H-dim}\;\omega$.

Finally we note that higher dimensional results for the dimension of harmonic measure can be found in [Bo87], [W95], and [LVV05].

In [BL05] the first author, together with Bennewitz, started the study of the dimension of a measure, here referred to as $p$ harmonic measure, associated with a positive $p$ harmonic function which vanishes on the boundary of certain domains in the plane. The study in [BL05] was continued in [L06]. Let $\mathbb{C}$ denote the complex plane and let $dA$ be Lebesgue measure on $\mathbb{C}.$ If $O\subset\mathbb{C}$ is open and $1\leq q\leq\infty,$ let $W^{1,q}(O)$ be the space of equivalence classes of functions $\hat{u}$ with distributional gradient $\nabla\hat{u}=(\hat{u}_{x},\hat{u}_{y}),$ both of which are $q$ th power integrable on $O.$ Let

$$\|\hat{u}\|_{1,q}=\|\hat{u}\|_{q}+\|\nabla\hat{u}\|_{q}$$

be the norm in $W^{1,q}(O)$ where $\|\cdot\|_{q}$ denotes the usual Lebesgue $q$ norm in $O.$ Let $C^{\infty}_{0}(O)$ be infinitely differentiable functions with compact support in $O$ and let $W^{1,q}_{0}(O)$ be the closure of $C^{\infty}_{0}(O)$ in the norm of $W^{1,q}(O).$ Let $\Omega\subset\mathbb{C}$ be a simply connected domain and suppose that the boundary of $\Omega$, $\partial\Omega$, is bounded and non empty. Let $N$ be a neighborhood of $\partial\Omega,\,p$ fixed, $1<p<\infty,$ and let $\hat{u}$ be a positive weak solution to the $p$ Laplace equation in $\Omega\cap N.$ That is, $\hat{u}\in W^{1,p}(\Omega\cap N)$ and

$$\int|\nabla\hat{u}|^{p-2}\,\langle\nabla\hat{u},\nabla\theta\rangle\,dA=0$$

(1.1)

whenever $\theta\in W^{1,p}_{0}(\Omega\cap N).$ Observe that if $\hat{u}$ is smooth and $\nabla\hat{u}\not=0$ in $\Omega\cap N,$ then $\,\nabla\cdot(|\nabla\hat{u}|^{p-2}\,\nabla\hat{u})\equiv 0,$ in the classical sense, where $\nabla\cdot$ denotes divergence. We assume that $\hat{u}$ has zero boundary values on $\partial\Omega$ in the Sobolev sense. More specifically if $\zeta\in C^{\infty}_{0}(N),$ then $\hat{u}\,\zeta\in W^{1,p}_{0}(\Omega\cap N).$ Extend $\hat{u}$ to $N\setminus\Omega$ by putting $\hat{u}\equiv 0$ on $N\setminus\Omega.$ Then $\hat{u}\in W^{1,p}(N)$ and it follows from (1.1), as in [HKM93], that there exists a positive finite Borel measure $\hat{\mu}$ on $\mathbb{C}$ with support contained in $\partial\Omega$ and the property that

$$\int|\nabla\hat{u}|^{p-2}\,\langle\nabla\hat{u},\nabla\phi\rangle\,dA=-\int\phi\,d\hat{\mu}$$

(1.2)

whenever $\phi\in C_{0}^{\infty}(N).$ We note that if $\partial\Omega$ is smooth enough, then $\,\,d\hat{\mu}=|\nabla\hat{u}|^{p-1}\,dH^{1}|_{\partial\Omega}.$ Note that if $p=2$ and if $\hat{u}$ is the Green function for $\Omega$ with pole at $x\in\Omega$ then the measure $\hat{\mu}$ coincides with harmonic measure at $x$, $\omega=\omega^{x}$, introduced above. We refer to $\hat{\mu}$ as the $p$ harmonic measure associated to $\hat{u}$. In [BL05], [L06] the Hausdorff dimension of the $p$ harmonic measure $\hat{\mu}$ is studied for general $p$, $1<p<\infty$, and to state results from [BL05], [L06] we next properly introduce the notions of Hausdorff measure and Hausdorff dimension. In particular, let points in the complex plan be denoted by $z=x+iy$ and put $B(z,r)=\{w\in\mathbb{C}:|w-z|<r\}$ whenever $z\in\mathbb{C}$ and $r>0.$ Let $d(E,F)$ denote the distance between the sets $E,F\subset\mathbb{C}$. If $\lambda>0$ is a positive function on $(0,r_{0})$ with ${\displaystyle\lim_{r\mbox{$\rightarrow$}0}\lambda(r)=0}$ define $H^{\lambda}$ Hausdorff measure on $\mathbb{C}$ as follows: For fixed $0<\delta<r_{0}$ and $E\subseteq\mathbb{R}^{2}$, let $L(\delta)=\{B(z_{i},r_{i})\}$ be such that $E\subseteq\bigcup\,B(z_{i},r_{i})$ and $0<r_{i}<\delta,~~i=1,2,..$. Set

$$\phi_{\delta}^{\lambda}(E)={\displaystyle\inf_{L(\delta)}}\sum\,\lambda(r_{i}).$$

Then

$$H^{\lambda}(E)={\displaystyle\lim_{\delta\rightarrow 0}}\,\,\phi_{\delta}^{\lambda}(E).$$

In case $\lambda(r)=r^{\alpha}$ we write $H^{\alpha}$ for $H^{\lambda}.$ We now define the Hausdorff dimension of the measure $\hat{\mu}$ introduced in (1.2) as

$$\mbox{H-dim}\;\hat{\mu}\,={\displaystyle\inf}\{\alpha:\mbox{ there exists $E$ Borel $\subset\partial\Omega$ with }H^{\alpha}(E)=0\mbox{ and }\hat{\mu}(E)=\hat{\mu}(\partial\Omega)\}.$$

In [BL05] the first author, together with Bennewitz, proved the following theorem.

In [L06] the results in [BL05] were improved at the expense of assuming more about $\partial\Omega$. In particular, we refer to [L06] for the definition of a $k$ quasi-circle. The following theorem is proved in [L06].

We note that Makarov in [Mak85] proved Theorem B for harmonic measure $\omega$, $p=2$, when $\Omega$ is simply connected. Moreover, in this case it suffices to take $A=6\sqrt{(\sqrt{24}-3)/{5}}$, see [HK07]. In this paper we continue the studies in [BL05] and [L06] and we prove the following theorem.

Note that Theorem 1 and the definition of $\mbox{H-dim}\;\hat{\mu}$ imply the following corollary.

In Lemma 2.4, stated below, we first show that it is enough to to prove Theorem 1 for a specific $p$ harmonic function $\hat{u}$ satisfying the hypotheses. Thus, we choose $z_{0}\in\Omega$ and let $u$ be the $p$ capacitary functions for $D=\Omega\setminus\overline{B}(z_{0},d(z_{0},\partial\Omega)/2).$ Then $u$ is $p$ harmonic in $D$ with continuous boundary values, $u\equiv 0$ on $\partial\Omega$ and $u\equiv 1$ on $\partial B(z_{0},d(z_{0},\partial\Omega)/2).$ Furthermore, to prove Theorem 1, we build on the tools and techniques developed in [BL05]. In particular, as noted in [BL05, sec. 7, Closing Remarks, problem 5], given the tools in [BL05] the main difficulty in establishing Theorem 1 is to prove the following result.

In fact, most of our effort in this paper is devoted to proving Theorem 2. Armed with Theorem 2 we then use arguments from [BL05] and additional measure-theoretic arguments to prove Theorem 1. To further appreciate and understand the importance of the type of estimate we establish in Theorem 2, we note that this type of estimate is also crucial in the recent work by the first and second author on the boundary behaviour, regularity and free boundary regularity for $p$ harmonic functions, $p\neq 2$, $1<p<\infty$, in domains in $\mathbf{R}^{n}$, $n\geq 2$, which are Lipschitz or which are well approximated by Lipschitz domains in the Hausdorff distance sense, see [LN07,LN,LN08a,LN08b]. Moreover, Theorem 2 seems likely to be an important step when trying to solve several problems for $p$ harmonic functions and $p$ harmonic measure, in planar simply-connected domains previously only studied in the case $p=2$, i.e., for harmonic functions and harmonic measure. In particular, we refer to [BL05, sec. 7, Closing Remarks] and [L06, Closing Remarks] for discussions of open problems.

The rest of the paper is organized as follows. In section 2 we list some basic local results for a positive $p$ harmonic function vanishing on a portion of $\partial\Omega.$ In section 3 we use these results to prove Theorem 1 under the assumption that Theorem 2 is valid. In sections 4 and 5 we then prove Theorem 2.

Finally the first author would like to thank Michel Zinsmeister for some helpful comments regarding the proof of (4.16).

In the sequel $c$ will denote a positive constant $\geq 1$ (not necessarily the same at each occurrence), which may depend only on $p,$ unless otherwise stated. In general, $c(a_{1},\dots,a_{n})$ denotes a positive constant $\geq 1,$ which may depend only on $p,a_{1},\dots,a_{n},$ not necessarily the same at each occurrence. $C$ will denote an absolute constant. $A\approx B$ means that $A/B$ is bounded above and below by positive constants depending only on $p.$ In this section, we will always assume that $\Omega$ is a bounded simply connected domain, $0<r<\mbox{ diam }\partial\Omega$ and $w\in\partial\Omega$. We begin by stating some interior and boundary estimates for $\tilde{u},$ a positive weak solution to the $p$ Laplacian in $B(w,4r)\cap\Omega$ with $\tilde{u}\equiv 0$ in the Sobolev sense on $\partial\Omega\cap B(w,4r).$ That is, $\tilde{u}\in W^{1,p}(B(w,4r)\cap\Omega)$ and (1.1) holds whenever $\theta\in W^{1,p}_{0}(B(w,4r)\cap\Omega).$ Also $\zeta\tilde{u}\in W^{1,p}_{0}(B(w,4r)\cap\Omega)$ whenever $\zeta\in C_{0}^{\infty}(B(w,4r)).$ Extend $\tilde{u}$ to $B(w,4r)$ by putting $\tilde{u}\equiv 0$ on $B(w,4r)\setminus\Omega.$ Then there exists a locally finite positive Borel measure $\tilde{\mu}$ with support $\subset B(w,4r)\cap\partial\Omega$ and for which (1.2) holds with $\hat{u}$ replaced by $\tilde{u}$ and $\phi\in C_{0}^{\infty}(B(w,4r)).$ Let ${\displaystyle\max_{B(z,s)}\tilde{u},\,\min_{B(z,s)}\tilde{u}}$ be the essential supremum and infimum of $\tilde{u}$ on $B(z,s)$ whenever $B(z,s)\subset B(w,4r).$ For references to proofs of Lemmas 2.1 - 2.3 (see [BL05]).

Using Lemma 2.3 we prove,

Now suppose that $\mu,\hat{\mu}$ are as in Lemma 2.4. Let $N_{1}$ be a neighborhood of $\partial\Omega$ with

$$\partial\Omega\subset N_{1}\subset\bar{N}_{1}\subset N.$$

Then from compactness and continuity of $\hat{u},u,$ there exists $\hat{M}<\infty$ such that

$$u\leq\hat{M}\hat{u}\leq\hat{M}^{2}u$$

(2.8)

on $\Omega\cap\partial N_{1}.$ From (2.8) and the boundary maximum principle for $p$ harmonic functions we conclude that (2.8) holds in $\Omega\cap N_{1}.$ In view of (2.8) and Lemma 2.3 we see there exists $\hat{r}>0,$ and a constant $b<\infty,$ such that

$$\mu(B(w,s))\leq b\hat{\mu}(B(w,2s))\leq b^{2}\mu(B(w,4s))$$

(2.9)

whenever $w\in\partial\Omega$ and $0<s\leq\hat{r}.$ We also note from Lemma 2.3 that supp $\mu$ = supp $\hat{\mu}=\partial\Omega.$

The proof of Lemma 2.4 is by contradiction. Let $E\subset\partial\Omega$ be a Borel set with $\hat{\mu}(E)=0.$ If $\mu(E)>0,$ then from properties of Borel measures, and with $\Gamma$ as in (2.5) with $\nu=\mu,$ we see there exists a compact set $K$ with

$$K\subset E\cap\Gamma\mbox{ and }\mu(K)>0.$$

(2.10)

Given $\epsilon>0$ there also exists an open set $O$ with

$$E\subset O\mbox{ and }\hat{\mu}(O)<\epsilon.$$

(2.11)

Moreover, we may suppose for each $z\in K$ that there is a $\rho=\rho(z)$ with $0<\rho(z)<\hat{r}/1000,$ $\bar{B}(z,100\rho(z))\subset O,$ and

$$\mu(B(z,100\rho))\leq 10^{10}\mu(B(z,\rho)).$$

(2.12)

Applying Vitali’s covering theorem we then get $\{B(z_{i},r_{i})\}$ with $z_{i}\in\partial\Omega,0<100r_{i}<\hat{r}$ and the property that

	$\displaystyle(a)$		(2.12) holds with $\rho=r_{i}$ for each $i,$
	$\displaystyle(b)$		$\displaystyle\ K\subset{\displaystyle\bigcup_{i}B(z_{i},100r_{i})}\subset O,$
	$\displaystyle(c)$		$\displaystyle\ B(z_{i},10r_{i})\cap B(z_{j},10r_{j})=\emptyset\mbox{ when $i\not=j$. }$		(2.13)

Using (2.9) and (2.11) - (2.13), it follows that

	$\displaystyle\mu(K)$	$\displaystyle\leq$	$\displaystyle\mu[\cup_{i}B(z_{i},100r_{i})]\leq\sum_{i}\mu[B(z_{i},100r_{i})]\leq 10^{10}\sum_{i}\mu[B(z_{i},r_{i})]$		(2.14)
		$\displaystyle\leq$	$\displaystyle 10^{10}b\sum_{i}\hat{\mu}[B(z_{i},10r_{i})]\leq 10^{10}\,b\,\hat{\mu}(O)\leq 10^{10}\,b\,\epsilon.$

Since $\epsilon$ is arbitrary we conclude that $\mu(K)=0,$ which contradicts (2.10). Thus $\mu$ is absolutely continuous with respect to $\hat{\mu}.$ Interchanging the roles of $\mu,\hat{\mu}$ we also get that $\hat{\mu}$ is absolutely continuous with respect to $\mu.$ Thus Lemma 2.4 is true. $\Box$

From Lemma 2.4 we see that it suffices to prove Theorem 1 with $\hat{u},\hat{\mu},$ replaced by $u,\mu.$ In this section we prove Theorem 1 for $u$ under the assumption that Theorem 2 is correct. Given Theorem 2 we can follow closely the argument in [BL05] from (6.9) on. However, our argument is necessarily somewhat more complicated, as in [BL05] we used the fact that $\mu$ was a doubling measure, which is not necessarily true when $\Omega$ is simply connected. We claim that it suffices to prove Theorem 1 when

$$z_{0}=0\mbox{ and }d(z_{0},\partial\Omega)=2.$$

(3.1)

Indeed, let $\tau=d(z_{0},\partial\Omega)/2$ and put $T(z)=z_{0}+\tau z.$ If $u^{\prime}(z)=u(T(z))$ for $z\in D,$ then since the $p$ Laplacian is invariant under translations, rotations, dilations, it follows that $u^{\prime}$ is $p$ harmonic in $T^{-1}(D).$ Let $\mu^{\prime}$ be the measure corresponding to $u^{\prime}.$ Then from (1.2) it follows easily that

$$\mu^{\prime}(E)=\tau^{p-2}\mu(T(E))\mbox{ whenever $E\subset\mathbb{R}^{n}$ is a Borel set. }$$

This equality clearly implies that $\mbox{H-dim}\;\mu^{\prime}=\mbox{H-dim}\;\mu.$ Thus we may assume that (3.1) holds. Then $B(0,2)\subset\Omega$ and $D=\Omega\setminus\bar{B}(0,1).$

Using Theorem 2 we have, for some $c=c(p)\geq 1$, that

$$c^{-1}\frac{u(z)}{d(z,\partial\Omega)}\leq|\nabla u(z)|\leq c\frac{u(z)}{d(z,\partial\Omega)}\mbox{ whenever $z\in D.$ }$$

(3.2)

Next set

$$v(x)=\left\{\begin{array}[]{l}\max(\log|\nabla u(x)|,0)\mbox{ when $1<p<2$ }\\ \max(-\log|\nabla u(x)|,0)\mbox{ when $p>2.$ }\end{array}\right.$$

Then in [BL05] it is shown that

$$\int_{\{x:u(x)=t\}}\,|\nabla u|^{p-1}\,\exp\left[\frac{w^{2}}{2c_{+}\log(1/t)}\right]\,dH^{1}x\,\leq\,2\,c_{+}$$

(3.3)

for some $c^{+}\geq 1.$ In [BL05], $c^{+}$ depends on $k,p,$ but only because the constant in (3.2) depends on $k,p.$ So, given Theorem 2, $c^{+}=c^{+}(p)$ in (3.3). Next let

$$\begin{array}[]{c}\xi(t)=2\sqrt{\,c_{+}\,\log(1/t)\,\log\log(1/t)}\mbox{ for $0<t<10^{-6},$ }\\ \\ F(t)=\{x:u(x)=t\mbox{ and }v(x)\,\geq\,\xi(t)\}.\end{array}$$

Then from (3.3) and weak type estimates we deduce

$$\,\int_{F(t)}\,|\nabla u|^{p-1}\,dH^{1}x\,\leq 2c_{+}\,[\log(1/t)]^{-2}.$$

(3.4)

Next for $A$ fixed with $|A|$ large, we define $\lambda$ as in Theorem 1. Let $a=\frac{|A|}{2\sqrt{c^{+}}}$ and note that

$$\lambda(r)=\left\{\begin{array}[]{l}r\,e^{a\xi(r)}\,\mbox{ when }1<p<2,\\ r\,e^{-a\xi(r)}\mbox{ when }p>2.\end{array}\right.$$

(3.5)

To prove Theorem 1 when either $1<p<2$ or $p>2,$ we intially allow $a$ to vary but will later fix it as a constant depending only on $p,$ satisfying several conditions. Fix $p,1<p<2,$ and let $K\subset\partial\Omega$ be a Borel set with $H^{\lambda}(K)=0.$ Let $K_{1}$ be the subset of all $z\in K$ with

$$\limsup_{r\mbox{$\rightarrow$}0}\frac{\mu(B(z,r))}{\lambda(r)}<\infty.$$

Then from the definition of $\lambda$ and a covering argument (see [Mat95, sec 6.9]), it is easily shown that $\mu(K_{1})=0.$ Thus to prove $\mu(K)=0,$ it suffices to show $\mu(E)=0$ when $E$ is Borel and is equal $\mu$ almost everywhere to the set of all points in $\partial\Omega$ for which

$$\limsup_{r\mbox{$\rightarrow$}0}\frac{\mu(B(z,r))}{\lambda(r)}=\infty.$$

(3.6)

Let $G$ be the set of all $z$ where $(3.6)$ holds. Given $0<r_{0}<10^{-100},$ we first show for each $z\in G$ that there exists $s=s(z),0<s/100<r_{0},$ such that

$$\mu(B(z,100s))\leq 10^{9}\mu(B(z,s))\mbox{ and }\lambda(100s)\leq\mu(B(z,s)).$$

(3.7)

In fact let $s\in(0,r_{0})$ be the first point starting from $r_{0}$ where

$$\frac{\mu(B(z,s))}{\lambda(s)}\geq 10^{20}\min\left\{\frac{\mu(B(z,r_{0}))}{\lambda(r_{0})},\,1\right\}.$$

From (3.6) we see that $s$ exists. Using $\lambda(100r)\leq 200\lambda(r),0<r<r_{0}/100,$ it is also easily checked that (3.7) holds. From (3.7) and Vitali again, we get $\{B(z_{i},r_{i})\}$ with $z_{i}\in G,0<100r_{i}<r_{0},$ and the property that

	$\displaystyle(a)$		(3.7) holds with $z=z_{i},s=r_{i},$ for each $i,$
	$\displaystyle(b)$		$\displaystyle\ G\subset{\displaystyle{\displaystyle\bigcup_{i}}B(z_{i},100r_{i})}$
	$\displaystyle(c)$		$\displaystyle\ B(z_{i},10r_{i})\cap B(z_{j},10r_{j})=\emptyset\mbox{ when $i\not=j$. }$		(3.8)

Let $t_{m}=2^{-m}$ for $m=1,2,\dots.$ Given $i,$ we claim there exists $w_{i}\in B(z_{i},5r_{i})$ and $m=m(i)$ with

	$\displaystyle(\alpha)$		$\displaystyle\ u(w_{i})=t_{m}\mbox{ and }d(w_{i},\partial\Omega)\approx r_{i}$
	$\displaystyle(\beta)$		$\displaystyle\ \mu[B(z_{i},10r_{i})]/r_{i}\approx[u(w_{i})/d(w_{i},\partial\Omega)]^{p-1}\approx|\nabla u(w)|^{p-1}$		(3.9 )
			whenever $w\in B(w_{i},d(w_{i},\partial\Omega)/2).$

In (3.9) all proportionality constants depend only on $p.$ To prove (3.9) choose $\zeta_{i}\in\partial B(z_{i},2r_{i})$ with $u(\zeta_{i})={\displaystyle\max_{\bar{B}(z_{i},2r_{i})}}u.$ Then $d(\zeta_{i},\partial\Omega)\approx r_{i},$ since otherwise, it would follow from Lemma 2.2 that $u(\zeta_{i})$ is small in comparison to ${\displaystyle\max_{\bar{B}(z_{i},5r_{i})}}u.$ However from (3.8) $(a)$ and Lemma 2.3, these two maximums are proportional with constants depending only on $p.$ Thus $d(\zeta_{i},\partial\Omega)\approx r_{i}.$ Using this fact, (3.2), (3.8) $(a),$ and Lemma 2.3, once again we get (3.9) $(\beta)$ with $w_{i}$ replaced by $\zeta_{i}.$ If $t_{m}\leq u(\zeta_{i})<t_{m-1}$ we let $w_{i}$ be the first point on the line segment connecting $\zeta_{i}$ to a point in $\partial\Omega\cap\partial B(\zeta_{i},d(\zeta_{i},\partial\Omega))$ where $u=t_{m}.$ From our construction, Harnack’s inequality, and Lemma 2.2 we see that (3.9) is true.

Using (3.8), (3.9), we deduce for $1<p<2$ that

$$v(z)=\log|\nabla u(z)|\geq\,a\,\xi(100r_{i})/\tilde{c}\mbox{ on }B(w_{i},d(w_{i},\partial\Omega)/2)$$

(3.10)

where $a$ is as in (3.5). Next we note that

$$H^{1}[B(w_{i},d(w_{i},\partial\Omega)/2)\cap\{z:u(z)=t_{m}\}]\geq d(w_{i},\partial\Omega)/2$$

(3.11)

as we see from the maximum principle for $p$ harmonic functions, a connectivity argument and basic geometry. Also, we can use (3.8) $(a)$ to estimate $t_{m}$ below in terms of $r_{i}$ and Lemma 2.2 to estimate $t_{m}$ above in terms of $r_{i}.$ Doing this we find for some $\beta=\beta(p),0<\beta<1,\bar{c}=\bar{c}(p),$ that

$$r_{i}\leq\bar{c}\,t_{m}^{\beta}\leq\bar{c}^{2}\,r_{i}^{\beta^{2}}\,.$$

(3.12)

Using (3.8)-(3.12) we conclude, for $a$ large enough, that

$$\mu[B(z_{i},10r_{i})]\,\leq\,c\,\int_{F(t_{m})\cap B(z_{i},10r_{i})}|\nabla u|^{p-1}\,dH^{1}.$$

(3.13)

Using (3.8), (3.12), (3.13), and (3.4) it follows for $c$ large enough that

	$\displaystyle\mu(G)$	$\displaystyle\leq$	$\displaystyle{\displaystyle\mu\left(\bigcup_{i}B(z_{i},100r_{i})\right)}\leq\,{\displaystyle 10^{9}\,\sum_{i}\,\mu[B(z_{i},10r_{i})]}$		(3.14)
		$\displaystyle\leq$	$\displaystyle c{\displaystyle\sum_{m=m_{0}}^{\infty}\int_{F(t_{m})}|\nabla u|^{p-1}dH^{1}x\,}\leq\,c^{2}{\displaystyle\sum_{m=m_{0}}^{\infty}}m^{-2}\,\leq c^{3}m_{0}^{-1}$

where $2^{-m_{0}\beta}=\bar{c}\,r_{0}^{\beta^{2}}.$ Since $r_{0}$ can be arbitrarily small we see from (3.14) that $\mu(G)=0.$ This equality and the remark above (3.6) yield $\mu(K)=0.$ Hence $\mu$ is absolutely continuous with respect to $H^{\lambda}$ and Theorem 1 is true for $1<p<2.$