$C^{1,\alpha }$-regularity for p-harmonic functions on SU(3) and semi-simple Lie groups

Abstract

In this paper, when $1<p<2$, we establish the $C^{1,\alpha }_{\,\textrm{loc}\,}$-regularity of weak solutions to the degenerate subelliptic p-Laplacian equation

$$\begin{aligned} \triangle _{{{\mathcal {H}}},p}u(x)=\sum \limits _{i=1}^6X^*_i(|{\nabla _{{{\mathcal {H}}}}u}|^{p-2}X_iu)=0 \end{aligned}$$

on SU(3) endowed with the horizontal vector fields $X_1,\dots ,X_6$. The result can be extended to a class of compact connected semi-simple Lie group.

Appendix

The following lemma is [13, Lemma 4.7], which will be used to prove Lemma 3.3.

Lemma 8.3

For a non-negative sequence $\{y_l\}_{l=0,1,2,\dots }$ , $y_{l+1}\le c_0 b^l_0 y^{1+\varepsilon }_l$ implies that

$$\begin{aligned} y_l\le c^{\frac{(1+\varepsilon )^l-1}{\varepsilon }}_0b^{\frac{(1 +\varepsilon )^l-1}{\varepsilon ^2}-\frac{l}{\varepsilon }}_0y^{(1+\varepsilon )^l}_0, \end{aligned}$$

where $c_0>0$ , $\varepsilon >0$ and $b_0>1$.

Moreover, if $y_0\le \theta =c^{-\frac{1}{\varepsilon }}_0b^{-\frac{1}{\varepsilon ^2}}_0,$ then $y_l\le \theta b^{-\frac{l}{\varepsilon }}_0.$ Consequently, $y_l\rightarrow 0$ as $l\rightarrow \infty$.

Proof of Lemma 3.3

For any $b\in (0,1)$, letting $\rho _h={\frac{\rho }{2}}+{\frac{\rho }{2^{h+1}}},\ \rho =\rho _h,\ \rho '=\rho _{h+1}$ and $k_h=k+bH-b^{h+1}H$ in (3.1), we have

$$\begin{aligned} \int _{A^+_{k_{h},\rho _{h+1}}}|\nabla _{{\mathcal {H}}}{u}|^2dx&\le {\frac{\gamma }{(\rho _h-\rho _{h+1})^2}}\int _{A^+_{k_{h}, \rho _h}}|u-k|^2dx+\chi ^2|{A^+_{k_{h},\rho _h}}|^{1-2/q}\nonumber \\&\le {\frac{2^{2h+4}\gamma }{\rho ^2}}|{A^+_{k_{h},\rho _h}}|H^2 +\chi ^2|{A^+_{k_{h},\rho _h}}|^{1-2/q}, \end{aligned}$$

(8.7)

where $h=0,1,2,\dots$ and $q>Q=10$.

The following inequality comes from [12, Lemma 2.3]

$$\begin{aligned} (l-k)|A^+_{l,\rho }|^{1-\frac{1}{Q}}\le \frac{C\rho |B_\rho |^{1 -1/Q}}{|B_\rho \setminus A^+_{k,\rho }|}\int _{A^+_{k,\rho } \setminus A^+_{l,\rho }} |{\nabla _{{{\mathcal {H}}}}}u|dx. \end{aligned}$$

(8.8)

Letting $l=k_{h+1}$, $k=k_{h}$ and $\rho =\rho _{h+1}$ in (8.8), we have

$$\begin{aligned} (k_{h+1}-k_{h})|A^+_{k_{h+1},\rho _{h+1}}|^{1-\frac{1}{Q}} \le \frac{C\rho _{h+1}|B_{\rho _{h+1}}|^{1-1/Q}}{|B_{\rho _{h+1}} \setminus A^+_{k_h,{\rho _{h+1}}}|}\int _{A^+_{k_h,{\rho _{h+1}}} \setminus A^+_{k_{h+1},{\rho _{h+1}}}} |{\nabla _{{{\mathcal {H}}}}}u|dx.\nonumber \\ \end{aligned}$$

(8.9)

The condition that $|{A^+_{k,\rho }}|\le \theta _1|B_\rho |$ shows that

$$\begin{aligned} |_{A^+_{k_{h},\rho _{h+1}}}|\le |A^+_{k,\rho }|\le {C}2^Q\theta _1|B_{\rho _{h+1}}|. \end{aligned}$$

From this, we choose $\theta _1$ small enough such that ${C}2^Q\theta _1\le 1/2$. Then (8.9) becomes

$$\begin{aligned} (b^{h+1}-b^{h+2})H|A^+_{k_{h+1},\rho _{h+1}}|^{1-\frac{1}{Q}} \le \frac{C\rho _{h+1}}{|B_{\rho _{h+1}}|^{1/Q}}\int _{A^+_{k_{h},\rho _{h+1}}}|\nabla _{{\mathcal {H}}}{u}|dx. \end{aligned}$$

(8.10)

Applying Hölder’s inequality to (8.10), then combining (8.7), from the assumption that $H=\mathop {\sup }\limits _{B_\rho }u(x)-k\ge \chi \rho ^{1-Q/q}$, we have

$$\begin{aligned} (b^{h+1}-b^{h+2})^2H^2|A^+_{k_{h+1},\rho _{h+1}}|^{2-\frac{2}{Q}}&\le {C}|{A^+_{k_{h},\rho _{h+1}}}|\int _{A^+_{k_{h},\rho _{h+1}}}|\nabla _{{\mathcal {H}}}{u}|^2dx\nonumber \\&\le {C}|{A^+_{k_{h},\rho _h}}|^{2-{\frac{2}{q}}}\left( \frac{2^{2h}\gamma }{\rho ^2}|{A^+_{k_{h},\rho _h}}|^{\frac{2}{q}}+\frac{\chi ^2}{H^2}\right) H^2\nonumber \\&\le {C}2^{2h}(1+\gamma )H^2\rho ^{{\frac{2Q}{q}}-2}|{A^+_{k_{h},\rho _h}}|^{2-{\frac{2}{q}}}. \end{aligned}$$

Here for any ball $B_\rho$ we use [24, Theorem V.4.1 on P69] to control its volume, that is, $C_1 \rho ^Q \le |B_\rho | \le C_2 \rho ^Q$. Thus

$$\begin{aligned} \left( \frac{|A^+_{k_{h+1},\rho _{h+1}}|}{\rho ^Q}\right) ^{1-{\frac{1}{Q}}} \le {C}2^{h}(1-b)^{-1}b^{-h-1}(1+\gamma )^{\frac{1}{2}}\left( \frac{| {A^+_{k_{h},\rho _h}}|}{\rho ^Q}\right) ^{1-{\frac{1}{q}}}. \end{aligned}$$

(8.11)

Denote

$$\begin{aligned} \mu _h=\frac{|{A^+_{k_{h},\rho _h}}|}{\rho ^Q}\ {{\textrm{and}}}\ \mu _0=\frac{|{A^+_{k,\rho }}|}{\rho ^Q}\le {C}\theta _1. \end{aligned}$$

From (8.11), by Lemma 8.3 with $y_l=\mu _h$, there exists $\theta _1=\theta _1(\gamma ,q,b)\in (0,1)$ such that $\mathop {\lim }\limits _{h\rightarrow \infty } \mu _h=0$, that is, $|A^+_{k+bH,\rho /2}|=0.$ $\square$

Proof of Lemma 3.4

The following inequality comes from [12, Lemma 2.3]

$$\begin{aligned} (l-k)|A^+_{l,\rho }|^{1-\frac{1}{Q}}\le \frac{C\rho |B_\rho |^{1-1/Q}}{|B_\rho \setminus A^+_{k,\rho }|}\int _{A^+_{k,\rho } \setminus A^+_{l,\rho }} |{\nabla _{{{\mathcal {H}}}}}u|dx. \end{aligned}$$

(8.12)

Denote

$$\begin{aligned} \mu _1=\mathop {\sup }\limits _{B_{\rho }}u(x)\ge {k},\ w_1=\mu _1-k\ {{\textrm{and}}}\ D_t=A^+_{\mu _1-\frac{w_1}{2^{t}},\rho /2}\setminus {A^+_{\mu _1-\frac{w_1}{2^{t+1}},\rho /2}}, t=1,2,\dots . \end{aligned}$$

Letting $l=\mu _1-\frac{w_1}{2^{t+1}}$, $k=\mu _1-\frac{w_1}{2^{t}}$ and $\rho \rightarrow \rho /2$ in (8.12), we have

$$\begin{aligned} {\frac{w_1}{2^{t+1}}}\left| {A^+_{\mu _1-\frac{w_1}{2^{t+1}},\rho /2}}\right| ^{1-1/Q} \le {\frac{C\rho |B_{\rho /2}|^{1-1/Q}}{|B_{\rho /2}\setminus {A^+_{\mu _1 -\frac{w_1}{2^{t}},\rho /2}}|}}\int _{D_t}|\nabla _{{\mathcal {H}}}{u}|dx,\ t=1,2,\dots .\nonumber \\ \end{aligned}$$

(8.13)

The assumption that $|A^-_{k,\rho /2}|\ge \tau |B_{\rho /2}|$ implies that

$$\begin{aligned} \left| {A^+_{\mu _1-\frac{w_1}{2^{t}},\rho /2}}\right| \le \left| {A^+_{\mu _1- \frac{w_1}{2},\rho /2}}\right| =\left| {A^+_{\frac{\mu _1}{2}+\frac{k}{2},\rho /2}}\right| \le \left| {A^+_{k,\rho /2}}\right| \le (1-\tau )|B_{\rho /2}|, \end{aligned}$$

which, together with Hölder’s inequality to (8.13), yields

$$\begin{aligned} \left( {\frac{w_1}{2^{t+1}}}\right) ^2\left| {A^+_{\mu _1-\frac{w_1}{2^{t+1}},\rho /2}}\right| ^{2-2/Q} \le {\frac{C}{\tau ^2}}|D_t|\int _{D_t}|\nabla _{{\mathcal {H}}}{u}|^2dx. \end{aligned}$$

(8.14)

Letting $k=\mu _1-\frac{w_1}{2^t}$, $\rho '\rightarrow \rho /2$ and $\rho \rightarrow \rho$ in (3.1), we have

$$\begin{aligned} \int _{A^+_{\mu _1-\frac{w_1}{2^{t}},\rho /2}}|\nabla _{{\mathcal {H}}}{u}|^2dx&\le {\frac{4\gamma }{(\rho )^2}} \int _{A^+_{\mu _1-\frac{w_1}{2^{t}},\rho }}\left| u-\left( \mu _1-\frac{w_1}{2^t} \right) \right| ^2dx+\chi ^2\left| {A^+_{\mu _1-\frac{w_1}{2^{t}},\rho }}\right| ^{1-2/q}\nonumber \\&\le {C}\left[ {\frac{\gamma }{\rho ^2}}\left| {A^+_{\mu _1-\frac{w_1}{2^{t}},\rho }}\right| ^{2/q}\!\! {\mathop {\sup }\limits _{B_{\rho }}\left| u{-}\left( \mu _1-\frac{w_1}{2^t}\right) \right| ^2} {+}\chi ^2\right] \left| {A^+_{\mu _1-\frac{w_1}{2^{t}},\rho }}\right| ^{1-2/q}\nonumber \\&\le {C}\left[ \gamma \rho ^{2Q/q-2}{\left( {\frac{w_1}{2^{t}}} \right) ^2}+\chi ^2\right] \left| {A^+_{\mu _1-\frac{w_1}{2^{t}},\rho }}\right| ^{1-2/q}. \end{aligned}$$

(8.15)

Below we may assume that

$$\begin{aligned} \chi \rho ^{1-Q/q}\le {w_1}/2^t=(\mu _1-k)/2^t, \end{aligned}$$

(8.16)

otherwise we have

$$\begin{aligned} 2^{-t_0}\mu _1-2^{-t_0}k<\chi \rho ^{1-Q/q}, \end{aligned}$$

that is,

$$\begin{aligned} 0<2^{-t_0}\left( k-{\mathop {\sup }\limits _{B_{\rho }}u(x)}\right) +\chi \rho ^{1-Q/q}. \end{aligned}$$

Thus (3.2) holds. Combining (8.16), (8.15) and (8.14), we have

$$\begin{aligned} \left| {A^+_{\mu _1-\frac{w_1}{2^{s-2}},\rho /2}}\right| ^{2-2/Q}\le \left| {A^+_{\mu _1-\frac{w_1}{2^{t+1}},\rho /2}}\right| ^{2-2/Q} \le {\frac{{C}(1+\gamma )}{\tau ^2}}|D_t|\rho ^{2Q/q-2}. \end{aligned}$$

(8.17)

Summing (8.17) with respect to t from 1 to $s-3$, we have

$$\begin{aligned} (s-2)\left| {A^+_{\mu _1-\frac{w_1}{2^{s-2}},\rho /2}}\right| ^{2-2/Q} \le {\frac{{C}(1+\gamma )}{\tau ^2}}|B_{\rho /2}|^{2-2/Q}. \end{aligned}$$

Here for any ball $B_\rho$ we use [24, Theorem V.4.1 on P69] to control its volume, that is, $C_1 \rho ^Q \le |B_\rho | \le C_2 \rho ^Q$. Thus

$$\begin{aligned} \left| {A^+_{\mu _1-\frac{w_1}{2^{s-2}},\rho /2}}\right| \le \left( \frac{C(1 +\gamma )}{\tau ^2(s-2)}\right) ^{Q/(2Q-2)}|B_{\rho /2}|. \end{aligned}$$

Denote

$$\begin{aligned} \theta _1=\left( \frac{C(1+\gamma )}{\tau ^2(s-2)}\right) ^{Q/(2Q-2)}. \end{aligned}$$

By (8.16) and Lemma 3.3 with $b=1/2$ and $k=\mu _1-\frac{w_1}{2^{s-2}}$, there exists $s=s(\gamma ,q,\tau )>0$ such that

$$\begin{aligned} \mathop {\sup }\limits _{B_{\rho /4}}u(x)\le \mu _1-2^{-s+1}w_1 =\mathop {\sup }\limits _{B_{\rho }}u(x)-2^{-s+1} \left( \mathop {\sup }\limits _{B_{\rho }}u(x)-k\right) . \end{aligned}$$

Thus (3.2) holds true. $\square$