Moser iteration for positive supersolutions: negative-power and logarithmic comparison

Scaling, bounded truncation, density and the energy estimate. Under $y=(x-x_0)\mathrm{/}R$, the principal divergence form and the weak supersolution inequality retain the same ellipticity bounds, while ball averages are invariant; it is enough to work on $B_1$. Fix $\epsilon >0$, put $U=u+\epsilon$, and for $p>0$ choose $\beta =-p-1<-1$. Then $U\ge \epsilon$, so $U^{\beta }$ and its weak gradient are bounded by constants (depending on $p,\epsilon$) times $1$ and $\mathrm{\mid }Du\mathrm{\mid }$, respectively. More explicitly, for $N>{\epsilon }^{\beta }$ the globally Lipschitz bounded truncation $P_{\epsilon ,N}(t):=\min \{(\max \{t,0\}+\epsilon {)}^{\beta },N\}$ satisfies $P_{\epsilon ,N}(u)=U^{\beta }$ a.e. The cutoff product ${\eta }^2{P}_{\epsilon ,N}(u)$ lies in $H_0^1(B_1)$ by the chain and product rules and compact-support zero extension; approximate it in $H_0^1$ by nonnegative smooth tests and use continuity of the form to pass the supersolution inequality to this test. Testing with ${\eta }^2{U}^{\beta }$ gives $(p+1){\int }_{B_1}{\eta }^2{U}^{-p-2}⟨ADU,DU⟩dx\le 2{\int }_{B_1}\mathrm{\mid }\eta \mathrm{\mid }{U}^{-p-1}\mathrm{\mid }⟨ADU,D\eta ⟩\mathrm{\mid }dx\mathrm{.}$ By Cauchy--Schwarz in the $A$-energy, $\mathrm{\mid }⟨ADU,D\eta ⟩\mathrm{\mid }\le ⟨ADU,DU{⟩}^{1\mathrm{/}2}⟨AD\eta ,D\eta {⟩}^{1\mathrm{/}2}\le M_a\mathrm{\mid }D\eta \mathrm{\mid }⟨ADU,DU{⟩}^{1\mathrm{/}2}$. Absorbing the resulting energy square root yields ${\int }_{B_1}{\eta }^2{U}^{-p-2}⟨ADU,DU⟩dx\le \frac{4M_a^2}{(p+1{)}^2}{\int }_{B_1}{U}^{-p}\mathrm{\mid }D\eta {\mathrm{\mid }}^{2}dx\mathrm{.}$ Ellipticity and $D(U^{-p\mathrm{/}2})=-(p\mathrm{/}2)U^{-p\mathrm{/}2-1}DU$ then give $\int {\eta }^2\mathrm{\mid }D(U^{-p\mathrm{/}2}){\mathrm{\mid }}^2\le \frac{M_a^2{p}^2}{(p+1{)}^2\theta }\int U^{-p}\mathrm{\mid }D\eta {\mathrm{\mid }}^2\le \frac{M_a^2}{\theta }\int U^{-p}\mathrm{\mid }D\eta {\mathrm{\mid }}^2$.

Local logarithmic oscillation. Put $w=\log U$ and $\mathrm{\ell }=\frac{1}{\mathrm{\mid }{B}_{7\mathrm{/}8}\mathrm{\mid }}{\int }_{B_{7\mathrm{/}8}}w$. The logarithmic Caccioppoli estimate [F5], with a smooth cutoff supported in $B_1$ and equal to one on $B_{7\mathrm{/}8}$, gives ${\int }_{B_{7\mathrm{/}8}}\mathrm{\mid }Dw{\mathrm{\mid }}^{2}dx\le C(n,\theta ,M_a)$; Poincare [F6] therefore gives $\mathrm{\parallel }w-\mathrm{\ell }{\mathrm{\parallel }}_{L^2(B_{7\mathrm{/}8})}\le C$. Cover $B_{3\mathrm{/}4}$ by finitely many axis-parallel cubes $Q_0$ of a fixed side $s_n>0$ so small that their closures lie in $B_{13\mathrm{/}16}$ and every concentric ball below lies in $B_1$. For each dyadic subcube $Q$ of side $s$, let $B$ be the concentric ball of radius $\sqrt{n}s$, which contains $Q$. The logarithmic estimate with a smooth cutoff equal to one on $B$ and supported in the concentric ball of radius $2\sqrt{n}s$ gives ${\int }_B\mathrm{\mid }Dw{\mathrm{\mid }}^2\le C{s}^{n-2}$. The ball Poincare inequality [F6] on $B$, together with $\mathrm{\mid }B\mathrm{\mid }\mathrm{/}\mathrm{\mid }Q\mathrm{\mid }=\mathrm{\mid }{B}_{\sqrt{n}}\mathrm{\mid }$, then yields $\frac{1}{\mathrm{\mid }Q\mathrm{\mid }}{\int }_Q\mathrm{\mid }w-w_Q\mathrm{\mid }\le K$, with $K=K(n,\theta ,M_a)\ge 1$ independent of $Q$, $\epsilon$ and $Q_0$.

The reverse-exponent iteration. Let $\kappa =n\mathrm{/}(n-2)$ for $n\ge 3$ and fix $\kappa =2$ for $n=2$, so the $L^{2\kappa }$ Sobolev inequality is available in both cases by [F3]. Combining step 1.1 with the product rule and a cutoff equal to one on $B_r$ and supported in $B_s$, $0<r<s\le 1$, yields $\mathrm{\parallel }{U}^{-1}{\mathrm{\parallel }}_{L^{p\kappa }(B_r)}\le (\frac{C}{s-r}{)}^{2\mathrm{/}p}\mathrm{\parallel }{U}^{-1}{\mathrm{\parallel }}_{L^p(B_s)},C=C(n,\theta ,M_a)\mathrm{.}$ For $p_j=p{\kappa }^j$ and $r_j=\rho +(1-\rho )2^{-j}$, apply this with $(p_j,r_{j+1},r_j)$ and multiply. The logarithm of the product is bounded by a constant multiple of ${\sum }_{j\ge 0}(1+j){\kappa }^{-j}<\mathrm{\infty }$, so ${\mathrm{ess\hspace{0.17em}sup}}_{B_{\rho }}{U}^{-1}\le C_1\mathrm{\parallel }{U}^{-1}{\mathrm{\parallel }}_{L^p(B_1)},C_1=C_1(n,\theta ,M_a,p,\rho )\mathrm{.}$ Taking reciprocals and inserting the volume factor gives $(\frac{1}{\mathrm{\mid }{B}_1\mathrm{\mid }}{\int }_{B_1}{U}^{-p}dx{)}^{-1\mathrm{/}p}\le C_2{\mathrm{ess\hspace{0.17em}inf}}_{B_{\rho }}U$. This is a positive-exponent iteration for $U^{-1}$; in particular the reverse-exponent range is $p_j=p{\kappa }^j>0$, with arbitrary starting $p>0$.

Bounded truncations, stopping cubes and factorial moments. For each $M>0$ set $X_M:=\min \{\mathrm{\mid }w-\mathrm{\ell }\mathrm{\mid },M\}$. It is a bounded $H^1$ truncation by the Lipschitz chain rule; its mean oscillation on every dyadic subcube of a covering cube $Q_0$ is at most $2K$. By [F8], dyadic averages differentiate $X_M$ almost everywhere, so the stopping cubes cover the relevant superlevel set up to a null set. For $g=X_M$ and every dyadic cube $Q$, ${\mathrm{avg}}_Q\mathrm{\mid }g-g_Q\mathrm{\mid }\le 2K$: compare first with the constant $\min \{\mathrm{\mid }{w}_Q-\mathrm{\ell }\mathrm{\mid },M\}$ using the $1$-Lipschitz scalar map, then with $g_Q$. Set $A=2^{n+3}K$. In each cube $Q$ select the maximal proper dyadic subcubes $P$ with ${\mathrm{avg}}_P\mathrm{\mid }g-g_Q\mathrm{\mid }>A$. They are disjoint and their total measure is at most $q\mathrm{\mid }Q\mathrm{\mid }$, where $q=2K\mathrm{/}A=2^{-(n+2)}$. Their immediate parents are not bad, so $\mathrm{\mid }{g}_P-g_Q\mathrm{\mid }\le {\mathrm{avg}}_P\mathrm{\mid }g-g_Q\mathrm{\mid }\le 2^{n}A$. Outside their union, dyadic differentiation gives $\mathrm{\mid }g-g_Q\mathrm{\mid }\le A$ a.e. Repeat the same selection inside each selected cube, recentering at its own mean; its mean oscillation is still at most $2K$. The generation-$m$ union has measure at most $q^m\mathrm{\mid }{Q}_0\mathrm{\mid }$, while outside it the accumulated mean differences and final good-set bound give $\mathrm{\mid }g-g_{Q_0}\mathrm{\mid }\le m{2}^{n}A$ for $m\ge 1$. Consequently there are dimensional constants $C_n,c_n>0$ such that $\frac{\mathrm{\mid }\{x\in Q_0:\mathrm{\mid }{X}_M-(X_M{)}_{Q_0}\mathrm{\mid }>t\}\mathrm{\mid }}{\mathrm{\mid }{Q}_0\mathrm{\mid }}\le C_n{e}^{-c_{n}t\mathrm{/}K}\mathrm{.}$ The layer-cake formula [F8] then yields $\frac{1}{\mathrm{\mid }{Q}_0\mathrm{\mid }}{\int }_{Q_0}\mathrm{\mid }{X}_M-(X_M{)}_{Q_0}{\mathrm{\mid }}^j\le C_{n}j!(K\mathrm{/}{c}_n{)}^j$ for every integer $j\ge 1$. The factorial cancels the denominator in the exponential series: its $j$th averaged term is at most $C_n(p_{0}K\mathrm{/}{c}_n{)}^j$. Choose $p_0:=\min \{1,c_n\mathrm{/}(2K)\}>0$. The geometric bound and monotone convergence of the nonnegative series give $\frac{1}{\mathrm{\mid }{Q}_0\mathrm{\mid }}{\int }_{Q_0}{e}^{p_0\mathrm{\mid }{X}_M-(X_M{)}_{Q_0}\mathrm{\mid }}dx\le C_n^{\mathrm{\prime }}\mathrm{.}$ By step 1.2 and the fixed cube size, $(X_M{)}_{Q_0}\le \frac{1}{\mathrm{\mid }{Q}_0\mathrm{\mid }}{\int }_{Q_0}\mathrm{\mid }w-\mathrm{\ell }\mathrm{\mid }\le C$ uniformly in $M$, hence $\frac{1}{\mathrm{\mid }{Q}_0\mathrm{\mid }}{\int }_{Q_0}{e}^{p_0{X}_M}dx\le e^{p_{0}C}{C}_n^{\mathrm{\prime }}$. Since $X_M\uparrow \mathrm{\mid }w-\mathrm{\ell }\mathrm{\mid }$, monotone convergence gives $\frac{1}{\mathrm{\mid }{Q}_0\mathrm{\mid }}{\int }_{Q_0}{e}^{p_0\mathrm{\mid }w-\mathrm{\ell }\mathrm{\mid }}dx\le C_3$. Summing over the finite cover and normalizing yields $\frac{1}{\mathrm{\mid }{B}_{3\mathrm{/}4}\mathrm{\mid }}{\int }_{B_{3\mathrm{/}4}}{e}^{p_0\mathrm{\mid }w-\mathrm{\ell }\mathrm{\mid }}dx\le C_4$, uniformly in $\epsilon$. The bounded negative-power test in step 1.1 was placed in $H_0^1$ by compact-support smooth density; here the bounded logarithm truncations ensure every oscillation estimate is finite before the monotone limit.

The product constant and the limit $\epsilon \downarrow 0$. Since $e^{\pm p_0(w-\mathrm{\ell })}\le e^{p_0\mathrm{\mid }w-\mathrm{\ell }\mathrm{\mid }}$, step 2.2 gives $\left(\frac{1}{\mathrm{\mid }{B}_{3\mathrm{/}4}\mathrm{\mid }}{\int }_{B_{3\mathrm{/}4}}{U}^{p_0}dx\right)\left(\frac{1}{\mathrm{\mid }{B}_{3\mathrm{/}4}\mathrm{\mid }}{\int }_{B_{3\mathrm{/}4}}{U}^{-p_0}dx\right)\le C_4^2,$ hence the second assertion with comparison constant $C_4^{2\mathrm{/}{p}_0}$. For the fixed exponent $p_0$, $U^{-p_0}\uparrow u^{-p_0}$ and $U^{p_0}\downarrow u^{p_0}$ as $\epsilon \downarrow 0$; dominated convergence for the positive moment (using $(u+1{)}^{p_0}\le u+1$ since $p_0\le 1$ on this bounded ball) and monotone convergence for the negative moment pass the product bound. Since $u>0$ a.e., the limiting positive moment is strictly positive, so the product bound also shows that this particular negative moment is finite. For an arbitrary exponent $p>0$ in the first assertion, monotone convergence passes $\mathrm{avg}{U}^{-p}$ with its extended value; interpret $(+\mathrm{\infty }{)}^{-1\mathrm{/}p}=0$, so the reciprocal negative-moment inequality remains valid without asserting finiteness. The same scaling as in step 1.1 restores arbitrary $R,x_0$; all constants depend only on the listed parameters, and no positive lower bound for $u$ is assumed.