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Weighted Sobolev-Poincaré inequalities for Grushin type operators. (English) Zbl 0822.46032
Let $$1\leq p\leq q<\infty$$ and $$n,m\geq 1$$, $$n+m= N$$. The inequality considered in the paper reads $\Biggl( {1\over {u(B)}} \int_ B | g(z)- \mu|^ q u(z) dz \Biggr) ^{1/q} \leq cr \Biggl( {1\over {v(B)}} \int_ B | \nabla_ \lambda g(z) |^ p v(z) dz \Biggr)^{1/ p}, \tag{1}$ with a constant $$c>0$$ independent of $$B$$ an $$g$$. Here, the symbol $$\nabla_ \lambda g(z)$$ stands for the “$$\lambda$$-gradient” of the function $$g$$ which is related to generalized Grushin differential operator $$\Delta_ x+ \lambda(x)^ 2 \Delta_ y$$, $$(x,y)= z\in \mathbb{R}^{n+m}$$, and defined by $$\nabla_ \lambda g(z)= (\nabla_ x g(z), \lambda (x) \nabla_ y g(z))$$. Further, $$B= B(z_ 0, r)$$ denotes the ball in $$\mathbb{R}^ N$$ with center $$z_ 0$$ and radius $$r$$ with respect to the metric $$\rho$$ which is natural associated with the vector fields $$X_ 1= \partial/ \partial x_ 1, \dots, X_ n= \partial/ \partial x_ n$$; $$Y_ 1= \lambda(x) \partial/ \partial y_ 1, \dots, Y_ m= \lambda(x) \partial/ \partial y_ m$$ by means of sub-unit curves. The weights $$u$$, $$v$$ are nonnegative and locally integrable, $$\mu= \mu (g, B(z_ 0, r))$$ is a suitable constant, and $$u(B)= \int_ B u(z)dz$$.
The function $$\lambda$$ is assumed to satisfy the following conditions: $$\lambda$$ is continuous, negative, $$\lambda>0$$ except for at most finite number of points (it is remarked that this condition can be slightly weakened), $$\lambda^ n$$ belongs to the class “strong $$A_ \infty$$” introduced by G. David and S. Semmes [Lect. Notes Pure Appl. Math. 122, 101-111 (1990; Zbl 0752.46014)], and $$\lambda$$ satisfies the condition $$RH_ \infty$$, $$r^{-n} \int_{| x-x_ 0| <r} \lambda (x)dx \sim\max \{\lambda (x)$$: $$| x-x_ 0 |< r\}$$. A simple example of such a function $$\lambda$$ is $$\lambda (x)= | x|^ \alpha$$, $$\alpha\geq 0$$. It is shown that the class “strong $$A_ \infty$$” contains $$w(z)= \rho(z, z_ 0)^ \alpha$$ for any $$\alpha>0$$. If $$\lambda=1$$, then absolute values of the Jacobians of quasiconformal mappings of $$\mathbb{R}^ N$$ are strong-$$A_ \infty$$ weights.
It is shown that if $$u$$ is a doubling weight, i.e., $$u(B(z, 2r))\leq cu(B (z,r))$$ for all $$z$$ and $$r$$ with $$c$$ independent of $$z$$ and $$r$$, then a necessary condition for (1) to hold is that ${{r(B)} \over {r(B_ 0)}} \Biggl[ {{u(B)} \over {u(B_ 0)}} \Biggr]^{1/q} \leq c\Biggl[ {{v(B)} \over {v(B_ 0)}} \Biggr]^{1/p}, \qquad B\subset c_ 1 B_ 0, \tag{2}$ for all balls $$B$$, $$B_ 0$$, where $$r(B)$$ denotes the radius of $$B$$ and $$c_ 1\leq 1$$ is a fixed constant.
To formulate the main results assume that $$\lambda$$ satisfies the conditions given above, and that $$w$$ is a weight satisfying the following condition: If $$\lambda (x_ 1)=0$$, then $$w(x, y)$$ is bounded as $$x\to x_ 1$$ uniformly in $$y$$ for $$y$$ in any bounded set.
Theorem I. Let $$1\leq p<q< \infty$$ and $$u$$, $$v$$ be a pair of weight functions satisfying (2) for all $$B_ 0$$, and let $$u$$ be doubling. If there exists a strong-$$A_ \infty$$ weight $$w$$ such that $$vw_{-(1- 1/N)}\in A^ p (w^{1-1/ N} dz)$$, then (1) holds for all $$B(z_ 0, r)$$ with $$\mu$$ equal to either the $$u$$-average of $$g$$ over $$B(z_ 0, r)$$ or the $$w\lambda_{m/ (N-1)}$$-average of $$g$$ over a central ball in $$B(z_ 0, r)$$. (The notion of a “central ball” is related to the Boman chain condition characterizing certain classes of open sets by means of covering with balls.)
Theorem II. Let $$1<p< \infty$$ and suppose there exists an $$s>1$$ and a strong-$$A_ \infty$$ weight $$w$$ such that the following conditions hold:
(i) $$(uw^{-(1- 1/N)} )^ s w^{1- 1/N}$$ is doubling;
(ii) $$(r(B)/ r(B_ 0))^ p [u(b_ 0)^{-1} (\int_ B w^{1- 1/N} dz)^{1- 1/s} (\int_ B (uw^{-(1- 1/N)})^ p w^{1- 1/N} dz)^{1/s}]\leq cv(B)/ v(B_ 0)$$ for all balls $$B$$, $$B_ 0$$ with $$B\subset c B_ 0$$;
(iii) $$vw^{-(1- 1/N)}\in A_ p (w^{1- 1/N} dz)$$.
Then (1) holds for all $$B(z_ 0,r)$$ with $$q=p$$ and $$\mu$$ as in Theorem I.
The core of the paper is devoted to the thorough proof of sufficient conditions. The authors did it in a way which unifies and further extends many results obtained before.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26D10 Inequalities involving derivatives and differential and integral operators
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