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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.

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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