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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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