Sufficient conditions are derived in order that there exist strong-type weighted norm inequalities for some off-centered maximal functions. The maximal functions are of Hardy-Littlewood and fractional types taken over starlike sets in R. The sufficient conditions are close to necessary and extend some previously known weak-type results.

We derive weighted Poincaré inequalities for vector fields which satisfy the Hörmander condition, including new results in the unweighted case. We also derive a new integral representation formula for a function in terms of the vector fields applied to the function. As a corollary of the ${L}^{1}$ versions of Poincaré’s inequality, we obtain relative isoperimetric inequalities.

The purpose of this paper is to derive norm inequalities for potentials of the form
Tf(x) = ∫_{(Rn)} f(y)K(x,y)dy, x ∈ R^{n},
when K is a Kernel which satisfies estimates like those that hold for the Green function associated with the degenerate elliptic equations studied in [3] and [4].

In this Note we prove a two-weight Sobolev-Poincaré inequality for the function spaces associated with a Grushin type operator. Conditions on the weights are formulated in terms of a strong ${A}_{\mathrm{\infty}}$ weight with respect to the metric associated with the operator. Roughly speaking, the strong ${A}_{\mathrm{\infty}}$ condition provides relationships between line and solid integrals of the weight. Then, this result is applied in order to prove Harnack's inequality for positive weak solutions of some degenerate elliptic equations....

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