A new characterization is given for the pairs of weight functions v, w for which the fractional maximal function is a bounded operator from ${L}_{v}^{p}\left(X\right)$ to ${L}_{w}^{q}\left(X\right)$ when 1 < p < q < ∞ and X is a homogeneous space with a group structure. The case when X is n-dimensional Euclidean space is included.

We derive two-weight weak type estimates for operators of potential type in homogeneous spaces. The conditions imposed on the weights are testing conditions of the kind first studied by E. T. Sawyer [4]. We also give some applications to strong type estimates as well as to operators on half-spaces.

We derive various integral representation formulas for a function minus a polynomial in terms of vector field gradients of the function of appropriately high order. Our results hold in the general setting of metric spaces, including those associated with Carnot-Carathéodory vector fields, under the assumption that a suitable ${L}^{1}$ to ${L}^{1}$ Poincaré inequality holds. Of particular interest are the representation formulas in Euclidean space and stratified groups, where polynomials exist and ${L}^{1}$ to ${L}^{1}$ Poincaré...

We prove two-weight norm estimates for fractional integrals and fractional maximal functions associated with starlike sets in Euclidean space. This is seen to include general positive homogeneous fractional integrals and fractional integrals on product spaces. We consider both weak type and strong type results, and we show that the conditions imposed on the weight functions are fairly sharp.

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.

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