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Weak type estimates for operators of potential type

Richard WheedenShiying Zhao — 1996

Studia Mathematica

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.

High order representation formulas and embedding theorems on stratified groups and generalizations

Guozhen LuRichard Wheeden — 2000

Studia Mathematica

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...

Some integral and maximal operators related to starlike sets

Sagun ChanilloDavid WatsonRichard Wheeden — 1993

Studia Mathematica

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.

Norm inequalities for off-centered maximal operators.

Richard L. Wheeden — 1993

Publicacions Matemàtiques

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.

Representation formulas and weighted Poincaré inequalities for Hörmander vector fields

Bruno FranchiGuozhen LuRichard L. Wheeden — 1995

Annales de l'institut Fourier

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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