Harnack's inequality and H\"older continuity for weak solutions of degenerate quasilinear equations with rough coefficients

We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form \begin{eqnarray} \text{div}\big(A(x,u,\nabla u)\big) = B(x,u,\nabla u)\text{ for }x\in\Omega\nonumber \end{eqnarray} as considered in our previous paper giving local boundedness of weak solutions. Here we derive a version of Harnack's inequality as well as local H\"older continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin and N. Trudinger for quasilinear equations, as well as ones for subelliptic linear equations obtained by Sawyer and Wheeden in their 2006 AMS memoir article.Comment: 39 page

Uncertainty principle estimates for vector fields

We derive weighted norm estimates for integral operators of potential type and for their related maximal operators. These operators are generalizations of the classical fractional integrals and fractional maximal functions. The norm estimates are derived in the context of a space of homogeneous type. The conditions required of the weight functions involve generalizations of the Fefferman-Phong "r-bump" condition. The results improve some earlier ones of the same kind, and they also extend to homogeneous spaces some estimates that were previously known to hold only in the classical Euclidean setting.Dirección General de Investigación Científica y TécnicaNational Science Foundatio

Potential operators, maximal functions, and generalizations of A∞

We derive weighted norm estimates which relate integral operators of potential type (fractional integrals) to corresponding maximal operators (fractional maximal operators). We also derive norm estimates for the maximal operators. The conditions that we impose on the weights involve A∞ conditions of “content type” which are weaker than the usual A∞ condition. The analysis is carried out in the context of spaces of homogeneous type.Dirección General de Investigación Científica y Técnic

A sum operator with applications to self-improving properties of Poincaré inequalities in metric spaces

We define a class of summation operators with applications to the self-improving nature of Poincaré-Sobolev estimates, in fairly general quasimetric spaces of homogeneous type. We show that these sum operators play the familiar role of integral operators of potential type (e.g., Riesz fractional integrals) in deriving Poincaré-Sobolev estimates in cases when representations of functions by such integral operators are not readily available. In particular, we derive norm estimates for sum operators and use these estimates to obtain improved Poincaré-Sobolev results.University of BolognaGruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (Istituto Nazionale di Alta Matematica

A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations

AbstractIn dimension n⩾3, we define a generalization of the classical two-dimensional partial Legendre transform, that reduces interior regularity of the generalized Monge–Ampère equation detD2u=k(x,u,Du) to regularity of a divergence form quasilinear system of special form. This is then used to obtain smoothness of C2,1 solutions, having n-1 nonvanishing principal curvatures, to certain subelliptic Monge–Ampère equations in dimension n⩾3. A corollary is that if k⩾0 vanishes only at nondegenerate critical points, then a C2,1 convex solution u is smooth if and only if the symmetric function of degree n-1 of the principal curvatures of u is positive, and moreover, u fails to be C3,1-2n+ɛ when not smooth

Boundedness of weak solutions of degenerate quasilinear equations with rough coefficients

We derive local boundedness estimates for weak solutions of a large class of second order quasilinear equations. The structural assumptions imposed on an equation in the class allow vanishing of the quadratic form associated with its principal part and require no smoothness of its coefficients. The class includes second order linear elliptic equations as studied by D. Gilbarg and N. S. Trudinger [1998] and second order subelliptic linear equations as studied by E. Sawyer and R. L. Wheeden [2006 and 2010]. Our results also extend ones obtained by J. Serrin [1964] concerning local boundedness of weak solutions of quasilinear elliptic equations

Weighted L^q estimates for derivatives of harmonic functions

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