Verbistky will investigate systematically an important class of superlinear integral and partial differential equations which serve as model problems in nonlinear analysis and also are common in applications. Verbitsky will combine a number of analytic techniques including discrete (wavelet-type) decompositions of linear and nonlinear operators, weighted norm inequalities with iterated weights and good control of the imbedding constants, quasi-metrics associated with Green's kernels, refined 3G-inequalities and nonlinear potential theory. An expected new development is that arbitrary nonnegative coefficients at the lower order terms and data can be handled without any a priori regularity assumptions. Essential to this approach is an introduction of new function spaces intrinsically associated with nonlinear problems, identification of their preduals and establishing basic imbedding theorems. Exact pointwise estimates of solutions and their gradients up to the boundary could be extremely valuable in applied problems. The proposed research develops new applications for modern methods of harmonic analysis and potential theory which previously were used extensively in linear analysis. Nonlinear techniques lead to more accurate mathematical models and numerical estimates for differential and integral equations with singular coefficients and data. They describe many environmental and technological phenomena with nonlinear sources, and are widely used in the studies of heat transfer, fluid flow, control theory, and stochastic processes.
