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Multilinear Littlewood-Paley estimates with applications to partial differential equations
E B Fabes1, D S Jerison, C E Kenig
1School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455.
Summary
This study introduces multilinear Littlewood-Paley estimates to solve complex partial differential equations. These estimates aid in analyzing elliptic operators and parabolic equations, advancing mathematical physics.
Area of Science:
- Mathematics
- Partial Differential Equations
- Harmonic Analysis
Background:
- Multilinear Littlewood-Paley theory is a powerful tool in harmonic analysis.
- Estimates for elliptic operators and parabolic equations are fundamental in PDEs.
- Existing methods may have limitations in certain contexts.
Purpose of the Study:
- To derive and apply novel multilinear Littlewood-Paley estimates.
- To address challenges in estimating the square root of elliptic operators.
- To analyze solutions of Cauchy problems for nondivergence-form parabolic equations.
Main Methods:
- Development of multilinear Littlewood-Paley inequalities.
- Application of these estimates to elliptic operators in divergence form.
- Utilizing the estimates for solutions to nondivergence-form parabolic equations.
Main Results:
- A collection of multilinear Littlewood-Paley estimates is obtained.
- Effective estimation of the square root of elliptic operators is achieved.
- Solutions to the Cauchy problem for nondivergence-form parabolic equations are estimated.
Conclusions:
- The derived multilinear Littlewood-Paley estimates provide a versatile framework.
- These estimates offer new approaches to significant problems in partial differential equations.
- The findings contribute to the understanding of operator theory and PDEs.
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