Weak Derivatives and Poincaré–Sobolev Estimates
First-order Poincaré-Sobolev estimates in Rn are inequalities showing how Lp norms of the gradient of a function control the function itself. For a sufficiently smooth function f , the first-order partial derivatives ∂f/∂xi, i= 1, . . . ,n, and the gradient ∇f of course have the usual meanings. When f is continuously differentiable and n> 1, the first-order Poincaré-Sobolev estimates that we will derive are fairly simple consequences of the subrepresentation formulas and norm estimates for fractional integrals proved in Chapter 14. A notable exception to the simplicity of their derivation occurs when p = 1, as we will see.