Fefferman's Inequality and Applications in Elliptic Partial Differential Equations

TL;DR

This paper extends Fefferman's inequalities to potentials in generalized Morrey and Stummel classes, showing that solutions have bounded mean oscillation and possess strong unique continuation properties, with specific counterexamples.

Contribution

It generalizes Fefferman's inequalities to broader potential classes and establishes strong unique continuation for solutions under these conditions.

Findings

Fefferman's inequalities are extended to generalized Morrey and Stummel potentials.

Logarithm of solutions belongs to BMO class under these potentials.

Counterexample shows some potentials do not satisfy strong unique continuation.

Abstract

In this paper we prove Fefferman's inequalities associated to potentials belonging to a generalized Morrey space $ L^{p,\varphi} $ or a Stummel class $ \tilde{S}_{\alpha,p} $. Our results generalize and extend Fefferman's inequalities obtained in \cite{CRR,CF,F,Z1}. We also show that the logarithmic of non-negative weak solution of second order elliptic partial differential equation, where its potentials are assumed in generalized Morrey spaces and Stummel classes, belongs to the bounded mean oscillation class. As a consequence, this elliptic partial differential equation has the strong unique continuation property. An example of an elliptic partial differential equation where its potential belongs to certain Morrey spaces or Stummel classes which does not satisfy the strong unique continuation is presented.