Determine the full solvability range for measurable-coefficient nondivergence equations

Determine whether the Sobolev estimate and unique solvability for one-dimensional nondivergence-form parabolic equations with bounded measurable uniformly elliptic coefficients hold throughout the exponent range p\in[3/2,3], and characterize the precise range if they do not.

Background

For the one-dimensional nondivergence-form parabolic equation u_t-a u_{xx}=f with a merely measurable and uniformly elliptic, the paper establishes solvability and W^{1,2}_p estimates in a neighborhood of p=2 and constructs counterexamples at exponents p_+=2/(1-\kappa) and p_-=2/(1+\kappa). Earlier work cited in the paper established nonuniqueness below 3/2 and nonsolvability above 3, but did not settle the intermediate range.

The unresolved issue is whether the entire interval [3/2,3] admits the corresponding estimates and solvability for general measurable coefficients. The present paper determines the optimal order, but not this full exponent range.