An endpoint version of uniform Sobolev inequalities

We prove an endpoint version of the uniform Sobolev inequalities in [C. E. Kenig, A. Ruiz and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 1987, 329-347]. It was known that strong type inequalities no longer hold at the endpoints; however, we show that restricted weak type inequalities hold there, which imply the earlier classical result by real interpolation. The key ingredient in our proof is a type of interpolation first introduced by Bourgain [J. Bourgain, Esitmations de certaines functions maximales, C. R. Acad. Sci. Paris 310 1985, 499-502]. We also prove restricted weak type Stein-Tomas restriction inequalities on some parts of the boundary of a pentagon, which completely characterizes the range of exponents for which the inequalities hold.