Convolution with affine arclength measures in the plane

We obtain an estimate for the L3/2,1(R2) L3(12) norm of a certain convolution operator. Let 0 be real-valued and smooth on an interval (a, b) C R. Define the measure A on I2 by f dA = f(t, q(t)) I(t) 1/3dt. J2 a We are interested in the LP(R2) Lq(R22) mapping properties of the operator given by convolution with A. The study of this operator was initiated by Drury ([D]), who used complex interpolation and certain integral estimates to obtain the optimal result IIA* fl3 0 and X(3) , 0 on (a, b). Then (1) IIA * XE1-3 (s) y(t) = (s t, 4(s) q(t)) is one-to-one. Thus rb rb / I XE(y (t)(s)) ql'(s) '(t)I ds dt if A C (t, b). To prove (3) we let IAu stand for the (one-dimensional) Lebesgue measure of A n (u, b) whenever t (4) J XA(s)(4 (s)(t))ds = XA(S) Q y(u)d du ds= (u[Au du. Also J XA(S) m) (S) 1/3ds = 1 XA(S) ?>/(S)i/3 11/31A 1-1/3d A () O"(s) s I (s) O" (s) lAslAs ds .b "(s) A 1 d) 2/3( Thus, it follows from (4) that (5) (/ A(s) (/ ) If 0 /bXA 1Al-/2 = ] y/2d -2 A1/2. J s Jo With this and the fact that 0" is nondecreasing, (5) yields (3) to complete the proof. REFERENCES [C] Y. Choi, Convolution operators with affine arclength measures on plane curves, J. Korean Math. Soc. 36 (1999), 193-207. CMP 99:09 [D] S. W. Drury, Degenerate curves and harmonic analysis, Math. Proc. Camb. Phil. Soc. 108 (1990), 89-96. MR 91h:42021 DEPARTMENT OF MATHEMATICS, FLORIDA STATE UNIVERSITY, TALLAHASSEE, FLORIDA 32306- 4510 E-mail address: oberlinQmath. fsu. edu This content downloaded from 157.55.39.244 on Fri, 17 Jun 2016 05:01:26 UTC All use subject to http://about.jstor.org/terms