Sobolev Estimates for Certain Singular Curves

Abstract

AbstractIn this paper we obtain some Sobolev estimates for the integral operator over singular curves (t, t m ) on R 2 form ≥ 2.Key words : Sobolev Estimates, Singular Curves, Boundedness 1. Introduction For m ≥ 2 we consider the integral operator,where σ is a smooth function with a compact supportnear the origin with σ(0) ≠ 0.For σ ≥ 0 and 1 < p < ∞ let L αp (R 2 ) denote the L p Sobolev space with the norm (R 2 ) = (R 2 ) It is well known that T maps L p (R 2 ) to L q (R 2 ) if 1/p-1/q =1/1+m and (1/p, 1/q) only if and belongs to theclosed triangle belongs with vertices (0,0), (1,1) and(2/1+m,1/1+m) (m/1+m, m−1/1+m), (see [1,3,4]). Thelocalized operator of T maps L p (R 2 ) to (R 2 ) if m/m−1<p < m in view of M. Christ in [2].The purpose of this paper is to determine the exactrange of (1/p, 1/q, α) for which T maps L p (R 2 ) toL αq (R 2 ) when 0 < α < 1/m and p < q. We shall provethe following :Theorem 1. Let 0 < α < 1/m. The operator T mapsL p (R 2 ) to L