Convolution operators with homogeneous singular measures on R^3 of polynomial type

URCIUOLO, MARTA

Journal of Ineq. in Pure and Applied Math.

Año: 2006 vol. 7 p. 1 - 89

Let ϕ(y₁,y₂)=y₂^{l}P(y₁,y₂) where P is a polynomial function of degree l such that P(1,0)≠0. Let μ_{δ} be the Borel measure on R³ defined by μ_{δ}(E)=∫_{V_{δ}}χ_{E}(x,ϕ(x))dx where    V_{δ}={x=(x₁,x₂)∈R²:|x₁|≤1,and |x₁|≤δ|x₂|}and let T_{μ_{δ}} be the convolution operator with the measure μ_{δ}. In this paper we explicitely describe the type set    E_{μ_{δ}}:={((1/p),(1/q))∈[0,1]×[0,1]:‖T_{μ_{δ}}‖_{p,q}<∞},for δ small enough.