𝐿^𝑝(⋅) − 𝐿^𝑞(⋅) estimates for convolution operators with singular measures supported on surfaces of half the ambient dimension

Abstract

Let α i , β i > 0 , 1 ≤ i ≤ n , and for t > 0 and x = ( x 1 , … , x n ) ∈ R n , let

and let

Let φ 1 , … , φ n be real functions in C ∞ ⁢ ( R n ∖ { 0 } ) such that φ = ( φ 1 , … , φ n ) is a homogeneous function with respect to these groups of dilations, i.e., φ ⁢ ( t ⋅ x ) = t ∘ φ ⁢ ( x ) . Let γ > 0 and let 𝜇 be the Borel measure in R 2 ⁢ n given by

Let T μ ⁢ f = μ ∗ f , f ∈ S ⁢ ( R 2 ⁢ n ) . In this paper, we study the boundedness of T μ from L p ⁢ ( ⋅ ) ⁢ ( R 2 ⁢ n ) into L q ⁢ ( ⋅ ) ⁢ ( R 2 ⁢ n ) for certain variable exponents p ⁢ ( ⋅ ) and q ⁢ ( ⋅ ) .