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Convolution estimates related to surfaces of half the ambient dimension. (English) Zbl 0739.42011

Let φ: n n be continuously differentiable, and define β: n 2n by β(x)=(x,φ(x)). Let λ n be the n-dimensional Lebesgue measure and σ=β(λ n ) the image of λ n by β. The authors study the validity of the convolution inequality

σ*f L 3 ( 2n ) Cf L 3/2 ( 2n ) ·

Let J(h):=inf x |det(φ ' (x+h)-φ ' (x))|. The authors assume that x(φ(x+h)-φ(x)) is injective unless J(h)=0. They then prove that the above convolution inequality is implied by a certain estimate for the exotic Riesz potential R α θ(x)=J(x-y) -1+α θ(y)dλ n (y), i.e.

R 1/2 θ L 6 ( n ) Cθ L 3/2 ( n ) ·

They discuss the validity of this estimate in a variety of situations.


MSC:
42B20Singular and oscillatory integrals, several variables