L^p(·)−L^q(·) Estimates for Some Convolution Operators with Singular Finite Measures

Abstract

Let μ be the surface measure on ℝⁿ⁺¹ supported on the graph of $\varphi ({y_1}, \ldots ,{y_n}) = {\left| {{y_1}} \right|^{{\beta _1}}} + \cdots + {\left| {{y_n}} \right|^{{\beta _n}}}$, (y₁, …, y_n) ∈ [−1, 1]ⁿ and let T_μbe the convolution operator given by

$${T_\mu }f(x) = \mu * f(x) = \int_{{\mathbb{R}^{n + 1}}} {f(x - y)\,d\mu (y).}$$

We obtain necessary conditions on the exponent functions p(·) and q(·) for the boundedness of T_μ from L^p(·) (Rⁿ⁺¹) into L^q(·) (Rⁿ⁺¹) and we also prove sufficient conditions for certain pairs (p(·), q(·)).