Marta Urciuolo

Weighted Inequalities for Integral Operators with almost Homogeneous Kernels

Let $m\in N$ and $a_{1},\dots,a_{m}$ be real numbers such that for each $i,$ $a_{i}\neq 0$ and $a_{i}\neq a_{j}$ if $i\neq j.$ In this paper we study integral operators of the form $$ Tf(x) =\int k_{1}(x-a_{1}y) \cdots k_{m}(x-a_{m}y) f(y)dy, $$
with $f,\varphi_{i,j}: R^n \to R$, $k_{i}(y) =\sum\limits_{j\in Z}2^{\frac{jn}{q_{i}}}\varphi_{i,j}(2^{j}y)$, $1\leq q_{i}<\infty$, $i=1,\dots,m$, $\frac{1}{q_{1}}+\cdots +\frac{1}{q_{m}}=1$.
If $\varphi _{i,j}$ satisfy certain uniform regularity conditions out of the origin, we obtain the boundedness of $T:L^{p}(w) \rightarrow L^{p}(w) $ for all power weights $w$ in adequate Muckenhoupt classes.

Weights, integral operators.

MSC 2000: 42B25, 42A50, 42B20