In this paper we study integral operators of the formwhere Ai are certain invertible matrices, αi > 0, 1 ≤ i ≤ m, α1 + ... + αm = n − α, 0 ≤ α < n. For 1 q = 1 p − α n we obtain the L p (R n , w p ) − L q (R n , w q ) boundedness for weights w in A(p, q) satisfying that there exists c > 0 such that w (Aix) ≤ cw (x) , a.e.x ∈ R n , 1 ≤ i ≤ m. Moreover we obtain the appropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman Type estimate for these operators.

In this paper we study integral operators with kernelswhere Ω : R → R are homogeneous functions of degree zero, satisfying a size and a Dini condition, A are certain invertible matrices, and / 1 + · · · + / = − α 0 ≤ α < . We obtain the appropriate weighted L -L estimate, the weighted BMO and weak type estimates for certain weights in A( ). We also give a Coifman type estimate for these operators. MSC:42B20, 42B25

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Abstract. Given certain n × n invertible matrices A 1 , ..., Am and 0 ≤ α < n, in this paper we obtain the H p(.) (R n ) → L q(.) (R n ) boundedness of the integral operator with kernel k(x, y) = |x − A 1 y| −α 1 ... |x − Amy| −αm , where α 1 +...+αm = n−α and p(.), q(.) are exponent functions satisfying log-Hölder continuity conditions locally and at infinity related byWe also obtain the H p(.) (R n ) → H q(.) (R n ) boundedness of the Riesz potential operator.