Weighted inequalities for fractional type operators with some homogeneous kernels

Abstract

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.

Weighted inequalities for some integral operators with rough kernels

“…In [11] we also obtain the weighted weak type (1 /( − α)) estimate for ∈ A(1 /( − α)) and satisfying (4). We also prove that if ∈ A( /α ∞) and satisfies (4) then…”

“…In [11] we obtain that the operator T is of weak-type (1 1) with respect to the Lebesgue measure. Thus taking 0 < δ < 1 and using Kolmogorov's inequality (see [7, Exercise 2.1.5, p. 91]) we get…”

“…It is well known [9] that if is a weight (i.e. is a non negative function and ∈ L 1 loc (R )) then M α is a bounded operator from L (R ) into L (R ) Throughout this paper we understand that for = ∞, E | | In [10,11] we consider Ω ≡ 1 and weights satisfying the following condition: There exists > 0 such that…”

“…The key argument to obtain the above stated results was the Coifman type estimate (see [11,Theorem 2…”

“…∈ R. In [11] we obtain weighted estimates for this kind of operator and certain weights satisfying (4), precisely as for the classical fractional integral operator I α with 0 < α < , or the singular integral operator with α = 0, we prove the L (R )-L (R ) boundedness of T α for weights ∈ A( ), 1 < < /α, 1/ = 1/ − α/ and 0 ≤ α < .…”