Advanced Search

In this paper we study integral operators with kernels $$K(x,y)=k_1(x-A_{1}y)\cdots k_m\left(x-A_{m}y\right),$$ $$k_i\left(x\right)={{\Omega }_i\left(x\right)/\phantom{{\Omega }_i\left(x\right)\left|x\right|}\left|x\right|}^{n/\phantom{n{q}_i}{q}_i}$$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, A i are certain invertible matrices, and n/q 1 +…+n/q m = n−α, 0 ≤ α < n. We obtain the appropriate weighted L p-L q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.

We prove weighted inequalities for commutators of one-sided singular integrals (given by a Calder’on-Zygmund kernel with support in $(-\infty ,0)$) with BMO functions. We give the one-sided version of the results in C. Pérez, Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function, J. Fourier Anal. Appl., vol. 3 (6), 1997, pages 743–756 and C. Pérez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., vol 128 (1), 1995, pages...

In this paper we study integral operators of the form $$Tf\left(x\right)=\int |x-a_1{y|}^{-{\alpha }_1}\cdots {|x-a_{m}y|}^{-{\alpha }_m}f\left(y\right)\mathrm{d}y,$$ ${\alpha }_1+\cdots +{\alpha }_m=n$. We obtain the $L^p\left(w\right)$ boundedness for them, and a weighted $(1,1)$ inequality for weights $w$ in $A_p$ satisfying that there exists $c\ge 1$ such that $w\left(a_{i}x\right)\le cw\left(x\right)$ for a.e. $x\in {ℝ}^n$, $1\le i\le m$. Moreover, we prove ${\parallel Tf\parallel }_{\mathrm{B}MO}\le c{\parallel f\parallel }_{\infty }$ for a wide family of functions $f\in L^{\infty }\left({ℝ}^n\right)$.

We consider two-weight estimates for singular integral operators and their commutators with bounded mean oscillation functions. Hörmander type conditions in the scale of Orlicz spaces are assumed on the kernels. We prove weighted weak-type estimates for pairs of weights (u,Su) where u is an arbitrary nonnegative function and S is a maximal operator depending on the smoothness of the kernel. We also obtain sufficient conditions on a pair of weights (u,v) for the operators to be bounded from $L^p\left(v\right)$ to...

In this paper we study the relationship between one-sided reverse Hölder classes $R{H}_r^{+}$ and the $A_p^{+}$ classes. We find the best possible range of $R{H}_r^{+}$ to which an $A_1^{+}$ weight belongs, in terms of the $A_1^{+}$ constant. Conversely, we also find the best range of $A_p^{+}$ to which a $R{H}_{\infty }^{+}$ weight belongs, in terms of the $R{H}_{\infty }^{+}$ constant. Similar problems for $A_p^{+}$, $1<p<\infty$ and $R{H}_r^{+}$, $1<r<\infty$ are solved using factorization.