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We obtain sharp weighted Lp estimates in the Rubio de Francia extrapolation theorem in terms of the Ap characteristic constant of the weight. Precisely, if for a given 1 < r < [infinity] the norm of a sublinear operator on Lr(w) is bounded by a function of the Ar characteristic constant of the weight w, then for p > r it is bounded on Lp(v) by the same increasing function of the Ap characteristic constant of v, and for p < r it is bounded on Lp(v) by the same increasing function of the r-1/p-1 power of the Ap characteristic constant of v. For some operators these bounds are sharp, but not always. In particular, we show that they are sharp for the Hilbert, Beurling, and martingale transforms.

We consider two weight $L^{p}\to L^{q}$-inequalities for dyadic shifts and the dyadic square function with general exponents $1, Comment: V2: 31 pages. Typos fixed. One mistake corrected in the proof of the main theorem 5.1, in the subsection "Deeply contained cubes". To appear in Studia Mathematica

We give a new proof of the sharp one weight $L^p$ inequality for any operator $T$ that can be approximated by Haar shift operators such as the Hilbert transform, any Riesz transform, the Beurling-Ahlfors operator. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators. Our method is flexible enough to prove the corresponding sharp one-weight norm inequalities for some operators of harmonic analysis: the maximal singular integrals associated to $T$, Dyadic square functions and paraproducts, and the vector-valued maximal operator of C. Fefferman-Stein. Also we can derive a very sharp two-weight bump type condition for $T$., We improve different parts of the first version, in particular we show the sharpness of our theorem for the vector-valued maximal function

We give a new proof of the sharp weighted $L^2$ inequality ||T||_{L^2(w)} \leq c [w]_{A_2} where $T$ is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift operators. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators., Comment: To appear in the Electronic Research Announcements in Mathematical Sciences

Using Bellman function techniques, we obtain the optimal dependence of the operator norms in $L^2(\mathbb{R})$ of the Haar multipliers $T_w^t$ on the corresponding $RH^d_2$ or $A^d_2$ characteristic of the weight $w$, for $t=1,\pm 1/2$. These results can be viewed as particular cases of estimates on homogeneous spaces $L^2(vd\sigma)$, for $\sigma$ a doubling positive measure and $v\in A^d_2(d\sigma)$, of the weighted dyadic square function $S_{\sigma}^d$. We show that the operator norms of such square functions in $L^2(v d\sigma)$ are bounded by a linear function of the $A^d_2(d\sigma )$ characteristic of the weight $v$, where the constant depends only on the doubling constant of the measure $\sigma$. We also show an inverse estimate for $S_{\sigma}^d$. Both results are known when $d\sigma=dx$. We deduce both estimates from an estimate for the Haar multiplier $(T_v^{\sigma})^{1/2}$ on $L^2(d\sigma)$ when $v\in A_2^d(d\sigma)$, which mirrors the estimate for $T_w^{1/2}$ in $L^2(\mathbb{R})$ when $w\in A^d_2$. The estimate for the Haar multiplier adapted to the $\sigma$ measure, $(T_v^{\sigma})^{1/2}$, is proved using Bellman functions. These estimates are sharp in the sense that the rates cannot be improved and be expected to hold for all $\sigma$, since the particular case $d\sigma=dx$, $v=w$, correspond to the estimates for the Haar multipliers $T^{1/2}_w$ proven to be sharp.