Related papers: A two weight local Tb theorem for the Hilbert tran…
A quantitative two weight theorem for the Hardy-Littlewood maximal operator is proved improving the known ones. As a consequence a new proof of the main results in [HP] and in [HPR12] is obtained which avoids the use of the sharp…
Using our T1 theorem with an energy side condition allowing common point masses, we extend our previous work in arXiv:1310.4484v3 on one measure supported on a line, to include regular C(1,delta) curves and to permit common point masses. In…
We develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll-Meyer's splitting theorem and a weaker Marino-Prodi perturbation type…
We develop a new general method to prove various non-doubling local Tb theorems. The method combines the non-homogeneous good lambda method of Tolsa, the big pieces Tb theorem of Nazarov-Treil-Volberg and a new change of measure argument…
The aim of this article is to explore in all remaining aspects the spectral theory of locally normal operators. In a previous article we proved the spectral theorem in terms of locally spectral measures. Here we prove the spectral theorem…
The Hilbert transform is essentially the \textit{only} singular operator in one dimension. This undoubtedly makes it one of the the most important linear operators in harmonic analysis. The Hilbert transform has had a profound bearing on…
Recent work of Lacey-Sawyer-Shen-Uriarte-Tuero and Lacey have established a conjecture of Nazarov-Treil-Volberg, giving a real-variable characterization of the two weight inequality for the Hilbert transform, provided the pair of weights do…
Two-weight criteria of various type for the Hardy-Littlewood maximal operator and singular integrals in variable exponent Lebesgue spaces defined on the real line are established.
We make progress on an interesting problem on the boundedness of maximal modulations of the Hilbert transform along the parabola. Namely, if we consider the multiplier arising from it and restrict it to lines, we prove uniform $L^p$ bounds…
We prove $L^p$ estimates for the shifted bilinear Hilbert transform, with a polylogarithmic bound in the size of the shift. As applications, we obtain $r$-variation estimates for bilinear ergodic averages in the sharp range $r > 2$, a sharp…
We prove a two weight theorem for alpha-fractional singular integrals in higher dimensions, assuming energy side conditions. We also show that reversal of the Energy Lemma fails for the vector Riesz transforms in the plane, as well as other…
The Hilbert transform has a linear bound in the $A_{2}$ characteristic on weighted $L^{2}$, \begin{equation*} \left\Vert H\right\Vert _{L^{2}(w)\rightarrow L^{2}(w)}\lesssim \left[ w \right] _{A_{2}}, \end{equation*} and we extend this…
Luna's etale slice theorem is a useful theorem for the local study of quotients by reductive algebraic groups. In this article, we show that the slice theorem can also be used to study local structures of invariant Hilbert schemes. By using…
Local $Tb$ theorems with $L^p$ type testing conditions, which are not scale invariant, have been studied widely in the case of the Lebesgue measure. Until very recently, local $Tb$ theorems in the non-homogeneous case had only been proved…
We obtain a necessary and sufficient condition on a polynomial $P(t_1,t_2)$ for the $\ell^{p}$ boundedness of the discrete double Hilbert transforms associated with $P(t)$ for $1 < p < \infty$. The proof is based on the multi-parameter…
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…
This paper proves two theorems. The first of these simplifies and lends clarity to the previous characterizations of the invariant subspaces of $S$, the operator of multiplication by the coordinate function $z$, on…
We prove a splitting theorem for globally hyperbolic, weighted spacetimes with metrics and weights of regularity $C^1$ by combining elliptic techniques for the negative homogeneity $p$-d'Alembert operator from our recent work in the smooth…
Weighted discrete Hilbert transforms $(a_n)_n \mapsto \big(\sum_n a_n v_n/(\lambda_j-\gamma_n)\big)_j$ from $\ell^2_v$ to $\ell^2_w$ are considered, where $\Gamma=(\gamma_n)$ and $\Lambda=(\lambda_j)$ are disjoint sequences of points in the…
If T is a fractional vector Riesz transform, 1<p<infinity, and sigma and omega are doubling measures, then the two weight L^{p} norm inequality holds if and only if the quadratic triple testing conditions of Hyt\"onen and Vuorinen hold. We…