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We obtain a two weight local Tb theorem for any elliptic and gradient elliptic fractional singular integral operator T on the real line, and any pair of locally finite positive Borel measures on the line. This includes the Hilbert transform and in a sense improves on the T1 theorem by the authors and M. Lacey., 120 pages, 3 figures, 56 pages of appendices. We clarify arguments, mostly in appendix B, especially regarding the backward Poisson testing condition. We thank the referee for a thorough and insightful report. Main results unchanged

We show that the energy conditions are not necessary for boundedness of Riesz transforms in dimension $n\geq 2$. In dimension $n=1$, we construct an elliptic singular integral operator $H_{\flat } $ for which the energy conditions are not necessary for boundedness of $H_{\flat }$. The convolution kernel $K_{\flat }\left ( x\right ) $ of the operator $H_{\flat }$ is a smooth flattened version of the Hilbert transform kernel $K\left ( x\right ) =\frac{1}{x}$ that satisfies ellipticity $ \vert K_{\flat }\left ( x\right ) \vert \gtrsim \frac{1}{\left \vert x\right \vert }$, but not gradient ellipticity $ \vert K_{\flat }^{\prime }\left ( x\right ) \vert \gtrsim \frac{1}{ \vert x \vert ^{2}}$. Indeed the kernel has flat spots where $K_{\flat }^{\prime }\left ( x\right ) =0$ on a family of intervals, but $K_{\flat }^{\prime }\left ( x\right ) $ is otherwise negative on $\mathbb{R}\setminus \left \{ 0\right \} $. On the other hand, if a one-dimensional kernel $K\left ( x,y\right ) $ is both elliptic and gradient elliptic, then the energy conditions are necessary, and so by our theorem in [30], the $T1$ theorem holds for such kernels on the line. This paper includes results from arXiv:16079.06071v3 and arXiv:1801.03706v2.

We prove that for certain positive operators $T$, such as the Hardy-Littlewood maximal function and fractional integrals, there is a constant $D>1$, depending only on the dimension $n$, such that the two weight norm inequality \begin{equation*} \int_{\mathbb{R}^{n}}T\left( f\sigma \right) ^{2}d\omega \leq C\int_{\mathbb{ R}^{n}}f^{2}d\sigma \end{equation*} holds for all $f\geq 0$ if and only if the (fractional) $A_{2}$ condition holds, and the restricted testing condition \begin{equation*} \int_{Q}T\left( 1_{Q}\sigma \right) ^{2}d\omega \leq C\left\ | Q\right\ |_{\sigma } \end{equation*} holds for all cubes $Q$ satisfying $\left\ | 2Q\right\ |_{\sigma }\leq D\left\ | Q\right\ |_{\sigma }$. If $T$ is linear, we require as well that the dual restricted testing condition \begin{equation*} \int_{Q}T^{\ast }\left( 1_{Q}\omega \right) ^{2}d\sigma \leq C\left\ | Q\right\ |_{\omega } \end{equation*} holds for all cubes $Q$ satisfying $\left\ | 2Q\right\ |_{\omega }\leq D\left\ | Q\right\ |_{\omega }$., Comment: This version also updates arXiv:1811.11032, 18 pages

This paper is a survey of recent developments in two weight testing theory arising in joint projects with Tuomas Hyötnen, Kangwei Li, Chun-Yen Shen, Ignacio Uriarte-Tuero, Robert Rahm, and Brett Wick. An hour lecture on this work was presented in a talk at the Alamo Symposium in San Antonio, Texas in June of 2019. Our focus is the half-century-old research area of two weight inequalities for Calderón-Zygmund and other classical operators \(T\).

In 1971 Fedi\u{i} proved the remarkable theorem that the linear second order partial differential operator in the plane with coefficients 1 and f^2 is hypoelliptic provided that f is smooth, vanishes at the origin and is positive otherwise. Variants of this result, with hypoellipticity replaced by continuity of weak solutions, were recently given by the authors, together with Cristian Rios and Ruipeng Shen, to infinitely degenerate elliptic divergence form equations where the nonnegative matrix A(x,u) has bounded measurable coefficients with trace roughly 1 and determinant comparable to f, and where F=ln(1/f) is essentially doubling. However, in the plane, these variants assumed additional geometric constraints on f, something not required in Fedi\u{i}'s theorem. In this paper we in particular remove these additional geometric constraints in the plane for homogeneous equations with F essentially doubling., Comment: 17 pages

The spherical maximal operator $$Af(x) = \mathop {sup}\limits_{t > 0} \left| {{A_t}f(x)} \right| = \mathop {sup}\limits_{t > 0} \left| \int{f(x - ty)d\sigma (y)} \right|$$ where σ is the surface measure on the unit sphere, is a classical object that appears in a variety of contexts in harmonic analysis, geometric measure theory, partial differential equation and geometric combinatorics. We establish Lp bounds for the Stein spherical maximal operator in the setting of compactly supported Borel measures μ, ν satisfying natural local size assumptions $$\mu (B(x,r)) \leqslant C{r {{s_\mu }}},v(B(x,r)) \leqslant C{r {{s_v}}}$$ . Taking the supremum over all t > 0 is not in general possible for reasons that are fundamental to the fractal setting, but we can obtain single scale (t ∈ [a, b] ⊂ (0,∞)) results. The range of possible Lp exponents is, in general, a bounded open interval where the upper endpoint is closely tied with the local smoothing estimates for Fourier Integral Operators. In the process, we establish L2(μ) → L2(ν) bounds for the convolution operator Tλf(x) = λ * (fμ), where λ is a tempered distribution satisfying a suitable Fourier decay condition. More generally, we establish a transference mechanism which yields Lp(μ) → Lp(ν) bounds for a large class of operators satisfying suitable Lp-Sobolev bounds. This allows us to effectively study the dimension of a blowup set ({x: Tf(x) = ∞}) for a wide class of operators, including the solution operator for the classical wave equation. Some of the results established in this paper have already been used to study a variety of Falconer type problems in geometric measure theory.

Let ( X , d , μ ) be a space of homogeneous type in the sense of Coifman and Weiss, i.e. d is a quasi metric on X and μ is a positive measure satisfying the doubling condition. Suppose that u and v are two locally finite positive Borel measures on ( X , d , μ ) . Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderon–Zygmund operator T from L 2 ( u ) to L 2 ( v ) in terms of the A 2 condition and two testing conditions. For every cube B ⊂ X , we have the following testing conditions, with 1 B taken as the indicator of B ‖ T ( u 1 B ) ‖ L 2 ( B , v ) ≤ T ‖ 1 B ‖ L 2 ( u ) , ‖ T ⁎ ( v 1 B ) ‖ L 2 ( B , u ) ≤ T ‖ 1 B ‖ L 2 ( v ) . The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

We show that the {\alpha}-fractional Bilinear Indicator/Cube Testing Constant arising in arXiv:1906.05602 is controlled by the classical fractional Muckenhoupt constant, provided the product measure {\sigma} x {\omega} is diagonally reverse doubling (in particular if it is reverse doubling) with exponent exceeding 2(n-{\alpha}). Moreover, this control is sharp within the class of diagonally reverse doubling product measures. When combined with the main results in arXiv:1906.05602, 1907.07571 and 1907.10734, the above control of BICT_{T^{{\alpha}}} for {\alpha}>0 yields a two weight T1 theorem for doubling weights with appropriate diagonal reverse doubling, i.e. the norm inequality for T^{{\alpha}} is controlled by cube testing constants and the {\alpha}-fractional one-tailed Muckenhoupt constants (without any energy assumptions), and also yields a corresponding cancellation condition theorem for the kernel of T^{{\alpha}}, both of which hold for arbitrary {\alpha}-fractional Calder\'on-Zygmund operators T^{{\alpha}}. We do not know if the analogous result for BICT_{H}({\sigma},{\omega}) holds for the Hilbert transform H in case {\alpha}=0, but we show that BICT_{H^{dy}}({\sigma},{\omega}) is not controlled by the Muckenhoupt condition for the dyadic Hilbert transform H^{dy} and doubling weights {\sigma},{\omega}., Comment: 15 pages

Let σ and ω be locally finite positive borel measures on rn with no common point masses, and let tα be a standard αα-fractional calderon–zygmund operator on rn with 0≤α

In this paper we construct a wavelet basis in weighted L^2 of Euclidean space possessing vanishing moments of a fixed order for a general locally finite positive Borel measure. The approach is based on a clever construction of Alpert in the case of Lebesgue measure that is appropriately modified to handle the general measures considered here. We then use this new wavelet basis to study a two-weight inequality for a general Calder\'on-Zygmund operator on the real line and show that under suitable natural conditions, including a weaker energy condition, the operator is bounded from one weighted L^2 space to another if certain stronger testing conditions hold on polynomials. An example is provided showing that this result is logically different than existing results in the literature., Comment: v2: 26 pages, typos corrected, Theorem changed to a Conjecture

This paper is a sequel to our paper Sawyer et al. (Revista Mat Iberoam 32(1):79–174, 2016). Let σ and ω be locally finite positive Borel measures on \(\mathbb{R}^{n}\) (possibly having common point masses), and let T α be a standard α-fractional Calderon-Zygmund operator on \(\mathbb{R}^{n}\) with 0 ≤ α < n. Suppose that \(\Omega: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) is a globally biLipschitz map, and refer to the images \(\Omega Q\) of cubes Q as quasicubes. Furthermore, assume as side conditions the \(\mathcal{A}_{2}^{\alpha }\) conditions, punctured A2 α conditions, and certain α -energy conditions taken over quasicubes. Then we show that T α is bounded from \(L^{2}\left (\sigma \right )\) to \(L^{2}\left (\omega \right )\) if the quasicube testing conditions hold for T α and its dual, and if the quasiweak boundedness property holds for T α .

Let σ and ω be locally finite positive Borel measures on \(\mathbb{R}^{n}\), let T α be a standard α-fractional Calderon-Zygmund operator on \(\mathbb{R}^{n}\) with 0 ≤ α < n, and assume as side conditions the \(\mathcal{A}_{2}^{\alpha }\) conditions, punctured A2 α conditions, and certain α -energy conditions. Then the weak boundedness property associated with the operator T α and the weight pair \(\left (\sigma,\omega \right )\), is ‘good-λ’ controlled by the testing conditions and the Muckenhoupt and energy conditions. As a consequence, assuming the side conditions, we can eliminate the weak boundedness property from Theorem 1 of Sawyer et al. (A two weight fractional singular integral theorem with side conditions, energy and k-energy dispersed. arXiv:1603.04332v2) to obtain that T α is bounded from \(L^{2}\left (\sigma \right )\) to \(L^{2}\left (\omega \right )\) if and only if the testing conditions hold for T α and its dual. As a corollary we give a simple derivation of a two weight accretive global Tb theorem from a related T1 theorem. The role of two different parameterizations of the family of dyadic grids, by scale and by translation, is highlighted in simultaneously exploiting both goodness and NTV surgery with families of grids that are common to both measures.

We investigate connections between potential theories on a Ahlfors-regular metric space X, on a graph G associated with X, and on the tree T obtained by removing the "horizontal edges" in G. Applications to the calculation of set capacity are given., 45 pages; presentation improved based on referee comments

In a previous paper we considered a class of infinitely degenerate quasilinear equations and derived a priori bounds for high order derivatives of solutions in terms of the Lipschitz norm. We now show that it is possible to obtain bounds just in terms of the supremum norm for a further subclass of such equations, and we apply the resulting estimates to prove that continuous weak solutions are necessarily smooth. We also obtain existence, uniqueness and interior regularity of solutions for the Dirichlet problem with continuous boundary data., 45 pages, 4 figures

It is well known that standard Calderon-Zygmund singular integral operators with the isotropic and non-isotropic homogeneities are bounded on the classical H(R) and non-isotropic H h(R ), respectively. In this paper, we develop a new Hardy space theory and prove that the composition of two Calderon-Zygmund singular integral operators with different homogeneities is bounded on this new Hardy space. It is interesting that such a Hardy space has surprisingly a multiparameter structure associated with the underlying mixed homogeneities arising from two singular integral operators under consideration. The Calderon-Zygmund decomposition and an interpolation theorem hold on such new Hardy spaces.

We prove that the multiplier algebra of the Drury-Arveson Hardy space $H_{n}^{2}$ on the unit ball in $\mathbb{C}^{n}$ has no corona in its maximal ideal space, thus generalizing the famous Corona Theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space $B_{p}^{\sigma}$ has the "baby corona property" for all $\sigma \geq 0$ and $1, Comment: v1: 70 pgs; v2: 73 pgs.; introduction expanded, clarified. v3: 73 pgs.; restriction in main result removed (see 9.2), arguments expanded (see 8.1.1). v4: 74 pgs.; systematic arithmetic misprints fixed on pgs. 37-48. v5: 76 pgs.; incorrect embedding corrected via Proposition 4. v6: 80 pgs.; main result extended to vector-valued setting. v7: 82 pgs.; typos removed.

We present results about spaces of holomorphic functions associated to the classical Dirichlet space. The spaces we consider have roles similar to the roles of $H^{1}$ and $BMO$ in the Hardy space theory and we emphasize those analogies., Comment: v1: 19 pages

We study the H∞(Bn) Corona problem ∑j=1Nfjgj=h and show it is always possible to find solutions f that belong to BMOA(Bn) for any n>1, including infinitely many generators N. This theorem improves upon both a 2000 result of Andersson and Carlsson and the classical 1977 result of Varopoulos. The former result obtains solutions for strictly pseudoconvex domains in the larger space H∞⋅BMOA with N=∞, while the latter result obtains BMOA(Bn) solutions for just N=2 generators with h=1. Our method of proof is to solve ∂¯-problems and to exploit the connection between BMO functions and Carleson measures for H2(Bn). Key to this is the exact structure of the kernels that solve the ∂¯ equation for (0,q) forms, as well as new estimates for iterates of these operators. A generalization to multiplier algebras of Besov–Sobolev spaces is also given.

We prove that generalized convex radial solutions to the generalized Monge-Ampère equation det D 2 u = f ( | x | 2 / 2 , u , | ∇ u | 2 / 2 ) \det D^2u = f(|x|^2/2,u,|\nabla u|^2/2) with f f smooth are always smooth away from the origin. Moreover, we characterize the global smoothness of these solutions in terms of the order of vanishing of f f at the origin.

For 0⩽σ

We consider planar curved strictly convex domains with no or very weak smoothness assumptions and prove sharp bounds for square-functions associated to the lattice point discrepancy., Revised version, to appear in Journal d'Analyse Mathematique

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 the special case of the Cauchy transform with one measure supported on the circle, this gives a slightly different conclusion than that in arXiv:1310.4820v4., Comment: 60 pages. A brief history is included in the introduction in this version, and the case of transverse surfaces has been removed due to a gap in the proof. All other results are unchanged with more typos corrected and some arguments clarified

The Krzy\.z conjecture concerns the largest values of the Taylor coefficients of a non-vanishing analytic function bounded by one in modulus in the unit disk. It has been open since 1968 even though information on the structure of extremal functions is available. The purpose of this paper is to collect various conditions that the coefficients of an extremal function (and various other quantities associated with it) should satisfy if the conjecture is true and to show that each one of these properties is equivalent to the conjecture itself. This may provide several possible starting points for future attempts at solving the problem., Comment: Condition (d) in the main Theorem of version 1 of this paper was erroneous. This has been corrected in the subsequent versions v2 and v3. Also, a proof of one implication in the Main Theorem has now been made shorter with respect to v2

We use a priori inequalities for quasilinear equations to obtain a regularity theorem for the Dirichlet problem for the Monge–Ampere equation, $$u_{xx}u_{yy}-(u_{xy})^{2}=k(x,y),$$ and the prescribed Gaussian curvature equation, $$u_{xx}u_{yy}-(u_{xy})^{2}=k(x,y)(1+u_{x}^{2}+u_{y}^{2})^{2},$$ where k(x,y) is close to a function of one variable alone when k is small, but permitted to vanish to infinite order.

The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show that the $L^2$ to $L^2$ inequality holds if and only if two L^2 to weak-L^2 inequalities hold. This is a corollary to a characterization in terms of a two-weight Poisson inequality, and a pair of testing inequalities on bounded functions., Final Version. To appear in Duke

We provea priori inequalities for non-subelliptic quasilinear equations related to the Monge-Ampere equation in two dimensions, for example, equations of the type 1 $$L_w = \partial _x^2 w + \partial _y \left[ {k\left( {x,w\left( {x,y} \right)\partial _y w} \right)} \right] = 0$$ .

In dimension n ⩾ 3 , we define a generalization of the classical two-dimensional partial Legendre transform, that reduces interior regularity of the generalized Monge–Ampere equation det D 2 u = k ( x , u , Du ) to regularity of a divergence form quasilinear system of special form. This is then used to obtain smoothness of C 2 , 1 solutions, having n - 1 nonvanishing principal curvatures, to certain subelliptic Monge–Ampere equations in dimension n ⩾ 3 . A corollary is that if k ⩾ 0 vanishes only at nondegenerate critical points, then a C 2 , 1 convex solution u is smooth if and only if the symmetric function of degree n - 1 of the principal curvatures of u is positive, and moreover, u fails to be C 3 , 1 - 2 n + ɛ when not smooth.

We prove various estimates for the mean square lattice point discrepancy for dilates of a convex body.

For alpha in [0,n) we demonstrate the failure of energy reversal for the vector of alpha-fractional Riesz transforms, and more generally for any vector of alpha-fractional convolution singular integrals having a kernel with vanishing integral on every great circle of the sphere., Comment: 24 pages. This version references a correct form of the T1 theorem, corrects typos and uses Bochner's theorem to complete the proof for the missing range of alpha, and also points out an easy extension to higher dimensions. arXiv admin note: text overlap with arXiv:1305.5104

We give a history of the Corona Problem in both the one variable and the several variable setting. We also describe connections with functional analysis and operator theory. A number of open problems are described.

We prove an alternate Toeplitz corona theorem for the algebras of pointwise kernel multipliers of Besov-Sobolev spaces on the unit ball in $\mathbb{C}^{n}$, and for the algebra of bounded analytic functions on certain strictly pseudoconvex domains and polydiscs in higher dimensions as well. This alternate Toeplitz corona theorem extends to more general Hilbert function spaces where it does not require the complete Pick property. Instead, the kernel functions $k_{x}\left(y\right)$ of certain Hilbert function spaces $\mathcal{H}$ are assumed to be invertible multipliers on $\mathcal{H}$, and then we continue a research thread begun by Agler and McCarthy in 1999, and continued by Amar in 2003, and most recently by Trent and Wick in 2009. In dimension $n=1$ we prove the corona theorem for the kernel multiplier algebras of Besov-Sobolev Banach spaces in the unit disk, extending the result for Hilbert spaces $H^\infty\cap Q_p$ by A. Nicolau and J. Xiao., Comment: v1: 34 pages. v2: 34 pages, typos corrected. v3: 35 pages, typos corrected, presentation improved. v4 35 pages, typos corrected and referee comments included. v5 35 pages, additional reference added and remark to prior related work included

These are the lecture notes generated for the CIMPA Summer School: Real and Complex Analysis with Applications to other Sciences that took place in Buea, Cameroon May 2 13, 2011. The topic of the series of lectures that were given focused on the Corona Problem in complex analysis. This problem can serve as a point of entry to numerous areas of analysis: complex analysis, harmonic analysis, operator theory, and real analysis. In fact, mastering many of the ideas that appear in the proof of the Corona Theorem will be immensely beneficial for students of analysis. There were three lectures given during the CIMPA school. Additionally, there were two problem sessions associated with the lectures so that the students could gain some mastery of the material being presented. The course notes first introduce most of the topics necessary for the Hardy space and then turned to the necessary background for the Corona problem. The topics covered in the course include the following • Definitions of these Spaces; • Computations of their Reproducing Kernels; • Definitions of their Carleson measures and Geometric Characterizations; • Corona Theorems. Based on my personal interests, we focused much more on the function theory and harmonic analysis aspects associated with these spaces.

Marcinkiewicz multipliers are [math] bounded for [math] on the Heisenberg group [math] , as shown by D. Müller, F. Ricci, and E. M. Stein. This is surprising in that these multipliers are invariant under a two-parameter group of dilations on [math] , while there is no two-parameter group of automorphic dilations on [math] . This lack of automorphic dilations underlies the failure of such multipliers to be in general bounded on the classical Hardy space [math] on the Heisenberg group, and also precludes a pure product Hardy space theory. ¶ We address this deficiency by developing a theory of flag Hardy spaces [math] on the Heisenberg group, [math] , that is in a sense “intermediate” between the classical Hardy spaces [math] and the product Hardy spaces [math] on [math] developed by A. Chang and R. Fefferman. We show that flag singular integral operators, which include the aforementioned Marcinkiewicz multipliers, are bounded on [math] , as well as from [math] to [math] , for [math] . We also characterize the dual spaces of [math] and [math] , and establish a Calderón–Zygmund decomposition that yields standard interpolation theorems for the flag Hardy spaces [math] . In particular, this recovers some [math] results of Müller, Ricci, and Stein (but not their sharp versions) by interpolating between those for [math] and [math] .

LetMf(x)=supt>0|f*δt(ψdσ)(x)|denote the maximal operator associated with surface measuredσon a smooth surfaceS. We prove that ifSis convex and has finite order contact with its tangent lines, then M is bounded onLp(Rn),p>2, if and only ifd(x,H)−1∈L1/loc(S) for all tangent planesHnot passing through the origin. LetM′f(x)=supt>0|f*δ′t(ψdσ)(x)|be the maximal operator associated with a nonisotropic dilationδ′tof surface measuredσ. We prove that M′ often behaves far better than M due to a rotational curvature in the time parametert.

We obtain necessary and sufficient conditions to characterize the boundedness of the composition of dyadic paraproduct operators., Comment: v1: 26 pages

A conjecture of Nazarov--Treil--Volberg on the two weight inequality for the Hilbert transform is verified. Given two non-negative Borel measures u and w on the real line, the Hilbert transform $H_u$ maps $L^2(u)$ to $L^2(w)$ if and only if the pair of measures of satisfy a Poisson $A_2$ condition, and dual collections of testing conditions, uniformly over all intervals. This strengthens a prior characterization of Lacey-Sawyer-Shen-Uriate-Tuero arxiv:1201.4319. The latter paper includes a `Global to Local' reduction. This article solves the Local problem., Comment: Final Version, to appear in Duke

In this paper, we study two-weight norm inequalities for operators of potential type in homogeneous spaces. We improve some of the results given in [6] and [8] by significantly weakening their hypotheses and by enlarging the class of operators to which they apply. We also show that corresponding results of Carleson type for upper half-spaces can be derived as corollaries of those for homogeneous spaces. As an application, we obtain some necessary and sufficient conditions for a large class of weighted norm inequalities for maximal functions under various assumptions on the measures or spaces involved.

For $ 1\le k, Comment: 20 pages