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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

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

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 .

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

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

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 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

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

For $ 1\le k, Comment: 20 pages

For 1, Comment: 41 pages, 1 figure

Let $\mathcal{D}$ be the classical Dirichlet space, the Hilbert space of holomorphic functions on the disk. Given a holomorphic symbol function $b$ we define the associated Hankel type bilinear form, initially for polynomials f and g, by $T_{b}(f,g):= < fg,b >_{\mathcal{D}} $, where we are looking at the inner product in the space $\mathcal{D}$. We let the norm of $T_{b}$ denotes its norm as a bilinear map from $\mathcal{D}\times\mathcal{D}$ to the complex numbers. We say a function $b$ is in the space $\mathcal{X}$ if the measure $d��_{b}:=| b^{\prime}(z)| ^{2}dA$ is a Carleson measure for $\mathcal{D}$ and norm $\mathcal{X}$ by $$ \Vert b\Vert_{\mathcal{X}}:=| b(0)| +\Vert | b^{\prime}(z)| ^{2}dA\Vert_{CM(\mathcal{D})}^{1/2}. $$ Our main result is $T_{b}$ is bounded if and only if $b\in\mathcal{X}$ and $$ \Vert T_{b}\Vert_{\mathcal{D\times D}}\approx\Vert b\Vert_{\mathcal{X}}. $$, v1: 29 pages

G. David, J.-L. Journe and S. Semmes have shown that if b1 and b2 are para-accretive functions on Rn, then the Tb theorem holds: A linear operator T with Calderon-Zygmund kernel is bounded on L2 if and only if Tb1 I BMO, T*b2 I BMO and Mb2TMb1 has the weak boundedness property. Conversely they showed that when b1 = b2 = b, para-accretivity of b is necessary for Tb Theorem to hold. In this paper we show that para-accretivity of both b1 and b2 is necessary for the Tb Theorem to hold in general. In addition, we give a characterization of para-accretivity in terms of the weak boundedness property and use this to give a sharp Tb Theorem for Besov and Triebel-Lizorkin spaces.

We show that the strong unique continuation property holds for the inequality | Δ u | ≤ | υ | | u | \left | {\Delta u} \right | \leq \left | \upsilon \right |\left | u \right | , where the potential υ ( x ) \upsilon (x) satisfies the C. Fefferman-Phong condition in a certain range of p p values. We also deal with the situation of u ( x ) u(x) vanishing at infinity. These are all consequences of appropriate Carleman inequalities.

We use scaling properties of convex surfaces of finite line type to derive new estimates for two problems arising in harmonic analysis. For Riesz means associated to such surfaces we obtain sharp $L^p$ estimates for $p > 4$, generalizing the Carleson-Sjölin theorem. Moreover we obtain estimates for the remainder term in the lattice point problem associated to convex bodies; these estimates are sharp in some instances involving sufficiently flat boundaries.

For 1 > p ⩽ q > ∞ 1 > p \leqslant q > \infty and w ( x ) w(x) , v ( x ) v(x) nonnegative functions on R n {{\mathbf {R}}^n} , we show that the weighted inequality \[ ( ∫ | T f | q w ) 1 / q ⩽ C ( ∫ f p v ) 1 / p {\left ( {\int {|Tf{|^q}w} } \right )^{1/q}} \leqslant C{\left ( {\int {{f^p}v} } \right )^{1/p}} \] holds for all f ⩾ 0 f \geqslant 0 if and only if both \[ ∫ [ T ( χ Q v 1 − p ′ ) ] q w ⩽ C 1 ( ∫ Q v 1 − p ′ ) q / p > ∞ \int {{{[T({\chi _Q}{v^{1 - p’}})]}^q}w \leqslant {C_1}{{\left ( {\int _Q {{v^{1 - p’}}} } \right )}^{q/p}} > \infty } \] and \[ ∫ [ T ( χ Q w ) ] p ′ v 1 − p ′ ⩽ C 2 ( ∫ Q w ) p ′ / q ′ > ∞ {\int {{{[T({\chi _Q}w)]}^{p’}}{v^{1 - p’}} \leqslant {C_2}\left ( {\int _Q w } \right )} ^{p’/q’}} > \infty \] hold for all dyadic cubes Q Q . Here T T denotes a fractional integral or, more generally, a convolution operator whose kernel K K is a positive lower semicontinuous radial function decreasing in | x | |x| and satisfying K ( x ) ⩽ C K ( 2 x ) K(x) \leqslant CK(2x) , x ∈ R n x \in {{\mathbf {R}}^n} . Applications to degenerate elliptic differential operators are indicated. In addition, a corresponding characterization of those weights v v on R n {{\mathbf {R}}^n} and w w on R + n + 1 {\mathbf {R}}_ + ^{n + 1} for which the Poisson operator is bounded from L p ( v ) {L^p}(v) to L q ( w ) {L^q}(w) is given.

Characterizations are obtained for those pairs of weight functions w , υ w,\upsilon for which the Hardy operator T f ( x ) = ∫ 0 x f ( s ) d s Tf(x) = \int _0^x {f(s)\;ds} is bounded from the Lorentz space L r , s ( ( 0 , ∞ ) , υ d x ) {L^{r,s}}((0,\infty ),\upsilon \,dx) to L p , q ( ( 0 , ∞ ) , w d x ) , 0 > p , q , r , s ⩽ ∞ {L^{p,q}}((0,\infty ),w\,dx),0 > p,q,r,s \leqslant \infty . The modified Hardy operators T η f ( x ) = x − η T f ( x ) {T_\eta }f(x) = {x^{ - \eta }}Tf(x) for η \eta real are also treated.

For 1 > p ⩽ q > ∞ , 0 > α > n 1 > p \leqslant q > \infty ,0 > \alpha > n and w ( x ) , υ ( x ) w(x),\upsilon (x) nonnegative weight functions on R n {R^n} we show that the weak type inequality \[ ∫ { T α f > λ } w ( x ) d x ⩽ A λ − q ( ∫ | f ( x ) | p υ ( x ) d x ) q / p \int _{\{ {T_\alpha }f > \lambda \} }\,w(x)\;dx \leqslant A{\lambda ^{ - q}}{\left ( \int |f(x){|^p}\;\upsilon (x)\;dx \right )^{q/p}} \] holds for all f ⩾ 0 f \geqslant 0 if and only if \[ ∫ Q [ T α ( χ Q w ) ( x ) ] p ′ υ ( x ) 1 − p ′ d x ⩽ B ( ∫ Q w ) p ′ / q ′ > ∞ \int _Q\,[{T_\alpha }({\chi _Q}w)\,(x)]^{p’}\upsilon (x)^{1 - p’}\,dx \leqslant B\left ( \int _Qw \right )^{p’/q’} > \infty \] for all cubes Q Q in R n {R^n} . Here T α {T_\alpha } denotes the fractional integral of order α , T α f ( x ) = ∫ | x − y | α − n f ( y ) d y \alpha ,{T_\alpha }f(x) = \int |x - y{|^{\alpha - n}}f(y)\,dy . More generally we can replace T α {T_\alpha } by any suitable convolution operator with radial kernel decreasing in | x | |x| .

On demontre que l'inegalite differentielle |Δu|≤v|u| a la propriete de prolongement unique se rapportant a l'espace de Sobolev H loc 2,1 (Ω), Ω⊂R n , n≤3, si v satisfait la condition: (K n loc )lim r → 0 sup x ∈ K ∫ |x−y

Subject to a range of side conditions, the two weight inequality for the Hilbert transform is characterized in terms of (1) a Poisson A_2 condition on the weights (2) A forward testing condition, in which the two weight inequality is tested on intervals (3) and a backwards testing condition, dual to (2). A critical new concept in the proof is an Energy Condition, which incorporates information about the distribution of the weights in question inside intervals. This condition is a consequence of the three conditions above. The Side Conditions are termed 'Energy Hypotheses'. At one endpoint they are necessary for the two weight inequality, and at the other, they are the Pivotal Conditions of Nazarov-Treil-Volberg. This new concept is combined with a known proof strategy devised by Nazarov-Treil-Volberg. A counterexample shows that the Pivotal Condition are not necessary for the two weight inequality., 60 pages, 1 figure. v3. An important revision: The Energy Condition is reformulated, a key concept of the proof, is reformulated. The main result is unchanged. v4. important display corrected. v6: The earlier versions incorrectly claimed a characterization, as was pointed out to us by S. Treil v7. Corrections in Section 5

We characterize Carleson measures for the analytic Besov spaces. The problem is first reduced to a discrete question involving measures on trees which is then solved. Applications are given to multipliers for the Besov spaces and to the determination of interpolating sequences. The discrete theorem is also applied to analysis of function space on trees.