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En este trabajo se analizan las variaciones semanales en el precio de 300 presentaciones de yerba mate publicadas en línea, para el período comprendido entre diciembre de 2015 y febrero del 2021, considerando atributos propios del producto, eventos temporales, costos y fluctuaciones cambiarias, recurriendo a estimaciones econométricas basadas en Efectos Aleatorios (EA) y Mínimos Cuadrados Generalizados. Factibles (MCGF). Los resultados derivados estas estimaciones, presentan significatividad para todos los grupos de variables explicativas consideradas. El impacto de cada una de ellas resulta ser desigual sobresaliendo el tipo de cambio oficial y logaritmo del precio promedio como las que mayor influencia tienen sobre la decisión de variar el precio de venta.

Patients with inborn errors of immunity (IEI) in Argentina were encouraged to receive licensed Sputnik, AstraZeneca, Sinopharm, Moderna, and Pfizer vaccines, even though most of the data of humoral and cellular responses combination on available vaccines comes from trials conducted in healthy individuals. We aimed to evaluate the safety and immunogenicity of the different vaccines in IEI patients in Argentina. The study cohort included adults and pediatric IEI patients (n = 118) and age-matched healthy controls (HC) (n = 37). B cell response was evaluated by measuring IgG anti-spike/receptor binding domain (S/RBD) and anti-nucleocapsid(N) antibodies by ELISA. Neutralization antibodies were also assessed with an alpha-S protein-expressing pseudo-virus assay. The T cell response was analyzed by IFN-γ secretion on S- or N-stimulated PBMC by ELISPOT and the frequency of S-specific circulating T follicular-helper cells (TFH) was evaluated by flow cytometry.No moderate/severe vaccine-associated adverse events were observed. Anti-S/RBD titers showed significant differences in both pediatric and adult IEI patients versus the age-matched HC cohort (p 0.05). Neutralizing antibodies were also significantly lower in the patient cohort than in age-matched HC (p 0.01). Positive S-specific IFN-γ response was observed in 84.5% of IEI patients and 82.1% presented S-specific TFH cells. Moderna vaccines, which were mainly administered in the pediatric population, elicited a stronger humoral response in IEI patients, both in antibody titer and neutralization capacity, but the cellular immune response was similar between vaccine platforms. No difference in humoral response was observed between vaccinated patients with and without previous SARS-CoV-2 infection.In conclusion, COVID-19 vaccines showed safety in IEI patients and, although immunogenicity was lower than HC, they showed specific anti-S/RBD IgG, neutralizing antibody titers, and T cell-dependent cellular immunity with IFN-γ secreting cells. These findings may guide the recommendation for a vaccination with all the available vaccines in IEI patients to prevent COVID-19 disease.

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.

La yerba es el ingrediente del mate, la infusión nacional de Argentina. El mate también es popular en otros países, especialmente de la misma región. Por ello, el objetivo de este trabajo consiste en caracterizar el mercado y su historia, y describir brevemente cómo la pandemia ha afectado su consumo y los precios. Este artículo aporta evidencia a los productores para diseñar su estrategia de marketing. Palabras clave: yerba mate, concentración de mercado, consumo, inflación

El objetivo de este trabajo es evaluar la dinámica semanal de precios del sector de galletitas en el marco de una economía con inflación moderada, tal como lo es la argentina. Empleando datos de frecuencia semanal, se estiman cinco versiones de la función que relaciona a la variación semanal de los precios de las galletitas con sus posibles determinantes. Cuatro de esas estimaciones han sido obtenidas mediante mínimos cuadrados generalizados (MCG), en tanto que la restante corresponde a una especificación con efectos aleatorios (EA). Se identificaron posibles influencias del nivel de concentración de la oferta, las características de cada producto (tamaño, sabor, tipo), la situación del mercado cambiario, ciertos efectos temporales y las variaciones en los precios de los insumos sobre la dinámica de precios de las galletitas. Sin embargo, solamente los efectos de las variaciones en los precios de los insumos se presentaron robustos. El signo de estos efectos robustos coincide siempre con el esperado, a excepción del caso de la harina de trigo The objective of this contribution is to evaluate the weekly price dynamics of the biscuit sector in the framework of an economy with moderate inflation, such as Argentina. Using weekly frequency data, five versions of the function that relate cookie price weekly variations with their possible determinants are estimated. Four of these estimates have been obtained using generalized least squares (GLS), while the rest corresponds to a specification with random effects (RE). Possible influences of the supply level concentration, the features of each product (size, flavour and type), the exchange market situation, certain temporary effects and the variations in the prices of inputs on the price dynamics of the cookies are identified. However, only the effects of changes in input prices were robust. The sign of these robust effects always coincides with the expected values, with the exception of the case of wheat flour. Universidad Pablo de Olavide

Let M be a compact d-dimensional Riemannian manifold without a boundary. Given a compact set E⊂M, we study the set of distances from the set E to a fixed point x∈E. This set is Δρx(E)={ρ(x,y):y∈E}, where ρ is the Riemannian metric on M. We prove that if the Hausdorff dimension of E is greater than d+12, then there exist many x∈E such that the Lebesgue measure of Δρx(E) is positive. This result was previously established by Peres and Schlag in the Euclidean setting. We give a simple proof of the Peres–Schlag result and generalize it to a wide range of distance type functions. Moreover, we extend our result to the setting of chains studied in our previous work and obtain a pinned estimate in this context.

Objective: We aimed to test the relevance of Deficiency of Adenosine Deaminase 2 in patients with antibody deficiency and describe the clinical picture of the disease in adulthood. Methods: We screened for DADA2 in a cohort of 181 patients with antibody deficiency with and without vascular lesions by the means of Next-Generation-Sequencing and targeted Sanger Sequencing. All mutations were confirmed by determining the ADA2 enzymatic activity levels in dried plasma spots. Clinical data and laboratory values were collected in a standardized format. Results: Following the diagnosis of two siblings of the index family, we identified nine additional affected patients with compound heterozygous or homozygous CECR1 mutations, containing six novel and four previously published mutations. The patients' age at evaluation ranged from 14 to 51 years with a median age of 22 years. Clinically, we saw a broad phenotype ranging from isolated antibody deficiency to recurrent strokes. All but one patient had low memory B-cells. Moreover, the B-cell function seemed to correlate with inflammation. Conclusions: Taken together, we found that DADA2 presents not only with vasculopathy but also with an immunodeficiency of the B-cell compartment. Therefore, patients with antibody deficiency should be screened for DADA2. Anti-TNF-alpha treatment might improve the immunological features over time and might be considered in patients without vascular manifestations but elevated inflammation markers. Conservative management of the less severely affected patients has so far proven to be the choice for our adolescent and adult DADA2-patients; however, in patients with severe cytopenias and bone marrow failure, HSCT should be considered. This article is protected by copyright. All rights reserved.

PURPOSE: Nitazoxanide was recently reported as having in vitro effectiveness against the rubella virus. Immunodeficiency-related vaccine-derived rubella occurs in some patients who have an inherited immunodeficiency and who received the MMR vaccine. This study investigated the in vivo effectiveness of nitazoxanide therapy. METHODS: This is a retrospective analysis of seven patients treated with nitazoxanide as salvage therapy for immunodeficiency-related vaccine-derived rubella infection. The patients were recruited from an ongoing rubella detection surveillance project. RESULTS: Seven patients with persistent rubella were treated with nitazoxanide and one demonstrated significant clinical improvement. Two additional patients exhibited diminished viral capsid production with one patient having transient slowing of progression. The cohort overall generally had low T cell counts and had a high burden of co-morbidities. There were three deaths. Two deaths were from PML and one was related to hematopoietic stem cell transplantation. CONCLUSIONS: Nitazoxanide has limited in vivo anti-viral effects for immunodeficiency-related vaccine-derived rubella. Most patients did not exhibit clinical improvement.

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

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 show that if E@?F"q^d, the d-dimensional vector space over the finite field with q elements, and |E|>=@rq^d, where q^-^1^2@?@r@?1, then E contains an isometric copy of at least c@r^d^-^1q^d^+^1^2 distinct (d+1)-point configurations.

Let α ∈ 2 (0, 1), K ≥ 1, and \( d = 2\frac{{1 + \alpha K}} {{1 + K}} \). Given a compact set E ⊂ ℂ, it is known that if \( \mathcal{H}^d (E) = 0 \), then E is removable for α-Holder continuous K-quasiregular mappings in the plane. The sharpness of the index d is shown with the construction, for any t > d, of a set E of Hausdorff dimension dim(E) = t which is not removable. In this paper, we improve this result and construct compact nonremovable sets E such that \( 0 < \mathcal{H}^d (E) < \infty \). For the proof, we give a precise planar K-quasiconformal mapping whose Holder exponent is strictly bigger than \( \frac{1} {K} \) and which exhibits extremal distortion properties.

We extend a well-known theorem of Jones and Makarov [8] on the singularity of boundary distortion of planar conformal mappings. Using a different technique, we recover the previous result and generalize the result to quasiconformal mappings of the unit ball \( \mathbb{B}^n \) ⊂_∝n, n ≥ 2. We also establish an estimate on the Hausdorff (gauge) dimension of the boundary of the image domain outside an exceptional set of given size on the sphere ∂\( \mathbb{B}^n \) and show that this estimate is essentially 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 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

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

In this paper we prove that if $\phi:\C\to\C$ is a $K$-quasiconformal map, with $K>1$, and $E\subset \C$ is a compact set contained in a ball $B$, then $$\frac{\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}(E)}{\diam(B)^{\frac2{K+1}}} \geq c^{-1} (\frac{\gamma(\phi(E))}{\diam(\phi(B))})^{\frac{2K}{K+1}},$$ where $\gamma$ stands for the analytic capacity and $\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}$ is a capacity associated to a non linear Riesz potential. As a consequence, if $E$ is not $K$-removable (i.e. removable for bounded $K$-quasiregular maps), it has positive capacity $\dot C_{frac{2K}{2K+1},\frac{2K+1}{K+1}}$. This improves previous results that assert that $E$ must have non $\sigma$-finite Hausdorff measure of dimension $2/(K+1)$. We also show that the indices $\frac{2K}{2K+1}$, $\frac{2K+1}{K+1}$ are sharp, and that Hausdorff gauge functions do not appropriately discriminate which sets are $K$-removable. So essentially we solve the problem of finding sharp "metric" conditions for $K$-removability., Comment: 57 pages; typos corrected

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

Two populations of human natural killer (NK) cells can be identified in peripheral blood. The majority are CD3(-)CD56(dim) cells while the minority exhibits a CD3(-)CD56(bright) phenotype. In vitro evidence indicates that CD56(bright) cells are precursors of CD56(dim) cells, but in vivo evidence is lacking. Here, we studied NK cells from a patient that suffered from a melanoma and opportunistic fungal infection during childhood. The patient exhibited a stable phenotype characterized by a reduction in the frequency of peripheral blood CD3(-)CD56(dim) NK cells, accompanied by an overt increase in the frequency and absolute number of CD3(-)CD56(bright) cells. These NK cells exhibited similar expression of perforin, CD57 and CD158, the major activating receptors CD16, NKp46, NKG2D, DNAM-1, and 2B4, as well as the inhibitory receptor CD94/NKG2A, on both CD56(bright) and CD56(dim) NK cells as healthy controls. Also, both NK cell subpopulations produced IFN-γ upon stimulation with cytokines, and CD3(-)CD56(dim) NK cells degranulated in response to cytokines or K562 cells. However, upon stimulation with cytokines, a substantial fraction of CD56(dim) cells failed to up-regulate CD57 and CD158, showed a reduction in the percentage of CD16(+) cells, and CD56(bright) cells did not down-regulate CD62L, suggesting that CD56(dim) cells could not acquire a terminally differentiated phenotype and that CD56(bright) cells exhibit a maturation defect that might result in a potential altered migration pattern. These observations, support the notion that NK cells of this patient display a maturation/activation defect that precludes the generation of mature NK cells at a normal rate accompanied by CD56(dim) NK cells that cannot completely acquire a terminally differentiated phenotype. Thus, our results provide evidence that support the concept that in vivo CD56(bright) NK cells differentiate into CD56(dim) NK cells, and contribute to further understand human NK cell ontogeny. Fil: Domaica, Carolina Ines. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Biología y Medicina Experimental (i); Argentina Fil: Fuertes, Mercedes Beatriz. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Biología y Medicina Experimental (i); Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Química Biológica; Argentina Fil: Uriarte, Ignacio. Gobierno de la Ciudad de Buenos Aires. Hospital General de Niños "Ricardo Gutierrez"; Argentina Fil: Girart, Maria Victoria. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Biología y Medicina Experimental (i); Argentina Fil: Sardañons, Jessica . Gobierno de la Ciudad de Buenos Aires. Hospital General de Niños "Ricardo Gutierrez"; Argentina Fil: Comas, Dorina Ileana . Gobierno de la Ciudad de Buenos Aires. Hospital General de Niños "Ricardo Gutierrez"; Argentina Fil: Di Giovanni, Daniela . Gobierno de la Ciudad de Buenos Aires. Hospital General de Niños "Ricardo Gutierrez"; Argentina Fil: Gaillard, María Isabel . Gobierno de la Ciudad de Buenos Aires. Hospital General de Niños "Ricardo Gutierrez"; Argentina Fil: Bezrodnik, Liliana. Gobierno de la Ciudad de Buenos Aires. Hospital General de Niños "Ricardo Gutierrez"; Argentina Fil: Zwirner, Norberto Walter . Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Biología y Medicina Experimental (i); Argentina. Universidad de Buenos Aires. Facultad de Medicina; Argentina

We present the converse to a higher-dimensional, scale-invariant version of the classical F. and M. Riesz theorem, proved by the first two authors. More precisely, for n ≥ 2, for an Ahlfors-David regular domain Ω ⊂ ℝn+1 which satisfies the Harnack chain condition plus an interior (but not exterior) corkscrew condition, we show that absolute continuity of the harmonic measure with respect to the surface measure on ∂Ω, with scale-invariant higher integrability of the Poisson kernel, is sufficient to imply quantitative rectifiability of ∂Ω., The first author was supported by NSF grants DMS-0801079 and DMS-1101244. The second author was supported by MICINN Grant MTM2010-16518, and by CSIC PIE 200850I015. The third author was partially supported by grants DMS-0901524, CAREER DMS- 1056965 (US NSF), Sloan Research Foundation, and MTM2010-16232, MTM2009-14694-C02-01 (Spain).

For 1, Comment: 41 pages, 1 figure

In this paper we prove the sharp distortion estimates for the quasiconformal mappings in the plane, both in terms of the Riesz capacities from non linear potential theory and in terms of the Hausdorff measures., Comment: 33 pages

S. Smirnov proved recently that the Hausdorff dimension of any K-quasicircle is at most 1+k^2, where k=(K-1)/(K+1). In this paper we show that if $\Gamma$ is such a quasicircle, then $H^{1+k^2}(B(x,r)\cap \Gamma)\leq C(k) r^{1+k^2}$ for all x in \C and r>0, where H^s stands for the s-Haudorff measure. On a related note we derive a sharp weak-integrability of the derivative of the Riemann map of a quasidisk., Comment: 15 pages

The classical Painlevé theorem tells us that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general $K$ -quasiregular mappings in planar domains, the corresponding critical dimension is ${2}/({K+1})$ . We show that when $K>1$ , unexpectedly one has improved removability. More precisely, we prove that sets $E$ of $\sigma$ -finite Hausdorff $({2}/({K+1}))$ -measure are removable for bounded $K$ -quasiregular mappings. On the other hand, $\dim(E) = {2}/({K+1})$ is not enough to guarantee this property. ¶ We also study absolute continuity properties of pullbacks of Hausdorff measures under $K$ -quasiconformal mappings: in particular, at the relevant dimensions $1$ and ${2}/({K+1})$ . For general Hausdorff measures ${\cal H}^t$ , $0 \lt t \lt 2$ , we reduce the absolute continuity properties to an open question on conformal mappings (see Conjecture 2.3)

We define two notions of discrete dimension based on the Minkowski and Hausdorff dimensions in the continuous setting. After proving some basic results illustrating these definitions, we apply this machinery to the study of connections between the Erdos and Falconer distance problems in geometric combinatorics and geometric measure theory, respectively., Comment: 27 pages

In his celebrated paper on area distortion for quasiconformal mappings, Astala showed optimal area distortion bounds and dimension distortion estimates for planar quasiconformal mappings. He asked (Question 4.4) whether a finer result held, namely absolute continuity of Hausdorff measures under push-forward by quasiconformal mappings. This was proved in one particular case relevant for removability questions, in joint work of Astala, Clop, Mateu, Orobitg and the author ("Distortion of Hausdorff measures and improved Painlev'e removability for bounded quasiregular mappings", Duke Math J., to appear [ACMOU]) (Theorem 1.1), the other cases remaining open. A related question that we left open in [ACMOU] (Question 4.2) (which was asked by Astala to the author before [ACMOU] in an equivalent form in a personal communication) is whether BMO removability for K-quasiregular mappings and ($L^{\infty}$) removability for K-quasiregular mappings are indeed different problems. In this paper we give a series of examples answering in the positive Question 4.2 in [ACMOU], at the same time proving sharpness in two different senses of Theorem 1.1 in [ACMOU], and also giving examples that would yield sharpness in those two different senses as well for the absolute continuity of Hausdorff measures under push-forward by quasiconformal mappings, were it to be proved., Comment: 38 pages, 2 figures; added references

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