Gradient of the single layer potential and quantitative rectifiability for general Radon measures

Altres ajuts: Acord transformatiu CRUE-CSICAltres ajuts: Gobierno Vasco IT-1247-19We identify a set of sufficient local conditions under which a significant portion of a Radon measure μ on with compact support can be covered by an uniformly n-rectifiable set, at the level of a ball such that . This result involves a flatness condition, formulated in terms of the so-called -number of B, and the -boundedness, as well as a control on the mean oscillation on the ball, of the operator. Here is the fundamental solution for a uniformly elliptic operator in divergence form associated with an matrix with Hölder continuous coefficients. This generalizes a work by Girela-Sarrión and Tolsa for the n-Riesz transform. The motivation for our result stems from a two-phase problem for the elliptic harmonic measure

On the density problem in the parabolic space

In this work we extend many classical results concerning the relationship between densities, tangents and rectifiability to the parabolic spaces, namely $\mathbb{R}^{n+1}$ equipped with parabolic dilations. In particular we prove a Marstrand-Mattila rectifiability criterion for measures of general dimension, we provide a characterisation through densities of intrinsic rectifiable measures, and we study the structure of $1$-codimensional uniform measures. Finally, we apply some of our results to the study of a quantitative version of parabolic rectifiability: we prove that the weak constant density condition for a $1$-codimensional Ahlfors-regular measure implies the bilateral weak geometric lemma

Campath, calcineurin inhibitor reduction and chronic allograft nephropathy (3C) study: background, rationale, and study protocol.

BACKGROUND: Kidney transplantation is the best treatment for patients with end-stage renal failure, but uncertainty remains about the best immunosuppression strategy. Long-term graft survival has not improved substantially, and one possible explanation is calcineurin inhibitor (CNI) nephrotoxicity. CNI exposure could be minimized by using more potent induction therapy or alternative maintenance therapy to remove CNIs completely. However, the safety and efficacy of such strategies are unknown. METHODS/DESIGN: The Campath, Calcineurin inhibitor reduction and Chronic allograft nephropathy (3C) Study is a multicentre, open-label, randomized controlled trial with 852 participants which is addressing two important questions in kidney transplantation. The first question is whether a Campath (alemtuzumab)-based induction therapy strategy is superior to basiliximab-based therapy, and the second is whether, from 6 months after transplantation, a sirolimus-based maintenance therapy strategy is superior to tacrolimus-based therapy. Recruitment is complete, and follow-up will continue for around 5 years post-transplant. The primary endpoint for the induction therapy comparison is biopsy-proven acute rejection by 6 months, and the primary endpoint for the maintenance therapy comparison is change in estimated glomerular filtration rate from baseline to 2 years after transplantation. The study is sponsored by the University of Oxford and endorsed by the British Transplantation Society, and 18 centers for adult kidney transplant are participating. DISCUSSION: Late graft failure is a major issue for kidney-transplant recipients. If our hypothesis that minimizing CNI exposure with Campath-based induction therapy and/or an elective conversion to sirolimus-based maintenance therapy can improve long-term graft function and survival is correct, then patients should experience better graft function for longer. A positive outcome could change clinical practice in kidney transplantation. TRIAL REGISTRATION: ClinicalTrials.gov, NCT01120028 and ISRCTN88894088.RIGHTS : This article is licensed under the BioMed Central licence at http://www.biomedcentral.com/about/license which is similar to the 'Creative Commons Attribution Licence'. In brief you may : copy, distribute, and display the work; make derivative works; or make commercial use of the work - under the following conditions: the original author must be given credit; for any reuse or distribution, it must be made clear to others what the license terms of this work are

Singular integrals, rectifiability and elliptic measure

La tesi investiga la relació entre les propietats geomètriques de les mesures en espais euclidians i el comportament de certs operadors integrals singulars corresponents. També proposem alguna aplicació a l’estudi d’EDPs el·líptiques. En primer lloc, es caracteritza la regularitat de les corbes planes de Jordan corda-arc que tenen transformades maximals de Cauchy associades que es poden afitar puntualment per la iteració de segon ordre de la funció maximal de Hardy-Littlewood a la corba, assumint una condició natural de fons de conformitat asimptòtica. En particular, resulta que aquestes corbes no necessàriament tenen tangents a cada punt, però es poden diferenciar gairebé pertot amb derivades en VMO. En el segon capítol estudiem les propietats d’una mesura que determinen si la seva transformada de Cauchy defineix un operador compacte a L ^ 2; determinem que la compacitat es pot caracteritzar per una convergència uniforme a zero de la densitat superior de la mesura. A continuació, investigem un equivalent en el context d’equacions el·líptiques de dos resultats importants i recents importants sobre la transformada de Riesz i la rectificabilitat uniforme. Sota un hipòtesi de continuïtat de Hölder per a la matriu que defineix l’operador uniforme el·líptic en forma de divergència demostrem, en col·laboració amb Laura Prat i Xavier Tolsa, que el gradient del potencial d’una sola capa associat amb una mesura n-Alhfors-David-regular amb suport compacte en R ^ (n + 1) és afitat a L ^ 2 si i només si la mesura és uniformement n-rectificable. Aquest resultat amplia l’important article de F. Nazarov, X. Tolsa i A. Volberg sobre la solució de l’anomenada problemàtica de David i Semmes en co-dimensió 1. Després, l’apliquem a un problema d’una-fase per a la mesura el·líptica. Sota la mateixa hipòtesi per a l’equació el·líptica establim un criteri local de rectificabilitat per a mesures de Radon que no són necessàriament regulars. El teorema es formula en termes d’un control de l’oscil·lació mitjana del gradient del potencial d’una sola capa. Això generalitza un resultat recent de D. Girela-Sarriòn i X. Tolsa. Aquest estudi constitueix un pas important per aconseguir un resultat de rectificabilitat per a un problema de dues-fases per a la mesura el·líptica.La tesis investiga la relación entre las propiedades geométricas de las medidas en espacios euclídeos y el comportamiento de ciertos operadores integrales singulares asociados. También mostramos alguna aplicación al estudio de EDPs elípticas. Primero, caracterizamos la regularidad de las curvas planas de Jordan cuerda-arco cuya transformada maximal de Cauchy puede ser dominada puntualmente por la iteración de segundo orden de la función maximal de Hardy-Littlewood en la curva, suponiendo una condición natural de conformidad asintótica de fondo. En particular, resulta que estas curvas no necesariamente tienen tangentes en cada punto, pero son diferenciables en casi todas partes con derivadas en VMO. Las condiciones que determinan si la transformada de Cauchy asociada a una medida define un operador compacto en L ^ 2 se estudian en el segundo capítulo; determinamos que la compacidad puede caracterizarse por una convergencia uniforme a cero de la densidad superior de la medida. Luego, investigamos un equivalente en el contexto de ecuaciones elípticas de dos importantes resultados recientes sobre la transformada de Riesz y la rectificabilidad uniforme. Bajo una suposición de continuidad de Hölder para la matriz que define el operador uniformemente elíptico en forma de divergencia demostramos, en colaboración con Laura Prat y Xavier Tolsa, que el gradiente del potencial de capa única asociado a una medida n-Alhfors-David regular con soporte compacto en R ^ (n + 1) está acotado en L ^ 2 si y sólo si la medida es uniformemente n-rectificable. Este resultado amplía el importante artículo de F. Nazarov, X. Tolsa y A. Volberg sobre la solución del problema de David y Semmes en co-dimensión 1, y lo aplicamos a un problema de una fase para la medida elíptica. Bajo la misma hipótesis para la ecuación elíptica, establecemos un criterio local de rectificabilidad para las medidas de Radon que no son necesariamente regulares. El teorema se formula en términos de un control de la oscilación media del gradiente del potencial de una sola capa. Esto generaliza un resultado reciente de D. Girela-Sarriòn y X. Tolsa. Este estudio constituye un paso importante para lograr un resultado de rectificabilidad para un problema de dos-fases para la medida elíptica.The thesis investigates the relation between the geometric properties of measures in Euclidean spaces and the behavior of certain associated singular integral operators. We also show some application to the study of elliptic PDEs. First, we characterize the regularity of the planar chord-arc Jordan curves whose associated maximal Cauchy transform can be pointwise dominated by the second-order iteration of the Hardy-Littlewood maximal function on the curve, assuming a natural background asymptotic conformality condition. In particular, it turns out that this curves do not necessarily have tangents at each point but they are differentiable almost everywhere with derivatives in VMO. The conditions on a measure that determine whether its associated Cauchy transform defines a compact operator on L^2 are studied in the second chapter; we determine that the compactness can be characterized by a uniform convergence to zero of the upper density of the measure. Then, we investigate an equivalent in the context of elliptic equations of two important recent results on Riesz transform and uniform rectifiabilty. Under a Hölder continuity assumption for the matrix defining the uniformly elliptic operator in divergence form we prove, in collaboration with Laura Prat and Xavier Tolsa, that the gradient of the single layer potential associated with a compactly supported n-Alhfors-David regular mesaure in R^(n+1) is bounded on L^2 if and only if the measure is uniformly n-rectifiable. This result extends the important article by F. Nazarov, X. Tolsa and A. Volberg on the solution of the so-called co-dimension 1 David and Semmes’ problem and we apply it to a one-phase problem for the elliptic measure. Under the same hypothesis for the elliptic equation we establish a local rectifiability criterion for Radon measures which are not necessarily regular. The theorem is formulated in terms of a control of the mean oscillation of the gradient of the single layer potential. This generalizes a recent result by D. Girela-Sarriòn and X. Tolsa. This study constitutes an important step to achieve a rectifiability result for a two-phase problem for the elliptic measure