Topics in Harmonic Analysis; commutators and directional singular integrals

This dissertation focuses on two main topics: commutators and maximal directional operators. Our first topic will also distinguish between two cases: commutators of singular integral operators and BMO functions and commutators of fractional integral operators and a BMO class that comes from changing the underlying measure. Commutators are not only interesting for its own sake, but they have been broadly studied because of their connection to PDEs. Our first result gives us a new way of characterizing the class BMO. Assuming that the commutator of the Hilbert transform in dimension 1 (or a Riesz transform in dimensions 2 and higher) and the symbol b satisfy an Llog L-type of modular inequality on the endpoint with constant B, we can bound the BMO norm of the symbol by a fixed multiple of B; thus providing an endpoint version of the classical result of Coifman, Rochberg and Weiss for commutators of Calderón-Zygmund operators and BMO. We also studied commutators of fractional integrals and BMO. In this case, we were interested in finding quantitave two-weights estimates for the iterated version of these operators. We extended the known sharp inequalities for the commutator of first order to the iterated case and also provided a new proof of the previous results. Lastly, we studied maximal directional operators. Specifically, we considered a singular integral operator that commutes with translations and studied the maximal directional operator that arises from it. We proved that for any subset of cardinality N of a lacunary set of directions we can bound the Lp(Rn)-norm of the operator by the sharp bound √log N, thus completing some previous results on the Hilbert transform on low dimensions

Topics in Harmonic Analysis; commutators and directional singular integrals.

74 p.La tesis presentada viene dividida en dos partes: la primer parte está dedicada al estudio de losconmutadores y la segunda a los operadores maximales direccionales. Dentro de la teoría de losconmutadores nos enfocamos en dos tipos de operadores diferentes: los que surgen de conmutarintegrales singulares con funciones de BMO y los que surgen de conmutar integrales fraccionarias conBMO. En el primer caso estudiamos la necesidad de BMO en el extremo en el contexto de la medida deLebesgue. En el segundo caso nos enfocamos en conseguir cotas en el contexto de dos pesos, tambiénincluyendo un resultado de necesidad de un espacio de BMO modificado para estas desigualdades.Finalmente la tesis incluye un resultado sobre las integrales singulares en direcciones, estos sonoperadores que surgen de considerar las integrales singulares actuando en rectas en un espacio euclídeode mayor dimensión. Nuestro resultado se enfoca en probar una cota óptima en términos de la cantidad dedirecciones para el caso en que el conjunto de direcciones tenga cierta estructura geométrica, que será"finite order lacunary"

On Bloom type estimates for iterated commutators of fractional integrals

In this paper we provide quantitative Bloom type estimates for iterated commutators of fractional integrals improving and extending results from [15]. We give new proofs for those inequalities relying upon a new sparse domination that we provide as well in this paper and also in techniques de- veloped in the recent paper [22]. We extend as well the necessity established in [15] to iterated commutators providing a new proof. As a consequence of the preceding results we recover the one weight estimates in [7, 1] and es- tablish the sharpness in the iterated case. Our result provides as well a new characterization of the BMO space

Directional square functions

Quantitative formulations of Fefferman's counterexample for the ball multiplier are naturally linked to square function estimates for conical and directional multipliers. In this article we develop a novel framework for these square function estimates, based on a directional embedding theorem for Carleson sequences and multi-parameter time-frequency analysis techniques. As applications we prove sharp or quantified bounds for Rubio de Francia type square functions of conical multipliers and of multipliers adapted to rectangles pointing along $N$ directions. A suitable combination of these estimates yields a new and currently best-known logarithmic bound for the Fourier restriction to an $N$-gon, improving on previous results of A. Cordoba. Our directional Carleson embedding extends to the weighted setting, yielding previously unknown weighted estimates for directional maximal functions and singular integrals.Comment: 49 pages, 4 figures. Submitted for publication. This article supersedes arXiv:1902.03644 by two of the authors. The results of arXiv:1902.03644 are extended in scope and improve

The Choices, Challenges, and Lessons Learned from a Multi-Method Social-Emotional / Character Assessment in and Out of School Time Setting

Out-of-School-Time (OST) programs are increasingly recognized as a venue to actively engage children and youth in character development activities, but little guidance exists as to how to assess individual children and youth in OST environments for the sake of evaluating their character development. This research brief uses an illustrative case study to reflect upon the experience of selecting and completing a strength-based, multi-modal social-emotional / character assessment that used a direct assessment and a multiple informant behavior rating scale in an OST setting. Insights derived from the case study reveal opportunities and challenges associated with each assessment modality. This paper shares lessons learned with those conducting individual assessments in OST environments and with those seeking to improve our capacity to complete screening, formative, and summative assessments of social-emotional and character constructs in OST youth development programs to help children

Solar-insolation-induced changes in the coma morphology of comet 67P/Churyumov-Gerasimenko. Optical monitoring with the Nordic Optical Telescope

Context. 67P/Churyumov-Gerasimenko (67P/C-G) is a short-period Jupiter family comet with an orbital period of 6.55 years. Being the target comet of ESA’s Rosetta mission, 67P/C-G has become one of the most intensively studied minor bodies of the Solar System. The Rosetta Orbiter and the Philae Lander have brought us unique information about the structure and activity of the comet nucleus, as well as its activity along the orbit, composition of gas, and dust particles emitted into the coma. However, as Rosetta stayed in very close proximity to the cometary nucleus (less than 500 km with a few short excursions reaching up to 1500 km), it could not see the global picture of a coma at the scales reachable by telescopic observations (103 - 105 km). Aims. In this work we aim to connect in-situ observations made by Rosetta with the morphological evolution of the coma structures monitored by the ground-based observations. In particular, we concentrate on causal relationships between the coma morphology and evolution observed with the Nordic Optical Telescope (NOT) in the Canary Islands, and the seasonal changes of the insolation and the activity of the comet observed by the Rosetta instruments. Methods. Comet 67P/C-G was monitored with the NOT in imaging mode in two colors. Imaging optical observations were performed roughly on a weekly basis, which provides good coverage of short- and long-term variability. With the three dimensional modeling of the coma produced by active regions on the Southern Hemisphere, we aim to qualify the observed morphology by connecting it to the activity observed by Rosetta. Results. During our monitoring program, we detected major changes in the coma morphology of comet 67P/C-G. These were longterm and long-lasting changes. They do not represent any sudden outburst or short transient event, but are connected to seasonal changes of the surface insolation and the emergence of new active regions on the irregular shaped comet nucleus. We have also found significant deviations in morphological changes from the prediction models based on previous apparitions of 67P/C-G, like the time delay of the morphology changes and the reduced activity in the Northern Hemisphere. According to our modeling of coma structures and geometry of observations, the changes are clearly connected with the activity in the Southern Hemisphere observed by the Rosetta spacecraft