LpL^p regularity for a class of averaging operators on the Heisenberg group

We prove $L^p_{comp}\to L^p_{s}$ boundedness for averaging operators associated to a class of curves in the Heisenberg group $\mathbb{H}^1$ via $L^2$ estimates for related oscillatory integrals and Bourgain-Demeter decoupling inequalities on the cone. We also construct a Sobolev space adapted to translations on the Heisenberg group to which these averaging operators map all $L^p$ functions boundedly.Comment: 28 Page

LpL^p Regularity Estimates for a Class of Integral Operators with Fold Blowdown Singularities

We prove sharp $L^p$ regularity results for a class of generalized Radon transforms for families of curves in a three-dimensional manifold associated to a canonical relation with fold and blowdown singularities. The proof relies on decoupling inequalities by Wolff and Bourgain-Demeter for plate decompositions of thin neighborhoods of cones and $L^2$ estimates for related oscillatory integrals.Comment: 31 pages. Reorganized the proof of Proposition 4. Revised version incorporating referee's suggestions. To appear in The Journal of Geometric Analysi

Accounting For Biotic Variability In Streams With Low Levels of Impervious Cover: The Role of Reach- and Watershed-Scale Factors

As landscapes become increasingly urbanized, there is an associated increase in impervious cover. Impervious surfaces, such as roads, rooftops, and parking lots contribute to the physical, hydrological, chemical, and biological alteration of stream systems. Biotic assemblages consistently degrade with increased watershed impervious cover; however, at low levels of impervious cover, these assemblages exhibit wide variability in biotic integrity. This study investigated which reach- and watershed-scale factors explained biotic condition (i.e., richness, flow traits, thermal traits, and tolerance for macroinvertebrates and fishes) at similar levels of low imperviousness. The primary objective was to identify factors that confer resistance for biota, such that they retain high biotic integrity at low levels of impervious cover, and, conversely, to determine which factors make biota more vulnerable to urban disturbance, such that they have low biotic integrity despite low levels of impervious cover. Forty sites were selected across Massachusetts within two narrow bands of impervious cover: 1–4% (n = 20) and 7–10% (n = 20). Models with reach-scale variables (reflecting habitat heterogeneity, flow, temperature, or water quality) or watershed-scale variables (representing natural characteristics, land use, flow alterations, and other measures of urbanization or impervious) explained additional variance compared to models with impervious cover alone. Reach-scale factors tended to explain more variance than watershed-scale factors for all biotic responses except fluvial fishes, with overall more variance explained for fish than macroinvertebrate assemblages. At the reach scale, colder water temperatures, higher dissolved oxygen, and more large wood were related to higher proportions of fluvial, coldwater, and intolerant fishes. For macroinvertebrates, warmer water temperature, smaller sediment size, and higher nitrate were related to higher macroinvertebrate richness and tolerance. At the watershed scale, air temperature emerged as an important predictor for both taxonomic groups and across response metrics; air temperature was highly correlated with high-elevation watersheds. Other important watershed-scale predictors were open water and dams, flow alteration, and other urban measures such as housing density, impervious in a 120-m buffer, and road crossings. Restoration should focus on strategies to reduce impacts that would degrade in-stream conditions that allow for higher biotic integrity, such as habitat heterogeneity, more large wood, and colder water temperatures. Similarly, watersheds should be prioritized for protection with those characteristics potentially more resistant to urban disturbance, such as high-elevation regions that retained high biotic integrity despite higher dam density, more road crossings, and more flow alteration

Diffuse Reflection of Light

This thesis describes an attempt to determine the feasibility of measuring the ozone content in the air by absorption of light over a long path