My research interests encompass two broad areas of analysis. The first area is in the intersection of geometric function theory, nonlinear potential theory and analysis on metric spaces. The second area consists of operator theoretic aspects of function theory, especially corona and interpolation problems.
In particular, in the first area, I have been studying sufficient conditions under which one would get strong A-infinity weights in the Euclidean setting and in metric measure spaces. I am also interested in Sobolev-Lorentz capacity and Besov capacity. I explore how Hausdorff measures and each of these capacities are related. More recently, in the second area, I have been studying corona and interpolation problems for Hilbert function spaces including the Drury-Arveson Hardy space, a space that occupies a central position in multivariate operator theory.
I start by giving more details about the area of operator function theory.
In recent work with Eric Sawyer and Brett Wick, we have obtained the corona theorem for the multiplier algebra of the Drury-Arveson Hardy space with infinitely many generators. This is the first generalization of Carleson's Corona theorem to a multiplier algebra of holomorphic functions in higher dimensions. Thus we settle the Corona problem for the Drury-Arveson space, which from the point of view of operator theory is the correct generalization of the Hardy space to several variables. This was an open question in this area that had been open for some time. Then we generalized both a theorem of Varopoulos and Andersson-Carlsson by obtaining BMO analytic solutions to the corona problem with infinitely many generators.
As far as the other domain of main interest is concerned, I studied strong A-infinity weights and capacities in different settings.
One of my main results shows that if the distributional gradient of a function, u, has small enough norm in a certain Besov or Morrey space, then the exponential of that function u is a strong A-infinity weight with data depending only on the dimensions involved in the definition of these spaces. This generalizes results proved by my adviser and his coauthors who proved similar results for spaces strictly contained in the above Besov space.
In order to prove my results, I developed a complete theory of Besov capacities. My theory includes a workable definition and basic properties of such capacities, including monotonicity, finite and countable subadditivity, convergence and truncation results and sharp estimates for the relative capacity of certain conductors. I also studied the Hausdorff dimension of null sets and quasicontinuity of functions with respect to Besov capacities.
I also generalized the above results about strong A-infinity weight to certain metric measure spaces. Those metric spaces are assumed to be Ahlfors Q-regular, geodesic, unbounded and to satisfy a (1,s) Poincare inequality for some s in (1,Q].
The notions discussed above about Besov and Sobolev capacity make sense in metric setting as well, so I generalized the above results to the metric setting.
I also studied capacities with respect to the Sobolev-Lorentz n,q norm, carrying out the above program.
In joint work with V. Maz'ya we proved a strong capacitary type inequality for the Sobolev-Lorentz p,q capacity, by relying on the superadditivity of this capacity.
In the metric setting, in joint work with M. Miranda Jr. we studied the Newtonian Lorentz metric spaces. We also studied the p,q-modulus of families of rectifiable curves and the global p,q-capacities associated with these spaces. Under some additional assumptions (namely, the space carries a doubling measure and satisfies a certain weak Poincare inequality) we showed that when q is in [1,p] the Lipschitz functions are dense in these spaces; moreover, in the same setting we showed that the p,q-capacity is Choquet provided that q is strictly greater than 1. We also provided a counterexample for the density result in the Euclidean setting when p is in (1,n] and q is infinite.
Strong A-infinity weights and scaling invariant Besov spaces,
appeared in Rev. Mat. Iberoamericana, 23 (2007), no. 3, 1067--1114.
Scaling invariant Sobolev-Lorentz capacity on Rn,
appeared in Indiana Univ. Math. J., 56 (2007), no. 6, 2641--2669.
Conductor inequalities and criteria for Sobolev-Lorentz two-weight inequalities
with Vladimir Maz'ya,
appeared in the Springer collection Sobolev Spaces in Mathematics II (2009) of the International Mathematical Series.
Besov capacity and Hausdorff measures in metric measure spaces,
appeared in Publ. Mat., 53 (2009), no. 1, 141--178.
Sobolev capacity and Hausdorff measures in metric measure spaces,
appeared in Ann. Acad. Sci. Fenn. Math., 34 (2009), no. 1, 179--194.
Strong A-infinity weights and Sobolev capacities in metric measure spaces,
appeared in Houston J. of Math., 35 (2009), no. 4, 1233--1249.
BMO Estimates for the H∞(B
appeared in J. Funct. Anal., 258 (2010), no. 11, 3818--3840.
The Corona Theorem for the Drury-Arveson Hardy space and other holomorphic
on the unit ball in Cn with Eric T. Sawyer and Brett D. Wick, appeared in Analysis and PDE, 4 (2011), no. 4, 499--550.
Newtonian Lorentz metric spaces with Michele Miranda Jr.,
appeared in Illinois J. of Math., 56 (2012), no. 2, 579--616.
Sobolev-Lorentz spaces in the Euclidean setting and counterexamples,
appeared in Nonlinear Analysis: Theory, Methods and Applications, 152 (2017), 149--182; preprint on http://arxiv.org/pdf/1605.08551.pdf and on http://cvgmt.sns.it/paper/3079/.
Sobolev-Lorentz capacity and its regularity in the Euclidean setting,
appeared in Ann. Acad. Sci. Fenn. Math., 44 (2019), no. 1, 537--568;
preprint on http://arxiv.org/pdf/1707.08873.pdf and on http://cvgmt.sns.it/paper/3532/.
12. Minimizers for the Sobolev-Lorentz p,q-capacity in the Euclidean setting, in preparation.
13. Newtonian Lorentz metric spaces with zero boundary values, with Michele Miranda Jr., in preparation.
1. Scaling Besov and Sobolev-Lorentz spaces and capacities in the Euclidean setting, Publishing House of the Romanian Academy, Bucharest, 2012, xii+114 pages.Comments and questions to firstname.lastname@example.org.