arXiv subjects

Tuomas Hytönen

Publications and source records attributed to Tuomas Hytönen.

61 records · Page 4Linked to original sources

arXiv2010

Stability in p of the H-infinity calculus of first-order systems in L^p

We study certain differential operators of the form AD arising from a first-order approach to the Kato square root problem. We show that if such operators are R-bisectorial in L^p, they remain R-bisectorial in L^q for all q close to p. In combination with our earlier results with Portal, which required such R-bisectoriality in different L^q spaces to start with, this shows that the R-bisectoriality in just one L^p actually implies bounded H-infinity calculus in L^q for all q close to p. We adapt the approach to related second-order results developed by Auscher, Hofmann and Martell, and also employ abstract extrapolation theorems due to Kalton and Mitrea.

math.FA↗

arXiv2009

The vector-valued non-homogeneous Tb theorem

The paper gives a Banach space -valued extension of the Tb theorem of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure, which only satisfies an upper control on the size of balls. Under the same assumptions as in their result, such operators are shown to be bounded on the Bochner spaces of functions with values in a Banach space with the unconditionality property of martingale differences (UMD). The new proof deals directly with all Lebesgue exponents p in the range 1<p<infinity, and relies on delicate estimates for the non-homogenous "Haar" functions, as well as McConnell's (1989) decoupling inequality for tangent martingale differences.

math.FA↗

arXiv2009

Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces

We prove a Tb theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound μ(B(x,r)) \le Cr^d. Our spaces are only assumed to satisfy the geometric doubling property: every ball of radius r can be covered by at most N balls of radius r/2. A key ingredient is the construction of random systems of dyadic cubes in such spaces.

math.FA↗

arXiv2009

Wavelet expansions for weighted, vector-valued BMO functions

We introduce a scale of weighted Carleson norms, which depend on an integrability parameter p, where p=2 corresponds to the classical Carleson measure condition. Relations between the weighed BMO norm of a vector-valued function f:R->X, and the Carleson norm of the sequence of its wavelet coefficients, are established. These extend the results of Harboure-Salinas-Viviani, also in the scalar-valued case when p is not 2.

math.FA↗

arXiv2008

The Banach space -valued BMO, Carleson's condition, and paraproducts

We define a scale of L^q Carleson norms, all of which characterize the membership of a function in BMO. The phenomenon is analogous to the John-Nirenberg inequality, but on the level of Carleson measures. The classical Carleson condition corresponds to the L^2 case in our theory. The result is applied to give a new proof for the L^p-boundedness of paraproducts with a BMO symbol. A novel feature of the argument is that all p are covered at once in a completely interpolation-free manner. This is achieved by using the L^1 Carleson norm, and indicates the usefulness of this notion. Our approach is chosen so that all these results extend in a natural way to the case of X-valued functions, where X is a Banach space with the UMD property.

math.FA↗

arXiv2008

On the norm of the Beurling-Ahlfors operator in several dimensions

The Lp operator norm of the generalized Beurling-Ahlfors transformation in n variables is at most (n/2+1)(p-1) for p>2. This improves on earlier results in all dimensions n>2. The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms.

math.CA↗

arXiv2007

Positive sesquilinear form measures and generalized eigenvalue expansions

Positive operator measures (with values in the space of bounded operators on a Hilbert space) and their generalizations, mainly positive sesquilinear form measures, are considered with the aim of providing a framework for their generalized eigenvalue type expansions. Though there are formal similarities with earlier approaches to special cases of the problem, the paper differs e.g. from standard rigged Hilbert space constructions and avoids the introduction of nuclear spaces. The techniques are predominantly measure theoretic and hence the Hilbert spaces involved are separable. The results range from a Naimark type dilation result to direct integral representations and a fairly concrete generalized eigenvalue expansion for unbounded normal operators.

math.FA↗