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A weak molecule condition for certain Triebel-Lizorkin spaces

Steve Hofmann — 1992

Studia Mathematica

A weak molecule condition is given for the Triebel-Lizorkin spaces Ḟ_p^{α,q}, with 0 < α < 1 and 1 < p, q < ∞. As an easy corollary, one may deduce, by atomic-molecular methods, a Triebel-Lizorkin space "T1" Theorem of Han and Sawyer, and Han, Jawerth, Taibleson and Weiss, for Calderón-Zygmund kernels K(x,y) which are not assumed to satisfy any regularity condition in the y variable.

On certain nonstandard Calderón-Zygmund operators

Steve Hofmann — 1994

Studia Mathematica

We formulate a version of the T1 theorem which enables us to treat singular integrals whose kernels need not satisfy the usual smoothness conditions. We also prove a weighted version. As an application of the general theory, we consider a class of multilinear singular integrals in n related to the first Calderón commutator, but with a kernel which is far less regular.

On singular integrals of Calderón-type in R and BMO.

Steve Hofmann — 1994

Revista Matemática Iberoamericana

We prove Lp (and weighted Lp) bounds for singular integrals of the form p.v.  ∫Rn E (A(x) - A(y) / |x - y|) (Ω(x - y) / |x - y|n) f(y) dy, where E(t) = cos t if Ω is odd, and E(t) = sin t if Ω is even, and where ∇ A ∈ BMO. Even in the case that Ω is smooth, the theory of singular integrals with rough kernels plays a key role in the proof. By...

The solution of the Kato problem in two dimensions.

Steve HofmannAlan McIntosh — 2002

Publicacions Matemàtiques

We solve, in two dimensions, the "square root problem of Kato". That is, for L ≡ -div (A(x)∇), where A(x) is a 2 x 2 accretive matrix of bounded measurable complex coefficients, we prove that L1/2: L1 2(R2) → L2(R2). [Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid),...

Gaussian estimates for fundamental solutions to certain parabolic systems.

Steve HofmannSeick Kim — 2004

Publicacions Matemàtiques

Auscher proved Gaussian upper bound estimates for the fundamental solutions to parabolic equations with complex coefficients in the case when coefficients are time-independent and a small perturbation of real coefficients. We prove the equivalence between the local boundedness property of solutions to a parabolic system and a Gaussian upper bound for its fundamental matrix. As a consequence, we extend Auscher's result to the time dependent case.

The L p Neumann problem for the heat equation in non-cylindrical domains

Steve HofmannJohn L. Lewis — 1998

Journées équations aux dérivées partielles

I shall discuss joint work with John L. Lewis on the solvability of boundary value problems for the heat equation in non-cylindrical (i.e., time-varying) domains, whose boundaries are in some sense minimally smooth in both space and time. The emphasis will be on the Neumann problem with data in L p . A somewhat surprising feature of our results is that, in contrast to the cylindrical case, the optimal results hold when p = 2 , with the situation getting progressively worse as p approaches 1 . In particular,...

Second order elliptic operators with complex bounded measurable coefficients in  L p , Sobolev and Hardy spaces

Steve HofmannSvitlana MayborodaAlan McIntosh — 2011

Annales scientifiques de l'École Normale Supérieure

Let  L be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with L , such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in  L p , Sobolev, and some new Hardy spaces naturally associated to  L . First, we show that the...

Square functions of Calderón type and applications.

Steve HofmannJohn L. Lewis — 2001

Revista Matemática Iberoamericana

We establish L and L bounds for a class of square functions which arises in the study of singular integrals and boundary value problems in non-smooth domains. As an application, we present a simplified treatment of a class of parabolic smoothing operators which includes the caloric single layer potential on the boundary of certain minimally smooth, non-cylindrical domains.

Carleson measures, trees, extrapolation, and T(b) theorems.

Pascal AuscherSteve HofmannCamil MuscaluTerence TaoChristoph Thiele — 2002

Publicacions Matemàtiques

The theory of Carleson measures, stopping time arguments, and atomic decompositions has been well-established in harmonic analysis. More recent is the theory of phase space analysis from the point of view of wave packets on tiles, tree selection algorithms, and tree size estimates. The purpose of this paper is to demonstrate that the two theories are in fact closely related, by taking existing results and reproving them in a unified setting. In particular we give a dyadic version of extrapolation...

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