Clifford algebras and the higher dimensional Cauchy integral
The solution of the Kato problem in two dimensions.
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),...
Comparison of joint spectra for certain classes of commuting operators
Second order elliptic operators with complex bounded measurable coefficients in , Sobolev and Hardy spaces
Let be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with , 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 , Sobolev, and some new Hardy spaces naturally associated to . First, we show that the...
Singular integral operators with non-smooth kernels on irregular domains.
Let χ be a space of homogeneous type. The aims of this paper are as follows. i) Assuming that T is a bounded linear operator on L2(χ), we give a sufficient condition on the kernel of T such that T is of weak type (1,1), hence bounded on Lp(χ) for 1 < p ≤ 2; our condition is weaker then the usual Hörmander integral condition. ii) Assuming that T is a bounded linear operator on L2(Ω) where Ω is a measurable...
Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces.
In the Fourier theory of functions of one variable, it is common to extend a function and its Fourier transform holomorphically to domains in the complex plane C, and to use the power of complex function theory. This depends on first extending the exponential function eixξ of the real variables x and ξ to a function eizζ which depends holomorphically on both the complex variables z and ζ . Our thesis is this. The natural analog in higher dimensions...
The monogenic functional calculus
A study is made of a symmetric functional calculus for a system of bounded linear operators acting on a Banach space. Finite real linear combinations of the operators have real spectra, but the operators do not necessarily commute with each other. Analytic functions of the operators are formed by using functions taking their values in a Clifford algebra.
Square roots of perturbed subelliptic operators on Lie groups
We solve the Kato square root problem for bounded measurable perturbations of subelliptic operators on connected Lie groups. The subelliptic operators are divergence form operators with complex bounded coefficients, which may have lower order terms. In this general setting we deduce inhomogeneous estimates. In case the group is nilpotent and the subelliptic operator is pure second order, we prove stronger homogeneous estimates. Furthermore, we prove Lipschitz stability of the estimates under small...
Global higher integrability of jacobians on bounded domains
Clifford algebras and the double-layer potential operator
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