Book contents
- Frontmatter
- Contents
- Foreword
- List of Main Lectures
- List of Participants
- A note on H. Ishihara and W. Takahashi modulus of convexity
- A property of non-strongly regular operators
- The entropy of convex bodies with ‘few’ extreme points
- Spaces of vector valued analytic functions and applications
- Notes on approximation properties in separable Banach spaces
- Moduli of complex convexity
- Grothendieck type inequalities and weak Hilbert spaces
- A weak topology characterization of l1 (m)
- Singular integral operators: a martingale approach
- Remarks about the interpolation of Radon-Nikodym operators
- Symmetric sequences in finite-dimensional normed spaces
- Some topologies on the space of analytic self-maps of the unit disk
- Minimal and strongly minimal Orlicz sequence spaces
- Type and cotype in Musielak-Orlicz spaces
- On the complex Grothendieck constant in the n-dimensional case
- Pathological properties and dichotomies for random quotients of finite-dimensional Banach spaces
- A note on a low M*-estimate
- The p1/p in Pisier's factorization theorem
- Almost differentiablity of convex functions in Banach spaces and determination of measures by their values on balls
- When E and E[E] are isomorphic
- A note on Gaussian measure of translates of balls
- Sublattices of M(X) isometric to M[0,1]
Singular integral operators: a martingale approach
Published online by Cambridge University Press: 03 May 2010
- Frontmatter
- Contents
- Foreword
- List of Main Lectures
- List of Participants
- A note on H. Ishihara and W. Takahashi modulus of convexity
- A property of non-strongly regular operators
- The entropy of convex bodies with ‘few’ extreme points
- Spaces of vector valued analytic functions and applications
- Notes on approximation properties in separable Banach spaces
- Moduli of complex convexity
- Grothendieck type inequalities and weak Hilbert spaces
- A weak topology characterization of l1 (m)
- Singular integral operators: a martingale approach
- Remarks about the interpolation of Radon-Nikodym operators
- Symmetric sequences in finite-dimensional normed spaces
- Some topologies on the space of analytic self-maps of the unit disk
- Minimal and strongly minimal Orlicz sequence spaces
- Type and cotype in Musielak-Orlicz spaces
- On the complex Grothendieck constant in the n-dimensional case
- Pathological properties and dichotomies for random quotients of finite-dimensional Banach spaces
- A note on a low M*-estimate
- The p1/p in Pisier's factorization theorem
- Almost differentiablity of convex functions in Banach spaces and determination of measures by their values on balls
- When E and E[E] are isomorphic
- A note on Gaussian measure of translates of balls
- Sublattices of M(X) isometric to M[0,1]
Summary
Introduction. Our purpose is to sketch a new approach to proving the boundedness of a vast class of linear operators which includes, e.g., the generalized Calderón-Zygmund operators discussed in [M]. The aproach is based on estimates of operator norms which come from applying recent results concerning the Lp-boundedness of martingale transforms.
In fact, the incentive for this work was the desire to extend some previously known boundedness results for operators acting in Lp-spaces of scalar-valued functions to the case of analogous spaces of X-valued Bochner measurable functions, where X is a Banach space. The recent results, due mainly to D. Burkholder and J. Bourgain, indicated that the class of the so-called UMD-spaces may be exactly the domain in which all results concerning Calderón–Zygmund integral operators and their generalizations remain valid. (Many Banach spaces which are important in classical analysis belong to that class.) Even the simplest singular integral operator, i.e., the Hilbert transform on the real line R, has the property that its natural extension to an operator acting on Lp (RX) where 1 < p < ∞ is a bounded linear map if and only if the Banach space X is a UMD-spa.ce (cf. [Bu2] and [Bo]).
In order to obtain this extension it was necessary to find such proofs which make no use of any result that does not hold in the UMD-setting (for instance, the Fourier transform should be avoided, because it is not bounded in Lp(R, X), unless X is isomorphic to a Hilbert space).
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- Geometry of Banach SpacesProceedings of the Conference Held in Strobl, Austria 1989, pp. 95 - 110Publisher: Cambridge University PressPrint publication year: 1991
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