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Variations on Bochner-Riesz multipliers in the plane

Daniele Debertol (2006)

Studia Mathematica

We consider the multiplier m μ defined for ξ ∈ ℝ by m μ ( ξ ) ≐ ( ( 1 - ξ ₁ ² - ξ ₂ ² ) / ( 1 - ξ ₁ ) ) μ 1 D ( ξ ) , D denoting the open unit disk in ℝ. Given p ∈ ]1,∞[, we show that the optimal range of μ’s for which m μ is a Fourier multiplier on L p is the same as for Bochner-Riesz means. The key ingredient is a lemma about some modifications of Bochner-Riesz means inside convex regions with smooth boundary and non-vanishing curvature, providing a more flexible version of a result by Iosevich et al. [Publ. Mat. 46 (2002)]. As an application, we show that the...

Wave equation and multiplier estimates on ax + b groups

Detlef Müller, Christoph Thiele (2007)

Studia Mathematica

Let L be the distinguished Laplacian on certain semidirect products of ℝ by ℝⁿ which are of ax + b type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form e i t √ L ψ ( √ L / λ ) for arbitrary time t and arbitrary λ > 0, where ψ is a smooth bump function supported in [-2,2] if λ ≤ 1 and in [1,2] if λ ≥ 1. As a corollary, we reprove a basic multiplier estimate of Hebisch and Steger [Math. Z. 245 (2003)] for this particular class of groups, and derive Sobolev...

Wavelet transform for functions with values in UMD spaces

Cornelia Kaiser, Lutz Weis (2008)

Studia Mathematica

We extend the classical theory of the continuous and discrete wavelet transform to functions with values in UMD spaces. As a by-product we obtain equivalent norms on Bochner spaces in terms of g-functions.

Weak-type (1,1) bounds for oscillatory singular integrals with rational phases

Magali Folch-Gabayet, James Wright (2012)

Studia Mathematica

We consider singular integral operators on ℝ given by convolution with a principal value distribution defined by integrating against oscillating kernels of the form e i R ( x ) / x where R(x) = P(x)/Q(x) is a general rational function with real coefficients. We establish weak-type (1,1) bounds for such operators which are uniform in the coefficients, depending only on the degrees of P and Q. It is not always the case that these operators map the Hardy space H¹(ℝ) to L¹(ℝ) and we will characterise those rational...

Weighted inequalities for commutators of one-sided singular integrals

María Lorente, María Silvina Riveros (2002)

Commentationes Mathematicae Universitatis Carolinae

We prove weighted inequalities for commutators of one-sided singular integrals (given by a Calder’on-Zygmund kernel with support in ( - ∞ , 0 ) ) with BMO functions. We give the one-sided version of the results in C. Pérez, Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function, J. Fourier Anal. Appl., vol. 3 (6), 1997, pages 743–756 and C. Pérez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., vol 128 (1), 1995, pages...

Weighted inequalities for square and maximal functions in the plane

Javier Duoandikoetxea, Adela Moyua (1992)

Studia Mathematica

We prove weighted inequalities for square functions of Littlewood-Paley type defined from a decomposition of the plane into sectors of lacunary aperture and for the maximal function over a lacunary set of directions. Some applications to multiplier theorems are also given.

Wolff's inequality for hypersurfaces.

Izabella Laba, Malabika Pramanik (2006)

Collectanea Mathematica

We extend Wolff's "local smoothing" inequality to a wider class of not necessarily conical hypersurfaces of codimension 1. This class includes surfaces with nonvanishing curvature, as well as certain surfaces with more than one flat direction. An immediate consequence is the Lp-boundedness of the corresponding Fourier multiplier operators.

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