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L p ( ) boundedness for the commutator of a homogeneous singular integral operator

Guoen Hu (2003)

Studia Mathematica

The commutator of a singular integral operator with homogeneous kernel Ω(x)/|x|ⁿ is studied, where Ω is homogeneous of degree zero and has mean value zero on the unit sphere. It is proved that Ω L ( l o g L ) k + 1 ( S n - 1 ) is a sufficient condition for the kth order commutator to be bounded on L p ( ) for all 1 < p < ∞. The corresponding maximal operator is also considered.

L p weighted inequalities for the dyadic square function

Akihito Uchiyama (1995)

Studia Mathematica

We prove that ʃ ( S d f ) p V d x C p , n ʃ | f | p M d ( [ p / 2 ] + 2 ) V d x , where S d is the dyadic square function, M d ( k ) is the k-fold application of the dyadic Hardy-Littlewood maximal function and p > 2.

L2 boundedness of a singular integral operator.

Dashan Fan, Yibiao Pan (1997)

Publicacions Matemàtiques

In this paper we study a singular integral operator T with rough kernel. This operator has singularity along sets of the form {x = Q(|y|)y'}, where Q(t) is a polynomial satisfying Q(0) = 0. We prove that T is a bounded operator in the space L2(Rn), n ≥ 2, and this bound is independent of the coefficients of Q(t).We also obtain certain Hardy type inequalities related to this operator.

Littlewood-Paley characterization of Hölder-Zygmund spaces on stratified Lie groups

Guorong Hu (2019)

Czechoslovak Mathematical Journal

We give a characterization of the Hölder-Zygmund spaces 𝒞 σ ( G ) ( 0 < σ < ) on a stratified Lie group G in terms of Littlewood-Paley type decompositions, in analogy to the well-known characterization of the Euclidean case. Such decompositions are defined via the spectral measure of a sub-Laplacian on G , in place of the Fourier transform in the classical setting. Our approach mainly relies on almost orthogonality estimates and can be used to study other function spaces such as Besov and Triebel-Lizorkin spaces...

Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets

Peter Sjögren, Per Sjölin (1981)

Annales de l'institut Fourier

Let E R be a closed null set. We prove an equivalence between the Littlewood-Paley decomposition in L p with respect to the complementary intervals of E and Fourier multipliers of Hörmander-Mihlin and Marcinkiewicz type with singularities on E . Similar properties are studied in R 2 for a union of rays from the origin. Then there are connections with the maximal function operator with respect to all rectangles parallel to these rays. In particular, this maximal operator is proved to be bounded on L p , 1 &lt; p &lt; ,...

Littlewood-Paley decompositions on manifolds with ends

Jean-Marc Bouclet (2010)

Bulletin de la Société Mathématique de France

For certain non compact Riemannian manifolds with ends which may or may not satisfy the doubling condition on the volume of geodesic balls, we obtain Littlewood-Paley type estimates on (weighted) L p spaces, using the usual square function defined by a dyadic partition.

Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2

Thierry Coulhon, Xuan Thinh Duong, Xiang Dong Li (2003)

Studia Mathematica

We study the weak type (1,1) and the L p -boundedness, 1 < p ≤ 2, of the so-called vertical (i.e. involving space derivatives) Littlewood-Paley-Stein functions and ℋ respectively associated with the Poisson semigroup and the heat semigroup on a complete Riemannian manifold M. Without any assumption on M, we observe that and ℋ are bounded in L p , 1 < p ≤ 2. We also consider modified Littlewood-Paley-Stein functions that take into account the positivity of the bottom of the spectrum. Assuming that...

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