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Lions-Peetre reiteration formulas for triples and their applications

Irina Asekritova, Natan Krugljak, Lech Maligranda, Lyudmila Nikolova, Lars-Erik Persson (2001)

Studia Mathematica

We present, discuss and apply two reiteration theorems for triples of quasi-Banach function lattices. Some interpolation results for block-Lorentz spaces and triples of weighted L p -spaces are proved. By using these results and a wavelet theory approach we calculate (θ,q)-spaces for triples of smooth function spaces (such as Besov spaces, Sobolev spaces, etc.). In contrast to the case of couples, for which even the scale of Besov spaces is not stable under interpolation, for triples we obtain stability...

Lipschitz continuity in Muckenhoupt 𝓐₁ weighted function spaces

Dorothee D. Haroske (2011)

Banach Center Publications

We study continuity envelopes of function spaces B p , q s ( ℝ ⁿ , w ) and F p , q s ( ℝ ⁿ , w ) where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.

Lipschitz spaces and Calderón-Zygmund operators associated to non-doubling measures.

José García-Cuerva, A. Eduardo Gatto (2005)

Publicacions Matemàtiques

In the setting of a metric measure space (X, d, μ) with an n-dimensional Radon measure μ, we give a necessary and sufficient condition for the boundedness of Calderón-Zygmund operators associated to the measure μ on Lipschitz spaces on the support of μ. Also, for the Euclidean space Rd with an arbitrary Radon measure μ, we give several characterizations of Lipschitz spaces on the support of μ, Lip(α,μ), in terms of mean oscillations involving μ. This allows us to view the "regular" BMO space of...

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 g-functions with rough kernels on homogeneous groups

Yong Ding, Xinfeng Wu (2009)

Studia Mathematica

Let 𝔾 be a homogeneousgroup on ℝⁿ whose multiplication and inverse operations are polynomial maps. In 1999, T. Tao proved that the singular integral operator with Llog⁺L function kernel on ≫ is both of type (p,p) and of weak type (1,1). In this paper, the same results are proved for the Littlewood-Paley g-functions on 𝔾

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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