On multilinear fractional integrals

Loukas Grafakos

Displaying similar documents to “On multilinear fractional integrals”

Trace inequalities for fractional integrals in grand Lebesgue spaces

Vakhtang Kokilashvili, Alexander Meskhi (2012)

Studia Mathematica

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rning the boundedness for fractional maximal and potential operators defined on quasi-metric measure spaces from $L^{p), θ} (X, μ)$ to $L^{q), q θ / p} (X, ν)$ (trace inequality), where 1 < p < q < ∞, θ > 0 and μ satisfies the doubling condition in X. The results are new even for Euclidean spaces. For example, from our general results D. Adams-type necessary and sufficient conditions guaranteeing the trace inequality for fractional maximal functions and potentials defined on so-called s-sets in ℝⁿ follow. Trace...

Bilinear fractional Hardy-type operators with rough kernels on central Morrey spaces with variable exponents

Hongbin Wang, Chenchen Niu (2024)

Czechoslovak Mathematical Journal

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We introduce a type of $n$ -dimensional bilinear fractional Hardy-type operators with rough kernels and prove the boundedness of these operators and their commutators on central Morrey spaces with variable exponents. Furthermore, the similar definitions and results of multilinear fractional Hardy-type operators with rough kernels are obtained.

Multi-Morrey spaces for non-doubling measures

Suixin He (2019)

Czechoslovak Mathematical Journal

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The spaces of multi-Morrey type for positive Radon measures satisfying a growth condition on $ℝ^{d}$ are introduced. After defining the spaces, we investigate the multilinear maximal function, the multilinear fractional integral operator and the multilinear Calderón-Zygmund operators, respectively, from multi-Morrey spaces to Morrey spaces.

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Ali Akbulut, Amil Hasanov (2016)

Open Mathematics

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In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels [...] TΩ,αA1,A2,…,Ak, $T_{Ω, α}^{A_{1}, A_{2}, ..., A_{k}},$ which is a generalization of the higher-order commutator of the rough fractional integral on the generalized weighted Morrey spaces Mp,ϕ (w). We find the sufficient conditions on the pair (ϕ1, ϕ2) with w ∈ Ap,q which ensures the boundedness of the operators [...] TΩ,αA1,A2,…,Ak, $T_{Ω, α}^{A_{1}, A_{2}, ..., A_{k}},$ from [...] Mp,φ1wptoMp,φ2wq $M_{p, ϕ_{1}} (w^{p}) to M_{p, ϕ_{2}} (w^{q})$ for 1 < p < q < ∞. In all cases the conditions...

Essential normality for certain finite linear combinations of linear-fractional composition operators on the Hardy space $H^{2}$

Mahsa Fatehi, Bahram Khani Robati (2012)

Czechoslovak Mathematical Journal

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In 1999 Nina Zorboska and in 2003 P. S. Bourdon, D. Levi, S. K. Narayan and J. H. Shapiro investigated the essentially normal composition operator $C_{ϕ}$ , when $ϕ$ is a linear-fractional self-map of $𝔻$ . In this paper first, we investigate the essential normality problem for the operator $T_{w} C_{ϕ}$ on the Hardy space $H^{2}$ , where $w$ is a bounded measurable function on $\partial 𝔻$ which is continuous at each point of $F (ϕ)$ , $ϕ \in 𝒮 (2)$ , and $T_{w}$ is the Toeplitz operator with symbol $w$ . Then we use these results and characterize the essentially...

Generalized $H_{m, m}^{m, 0}$ function fractional integration operators in some classes of analytic functions

V. Kiryakova (1988)

Matematički Vesnik

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Multiplier transformations on $H^{p}$ spaces

Daning Chen, Dashan Fan (1998)

Studia Mathematica

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The authors obtain some multiplier theorems on $H^{p}$ spaces analogous to the classical $L^{p}$ multiplier theorems of de Leeuw. The main result is that a multiplier operator ${(T f)}^{(} x) = λ (x) f ̂ (x)$ $(λ \in C (ℝ^{n}))$ is bounded on $H^{p} (ℝ^{n})$ if and only if the restriction ${λ (ε m)}_{m \in Λ}$ is an $H^{p} (T^{n})$ bounded multiplier uniformly for ε>0, where Λ is the integer lattice in $ℝ^{n}$ .

Boundedness of one-sided fractional integrals in the one-sided Calderón-Hardy spaces

Alejandra Perini (2011)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we study the mapping properties of the one-sided fractional integrals in the Calderón-Hardy spaces $ℋ_{q, α}^{p, +} (ω)$ for $0 < p \leq 1$ , $0 < α < \infty$ and $1 < q < \infty$ . Specifically, we show that, for suitable values of $p, q, γ, α$ and $s$ , if $ω \in A_{s}^{+}$ (Sawyer’s classes of weights) then the one-sided fractional integral $I_{γ}^{+}$ can be extended to a bounded operator from $ℋ_{q, α}^{p, +} (ω)$ to $ℋ_{q, α + γ}^{p, +} (ω)$ . The result is a consequence of the pointwise inequality $N_{q, α + γ}^{+} (I_{γ}^{+} F; x) \leq C_{α, γ} N_{q, α}^{+} (F; x),$ where $N_{q, α}^{+} (F; x)$ denotes the Calderón maximal function.

On Sobolev spaces of fractional order and ε-families of operators on spaces of homogeneous type

A. Gatto, Stephen Vági (1999)

Studia Mathematica

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We introduce Sobolev spaces $L_{α}^{p}$ for 1 < p < ∞ and small positive α on spaces of homogeneous type as the classes of functions f in $L^{p}$ with fractional derivative of order α, $D^{α} f$ , as introduced in [2], in $L^{p}$ . We show that for small α, $L_{α}^{p}$ coincides with the continuous version of the Triebel-Lizorkin space $F_{p}^{α, 2}$ as defined by Y. S. Han and E. T. Sawyer in [4]. To prove this result we give a more general definition of ε-families of operators on spaces of homogeneous type, in which the identity...

Harmonic interpolating sequences, $L^{p}$ and BMO

John B. Garnett (1978)

Annales de l'institut Fourier

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Let $(z_{ν})$ be a sequence in the upper half plane. If $1 < p \leq \infty$ and if $y_{ν}^{1 / p} f (z_{ν}) = a_{ν}, ν = 1, 2, ... (*)$ has solution $f (z)$ in the class of Poisson integrals of $L^{p}$ functions for any sequence $(a_{ν}) \in ℓ^{p}$ , then we show that $(z_{ν})$ is an interpolating sequence for $H^{\infty}$ . If $f (z_{ν}) = a_{ν}$ , $ν = 1, 2, ...$ has solution in the class of Poisson integrals of BMO functions whenever $(a_{ν}) \in ℓ^{\infty}$ , then $(z_{ν})$ is again an interpolating sequence for $H^{\infty}$ . A somewhat more general theorem is also proved and a counterexample for the case $p \leq 1$ is described.