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

Hongbin Wang; Chenchen Niu

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Fractional Hardy-Sobolev-Maz'ya inequality for domains

Bartłomiej Dyda, Rupert L. Frank (2012)

Studia Mathematica

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We prove a fractional version of the Hardy-Sobolev-Maz’ya inequality for arbitrary domains and $L^{p}$ norms with p ≥ 2. This inequality combines the fractional Sobolev and the fractional Hardy inequality into a single inequality, while keeping the sharp constant in the Hardy inequality.

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...

Hardy inequality of fractional order.

Gurka, Petr, Opic, Bohumír (2008)

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Fractional Hardy inequalities and visibility of the boundary

Lizaveta Ihnatsyeva, Juha Lehrbäck, Heli Tuominen, Antti V. Vähäkangas (2014)

Studia Mathematica

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We prove fractional order Hardy inequalities on open sets under a combined fatness and visibility condition on the boundary. We demonstrate by counterexamples that fatness conditions alone are not sufficient for such Hardy inequalities to hold. In addition, we give a short exposition of various fatness conditions related to our main result, and apply fractional Hardy inequalities in connection with the boundedness of extension operators for fractional Sobolev spaces.

A Proof of the Hardy-Littlewood Theorem on Fractional Integration and a Generalization

Miroslav Pavlović (1996)

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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...

Fractional Hardy inequality with a remainder term

Bartłomiej Dyda (2011)

Colloquium Mathematicae

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We prove a Hardy inequality for the fractional Laplacian on the interval with the optimal constant and additional lower order term. As a consequence, we also obtain a fractional Hardy inequality with the best constant and an extra lower order term for general domains, following the method of M. Loss and C. Sloane [J. Funct. Anal. 259 (2010)].

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

V. Kiryakova (1988)

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Suitable domains to define fractional integrals of Weyl via fractional powers of operators

Celso Martínez, Antonia Redondo, Miguel Sanz (2011)

Studia Mathematica

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We present a new method to study the classical fractional integrals of Weyl. This new approach basically consists in considering these operators in the largest space where they make sense. In particular, we construct a theory of fractional integrals of Weyl by studying these operators in an appropriate Fréchet space. This is a function space which contains the $L^{p} (ℝ)$ -spaces, and it appears in a natural way if we wish to identify these fractional operators with fractional powers of a suitable...

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.

Existence results for systems of conformable fractional differential equations

Bouharket Bendouma, Alberto Cabada, Ahmed Hammoudi (2019)

Archivum Mathematicum

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In this article, we study the existence of solutions to systems of conformable fractional differential equations with periodic boundary value or initial value conditions. where the right member of the system is $L_{α}^{1}$ -carathéodory function. We employ the method of solution-tube and Schauder’s fixed-point theorem.