Fractional integrals and their commutators on martingale Orlicz spaces

Arai, Ryutaro; Nakai, Eiichi; Sadasue, Gaku

doi:10.1016/j.jmaa.2020.123991

Cited by 14 publications

(13 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The fractional maximal operator M α generates the Hardy-Littlewood operator and is bounded from L q(•) to L p(•) , whenever 0 ≤ α < 1, 1 < q − ≤ q + ≤ 1/α and 1/ p(•) = 1/q(•) − α (see Capone et al [2], Cruz-Uribe and Fiorenza [5] or Kokilashvili and Meskhi [22]). The fractional integral operator was investigated in Diening [7], Ephremidze et al [9], Mizuta and Shimomura [28], Rafeiro and Samko [35], Izuki et al [15] and, for martingales, in Arai et al [1], Hao et al [13,20], Jiao et al [17] and Sadasue [36].…”

Section: Introductionmentioning

confidence: 99%

New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents

Weisz

2022

Fract Calc Appl Anal

View full text Add to dashboard Cite

We generalize the usual Doob maximal operator as well as the fractional maximal operator and introduce $$M_{\gamma ,s,\alpha }$$ M γ , s , α , a new fractional maximal operator for martingales. We prove that under the log-Hölder continuity condition of the variable exponents $$p(\cdot )$$ p ( · ) and $$q(\cdot )$$ q ( · ) , the maximal operator $$M_{\gamma ,s,\alpha }$$ M γ , s , α is bounded from the variable Lebesgue space $$L_{q(\cdot )}$$ L q ( · ) to $$L_{p(\cdot )}$$ L p ( · ) and from the variable Hardy space $$H_{q(\cdot )}$$ H q ( · ) to $$L_{p(\cdot )}$$ L p ( · ) , whenever $$0 \le \alpha <1$$ 0 ≤ α < 1 , $$0<q_-\le q_+ \le 1/\alpha $$ 0 < q - ≤ q + ≤ 1 / α , $$0<\gamma ,s<\infty $$ 0 < γ , s < ∞ , $$1/p(\cdot )= 1/q(\cdot )- \alpha $$ 1 / p ( · ) = 1 / q ( · ) - α and $$1/p_- - 1/p_+ < \gamma +s$$ 1 / p - - 1 / p + < γ + s . Moreover, for $$\alpha =0$$ α = 0 , the operator $$M_{\gamma ,s,0}$$ M γ , s , 0 generates equivalent quasi-norms on the Hardy spaces $$H_{p(\cdot )}$$ H p ( · ) .

show abstract

Section: Introductionmentioning

confidence: 99%

New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents

Weisz

2022

Fract Calc Appl Anal

View full text Add to dashboard Cite

show abstract

“…One basic difference to the Volterra-kernel approach is that, starting with a (sub-, super-) martingale ϕ, we again obtain a (sub-, super-) martingale I α t ϕ. This second approach was used in [23, Definition 4.2], [24, Section 4], and [2], and relates to fractional integral transforms of martingales (see, for example, [3]). This corresponds to equation (3.3) of our Proposition 3.8.…”

Section: Riemann-liouville Type Operatorsmentioning

confidence: 99%

“…Then we obtain an Y -consistent function by where we use EKpt, y `Yt´s q " Kps, yq. (2a) If we have y n , y P Q with y n Ñ y, then we take ε " εpt, yq ą 0 from assumption (3) and obtain lim n Hpt, y n q " Hpt, yq by the uniform integrability imposed in (3) and assumption (2).…”

Section: Qzt0umentioning

confidence: 99%

On Riemann-Liouville type operators, BMO, gradient estimates in the Lévy-Itô space, and approximation

Geiss,

Nguyen

2020

Preprint

View full text Add to dashboard Cite

In this article we discuss in a stochastic framework the interplay between Riemann-Liouville operators applied to càdlàg processes, real interpolation, weighted bounded mean oscillation, estimates for gradient processes on the Lévy-Itô space, and the connection to an approximation problem for stochastic integrals. We prove upper and lower bounds for gradient processes appearing in a Brownian setting within the Feynman-Kac theory for parabolic PDEs and in the setting of Lévy processes. The upper bounds are formulated by BMO-conditions on the fractional integrated gradient, the lower bounds are formulated in terms of oscillatory quantities. In the case of Lévy processes we are concerned with a gradient process with values in a Hilbert space where the regularity of this process depends on the direction within this Hilbert space. Moreover, it turns out that certain Hölder properties of terminal functions transfer into a singularity in time that can be compensated by Riemann-Liouville operators.

show abstract

“…We will state the definitions of the Young function and the Orlicz space in Section 5. For the generalized fractional integral operator I ρ , see also [1,2,15,18,26,30,31,40] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Generalized Fractional Integral Operators Based on Symmetric Markovian Semigroups with Application to the Heisenberg Group

Amagai¹,

Nakai

Sadasue

2023

Taiwanese J. Math.

Self Cite

View full text Add to dashboard Cite

It is known that the fractional integral operator I α based on a symmetric Markovian semigroup with Varopoulos dimension d is bounded from L p to L q , if 0 < α < d, 1 < p < q < ∞ and −d/p + α = −d/q, like the usual fractional integral operator defined on the d dimensional Euclidean space. We introduce generalized fractional integral operators based on symmetric Markovian semigroups and extend the L p -L q boundedness to Orlicz spaces. We also apply the result to the semigroup associated with the diffusion process generated by the sub-Laplacian on the Heisenberg group. Moreover, we show necessary and sufficient conditions for the boundedness of the generalized fractional integral operator on the space of homogeneous type and apply them to the Heisenberg group.

show abstract

Fractional integrals and their commutators on martingale Orlicz spaces

Cited by 14 publications

References 14 publications

New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents

New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents

On Riemann-Liouville type operators, BMO, gradient estimates in the Lévy-Itô space, and approximation

Generalized Fractional Integral Operators Based on Symmetric Markovian Semigroups with Application to the Heisenberg Group

Contact Info

Product

Resources

About