Boundedness of dyadic maximal operators on variable Lebesgue spaces

Weisz, Ferenc

doi:10.1007/s43036-020-00071-9

Cited by 4 publications

(4 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Later we generalized this result to 1 < p − ≤ p + ≤ ∞ in [45]. In [16,43,44,46], we investigated more general maximal operators for dyadic martingales and verified that they are bounded on L p(•) if 1 < p − ≤ p + < ∞. These operators were the key points in the proof of the boundedness of the maximal Fejér operator of the Walsh-Fourier series from the variable Hardy space H p(•) to L p(•) (see [16,40,46]).…”

Section: Introductionmentioning

confidence: 79%

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

Weisz

2022

Fract Calc Appl Anal

Self Cite

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: 79%

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

Weisz

2022

Fract Calc Appl Anal

Self Cite

View full text Add to dashboard Cite

show abstract

“…Theorem 4.1 holds for the operators in Examples 5.1 and 5.2 and Theorem 4.3 holds for Examples 5.3, 5.4 and 5.5. Under the additional condition p + < ∞, Theorem 4.3 was proved in [11,35] for the next special operators.…”

Section: Some Special Maximal Operatorsmentioning

confidence: 94%

“…In [36], we generalized this result to 1 < p − ≤ p + ≤ ∞. In [11,35], we investigated four more dyadic maximal operators and show that they are bounded on L p(•) if 1 < p − ≤ p + < ∞. The boundedness of these operators was the key point in proving the boundedness of the maximal Fejér, Cesàro and Riesz operators of the Walsh-Fourier series from the variable Hardy space H p(•) to L p(•) (see [11,33]).…”

Section: Introductionmentioning

confidence: 91%

See 1 more Smart Citation

Characterizations of Variable Martingale Hardy Spaces Via Maximal Functions

Weisz

2021

Fract Calc Appl Anal

View full text Add to dashboard Cite

We introduce a new type of dyadic maximal operators and prove that under the log-Hölder continuity condition of the variable exponent p(⋅), it is bounded on L p(⋅) if 1 < p − ≤ p + ≤ ∞. Moreover, the space generated by the L p(⋅)-norm (resp. the L p(⋅), q -norm) of the maximal operator is equivalent to the Hardy space H p(⋅) (resp. to the Hardy-Lorentz space H p(⋅), q ). As special cases, our maximal operator contains the usual dyadic maximal operator and four other maximal operators investigated in the literature.

show abstract