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Media type:
E-Article
Title:
New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents
Contributor:
Weisz, Ferenc
Published:
Springer Science and Business Media LLC, 2023
Published in:
Fractional Calculus and Applied Analysis, 26 (2023) 1, Seite 1-31
Language:
English
DOI:
10.1007/s13540-022-00121-4
ISSN:
1311-0454;
1314-2224
Origination:
Footnote:
Description:
AbstractWe 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 ( · ) .