Claudia Capone

The fractional maximal operator and fractional integrals on variable

We prove that if the exponent function p(·) satisfies log-Hölder continuity conditions locally and at infinity, then the fractional maximal operator Mα, 0 <α< n, maps Lp(·) to Lq(·), where 1 p(x)− 1 q(x) = αn . We also prove a weak-type inequality corresponding to the weak (1, n/(n − α)) inequality for Mα. We build upon earlier work on the Hardy-Littlewood maximal operator by Cruz-Uribe, Fiorenza and Neugebauer [3]. As a consequence of these results for Mα, we show that the fractional integral operator Iα satisfies the same norm inequalities. These in turn yield a generalization of the Sobolev embedding theorem to variable Lp spaces.

L^p

We consider a generalized version of the small Lebesgue spaces, introduced in [5] as the associate spaces of the grand Lebesgue spaces. We find a simplified expression for the norm, prove relevant properties, compute the fundamental function and discuss the comparison with the Orlicz spaces.

spaces.

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On small Lebesgue spaces

— Let W;W 0 HR be bounded domains and let f : W ! W 0 be a bi-Sobolev mapping. We provide regularity properties for the inverse map f 1 under suitable assumptions on q-distortion function of f .

On extrapolation blowups in the

We give a decomposition for the dual space of some Banach Function Spaces as the Zygmund space EXP𝛼 of the exponential integrable functions, the Marcinkiewicz space 𝐿𝑝,∞, and the Grand Lebesgue Space 𝐿𝑝),𝜃.

On the regularity theory of bi-Sobolev mappings

Yanos extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operatorOpen image in new window acting continuously inOpen image in new window forOpen image in new window close toOpen image in new window and/or takingOpen image in new window intoOpen image in new window asOpen image in new window and/orOpen image in new window with norms blowing up at speedOpen image in new window and/orOpen image in new window,Open image in new window. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens ifOpen image in new window asOpen image in new window. The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs forOpen image in new window . We also touch the problem of comparison of results in various scales of spaces.Yanos extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operator acting continuously in for close to and/or taking into as and/or with norms blowing up at speed and/or,. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens if as. The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs for . We also touch the problem of comparison of results in various scales of spaces.

A Decomposition of the Dual Space of Some Banach Function Spaces

We prove continuity of the Riesz potential operator in optimal couples of rearrangement invariant function spaces defined in R with the Lebesgue measure. An application is given to the Hardy-Littlewood maximal operator.

On extrapolation blowups in the scale

Abstract In this note we prove a modular variable Orlicz inequality for the local maximal operator. This result generalizes several Orlicz and variable exponent modular inequalities that have appeared previously in the literature.

The Riesz Potential Operator in Optimal Couples of Rearrangement Invariant Spaces

Yanos extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operatorOpen image in new window acting continuously inOpen image in new window forOpen image in new window close toOpen image in new window and/or takingOpen image in new window intoOpen image in new window asOpen image in new window and/orOpen image in new window with norms blowing up at speedOpen image in new window and/orOpen image in new window,Open image in new window. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens ifOpen image in new window asOpen image in new window. The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs forOpen image in new window . We also touch the problem of comparison of results in various scales of spaces.Yanos extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operator acting continuously in for close to and/or taking into as and/or with norms blowing up at speed and/or,. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens if as. The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs for . We also touch the problem of comparison of results in various scales of spaces.