Weighted norm inequalities for the maximal operator on variable Lebesgue spaces

“…By use of [, Lemmas 3.1 and 3.4], we have the following result. Lemma If

w \in A_{p (\cdot)}

, then there exist

δ_{1} > 0

and

δ_{2} > 0

such that

C^{- 1} r^{- δ_{1}} \leq \frac{μ (B)}{μ (r B)} \leq C r^{- δ_{2}} italicfor r > 1 a n d b a l l s B .

…”

Section: Preliminariesmentioning

confidence: 99%

L^{p (\cdot)} ({boldR}^{n}; μ)

(see [, Theorem 1.5]).…”

Section: Introductionmentioning

confidence: 99%

“…The weighted version was given by [, Theorem 3.2]. The purpose of this paper is to prove weighted inequalities for the maximal operator on weighted Morrey spaces

L^{p (\cdot), ν} ({boldR}^{n}; μ)

of variable exponent, as an extension of . As an application of the boundedness of the maximal operator, we establish weighted Sobolev's inequality for Riesz potentials.…”

Section: Introductionmentioning

confidence: 99%

“…Following Cruz‐Uribe, Fiorenza and Neugebauer , we say that a positive measurable function w on

{boldR}^{n}

is an

A_{p (\cdot)}

weight (written as

w \in A_{p (\cdot)})

if there exists a positive constant

C_{p}

such that

∥ w χ_{B} ∥_{L^{p (\cdot)} ({boldR}^{n})} {(‖‖, w^{- 1}, χ_{B})}_{L^{p^{'} (\cdot)} ({boldR}^{n})} \leq C_{p} | B |

for all balls B , where

p^{'} (\cdot)

is the conjugate exponent defined pointwise by

\frac{1}{p (x)} + \frac{1}{p^{'} (x)} = 1 .

…”

Section: Introductionmentioning

confidence: 99%

“…They proved the boundedness of the maximal operator in the weighted space

L^{p (\cdot)} ({boldR}^{n}; μ)

(see [, Theorem 1.5]):…”

Section: Introductionmentioning

confidence: 99%

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Weighted Morrey spaces of variable exponent and Riesz potentials

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Shimomura

2014

Weighted mixed‐norm inequalities through extrapolation

Duoandikoetxea

Oruetxebarria

2019

We prove that operators satistying weighted inequalities with radial weights are bounded in mixed-norm spaces of radial-angular type, even with a weight in the radial part. This is achieved by using the usual extrapolation methods, adapted to the radial setting. All the versions of the extrapolation theorem can be adapted to this setting, and in particular we get results in variable Lebesgue spaces and also for multilinear operators. Furthermore, quantitative estimates are obtained with this approach, but their sharpness remains an open question. K E Y W O R D S extrapolation of weights, mixed-norm spaces, Muckenhoupt weights, variable exponent M S C ( 2 0 1 0 ) 42B20, 42B25, 42B35 1482

Weighted estimates for square functions associated with operators

Wen

Shen

Sun

2023

Let L be a non‐negative self‐adjoint operator on . Suppose that the kernels of the analytic semigroup satisfy the upper bound related to a critical function ρ but without any assumptions of smooth conditions on spacial variables. In this paper, we consider the weighted inequalities for square functions associated with L, which include the vertical square functions, the conical square functions and the Littlewood–Paley g‐functions. A new bump condition related to the critical function is given for the two‐weighted boundedness of square functions associated with L. Besides, we also prove the weighted inequalities for square functions associated with L on weighted variable Lebesgue spaces with new classes of weights considered in [5]. As applications, our results can be applied to magnetic Schrödinger operator, Laguerre operators.