Extrapolation for Classes of Weights Related to a Family of Operators and Applications

Bongioanni, Bruno; Cabral, Adrián; Harboure, Eleonor

doi:10.1007/s11118-012-9313-x

Cited by 30 publications

(36 citation statements)

References 16 publications

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“…Following the proof of [, Proposition 3] we can write

M_{ρ}^{θ} f (x) \leq M_{ρ}^{loc} f (x) + M_{ρ}^{θ, 2} f (x),

where

M_{ρ}^{θ, 2} f (x) = \underset{r > ρ (x)}{trueprefixsup} {(() 1 + \frac{r}{ρ (x)})}^{- θ} \frac{1}{| B (x, r) |} \int_{B (x, r)} | f | .

As we mentioned above (see [, Theorem 1])

M_{ρ}^{loc}

is of weak type (1, 1) for

w \in A_{1}^{ρ, loc}

and since

A_{1}^{ρ} \subset A_{1}^{ρ, loc}

, it is enough to bound

M_{ρ}^{θ, 2} f

. As in , for

x \in Q_{k}

, setting

Q_{k}^{j} = 2^{j} Q_{k}

, we have

\begin{matrix} true \\ right M_{ρ}^{θ, 2} f (x) & ≲ & left sup_{j \geq 1} 2^{- j θ} \frac{1}{| Q_{k}^{j} |} \int_{Q_{k}^{j}} | f | ≲ \sum_{j \geq 1} \frac{2^{- j (θ - σ)}}{w (Q_{k}^{j})} \int_{Q_{k}^{j}} ... \end{matrix}

…”

Section: Weighted Hardy Spaces Associated With ρmentioning

confidence: 90%

“…Classes

A_{p}^{ρ}

are intimately connected with the family of maximal operators

M_{ρ}^{θ}

, defined by

M_{ρ}^{θ} f (x) = \underset{r > 0}{trueprefixsup} {(() 1 + \frac{r}{ρ (x)})}^{- θ} \frac{1}{| B (x, r) |} \int_{B (x, r)} | f |,

for

θ > 0

. In fact, for

1 < p < \infty

they are bounded on

L^{p} (w)

, provided

w \in A_{p}^{ρ}

(see [, Proposition 3]).…”

Section: Classes Of Weights Related To ρmentioning

confidence: 99%

“…Using results from and we have the following proposition concerning strong weighted inequalities for operators in

S_{0}^{ρ}

and

S_{s}^{ρ}

. Proposition (i)If

T \in {scriptS}_{0}^{ρ}

, then T is bounded on

L^{p} (w)

with

1 < p < \infty

and any

w \in A_{p}^{ρ}

. (ii)If

T \in {scriptS}_{s}^{ρ}

, then T is bounded on

L^{p} (w)

with

1 < p < s

and any weight w such that

w^{- \frac{1}{p - 1}} \in A_{p^{'} / s^{'}}^{ρ}

. …”

Section: Weak Type (11) Of Singular Integrals Of ρ‐Typementioning

confidence: 99%

“…Consequently,

\int_{R^{d}} | T^{*} {f (x) |}^{q} v (x) d x \leq C \int_{R^{d}} {| M_{ρ, s_{1}^{'}}^{θ} f (x) |}^{q} v (x) d x,

for any

θ > 0

and every

v \in A_{\infty}^{ρ, loc} = ⋃_{q \geq 1} A_{q}^{ρ, loc}

, where

M_{ρ, r}^{θ} f = {(M_{ρ}^{θ} (f^{r}))}^{1 / r}

. By [, Proposition 3] the operator

M_{ρ, s_{1}^{'}}^{θ}

is bounded in

L^{q} (v)

for

v \in A_{q / s_{1}^{'}}^{ρ}

with

q > s_{1}^{'}

. Inequality and a duality argument imply that T is bounded on

L^{q} (v)

1 < q < s_{1}

, for

v^{- \frac{1}{q - 1}} \in A_{q^{'} / s_{1}^{'}}^{ρ}

, particularly on

L^{p} (w)

.…”

Section: Weak Type (11) Of Singular Integrals Of ρ‐Typementioning

confidence: 99%

“…As we mentioned before, in ,

L^{p}

‐weighted inequalities were studied for the operator

M_{ρ}^{θ}

whenever

1 < p < \infty

. Following the ideas developed there, it is not hard to see the behavior of

M_{ρ}^{θ}

for

p = 1

.…”

Section: Weighted Hardy Spaces Associated With ρmentioning

confidence: 99%

See 4 more Smart Citations

Schrödinger type singular integrals: weighted estimates for p=1

2016

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A critical radius function ρ assigns to each x ∈ R d a positive number in a way that its variation at different points is somehow controlled by a power of the distance between them. This kind of function appears naturally in the harmonic analysis related to a Schrödinger operator − + V with V a non-negative potential satisfying some specific reverse Hölder condition. For a family of singular integrals associated with such critical radius function, we prove boundedness results in the extreme case p = 1. On one side we obtain weighted weak (1, 1) results for a class of weights larger than Muckenhoupt class A 1 . On the other side, for the same weights, we prove continuity from appropriate weighted Hardy spaces into weighted L 1 . To achieve the latter result we define weighted Hardy spaces by means of a ρ-localized maximal heat operator. We obtain a suitable atomic decomposition and a characterization via ρ-localized Riesz Transforms for these spaces. For the case of ρ derived from a Schrödinger operator, we obtain new estimates for many of the operators appearing in [27].

show abstract

“…Following the proof of [, Proposition 3] we can write

M_{ρ}^{θ} f (x) \leq M_{ρ}^{loc} f (x) + M_{ρ}^{θ, 2} f (x),

where

M_{ρ}^{θ, 2} f (x) = \underset{r > ρ (x)}{trueprefixsup} {(() 1 + \frac{r}{ρ (x)})}^{- θ} \frac{1}{| B (x, r) |} \int_{B (x, r)} | f | .

As we mentioned above (see [, Theorem 1])

M_{ρ}^{loc}

is of weak type (1, 1) for

w \in A_{1}^{ρ, loc}

and since

A_{1}^{ρ} \subset A_{1}^{ρ, loc}

, it is enough to bound

M_{ρ}^{θ, 2} f

. As in , for

x \in Q_{k}

, setting

Q_{k}^{j} = 2^{j} Q_{k}

, we have

\begin{matrix} true \\ right M_{ρ}^{θ, 2} f (x) & ≲ & left sup_{j \geq 1} 2^{- j θ} \frac{1}{| Q_{k}^{j} |} \int_{Q_{k}^{j}} | f | ≲ \sum_{j \geq 1} \frac{2^{- j (θ - σ)}}{w (Q_{k}^{j})} \int_{Q_{k}^{j}} ... \end{matrix}

…”

Section: Weighted Hardy Spaces Associated With ρmentioning

confidence: 90%

“…Classes

A_{p}^{ρ}

are intimately connected with the family of maximal operators

M_{ρ}^{θ}

, defined by

M_{ρ}^{θ} f (x) = \underset{r > 0}{trueprefixsup} {(() 1 + \frac{r}{ρ (x)})}^{- θ} \frac{1}{| B (x, r) |} \int_{B (x, r)} | f |,

for

θ > 0

. In fact, for

1 < p < \infty

they are bounded on

L^{p} (w)

, provided

w \in A_{p}^{ρ}

(see [, Proposition 3]).…”

Section: Classes Of Weights Related To ρmentioning

confidence: 99%

“…Using results from and we have the following proposition concerning strong weighted inequalities for operators in

S_{0}^{ρ}

and

S_{s}^{ρ}

. Proposition (i)If

T \in {scriptS}_{0}^{ρ}

, then T is bounded on

L^{p} (w)

with

1 < p < \infty

and any

w \in A_{p}^{ρ}

. (ii)If

T \in {scriptS}_{s}^{ρ}

, then T is bounded on

L^{p} (w)

with

1 < p < s

and any weight w such that

w^{- \frac{1}{p - 1}} \in A_{p^{'} / s^{'}}^{ρ}

. …”

Section: Weak Type (11) Of Singular Integrals Of ρ‐Typementioning

confidence: 99%

“…Consequently,

\int_{R^{d}} | T^{*} {f (x) |}^{q} v (x) d x \leq C \int_{R^{d}} {| M_{ρ, s_{1}^{'}}^{θ} f (x) |}^{q} v (x) d x,

for any

θ > 0

and every

v \in A_{\infty}^{ρ, loc} = ⋃_{q \geq 1} A_{q}^{ρ, loc}

, where

M_{ρ, r}^{θ} f = {(M_{ρ}^{θ} (f^{r}))}^{1 / r}

. By [, Proposition 3] the operator

M_{ρ, s_{1}^{'}}^{θ}

is bounded in

L^{q} (v)

for

v \in A_{q / s_{1}^{'}}^{ρ}

with

q > s_{1}^{'}

. Inequality and a duality argument imply that T is bounded on

L^{q} (v)

1 < q < s_{1}

, for

v^{- \frac{1}{q - 1}} \in A_{q^{'} / s_{1}^{'}}^{ρ}

, particularly on

L^{p} (w)

.…”

Section: Weak Type (11) Of Singular Integrals Of ρ‐Typementioning

confidence: 99%

“…As we mentioned before, in ,

L^{p}

‐weighted inequalities were studied for the operator

M_{ρ}^{θ}

whenever

1 < p < \infty

. Following the ideas developed there, it is not hard to see the behavior of

M_{ρ}^{θ}

for

p = 1

.…”

Section: Weighted Hardy Spaces Associated With ρmentioning

confidence: 99%

See 3 more Smart Citations

Schrödinger type singular integrals: weighted estimates for p=1

2016

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show abstract

Weighted Inequalities for Schrödinger Type Singular Integrals

Bongioanni

Harboure

Quijano

2018

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Mixed Inequalities for Operators Associated to Critical Radius Functions with Applications to Schrödinger Type Operators

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Extrapolation for Classes of Weights Related to a Family of Operators and Applications

Cited by 30 publications

References 16 publications

Schrödinger type singular integrals: weighted estimates for p=1

Schrödinger type singular integrals: weighted estimates for p=1

Weighted Inequalities for Schrödinger Type Singular Integrals

Mixed Inequalities for Operators Associated to Critical Radius Functions with Applications to Schrödinger Type Operators

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