Multipliers and imaginary powers of the Schrödinger operators characterizing UMD Banach spaces

Fariña

Mathematische Nachrichten

2015

Self Cite

In this paper we define square functions (also called Littlewood-Paley-Stein functions) associated with heat semigroups for Schrödinger and Laguerre operators acting on functions which take values in UMD Banach spaces. We extend classical (scalar) L p -boundedness properties for the square functions to our Banach valued setting by using γ -radonifying operators. We also prove that these L p -boundedness properties of the square functions actually characterize the Banach spaces having the UMD property.

“…Plancherel's theorem implies that

H^{i γ}

is bounded from

L^{2} (R^{n})

into itself. Moreover,

H^{i γ}

is an spectral multiplier of Laplace transform type ([, p. 121]) associated with the Hermite operator and

H^{i γ}

can be extended from

L^{2} (R^{n}) \cap L^{p} (R^{n})

L^{p} (R^{n})

as a bounded operator from

L^{p} (R^{n})

into itself, for every

1 < p < \infty

([, Theorem 1.1], [, Theorem 3]) . Let B be a Banach space.…”

Section: Proof Of Theorem For the Hermite Operatormentioning

confidence: 99%

“…If

1 < p < \infty

we can define in a natural way

H^{i γ}

L^{p} (R^{n}) \otimes B

as a linear operator from

L^{p} (R^{n}) \otimes B

into itself. In [, Theorem 1.2] (see also [, Theorem 3]) it was established that B is UMD if and only if

H^{i γ}

γ \in R ∖ {0}

, can be extended from

L^{p} (R^{n}) \otimes B

L^{p} (R^{n}, boldB)

as a bounded operator from

L^{p} (R^{n}, boldB)

into itself for some (equivalently, for every)

1 < p < \infty

.…”

Section: Proof Of Theorem For the Hermite Operatormentioning

confidence: 99%

“…From ( b ) and Proposition it follows, for every

f \in span {{boldh}_{k}}_{k \in {boldN}^{n}} \otimes B

\begin{matrix} true \\ right ∥ H^{i γ} {(f) ∥}_{L^{p} ({boldR}^{n}, B)} & \leq & left C {‖G_{scriptH, boldB}^{1} ({scriptH}^{i γ} (f))‖}_{L^{p} ({boldR}^{n}, γ (H, boldB))} \\ \leq & left C {‖G_{scriptH, boldB}^{2} (f)‖}_{L^{p} ({boldR}^{n}, γ (H, boldB))} \leq C {∥ f ∥}_{L^{p} ({boldR}^{n}, B)} . \end{matrix}

Since

span {{boldh}_{k}}_{k \in {boldN}^{n}} \otimes B

is a dense subspace in

L^{p} (R^{n}, boldB)

H^{i γ}

can be extended to

L^{p} (R^{n}, boldB)

as a bounded operator from

L^{p} (R^{n}, boldB)

into itself. From [, Theorem 1.2] ([, Theorem 3]) we deduce that B is UMD.…”

Section: Proof Of Theorem For the Hermite Operatormentioning

confidence: 99%

“…We assume that

n \geq 3

and that the potential function V satisfies the reverse Hölder's inequality where

s > n / 2

. In the proof of

(c) \Rightarrow (a)

we will use [, Theorem 3] where UMD Banach spaces are characterized by the

L^{p}

‐boundedness properties of the imaginary power

L^{i γ}

γ \in R ∖ {0}

, of the Schrödinger operator

scriptL

.…”

Section: Proof Of Theorem For the Schrödinger Operatormentioning

confidence: 99%

“…Then, we use [, Theorem 4.2]. To see that the equivalences in ( b ), ( c ) and ( d ) imply that

boldB

is UMD, we have taken into account that the UMD Banach spaces are characterized by the

L^{p}

‐boundedness properties of the imaginary powers

H^{i γ}

L^{i γ}

L_{α}^{i γ}

γ > 0

scriptH

scriptL

and

L_{α}

, respectively ([, Theorem 1.2] and [, Theorem 3]).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

UMD Banach spaces and square functions associated with heat semigroups for Schrödinger, Hermite and Laguerre operators

Fariña

Mathematische Nachrichten

2015

Self Cite

UMD-Valued Square Functions Associated with Bessel Operators in Hardy and BMO Spaces

Integr. Equ. Oper. Theory

2014

Self Cite

We consider Banach valued Hardy and BMO spaces in the Bessel setting. Square functions associated with Poisson semigroups for Bessel operators are defined by using fractional derivatives. If B is a UMD Banach space we obtain for B-valued Hardy and BMO spaces equivalent norms involving γ-radonifying operators and square functions. We also establish characterizations of UMD Banach spaces by means of Hardy and BMO-boundedness properties of g-functions associated to Bessel-Poisson semigroup.Mathematics Subject Classification. Primary 46E40, 42B25; Secondary 42A50, 42B35.

Conical square functions associated with Bessel, Laguerre and Schrödinger operators in UMD Banach spaces

Fariña

Journal of Mathematical Analysis and Applications

2017

Self Cite

In this paper we consider conical square functions in the Bessel, Laguerre and Schrödinger settings where the functions take values in UMD Banach spaces. Following a recent paper of Hytönen, van Neerven and Portal [31], in order to define our conical square functions, we use γ-radonifying operators. We obtain new equivalent norms in the Lebesgue-Bochner spaces L p ((0, ∞), B) and L p (R n , B), 1 < p < ∞, in terms of our square functions, provided that B is a UMD Banach space. Our results can be seen as Banach valued versions of known scalar results for square functions.