On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space

Sawano, Yoshihiro; Wadade, Hidemitsu

doi:10.1007/s00041-012-9223-8

Cited by 67 publications

(55 citation statements)

References 39 publications

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“…In [54], it was shown that this function space E 0 pq2 (R n ) is equivalent to the Hardy-Morrey space defined by Jia and Wang in [22]. See [1,20,22,63] for more about Hardy Morrey spaces. In [32,Theorem 4.2], Mazzucato proved that the Triebel-Lizorkin space E 0 pq2 (R n ) with 1 < q ≤ p < ∞ is equivalent to the Morrey space M p q (R n ).…”

Section: 2mentioning

confidence: 99%

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Generalized Morrey spaces and trace operator

2015

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Abstract. The theory of generalized Besov-Morrey spaces and generalized Triebel-LizorkinMorrey spaces is developed. Generalized Morrey spaces, which T. Mizuhara and E. Nakai proposed, are equipped with a parameter and a function. The trace property is one of the main focuses of the present paper, which will clarify the role of the parameter of generalized Morrey spaces. The quarkonial decomposition is obtained as an application of atomic decomposition. In the end, the relation between the function spaces dealt in the present paper and the foregoing researches is discussed.

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Section: 2mentioning

confidence: 99%

“…(1) Let r = 1 in Proposition 7.3. The authors in [63] showed that (3) One can not replace min(1, r) by 1 in (7.10) when r ∈ (0, 1). Assume to the contrary that this is possible.…”

Section: 2mentioning

confidence: 99%

Generalized Morrey spaces and trace operator

2015

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“…Take a function ϕ belonging to the schwartz class

S ({boldR}^{n})

, which is non‐degenerate in the sense that

\int_{R^{n}} ϕ (x) d x \neq 0 .

Then for the tempered distribution

f \in {scriptS}^{'}

, define

{∥ f ∥}_{H {scriptM}_{q}^{p}} : = {(‖‖, \underset{j \in Z}{trueprefixsup}, | ϕ_{j} * f |)}_{M_{q}^{p}},

where

ϕ_{j} : = 2^{j n} ϕ (2^{j} \cdot)

for

j \in Z

. We define the norm of the Sobolev–Morrey space

H {scriptM}_{p, q}^{s}

for

s \in R

0 < q \leq p < \infty

by using the Hardy–Morrey space as follows:

{∥ f ∥}_{H {scriptM}_{p, q}^{s}} : = {(‖‖, {(1 - Δ)}^{\frac{s}{2}}, f)}_{H {scriptM}_{q}^{p}} .

In , Sawano and Wadade obtained the following result: Proposition [32, Theorem 5.1] Let

1 < q < p < \infty

and

φ (t) = {(1 + t^{n})}^{\frac{1}{p}} {((), log, (e + \frac{1}{t^{n}}))}^{...}

…”

Section: Introductionmentioning

confidence: 99%

Generalized weighted Morrey spaces and classical operators

Nakamura

2016

Mathematische Nachrichten

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Hardy-Littlewood maximal operator, generalized fractional maximal operator, generalized fractional integral operator, singular integral operator MSC (2010) Primary: 42B25, Secondary: 42B35We introduce the notion of generalized weighted Morrey spaces and investigate the boundedness of some operators in these spaces, such as the Hardy-Littlewood maximal operator, generalized fractional maximal operators, generalized fractional integral operators, and singular integral operators. We also study their boundedness in the vector-valued setting.

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“…We refer to [36] for a counterexample showing that (1.5) is no longer true for α = d p . Meanwhile, the function q(·) can be used to describe the Hardy-Littlewood maximal operator control in very subtle settings.…”

Section: ( ) P(·)q(·) (X ·) Is Convex On [0 ∞) For Every X ∈ Xmentioning

confidence: 99%

Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents

Sawano

Shimomura

2013

Collect. Math.

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The target is potential theory in connection with Morrey spaces on general metric measure spaces. The present paper is oritented to investigating Sobolev's inequality, Trudinger exponential integrability and continuity for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents. A counterexample shows that our results are reasonable. In addition to the example above, what is new about this paper is that everything can be developed once the underlying measure does not charge any point.

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On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space

Cited by 67 publications

References 39 publications

Generalized Morrey spaces and trace operator

Generalized Morrey spaces and trace operator

Generalized weighted Morrey spaces and classical operators

Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents

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