The Kato-Ponce Inequality

Grafakos, Loukas; Oh, Seungly

doi:10.1080/03605302.2013.822885

Cited by 217 publications

(153 citation statements)

References 20 publications

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“…As we did for 1 , 2 , we may observe that in the case in which 1 4 takes a value in an interval which is not reduced to a singleton we can assume that 4 is different from a finite number of given values (for example 1 4 = 0, 2 from the relation 1 2 = 1 3 + 1 4 ). On the other hand we must pay attention to the case in which we have only one possible choice for 4 .…”

Section: Existence Of Parameters In Theorem 11mentioning

confidence: 73%

“…Thus, we get a second condition for 4 . Ensuring that we have a nonempty range for the parameter 4 we get on the one hand the possibility to find a proper 3 and on the other hand we guarantee the possibility to choose in a proper way 6 , and then, 5 . So, finally, we check the resulting condition on coming from the requirement that the range for admissible 4 becomes nonempty, that is,…”

Section: Existence Of Parameters In Theorem 11mentioning

confidence: 99%

“…The second inequality to which we refer allows to estimate the norm of a product in

H^{σ}

. For its proof we refer to . Proposition Let us assume

σ > 0

and

1 \leq r \leq \infty, 1 < p_{1}, p_{2}, q_{1}, q_{2} \leq \infty

satisfying the relation

\frac{1}{r} = \frac{1}{p_{1}} + \frac{1}{p_{2}} = \frac{1}{q_{1}} + \frac{1}{q_{2}} .

Then it holds the following fractional Leibniz rule:

\begin{matrix} true \\ right {false∥ false| D false|}^{σ} {(u v) false∥}_{L^{r}} ≲ {false∥ false| D false|}^{σ} {u false∥}_{L^{p_{1}}} {false∥ v false∥}_{L^{p_{2}}} + {false∥ u false∥}_{L^{q_{1}}} {false∥ false| D false|}^{σ} {v false∥}_{L^{q_{2}}} \end{matrix}

for any u and v such that the norms of the right‐hand side exist.…”

Section: Tools From Harmonic Analysismentioning

confidence: 99%

“…The second inequality to which we refer allows to estimate the norm of a product in . For its proof we refer to [4]. .…”

Section: Fractional Leibniz Rulementioning

confidence: 99%

“…where 3 , 4 satisfy the relation 1 2 = −1 3 + 1 4 . By fractional Gagliardo-Nirenberg inequality one gets…”

Section: Case >mentioning

confidence: 99%

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Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation, II

Palmieri

Reissig

2018

Mathematische Nachrichten

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This paper is a continuation of our recent paper . We will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove global existence (in time) of small data energy solutions assuming suitable regularity on the L2 scale with additional L1 regularity for the data. In order to deal with this L2 regularity in the non‐linear part, we will develop and employ some tools from Harmonic Analysis.

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Section: Existence Of Parameters In Theorem 11mentioning

confidence: 73%

Section: Existence Of Parameters In Theorem 11mentioning

confidence: 99%

“…The second inequality to which we refer allows to estimate the norm of a product in

H^{σ}

. For its proof we refer to . Proposition Let us assume

σ > 0

and

1 \leq r \leq \infty, 1 < p_{1}, p_{2}, q_{1}, q_{2} \leq \infty

satisfying the relation

\frac{1}{r} = \frac{1}{p_{1}} + \frac{1}{p_{2}} = \frac{1}{q_{1}} + \frac{1}{q_{2}} .

Then it holds the following fractional Leibniz rule:

\begin{matrix} true \\ right {false∥ false| D false|}^{σ} {(u v) false∥}_{L^{r}} ≲ {false∥ false| D false|}^{σ} {u false∥}_{L^{p_{1}}} {false∥ v false∥}_{L^{p_{2}}} + {false∥ u false∥}_{L^{q_{1}}} {false∥ false| D false|}^{σ} {v false∥}_{L^{q_{2}}} \end{matrix}

for any u and v such that the norms of the right‐hand side exist.…”

Section: Tools From Harmonic Analysismentioning

confidence: 99%

“…The second inequality to which we refer allows to estimate the norm of a product in . For its proof we refer to [4]. .…”

Section: Fractional Leibniz Rulementioning

confidence: 99%

“…where 3 , 4 satisfy the relation 1 2 = −1 3 + 1 4 . By fractional Gagliardo-Nirenberg inequality one gets…”

Section: Case >mentioning

confidence: 99%

See 3 more Smart Citations

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation, II

Palmieri

Reissig

2018

Mathematische Nachrichten

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show abstract

Weighted Kato–Ponce inequalities for multiple factors

Douglas,

Grafakos

2024

Mathematische Nachrichten

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Global solution for semilinear σ$$ \sigma $$‐evolution models with memory term

Xie,

Yang

2024

Math Methods in App Sciences

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In this paper, the initial value problem of the semilinear ‐evolution equations with a memory term is concerned. Firstly, using the energy method in the Fourier space, the decay estimates for the solutions to the corresponding linear problem are established. Additionally, assuming small initial data in suitable time‐weighted Sobolev spaces, the global‐in‐time existence of the solutions to the semilinear issue is proved by contraction mapping. Finally, the decay estimates of solutions are obtained under the additional regularity assumption on the initial data.

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The Kato-Ponce Inequality

Cited by 217 publications

References 20 publications

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation, II

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation, II

Weighted Kato–Ponce inequalities for multiple factors

Global solution for semilinear σ$$ \sigma $$‐evolution models with memory term

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