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Journal articles on the topic 'Local Hardy spaces'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

Betancor, Jorge J., and Wendolín Damián. "Anisotropic Local Hardy Spaces." Journal of Fourier Analysis and Applications 16, no. 5 (February 18, 2010): 658–75. http://dx.doi.org/10.1007/s00041-010-9121-x.

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2

Ding, Wei, and Feng Yu. "Dual Spaces of Multiparameter Local Hardy Spaces." Journal of Function Spaces 2021 (December 3, 2021): 1–12. http://dx.doi.org/10.1155/2021/9619925.

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In this paper, we study the duality theory of the multiparameter local Hardy spaces h p ℝ n 1 × ℝ n 2 , and we prove that h p ℝ n 1 × ℝ n 2 ∗ = cm o p ℝ n 1 × ℝ n 2 , where cm o p ℝ n 1 × ℝ n 2 are defined by discrete Carleson measure. Moreover, we discuss the relationship among cm o p ℝ n 1 × ℝ n 2 , Li p p ℝ n 1 × ℝ n 2 , and rectangle cm o rect p ℝ n 1 × ℝ n 2 .

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3

Xia, Runlian, and Xiao Xiong. "Operator-valued local Hardy spaces." Journal of Operator Theory 82, no. 2 (September 15, 2019): 383–443. http://dx.doi.org/10.7900/jot.2018jun02.2191.

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This paper gives a systematic study of operator-valued local\break Hardy spaces, which are localizations of the Hardy spaces defined by Mei. We prove the h1-bmo duality and the hp-hq duality for any conjugate pair (p,q) when p∈(1,∞). We show that h1(Rd,M) and bmo(Rd,M) are also good endpoints of Lp(L∞(Rd)¯¯¯¯⊗M) for interpolation. We obtain the local version of Calder\'on--Zygmund theory, and then deduce that the Poisson kernel in our definition of the local Hardy norms can be replaced by any reasonable test function. Finally, we establish the atomic decomposition of the local Hardy space hc1(Rd,M).

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4

Ding, Wei, Guozhen Lu, and YuePing Zhu. "Multi-parameter local Hardy spaces." Nonlinear Analysis 184 (July 2019): 352–80. http://dx.doi.org/10.1016/j.na.2019.02.014.

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5

Tang, Lin. "The weighted weak local Hardy spaces." Rocky Mountain Journal of Mathematics 44, no. 1 (February 2014): 297–315. http://dx.doi.org/10.1216/rmj-2014-44-1-297.

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6

Hounie, J., and Rafael Augusto dos Santos Kapp. "Pseudodifferential Operators on Local Hardy Spaces." Journal of Fourier Analysis and Applications 15, no. 2 (April 22, 2008): 153–78. http://dx.doi.org/10.1007/s00041-008-9021-5.

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7

Peloso, Marco M., and Silvia Secco. "Local Riesz transforms characterization of local Hardy spaces." Collectanea mathematica 59, no. 3 (October 2008): 299–320. http://dx.doi.org/10.1007/bf03191189.

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8

Yang, Dachun. "Local hardy and BMO spaces on non-homogeneous spaces." Journal of the Australian Mathematical Society 79, no. 2 (October 2005): 149–82. http://dx.doi.org/10.1017/s1446788700010430.

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AbstractLet µ be Radon measure on Rd which may be non doubling. The only condition that µ must satisfy is the size condition µ(B(x, r)) ≤ Crn for some fixed n є (0, d). Recently, Tolsa introduced the spaces RBMO(µ) and Hatb1.∞ (µ), which, in some ways, play the role of the classical spaces BMO and H1 in case u is a doubling measure. In this paper, the author considers the local versions of the spaces RBMO(µ) and Hatb1.∞ (µ) in the sense of Goldberg and establishes the connections between the spaces RBMO(µ) and Hatb1.∞ (µ) with their local versions. An interpolation result of linear operators is also given.

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9

Zhou, Guangcai, and Weixing Zheng. "LOCAL HARDY SPACES ON HEISENBERG GROUP OVER LOCAL FIELDS." Acta Mathematica Scientia 16, no. 2 (April 1996): 129–41. http://dx.doi.org/10.1016/s0252-9602(17)30788-9.

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10

Bloom, Walter R., and Zengfu Xu. "Local Hardy spaces on Chébli-Trimèche hypergroups." Studia Mathematica 133, no. 3 (1999): 197–230. http://dx.doi.org/10.4064/sm-133-3-197-230.

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11

Tang, Lin. "Weighted local Hardy spaces and their applications." Illinois Journal of Mathematics 56, no. 2 (2012): 453–95. http://dx.doi.org/10.1215/ijm/1385129959.

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12

Daly, James E. "Factorization of Hardy spaces on local fields." Mathematische Annalen 282, no. 2 (June 1988): 243–50. http://dx.doi.org/10.1007/bf01456973.

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13

Ding, Wei, Guozhen Lu, and Yueping Zhu. "Discrete Littlewood–Paley–Stein characterization of multi-parameter local Hardy spaces." Forum Mathematicum 31, no. 6 (November 1, 2019): 1467–88. http://dx.doi.org/10.1515/forum-2019-0038.

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AbstractIn our recent work [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380], the multi-parameter local Hardy space {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} has been introduced by using the continuous inhomogeneous Littlewood–Paley–Stein square functions. In this paper, we will first establish the new discrete multi-parameter local Calderón’s identity. Based on this identity, we will define the local multi-parameter Hardy space {h_{\mathrm{dis}}^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} by using the discrete inhomogeneous Littlewood–Paley–Stein square functions. Then we prove that these two multi-parameter local Hardy spaces are actually the same. Moreover, the norms of the multi-parameter local Hardy spaces under the continuous and discrete Littlewood–Paley–Stein square functions are equivalent. This discrete version of the multi-parameter local Hardy space is also critical in establishing the duality theory of the multi-parameter local Hardy spaces.

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14

Wang, Hongbin, and Zongguang Liu. "Local Herz-type Hardy spaces with variable exponent." Banach Journal of Mathematical Analysis 9, no. 4 (2015): 359–78. http://dx.doi.org/10.15352/bjma/09-4-17.

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15

Kobayashi, Masaharu, Akihiko Miyachi, and Naohito Tomita. "Embedding relations between local Hardy and modulation spaces." Studia Mathematica 192, no. 1 (2009): 79–96. http://dx.doi.org/10.4064/sm192-1-7.

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16

Weisz, Ferenc. "Multi-dimensional Fejér summability and local Hardy spaces." Studia Mathematica 194, no. 2 (2009): 181–95. http://dx.doi.org/10.4064/sm194-2-5.

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17

Bogdan, Krzysztof, Bartłomiej Dyda, and Tomasz Luks. "On Hardy spaces of local and nonlocal operators." Hiroshima Mathematical Journal 44, no. 2 (July 2014): 193–215. http://dx.doi.org/10.32917/hmj/1408972907.

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18

Weisz, Ferenc. "Local Hardy spaces and summability of Fourier transforms." Journal of Mathematical Analysis and Applications 362, no. 2 (February 2010): 275–85. http://dx.doi.org/10.1016/j.jmaa.2009.08.003.

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19

Zhu, Hua, and Lin Tang. "Weighted local Hardy spaces associated to Schrödinger operators." Illinois Journal of Mathematics 60, no. 3-4 (2016): 687–738. http://dx.doi.org/10.1215/ijm/1506067287.

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20

Mizuhara, Takahiro. "Notes on Embedding Theorems for Local Hardy spaces." Mathematische Nachrichten 165, no. 1 (1994): 231–44. http://dx.doi.org/10.1002/mana.19941650115.

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21

Xiao, Jiang Wei, Yin Sheng Jiang, and Wen Hua Gao. "Bilinear pseudo-differential operators on local hardy spaces." Acta Mathematica Sinica, English Series 28, no. 2 (January 15, 2012): 255–66. http://dx.doi.org/10.1007/s10114-012-0283-0.

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22

Zhu, Hua. "Weighted Weak Local Hardy Spaces Associated with Schrödinger Operators." Journal of Function Spaces 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/490259.

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We characterize the weighted weak local Hardy spacesWhρp(ω)related to the critical radius functionρand weightsω∈A∞ρ,∞(Rn)which locally behave as Muckenhoupt’s weights and actually include them, by the atomic decomposition. As an application, we show that localized Riesz transforms are bounded on the weighted weak local Hardy spaces.

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23

Song, Futao, and Na Ju. "Double Points Local Hardy-Littlewood Maximal Operator." Abstract and Applied Analysis 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/637314.

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A double points local Hardy-Littlewood maximal operatorMa,b,k,locis defined and investigated in Euclidean spaces. It is proved thatMa,b,k,locis bounded onLpwwhenp>1and fromL1wtoL1,∞wwith weight functionw∈Aa,b,k,loc, the class of double points localApweights which is larger than the MuckenhouptApclass and the localApweights defined by Lin and Stempak.

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24

He, Ziyi, Dachun Yang, and Wen Yuan. "Real‐variable characterizations of local Hardy spaces on spaces of homogeneous type." Mathematische Nachrichten 294, no. 5 (April 2021): 900–955. http://dx.doi.org/10.1002/mana.201900320.

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25

Gong, RuMing, Ji Li, and LiXin Yan. "A local version of Hardy spaces associated with operators on metric spaces." Science China Mathematics 56, no. 2 (May 30, 2012): 315–30. http://dx.doi.org/10.1007/s11425-012-4428-5.

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26

Cao, Jun, Svitlana Mayboroda, and Dachun Yang. "Local Hardy spaces associated with inhomogeneous higher order elliptic operators." Analysis and Applications 15, no. 02 (January 25, 2017): 137–224. http://dx.doi.org/10.1142/s0219530515500189.

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Let [Formula: see text] be a divergence form inhomogeneous higher order elliptic operator with complex bounded measurable coefficients. In this paper, for all [Formula: see text] and [Formula: see text] satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces [Formula: see text] associated with [Formula: see text], which coincide with Goldberg’s local Hardy spaces [Formula: see text] for all [Formula: see text] when [Formula: see text] (the Laplace operator). The authors also establish a real-variable theory of [Formula: see text], which includes their characterizations in terms of the local molecules, the square functions or the maximal functions, the complex interpolation and dual spaces. These real-variable characterizations on the local Hardy spaces are new even when [Formula: see text] (the divergence form homogeneous second-order elliptic operator). Moreover, the authors show that [Formula: see text] coincides with the Hardy space [Formula: see text] associated with the operator [Formula: see text] for all [Formula: see text], where [Formula: see text] is some positive constant depending on the ellipticity and the off-diagonal estimates of [Formula: see text]. As an application, the authors establish some mapping properties for the local Riesz transforms [Formula: see text] on [Formula: see text], where [Formula: see text] and [Formula: see text].

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27

Ho, Hon Ming. "On block decomposition of local Hardy-type amalgam spaces." International Journal of Mathematical Analysis 15, no. 3 (2021): 121–38. http://dx.doi.org/10.12988/ijma.2021.912195.

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28

Gogatishvili, A., A. Danelia, and T. Kopaliani. "Local Hardy--Littlewood maximal operator in variable Lebesgue spaces." Banach Journal of Mathematical Analysis 8, no. 2 (2014): 229–44. http://dx.doi.org/10.15352/bjma/1396640066.

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29

Aykol, C., Z. O. Azizova, and J. J. Hasanov. "Weighted Hardy operators in local generalized Orlicz-Morrey spaces." Carpathian Mathematical Publications 13, no. 2 (November 6, 2021): 522–33. http://dx.doi.org/10.15330/cmp.13.2.522-533.

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In this paper, we find sufficient conditions on general Young functions $(\Phi, \Psi)$ and the functions $(\varphi_1,\varphi_2)$ ensuring that the weighted Hardy operators $A_\omega^\alpha$ and ${\mathcal A}_\omega^\alpha$ are of strong type from a local generalized Orlicz-Morrey space $M^{0,\,loc}_{\Phi,\,\varphi_1}(\mathbb R^n)$ into another local generalized Orlicz-Morrey space $M^{0,\,loc}_{\Psi,\,\varphi_2}(\mathbb R^n)$. We also obtain the boundedness of the commutators of $A_\omega^\alpha$ and ${\mathcal A}_\omega^\alpha$ from $M^{0,\,loc}_{\Phi,\,\varphi_1}(\mathbb R^n)$ to $M^{0,\,loc}_{\Psi,\,\varphi_2}(\mathbb R^n)$.

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30

Komori-Furuya, Yasuo. "Commutators of pseudo-differential operators on local Hardy spaces." Acta Scientiarum Mathematicarum 77, no. 3-4 (December 2011): 489–501. http://dx.doi.org/10.1007/bf03643930.

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31

Carbonaro, Andrea, Alan McIntosh, and Andrew J. Morris. "Local Hardy Spaces of Differential Forms on Riemannian Manifolds." Journal of Geometric Analysis 23, no. 1 (May 24, 2011): 106–69. http://dx.doi.org/10.1007/s12220-011-9240-x.

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32

Weisz, Ferenc. "Restricted summability of Fourier transforms and local Hardy spaces." Acta Mathematica Sinica, English Series 26, no. 9 (August 15, 2010): 1627–40. http://dx.doi.org/10.1007/s10114-010-9529-x.

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33

Peloso, Marco M., and Silvia Secco. "BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON HARDY SPACES." Proceedings of the Edinburgh Mathematical Society 51, no. 2 (June 2008): 443–63. http://dx.doi.org/10.1017/s001309150500012x.

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AbstractFor $0\ltp\le1$, let $h^p(\mathbb{R}^n)$ denote the local Hardy space. Let $\mathcal{F}$ be a Fourier integral operator defined by the oscillatory integral$$ \mathcal{F}f(x)=\iint_{\mathbb{R}^{2n}}\exp(2\pi\mathrm{i}(\phi(x,\xi)-y\cdot\xi))b(x,y,\xi)f(y)\,\mathrm{d} y\,\mathrm{d}\xi, $$where $\phi$ is a $\mathcal{C}^\infty$ non-degenerate real phase function, and $b$ is a symbol of order $\mu$ and type $(\rho,1-\rho)$, $\sfrac12\lt\rho\le1$, vanishing for $x$ outside a compact set of $\mathbb{R}^n$. We show that when $p\le1$ and $\mu\le-(n-1)(1/p-1/2)$ then $\mathcal{F}$ initially defined on Schwartz functions in $h^p(\mathbb{R}^n)$ extends to a bounded operator $\mathcal{F}:h^p(\mathbb{R}^n)\rightarrow h^p(\mathbb{R}^n)$. The range of $p$ and $\mu$ is sharp. This result extends to the local Hardy spaces the seminal result of Seeger \et for the $L^p$ spaces. As immediate applications we prove the boundedness of smooth Radon transforms on hypersurfaces with non-vanishing Gaussian curvature on the local Hardy spaces.Finally, we prove a local version for the boundedness of Fourier integral operators on local Hardy spaces on smooth Riemannian manifolds of bounded geometry.

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34

Kato, Tomoya. "The inclusion relations between α-modulation spaces and Lp-Sobolev spaces or local Hardy spaces." Journal of Functional Analysis 272, no. 4 (February 2017): 1340–405. http://dx.doi.org/10.1016/j.jfa.2016.12.002.

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35

Samko, S. G., and S. M. Umarkhadzhiev. "Локальные гранд пространства Лебега." Владикавказский математический журнал, no. 4 (December 23, 2021): 96–108. http://dx.doi.org/10.46698/e4624-8934-5248-n.

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We introduce ``local grand'' Lebesgue spaces $L^{p),\theta}_{x_0,a}(\Omega)$, $0<p<\infty,$ $\Omega \subseteq \mathbb{R}^n$, where the process of ``grandization'' relates to a single point $x_0\in \Omega$, contrast to the case of usual known grand spaces $L^{p),\theta}(\Omega)$, where ``grandization'' relates to all the points of $\Omega$. We define the space $L^{p),\theta}_{x_0,a}(\Omega)$ by means of the weight $a(|x-x_0|)^{\varepsilon p}$ with small exponent, $a(0)=0$. Under some rather wide assumptions on the choice of the local ``grandizer'' $a(t)$, we prove some properties of these spaces including their equivalence under different choices of the grandizers $a(t)$ and show that the maximal, singular and Hardy operators preserve such a ``single-point grandization'' of Lebesgue spaces $L^p(\Omega)$, $1<p<\infty$, provided that the lower Matuszewska--Orlicz index of the function $a$ is positive. A Sobolev-type theorem is also proved in local grand spaces under the same condition on the grandizer.

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36

Huang, Jizheng, and Yu Liu. "Molecular Characterization of Hardy Spaces Associated with Twisted Convolution." Journal of Function Spaces 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/326940.

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37

Ruzhansky, Michael, Durvudkhan Suragan, and Nurgissa Yessirkegenov. "Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces." Fractional Calculus and Applied Analysis 21, no. 3 (June 26, 2018): 577–612. http://dx.doi.org/10.1515/fca-2018-0032.

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AbstractWe analyze local (central) Morrey spaces, generalized local (central) Morrey spaces and Campanato spaces on homogeneous groups. The boundedness of the Hardy-Littlewood maximal operator, Bessel-Riesz operators, generalized Bessel-Riesz operators and generalized fractional integral operators in generalized local (central) Morrey spaces on homogeneous groups is shown. Moreover, we prove the boundedness of the modified version of the generalized fractional integral operator and Olsen type inequalities in Campanato spaces and generalized local (central) Morrey spaces on homogeneous groups, respectively. Our results extend results known in the isotropic Euclidean settings, however, some of them are new already in the standard Euclidean cases.

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38

CHEN, DANING. "TRANSFERENCE OF MAXIMAL MULTIPLIER OPERATORS ON LOCAL HARDY-LORENTZ SPACES." Chinese Annals of Mathematics 25, no. 01 (January 2004): 111–18. http://dx.doi.org/10.1142/s0252959904000111.

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39

Yang, DaChun, and SiBei Yang. "Local Hardy spaces of Musielak-Orlicz type and their applications." Science China Mathematics 55, no. 8 (February 28, 2012): 1677–720. http://dx.doi.org/10.1007/s11425-012-4377-z.

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40

Samko, Natasha. "Weighted Hardy operators in the local generalized vanishing Morrey spaces." Positivity 17, no. 3 (September 9, 2012): 683–706. http://dx.doi.org/10.1007/s11117-012-0199-z.

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41

Bloom, Walter R., and Zengfu Xu. "Fourier Multipliers For Local Hardy Spaces On Chébli-Trimèche Hypergroups." Canadian Journal of Mathematics 50, no. 5 (October 1, 1998): 897–928. http://dx.doi.org/10.4153/cjm-1998-047-9.

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AbstractIn this paper we consider Fourier multipliers on local Hardy spaces hp (0 < p ≤ 1) for Chébli-Trimèche hypergroups. The molecular characterization is investigated which allows us to prove a version of Hörmander’s multiplier theorem.

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42

AYKOL, CANAY, ZULEYXA O. AZIZOVA, and ,. JAVANSHIR HASANOV. "GENERALIZED WEIGHTED HARDY OPERATORS AND THEIR COMMUTATORS IN THE LOCAL ”COMPLEMENTARY” GENERALIZED VARIABLE EXPONENT WEIGHTED MORREY SPACES." Journal of Mathematical Analysis 13, no. 1 (March 30, 2022): 14–27. http://dx.doi.org/10.54379/jma-2022-1-2.

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In this paper we study the boundedness of weighted Hardy operators Hα u and Hα u in local ”complementary” generalized variable exponent weighted Morrey spaces {M p(·),ω,ϕ {0} , characterized by variable exponent p(x), a general function ω(r) and a weight ϕ. We also study the boundedness of the commutators of Hα u and Hα u in the spaces {M p(·),ω,ϕ {0} .

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43

Bloom, Walter R., and Zengfu Xu. "Pseudo differential operators on local Hardy spaces on chébli-teimèche hypergroups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 68, no. 2 (April 2000): 202–30. http://dx.doi.org/10.1017/s1446788700001956.

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44

Mirotin, A. R. "Hausdorff Operators on Real Hardy Spaces H1 Over Homogeneous Spaces with Local Doubling Property." Analysis Mathematica 47, no. 2 (May 31, 2021): 385–403. http://dx.doi.org/10.1007/s10476-021-0087-5.

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45

Bui, The Anh, Xuan Thinh Duong, and Fu Ken Ly. "Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type." Transactions of the American Mathematical Society 370, no. 10 (July 5, 2018): 7229–92. http://dx.doi.org/10.1090/tran/7289.

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46

Chen, Jiao, Wei Ding, and Guozhen Lu. "Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces." Forum Mathematicum 32, no. 4 (July 1, 2020): 919–36. http://dx.doi.org/10.1515/forum-2019-0319.

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AbstractAfter the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are {L^{p}({\mathbb{R}^{n}})} bounded for {1<p<\infty}, but only bounded on local Hardy spaces {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for {0<p\leq 1}. Though much work has been done on the {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {1<p<\infty} and Hardy {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of {0<p\leq 1}. The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

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47

Lee, Ming-Yi, Chin-Cheng Lin, and Ying-Chieh Lin. "The continuity of pseudo-differential operators on weighted local Hardy spaces." Studia Mathematica 198, no. 1 (2010): 69–77. http://dx.doi.org/10.4064/sm198-1-4.

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48

Yang, Dachun, and Sibei Yang. "Weighted local Orlicz–Hardy spaces with applications to pseudo-differential operators." Dissertationes Mathematicae 478 (2011): 1–78. http://dx.doi.org/10.4064/dm478-0-1.

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49

Liyuan, Chen, and Sun Qiyu. "Behaviour of an oscillatory singular integral on weighted local Hardy spaces." Acta Mathematica Sinica 13, no. 3 (July 1997): 305–20. http://dx.doi.org/10.1007/bf02560010.

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50

Chen, Jiecheng, Dashan Fan, Lijing Sun, and Chunjie Zhang. "Estimates for Unimodular Multipliers on Modulation Hardy Spaces." Journal of Function Spaces and Applications 2013 (2013): 1–16. http://dx.doi.org/10.1155/2013/982753.

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It is known that the unimodular Fourier multiplierseit|Δ|α/2,α>0,are bounded on all modulation spacesMp,qsfor1≤p,q≤∞. We extend such boundedness to the case of all0<p,q≤∞and obtain its asymptotic estimate astgoes to infinity. As applications, we give the grow-up rate of the solution for the Cauchy problems for the free Schrödinger equation with the initial data in a modulation space, as well as some mixed norm estimates. We also study theMp1,qs→Mp2,qsboundedness for the operatoreit|Δ|α/2, for the case0<α≤2andα≠1.Finally, we investigate the boundedness of the operatoreit|Δ|α/2forα>0and obtain the local well-posedness for the Cauchy problem of some nonlinear partial differential equations with fundamental semigroupeit|Δ|α/2.

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