Riv. Mat. Univ. Parma, Vol. 9, No. 1, 2018
Kwok-Pun Ho
[a]
Mixed norm Lebesgue spaces with variable exponents and applications
Pages: 21-44
Received: 23 October 2017
Accepted: 25 May 2018
Mathematics Subject Classification (2010): 42B35, 42B20, 46E30, 31C05.
Keywords: Mixed-norm spaces, Lebesgue spaces with variable exponent, extrapolation, product domains,
Calderón-Zygmund operators, Littlewood-Paley operators, bi-harmonic functions, Ricci-Stein singular integrals, BMO.
Author address:
[a]: The Education University of Hong Kong, 10 Lo Ping Road, Tai Po Hong Kong, China
Full Text (PDF)
Abstract:
In this paper, we introduce the mixed norm Lebesgue spaces with variable exponent.
We use this family of function spaces to study Calderón-Zygmund operators on product domains,
the Littlewood-Paley operators associated with family of disjoint rectangles,
the non-tangential maximal
function and the area function for bi-harmonic functions,
the Ricci-Stein singular integrals and
the characterizations of the function space of bounded mean oscillation.
References
- [1]
-
R. Algervik and V. Kolyada, On Fournier-Gagliardo mixed norm spaces,
Ann. Acad. Sci. Fenn. Math. 36 (2011), 493-508.
MR2865509
- [2]
-
A. Benedek and R. Panzone, The space Lp, with mixed norm,
Duke Math. J. 28 (1961), 301-324.
MR0126155
- [3]
-
C. Bennett and R. Sharpley,
Interpolation of operators, Pure Appl. Math., 129, Academic Press, Boston, MA, 1988.
MR0928802
- [4]
-
A. P. Blozinski,
Multivariate rearrangements and Banach function spaces with mixed norms,
Trans. Amer. Math. Soc. 263 (1981), 149-167.
MR590417
- [5]
-
D. V. Cruz-Uribe and A. Fiorenza,
Variable Lebesgue spaces, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Heidelberg, 2013.
MR3026953
- [6]
-
L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces,
Bull. Sci. Math. 129 (2005), 657-700.
MR2166733
- [7]
-
L. Diening, P. Harjulehto, P. Hästö and M. Ružička,
Lebesgue and Sobolev spaces with variable exponents,
Lecture Notes in Math., 2017, Springer, Heidelberg, 2011.
MR2790542
- [8]
-
R. Fefferman, Ap weights and singular integrals,
Amer. J. Math. 110(1988), 975-987.
MR961502
- [9]
-
R. Fefferman and J. Pipher,
Multiparameter operators and sharp weighted inequalities,
Amer. J. Math. 119 (1997), 337-369.
MR1439553
- [10]
-
D. L. Fernandez,
Lorentz spaces, with mixed norms,
J. Functional Analysis 25 (1977), 128-146.
MR0482127
- [11]
-
J. J. F. Fournier,
Mixed norms and rearrangements: Sobolev's inequality and Littlewood's inequality,
Ann. Mat. Pura Appl. (4) 148 (1987), 51-76.
MR0932758
- [12]
-
J. García-Cuerva and J. L. Rubio de Francia,
Weighted norm inequalities and related topics,
North-Holland Publishing Co., Amsterdam, 1985.
MR0807149
- [13]
-
L. Grafakos},
Modern Fourier analysis,
2nd edition, Grad. Texts in Math., 250, Springer, New York, 2009.
MR2463316
- [14]
-
W. Grey and G. Sinnamon,
The inclusion problem for mixed norm spaces,
Trans. Amer. Math. Soc. 368 (2016), 8715-8736.
MR3551586
- [15]
-
R. F. Gundy and E. M. Stein,
Hp theory for the poly-disc,
Proc. Nat. Acad. Sci. U.S.A. 76 (1979), 1026-1029.
MR0524328
- [16]
-
K.-P. Ho,
Characterization of BMO in terms of rearrangement-invariant Banach function spaces,
Expo. Math. 27 (2009), 363-372.
MR2567029
- [17]
-
K.-P. Ho,
Characterizations of BMO by Ap weights and p-convexity,
Hiroshima Math. J. 41 (2011), 153-165.
MR2849152
- [18]
-
K.-P. Ho,
Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces,
Anal. Math. 38 (2012), 173-185.
MR2958353
- [19]
-
K.-P. Ho,
John-Nirenberg inequalities on Lebesgue spaces with variable exponents,
Taiwanese J. Math. 18 (2014), 1107-1118.
MR3245432
- [20]
-
K.-P. Ho,
Vector-valued John-Nirenberg inequalities and vector-valued mean oscillations characterization of BMO,
Results Math. 70 (2016), 257-270.
MR3535006
- [21]
-
K.-P. Ho, Strong maximal operator on mixed-norm spaces,
Ann. Univ. Ferrara Sez. VII Sci. Mat. 62 (2016), 275-291.
MR3570358
- [22]
-
K.-P. Ho,
Singular integral operators, John-Nirenberg inequalities and Triebel-Lizorkin type spaces on weighted Lebesgue spaces with variable exponents,
Rev. Un. Mat. Argentina 57 (2016), 85-101.
MR3519286
- [23]
-
K.-P. Ho,
Extrapolation, John-Nirenberg inequalities and characterizations of BMO in terms of Morrey type spaces,
Rev. Mat. Complut. 30 (2017), 487-505.
MR3690751
- [24]
-
M. Izuki and Y. Sawano,
Variable Lebesgue norm estimates for BMO functions,
Czechoslovak Math. J. 62 (2012), 717-727.
MR2984631
- [25]
-
M. Izuki, Y. Sawano and Y. Tsutsui,
Variable Lebesgue norm estimates for BMO functions, II,
Anal. Math. 40 (2014), 215-230.
MR3240224
- [26]
-
M. Izuki, Another proof of characterization of BMO via Banach function spaces,
Rev. Un. Mat. Argentina 57 (2016), 103-109.
MR3519287
- [27]
-
M. Izuki and Y. Sawano,
Characterization of BMO via ball Banach function spaces,
Vestn. St.-Peterbg. Univ. Mat. Mekh. Astron. 4(62) (2017), 78-86.
MR3664632
- [28]
-
J.-L. Journé,
Calderón-Zygmund operators on product spaces,
Rev. Mat. Iberoamericana 1 (1985), 55-91.
MR0836284
- [29]
-
T. Kopaliani,
A note on strong maximal operator in Lp(·)(ℝ n) spaces,
Proc. A. Razmadze Math. Inst. 145 (2007), 43-46.
MR2387444
- [30]
-
H. Lin, Some weighted inequalities on product domains,
Trans. Amer. Math. Soc. 318 (1990), 69-85.
MR0970269
- [31]
-
W. A. J. Luxemburg,
On the measurability of a function which occurs in a paper by A. C. Zaanen,
Indag. Math. 20 (1958), 259-265.
MR0096765
- [32]
-
M. Milman, Embeddings of L(p,q) spaces and Orlicz spaces with mixed norms, Notas de Math. 13, Univ. de Los Andes, 1977.
http://www.ciens.ula.ve/matematica/publicaciones/notas_matematicas/lista_notas.pdf
- [33]
-
M. Milman,
Notes on interpolation of mixed norm spaces and applications,
Quart. J. Math. Oxford Ser. (2) 42 (1991), 325-334.
MR1120993
- [34]
-
F. Ricci and E. M. Stein,
Multiparameter singular integrals and maximal functions,
Ann. Inst. Fourier (Grenoble) 42 (1992), 637-670.
MR1182643
- [35]
-
J. L. Rubio de Francia,
A Littlewood-Paley inequality for arbitrary intervals,
Rev. Mat. Iberoamericana 1 (1985), 1-14.
MR0850681
- [36]
-
S. Sato, Weighted inequalities on product domains,
Studia Math. 92 (1989), 59-72.
MR0984850
- [37]
-
Y. Sawano, K.-P. Ho, D. Yang and S. Yang,
Hardy spaces for ball quasi-Banach function spaces,
Dissertationes Math. 525 (2017), 1-102.
MR3687096
- [38]
-
E. M. Stein,
Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals,
Princeton University Press, Princeton, NJ, 1993.
MR1232192
- [39]
-
M. Väth,
Some measurability results and applications to spaces with mixed family-norm,
Positivity 10 (2006), 737-753.
MR2280647
Home Riv.Mat.Univ.Parma