Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces

Yoshihiro Mizuta; Tetsu Shimomura

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 4, page 1201-1217
  • ISSN: 0011-4642

Abstract

top
Our aim is to establish Sobolev type inequalities for fractional maximal functions M ℍ , ν f and Riesz potentials I ℍ , α f in weighted Morrey spaces of variable exponent on the half space ℍ . We also obtain Sobolev type inequalities for a C 1 function on ℍ . As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents Φ ( x , t ) = t p ( x ) + ( b ( x ) t ) q ( x ) , where p ( · ) and q ( · ) satisfy log-Hölder conditions, p ( x ) < q ( x ) for x ∈ ℍ , and b ( · ) is nonnegative and Hölder continuous of order θ ∈ ( 0 , 1 ] .

How to cite

top

Mizuta, Yoshihiro, and Shimomura, Tetsu. "Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces." Czechoslovak Mathematical Journal 73.4 (2023): 1201-1217. <http://eudml.org/doc/299564>.

@article{Mizuta2023,
abstract = {Our aim is to establish Sobolev type inequalities for fractional maximal functions $M_\{\mathbb \{H\},\nu \}f$ and Riesz potentials $I_\{\mathbb \{H\},\alpha \}f$ in weighted Morrey spaces of variable exponent on the half space $\mathbb \{H\}$. We also obtain Sobolev type inequalities for a $C^1$ function on $\mathbb \{H\}$. As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents $\Phi (x,t) = t^\{p(x)\} + (b(x) t)^\{q(x)\}$, where $p(\cdot )$ and $q(\cdot )$ satisfy log-Hölder conditions, $p(x)<q(x)$ for $x \in \{\mathbb \{H\}\} $, and $b(\cdot )$ is nonnegative and Hölder continuous of order $\theta \in (0,1]$.},
author = {Mizuta, Yoshihiro, Shimomura, Tetsu},
journal = {Czechoslovak Mathematical Journal},
keywords = {variable exponent; fractional maximal function; Riesz potential; Sobolev's inequality; weighted Morrey space; double phase functional},
language = {eng},
number = {4},
pages = {1201-1217},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces},
url = {http://eudml.org/doc/299564},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Mizuta, Yoshihiro
AU - Shimomura, Tetsu
TI - Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 4
SP - 1201
EP - 1217
AB - Our aim is to establish Sobolev type inequalities for fractional maximal functions $M_{\mathbb {H},\nu }f$ and Riesz potentials $I_{\mathbb {H},\alpha }f$ in weighted Morrey spaces of variable exponent on the half space $\mathbb {H}$. We also obtain Sobolev type inequalities for a $C^1$ function on $\mathbb {H}$. As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents $\Phi (x,t) = t^{p(x)} + (b(x) t)^{q(x)}$, where $p(\cdot )$ and $q(\cdot )$ satisfy log-Hölder conditions, $p(x)<q(x)$ for $x \in {\mathbb {H}} $, and $b(\cdot )$ is nonnegative and Hölder continuous of order $\theta \in (0,1]$.
LA - eng
KW - variable exponent; fractional maximal function; Riesz potential; Sobolev's inequality; weighted Morrey space; double phase functional
UR - http://eudml.org/doc/299564
ER -

References

top
  1. Adams, D. R., 10.1215/S0012-7094-75-04265-9, Duke Math. J. 42 (1975), 765-778. (1975) Zbl0336.46038MR0458158DOI10.1215/S0012-7094-75-04265-9
  2. Adams, D. R., Hedberg, L. I., 10.1007/978-3-662-03282-4, Grundlehren der Mathematischen Wissenschaften 314. Springer, Berlin (1995). (1995) Zbl0834.46021MR1411441DOI10.1007/978-3-662-03282-4
  3. Almeida, A., Hasanov, J., Samko, S., 10.1515/GMJ.2008.195, Georgian Math. J. 15 (2008), 195-208. (2008) Zbl1263.42002MR2428465DOI10.1515/GMJ.2008.195
  4. Baroni, P., Colombo, M., Mingione, G., 10.1090/spmj/1392, St. Petersbg. Math. J. 27 (2016), 347-379. (2016) Zbl1335.49057MR3570955DOI10.1090/spmj/1392
  5. Baroni, P., Colombo, M., Mingione, G., 10.1007/s00526-018-1332-z, Calc. Var. Partial Differ. Equ. 57 (2018), Article ID 62, 48 pages. (2018) Zbl1394.49034MR3775180DOI10.1007/s00526-018-1332-z
  6. Byun, S.-S., Lee, H.-S., 10.1016/j.jmaa.2020.124015, J. Math. Anal. Appl. 501 (2021), Article ID 124015, 31 pages. (2021) Zbl1467.35064MR4258791DOI10.1016/j.jmaa.2020.124015
  7. Capone, C., Cruz-Uribe, D., Fiorenza, A., 10.4171/RMI/511, Rev. Mat. Iberoam. 23 (2007), 743-770. (2007) Zbl1213.42063MR2414490DOI10.4171/RMI/511
  8. Colombo, M., Mingione, G., 10.1007/s00205-015-0859-9, Arch. Ration. Mech. Anal. 218 (2015), 219-273. (2015) Zbl1325.49042MR3360738DOI10.1007/s00205-015-0859-9
  9. Colombo, M., Mingione, G., 10.1007/s00205-014-0785-2, Arch. Ration. Mech. Anal. 215 (2015), 443-496. (2015) Zbl1322.49065MR3294408DOI10.1007/s00205-014-0785-2
  10. Cruz-Uribe, D. V., Fiorenza, A., 10.1007/978-3-0348-0548-3, Applied and Numerical Harmonic Analysis. Birkhäuser, New York (2013). (2013) Zbl1268.46002MR3026953DOI10.1007/978-3-0348-0548-3
  11. Cruz-Uribe, D., Fiorenza, A., Neugebauer, C. J., 10.1016/j.jmaa.2012.04.044, J. Math. Anal. Appl. 394 (2012), 744-760. (2012) Zbl1298.42021MR2927495DOI10.1016/j.jmaa.2012.04.044
  12. Diening, L., Harjulehto, P., Hästö, P., Růžička, M., 10.1007/978-3-642-18363-8, Lecture Notes in Mathematics 2017. Springer, Berlin (2011). (2011) Zbl1222.46002MR2790542DOI10.1007/978-3-642-18363-8
  13. Fazio, G. Di, Ragusa, M. A., Commutators and Morrey spaces, Boll. Unione Mat. Ital., VII. Ser., A 5 (1991), 323-332. (1991) Zbl0761.42009MR1138545
  14. Haj{ł}asz, P., Koskela, P., 10.1090/memo/0688, Memoirs of the American Mathematical Society 688. AMS, Providence (2000). (2000) Zbl0954.46022MR1683160DOI10.1090/memo/0688
  15. Hästö, P., Ok, J., 10.1016/j.jde.2019.03.026, J. Differ. Equations 267 (2019), 2792-2823. (2019) Zbl1420.35087MR3953020DOI10.1016/j.jde.2019.03.026
  16. Kinnunen, J., Lindqvist, P., 10.1515/crll.1998.095, J. Reine Angew. Math. 503 (1998), 161-167. (1998) Zbl0904.42015MR1650343DOI10.1515/crll.1998.095
  17. Kinnunen, J., Saksman, E., 10.1112/S0024609303002017, Bull. Lond. Math. Soc. 35 (2003), 529-535. (2003) Zbl1021.42009MR1979008DOI10.1112/S0024609303002017
  18. Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T., 10.1016/j.bulsci.2012.03.008, Bull. Sci. Math. 137 (2013), 76-96. (2013) Zbl1267.46045MR3007101DOI10.1016/j.bulsci.2012.03.008
  19. Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T., 10.1515/forum-2018-0077, Forum Math. 31 (2019), 517-527. (2019) Zbl1423.46049MR3918454DOI10.1515/forum-2018-0077
  20. Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T., 10.1080/17476933.2010.504837, Complex Var. Elliptic Equ. 56 (2011), 671-695. (2011) Zbl1228.31004MR2832209DOI10.1080/17476933.2010.504837
  21. Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T., 10.1007/s13163-011-0074-7, Rev. Mat. Complut. 25 (2012), 413-434. (2012) Zbl1273.31005MR2931419DOI10.1007/s13163-011-0074-7
  22. Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T., 10.1016/j.na.2020.111827, Nonlinear Anal., Theory Methods Appl., Ser. A 197 (2020), Article ID 111827, 18 pages. (2020) Zbl1441.31004MR4073513DOI10.1016/j.na.2020.111827
  23. Mizuta, Y., Ohno, T., Shimomura, T., 10.7153/mia-2018-21-30, Math. Inequal. Appl. 21 (2018), 433-453. (2018) Zbl1388.31009MR3776085DOI10.7153/mia-2018-21-30
  24. Mizuta, Y., Ohno, T., Shimomura, T., 10.1016/j.jmaa.2020.124360, J. Math. Anal. Appl. 501 (2021), Article ID 124360, 16 pages. (2021) Zbl1478.46028MR4258801DOI10.1016/j.jmaa.2020.124360
  25. Mizuta, Y., Shimomura, T., 10.2969/jmsj/06020583, J. Math. Soc. Japan 60 (2008), 583-602. (2008) Zbl1161.46305MR2421989DOI10.2969/jmsj/06020583
  26. Mizuta, Y., Shimomura, T., 10.1016/j.jmaa.2020.124133, J. Math. Anal. Appl. 501 (2021), Article ID 124133, 17 pages. (2021) Zbl1478.46037MR4258797DOI10.1016/j.jmaa.2020.124133
  27. Mizuta, Y., Shimomura, T., 10.1007/s11117-021-00810-z, Positivity 25 (2021), 1131-1146. (2021) Zbl1481.46030MR4274309DOI10.1007/s11117-021-00810-z
  28. Mizuta, Y., Shimomura, T., 10.1002/mma.7425, Math. Methods Appl. Sci. 45 (2022), 8631-8654. (2022) MR4475228DOI10.1002/mma.7425
  29. Mizuta, Y., Shimomura, T., Sobolev type inequalities for fractional maximal functions and Riesz potentials in half spaces, Available at https://arxiv.org/abs/2305.13708 (2023), 22 pages. (2023) MR4274309
  30. C. B. Morrey, Jr., 10.1090/S0002-9947-1938-1501936-8, Trans. Am. Math. Soc. 43 (1938), 126-166. (1938) Zbl0018.40501MR1501936DOI10.1090/S0002-9947-1938-1501936-8
  31. Ragusa, M. A., Tachikawa, A., 10.1515/anona-2020-0022, Adv. Nonlinear Anal. 9 (2020), 710-728. (2020) Zbl1420.35145MR3985000DOI10.1515/anona-2020-0022
  32. Sawano, Y., Shimomura, T., 10.1007/s00025-021-01490-7, Result. Math. 76 (2021), Article ID 188, 22 pages. (2021) Zbl1479.42055MR4305494DOI10.1007/s00025-021-01490-7

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.