Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
vector-valued space; $L^p(\ell ^1)$ capacity; vector-valued extension; boundedness
vector-valued space; $L^p(\ell ^1)$ capacity; vector-valued extension; boundedness
Summary:
This paper is devoted to the study of a class of vector-valued convolution operators. A fundamental problem in this context is to characterize the nonnegative measures $\mu $ for which such operators are bounded from $L^p(\ell ^1)$ to $L^q(\ell ^1; \mu )$. Inspired by classical capacity theory, we introduce a novel notion of $L^p(\ell ^1)$ capacity specifically adapted to the vector-valued setting. By developing this capacity approach, we establish complete necessary and sufficient conditions on the measure $\mu $ for the boundedness of the operators. Our results provide a unified characterization that covers both cases $p \leq q$ and $q < p$. The findings yield a powerful tool for analyzing mapping properties of vector-valued operators in weighted spaces.
References:
[1] Abbondandolo, A., Benedetti, G., Edtmair, O.: Symplectic capacities of domains close to the ball and Banach-Mazur geodesics in the space of contact forms. Duke Math. J. 174 (2025), 1567-1646. DOI 10.1215/00127094-2024-0066 | MR 4916111 | Zbl 08072318
[2] Adams, D. R., Hedberg, L. I.: Function Spaces and Potential Theory. Grundlehren der mathematischen Wissenschaften 314. Springer, Berlin (1996). DOI 10.1007/978-3-662-03282-4 | MR 1411441 | Zbl 0834.46021
[3] Adams, D. R., Xiao, J.: Elliptic-to-parabolic Morrey spaces-potentials-capacities with applications to certain evolution PDE. J. Lond. Math. Soc., II. Ser. 111 (2025), Article ID e70131, 38 pages. DOI 10.1112/jlms.70131 | MR 4889441 | Zbl 1566.31012
[4] Addona, D., Leone, V., Lorenzi, L., Rhandi, A.: $L^p$ maximal regularity for vector-valued Schrödinger operators. J. Math. Pures Appl. (9) 187 (2024), 171-206. DOI 10.1016/j.matpur.2024.05.010 | MR 4755024 | Zbl 1542.35131
[5] Alvarez-Romero, I., Barrios, B., Betancor, J. J.: Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted $L^p$ maximal inequalities. Commun. Contemp. Math. 27 (2025), Article ID 2450010, 28 pages. DOI 10.1142/S021919972450010X | MR 4835294 | Zbl 1553.42023
[6] Cejas, M. E., Li, K., Pérez, C., Rivera-Ríos, I. P.: Vector-valued operators, optimal weighted estimates and the $C_p$ condition. Sci. China, Math. 63 (2020), 1339-1368. DOI 10.1007/s11425-018-9404-4 | MR 4119561 | Zbl 1450.42011
[7] Chen, H., Liang, S., Tsai, T.-P.: Poisson kernel and blow-up of the second derivatives near the boundary for Stokes equations with Navier boundary condition. Int. Math. Res. Not. 2025 (2025), Article ID rnaf204, 31 pages. DOI 10.1093/imrn/rnaf204 | MR 4931789 | Zbl 08103272
[8] Chen, J., Wang, M.: Weighted norm inequalities, embedding theorems and integration operators on vector-valued Fock spaces. Math. Z. 307 (2024), Article ID 36, 30 pages. DOI 10.1007/s00209-024-03514-8 | MR 4747594 | Zbl 1543.30157
[9] Duman, O., Erkus-Duman, E.: Nonlinear approximation of vector-valued functions by Shepard operators based on max-product and max-min operations. Fuzzy Sets Syst. 509 (2025), Article ID 109332, 14 pages. DOI 10.1016/j.fss.2025.109332 | MR 4873708 | Zbl 1573.41009
[10] Fu, X., Xiao, J., Xiong, Q.: Toward weighted Lorentz-Sobolev capacities from Caffarelli-Silvestre extensions. J. Geom. Anal. 34 (2024), Article ID 124, 85 pages. DOI 10.1007/s12220-024-01569-x | MR 4716761 | Zbl 1536.35163
[11] Hoang, C., Moen, K., Moreno, C. P.: New pointwise bounds by Riesz potential type operators. J. Funct. Anal. 289 (2025), Article ID 111060, 35 pages. DOI 10.1016/j.jfa.2025.111060 | MR 4916065 | Zbl 08065948
[12] Isralowitz, J., Pott, S., Rivera-Ríos, I. P.: Sharp $A_1$ weighted estimates for vector-valued operators. J. Geom. Anal. 31 (2021), 3085-3116. DOI 10.1007/s12220-020-00385-3 | MR 4225835 | Zbl 1484.47069
[13] Jiang, Z.-j.: Weighted composition operators between vector-valued Bloch-type spaces on the polydisk. AIMS Math. 10 (2025), 18957-18982. DOI 10.3934/math.2025847 | MR 4950865
[14] Kadak, U., Costarelli, D., Coroainu, L.: Neural network operators of generalized fractional integrals equipped with a vector-valued function. Chaos Solitons Fractals 177 (2023), Article ID 114272, 12 pages. DOI 10.1016/j.chaos.2023.114272 | MR 4668224
[15] Karvonen, T., Särkkä, S.: Gaussian kernel quadrature at scaled Gauss-Hermite nodes. BIT 59 (2019), 877-902. DOI 10.1007/s10543-019-00758-3 | MR 4032891 | Zbl 1437.65012
[16] Kruse, K.: Weighted spaces of vector-valued functions and the $\epsilon$-product. Banach J. Math. Anal. 14 (2020), 1509-1531. DOI 10.1007/s43037-020-00072-z | MR 4133638 | Zbl 1539.46025
[17] Li, G., Wang, G., Zhang, L.: A capacity associated with the weighted Lebesgue space and its applications. J. Funct. Spaces 2022 (2022), Article ID 1257963, 11 pages. DOI 10.1155/2022/1257963 | MR 4392088 | Zbl 1495.46021
[18] Li, P., Zhai, Z.: Application of capacities to space-time fractional dissipative equations. I. Regularity and the blow-up set. Can. J. Math. 75 (2023), 1904-1956. DOI 10.4153/S0008414X22000566 | MR 4658184 | Zbl 1531.35360
[19] Li, P., Zhai, Z.: Application of capacities to space-time fractional dissipative equations. II. Carleson measure characterization for $L^q(\Bbb{R}^{n+1}_+,\mu)$-extension. Adv. Nonlinear Anal. 11 (2022), 850-887. DOI 10.1515/anona-2021-0232 | MR 4379604 | Zbl 1486.31014
[20] Li, X. Y., Wu, B. Y.: Fast and highly accurate reproducing kernels based approach for Poisson equation. Appl. Math. Lett. 171 (2025), Article ID 109636, 5 pages. DOI 10.1016/j.aml.2025.109636 | MR 4919679 | Zbl 08123475
[21] Maz'ya, V., Netrusov, Y.: Some counterexamples for the theory of Sobolev spaces on bad domains. Potential Anal. 4 (1995), 47-65. DOI 10.1007/BF01048966 | MR 1313906 | Zbl 0819.46023
[22] Oi, S.: Isometries and Hermitian operators on spaces of vector-valued Lipschitz maps. J. Inst. Math. Jussieu 23 (2024), 1857-1874. DOI 10.1017/S1474748023000415 | MR 4803688 | Zbl 1558.46014
[23] Takeuchi, T.: On the Chemin-Lerner-Lorentz space: Completeness, real interpolation, and estimates of the heat semigroup. J. Fourier Anal. Appl. 31 (2025), Article ID 27, 50 pages. DOI 10.1007/s00041-025-10160-z | MR 4900216 | Zbl 1571.35291
[24] Tsagris, M., Papastamoulis, P., Kato, S.: Directional data analysis: Spherical Cauchy or Poisson kernel-based distribution?. Stat. Comput. 35 (2025), Article ID 51, 21 pages. DOI 10.1007/s11222-025-10583-0 | MR 4866645 | Zbl 1556.62037
[25] Wang, Z., Huang, J., Liu, Y., Li, P.: Capacities and blow-up set of extension problem related to fractional operators on metric measure spaces. J. Geom. Anal. 35 (2025), Article ID 105, 62 pages. DOI 10.1007/s12220-025-01933-5 | MR 4867298 | Zbl 1560.42012
[26] Wen, Y., Wu, H.: Weighted variational inequalities for heat semigroups associated with Schrödinger operators related to critical radius functions. J. Fourier Anal. Appl. 31 (2025), Article ID 30, 39 pages. DOI 10.1007/s00041-025-10166-7 | MR 4903732 | Zbl 1571.42035
[27] Würth, A., Binois, M., Goatin, P.: Traffic prediction by combining macroscopic models and Gaussian processes. Appl. Math. Modelling 150 (2026), Article ID 116397, 14 pages. DOI 10.1016/j.apm.2025.116397 | MR 4951878 | Zbl 08129638
[28] Xue, Q., Yabuta, K., Yan, J.: Weighted Fréchet-Kolmogorov theorem and compactness of vector-valued multilinear operators. J. Geom. Anal. 31 (2021), 9891-9914. DOI 10.1007/s12220-021-00630-3 | MR 4303944 | Zbl 1471.42040
[29] Zhao, N., Liu, Y.: Besov space via heat semigroup on Carnot group and its capacity. J. Geom. Anal. 34 (2024), Article ID 353, 57 pages. DOI 10.1007/s12220-024-01817-0 | MR 4801761 | Zbl 1550.30024
Search
Partner of
